COMPUTER SIMULATION OF 

WHITE PINE BLISTER RUST EPIDEMICS 

I. MODEL FORMULATION 

Ceral I. McDonald, Raymond J. Hoff, and William R. Wykoff 

USDA Forest Service Research Paper INT -258 

Intermountain Forest and Range Experiment Station 

U.S. Department of Agriculture, Forest Service 

This file was created by scanning the printed publication.

Errors identified by the software have been corrected;

however, some errors may remain.

RESEARCH SUMMARY 

The major problem of managing white pines in North America 

is the blister rust disease caused by the introduced stern rust 

Cronartium ribicola. Populations of white pines exhibiting 

differing degrees of resistance as well as some proven management options are available for use in control strategies. But, 

since adaptative flexibility is of prime concern, a compromise 

must be reached between degree of resistance to a particular 

pest on the one hand and all the other factors of the environment 

on the other. An effective route to this compromise is integration 

of all genetic, biologic, and conventional control methodologies 

so as to maximize the number of management actions that work in 

a common direction. 

Computer simulation is an efficient way to organize and 

analyze the knowledge available about complex systems. The 

research reported herein is an attempt at organizing knowledge 

about the complex blister rust system into a simulator that 

will ultimately supply site-by-site and stand-by-stand prescriptions of integrated management options. The model was formulated 

with the aid of a continuous systems simulation compiler, called 

SIMCOMP, that allows biologists having a minimal command of 

computer programing to maximize their input into model formulation. 

The step-by-step formulation of the blister rust model is 

presented to illustrate model rationale and how use of SIMCOMP 

can facilitate incorporation of biologic knowledge. The detailed 

description also provides flow diagrams and quantitative 

relationships for implementation of the programing phase. Procedures for verification'of the completed simulation and needs 

for additional research are also discussed. 
CONTENTS 

INTRODUCTION . • 

The Blister Rust Epidemic • 

THE OVERALL MODEL 

SIMCOMP Method of Simulation 

DIAGRru~1ATIC AND WORD MODELS 

Ribes Density Submodel. 

Ribes Infection Submodel. 

Pine Host Target Submodel . . 

Pine Infection Submodel . . 

Stand Infection Submodel .•.•.. 

DISCUSSION 

verification. . 

Additional Research . 

PUBLICATIONS CITED . 

APPENDIXES . 

APPENDIX I. Quantitative Models 

APPENDIX II. Function Height .. 

APPENDIX III. Subroutine Spores. 

APPENDIX IV. Subroutine Girdle. 

Page 

1 

2 

2 

2 

5 

6 

8 

28 

34 

54 

65 

65 

66 

73 

79 

81 

121 

127 

133 
USDA Forest Service 

Research Paper INT-258 

January 1981 

COMPUTER SIMULATION OF 

WHITE PINE BLISTER RUST EPIDEMICS 

I. MODEL FORMULATION 

C.1. McDonald, R. J. Hoff, and W. Wykoff 

INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION 

U.S. Department of Agriculture 

Forest Service 

Ogden, Utah 84401 
THE AUTHORS 

GERAL I. McDONALD is principal plant pathologist on the station's 

genetics and pest resistance research work unit at Moscow, 

Idaho. Dr. McDonald received his B.S. in forest management 

(1963) and Ph.D. in plant pathology (1969) from Washington 

State University, Pullman. He has been investigating the 

genetic interaction of the blister rust organism and its hosts 

as well as epidemiology of blister rust since he joined the 

Station in 1966. 

RAYMOND J. HOFF is principal plant geneticist on the station's 

genetics and pest resistance research work unit at Moscow, 

Idaho. Dr. Hoff received a B.A. in biology (1957) from 

Western washington state University and a Ph.D. in botany 

(1968) from Washington State University, Pullman. He has 

been working on the development of western white pine resistance to blister rust and other pest problems with other conifers. 

WILLIAM R. WYKOFF is a research forester on the Station's quantitative analysis of forest management practices and resources 

for planning and control research work unit at Moscow, Idaho. 

Mr. Wykoff received his B.S. in forest management (1970) from 

the University of Minnesota, St. Paul, and his M.S. in forest 

management (1975) from Washington state University, Pullman. 

,Since joining the Station in 1974, he has worked on the development of tree growth models, and the implementation of these 

models in a stand development simulation program. 

ACKN OWLEDGMENTS 

The authors would like to express their sincere appreciation 

to those who contributed valuable technical assistance. Special 

thanks to Dr. Ralph Williams, USDA Forest Service, Region 1; Dr. 

Everett Hansen, oregon State University; and Dr. Neil Martin and 

Dr. David Hamilton, USDA Forest Service, Intermountain Station. 
INTRODUCTION 

The application of computer simulation to the quantitative analysis of plant 

disease epidemics has received increased attention in recent years (Waggoner and 

Horsfall 1969; Zadoks 1971, 1972; Kranz 1974; Waggoner 1974; Strand and Roth 1976; 

Rouse and Baker 1978). Computer simulation requires quantification of pest propagules, 

host target, and weather variables. These measurements are then used to formulate 

equations that relate the biotic and abiotic variables. 

Any specific modeling effort in biology is directed at one of the following four 

broad objectives: (1) compilation and summarization of the knowledge about a certain 

subject; (2) iilustration or exemplification in support of a provisional theory; (3) 

illustration or exemplification to show the inadequacies or incorrectness of a 

provisional theory; and (4) selection of the most reasonable theory from a series of 

theories about a certain phenomenon (Reddingius 1974). White pine blister rust is an 

excellent candidate for the application of general objective (1). This epidemic is a 

complex biological phenomenon that has been intensely studied over many years producing 

a great amount of data. Thus, our modeling objective is to bring together the 

available knowledge to try to understand how blister rust epidemics behave under 

various natural and man-influenced circumstances. The ultimate objective of this 

effort is to develop low cost, integrated rust management plans. The objectives of 

this paper are to (1) describe the components of the blister rust system, and (2) to 

present a model of blister rust epidemics. Details of programing the model will be 

published in a separate Station General Technical Report. 

Cronartium ribicola is a heteroecious full-cycle rust that infects to varying 

degrees all species of Pinus subgenus Strobus, section Strobus, and most species of 

Ribes. 

In most of the forested areas of the world located in the north temperate zone, 

Ribes and Pinus species grow together. Currently the rust is present in most localities, but in the warmer drier regions, the disease is found infrequently. 

Aeciospores are produced on pine in the spring when the Ribes plants are leafing 

out. The spores are windborne to the developing leaves where they germinate and 

infect the Ribes leaves. Infections begin producing urediospores in about 2 weeks. 

The urediospores reinfect Ribes leaves throughout the summer any time temperature and 

moisture conditions are favorable and viable spores are present. In northern Idaho, 

teliospores are present from July 1 to October 1 most years. They generally germinate 

immediately upon the presence of any moist period of at least 12 hours duration that 

occurs while viable teliospores are present. The sporidia produced by teliospore 

germination are windborne to the pine needles where they immediately germinate and 

infect the pine through stomata. Over the next 1 to 3 years, the fungus grows down 

the needle and penetrates the stem or branch. About 1 year after penetration of the 

stem, pycniospores are produced. The function of these spores has not been definitely 

shown, but they are believed to be spermatia--it is not known if the fungus is 

heterothallic or homothallic (McDonald 1978). The aeciospores are generally produced 

the year following pycnia production. 

1 
The fungus produces fusiform swellings and cankers on the branches and stems of 

susceptible trees. The cankers grow parallel (up and down) with the stems and 

branches at a rate of about 5-13 cm each way each year. The growth around the stem 

is about 3 cm each way each year, making the stem encirclement rate about 6 cm per 

year. Factors such as site, genotype of the host, and genotype of the rust may cause 

variation in the encirclement rate. Thus, the time needed to produce a canker that 

girdles a stem varies with the growth rate of the tree, growth rate of the rust, the 

diameter of the tree at the point that the fungus reached the stem, and possibly 

other genetic and environmental factors. This period can vary from a few weeks on 

small seedlings to 40 years or more on mature trees. After trees are girdled (even 

seedlings) it takes 3 to 10 years for the tree to die. 

In many cases, trees do not die entirely. Tops are lost; branches die; and 

growth slows down as a result. But, it can be generally sa~d that there is little 

effect on growth until shortly before death. 

I-Ieavy infection can cause defoliation of Ribes plants, but, in most areas in 

most years, the infection is not of sufficient magnitude to cause obvious damage to 

the Ribes population. 

The Blister Rust Epidemic 

The rust was introduced into the northeastern United States about 1900 on P. 

strobus seedlings shipped from France (Mielke 1943). The disease quickly spread 

through the range of P. strobus. 

The rust was introduced into western North America about 1916. This introduction 

was apparently also on infected P. strobus seedlings from France (Hahn 1949). The 

disease quickly spread throughout the extensive stands of western white pine in the 

Cascade and Rocky Mountains. The economic management of P. monticola is presently 

out of the question without some sort of control strategy (Ketcham and others 1968). 

THE OVERALL MODEL 

The two approaches to ecosystem modeling that have been commonly employed are 

analytical models and simulation models (Hall and Day 1977). A simulation ~odel was 

used for the white pine blister rust system because of our desire to incorporate as 

much biologic knowledge as possible. 

Box-and-arrow diagrams and word models were used to define the system and then 

both mechanistic and descriptive equations were created to describe the relations 

between state variables or "boxes" (see also Hall and Day 1977). The rationale of 

box-and-arrow diagrams is illustrated in figure 1. Each compartment is a state 

variable and the movement of quantities (needles, spores, Ribes bushes, infections, 

trees, and so on) from one box to another is called a flow. The flows are controlled 

by forcing functions, donor-controlled interaction, and donor-recipient controlled 

interaction. State variables will be represented by numbers and(or) numbers and XIS. 

Flows will be represented by J's. 

SIMCOMP Method of Simulation 

The computer programing system that was used is called SIMCOMP version 2.1 

(Gustafson and Innis 1973). Their system is based on "box-and-arrow" flow diagraming 

as shown in figure 1. 

2 
TNIB2 C 11B2 

1 nfection 51 tree 1 nfections/ tree 

X (12) 

Canker 

Appearance 

Function 

X (13) 

Figure l.--Rationale of the box-and-arrow method of diagramming simulation models illustrated with flow J(12+l3) from state variable X(12) to state variable X(13). Solid 

line from box to box shows flow of matter and arrow shows direction of flow. Dotted 

line shows flow of information that controls flow of matter through butterfly valve. 

The SIMCOMP approach has worked very well for us in providing guidance to the 

biologists (Geral I McDonald and R. J. Hoff) for development of the flow diagrams, 

and the SIMCOMP framework greatly facilitated communication between the biologists 

and the systems analyst (William R. Wykoff). Keep in mind as you read that the model 

was formulated to be programed by the SIMCOMP compiler. 

Since the blister rust system is complex, submodels were developed that group 

state variables composed of the same materials, like bushes per hectare or infections 

per tree. The amount of these materials may be increased, decreased, or left the 

same as time changes. In some cases, the time change from state variable to state 

variable occurs on an annual cycle and, in other cases, many state variable changes 

occur within one annual cycle. 

The model was organized into five separate submodels (fig. 2): 

A. Ribes density (bushes per hectare) 

B. Ribes infection (spores of infections per square centimeter of leaf surface) 

C. Pine target (square centimeters of infective surface per tree) 

D. Pine infection (infections per tree) 

E. Stand infection (trees per hectare) 

The model is developed with word and diagrammatic definitions of each flow 

within each submodel. Four appendixes provide the derivation of the mathematical 

formulas that specify the material and informational flows and the construction of 

some specialized functions and subroutines. These derivations were approached from 

the viewpoint of explaining natural processes rather than the expectation of a simple 

statistical fit (Jowett and others 1974). But, in many cases, lack of understanding 

forced us to accept a statistical fit or even an assumed relationship. 

Figure 2 shows the relationships among the various components. The Ribes density 

submodel computes the number of bushes per hectare as a function of light intensity, 

which is in turn computed from the Total number of Living Trees per hectare (TLT) and 

Current Tree Height (CTH). The level of Ribes eradication can be imposed as a 

control measure or set at a given value to simulate some natural restriction on Ribes 

density. This submodel computes Ribes Density for each of the four major species in 

northern Idaho to produce four outputs: RD(l), RD(2), RD(3), and RD(4) (see table 1 

for variable definitions). 

3 
.j::. 

Rlbes bushes 

ROI 

Basidiospares 

QBAB 

Reactivatedcanker Trees 

RSCIAS 

RBCIAS 

Damaged Tree. 

SCGS 

BCISG 

BCSG 

SCIGS 

Figure 2.--Flow diagram of white pine blister rust model. Rectangular boxes contain computed variables; paralle

grams contain inputs and outputs; and connectors indicate user-controlled variables defined as follows: conne

table 1 variables; connector 2 - SITE and TIME from table 1; connector 3 - variables from table 2; connector 4 

GPFi and TARGETi of table 2; connector 5 - OIP(OAAl), TCW i and TSO of table 3; connector 6 - table 9 variables

connector 7 - table 12 variables; connector 8 - table 17 variables; and connector 9 - FER and ICIAS of table 1
The Ribes infection submodel accepts user controlled (input) variables defined 

in table 3. (Tables numbered out of sequence are found under respective submodel 

discussions.) The output is the number of basidiospores per square centimeter 

available at the pine tree (QBSAT). The input variables are grouped into weather 

parameters, Ribes-species-specific-rust-growth parameters, and rust growth parameters. 

In all, 21 variables influence the simulation of rust development on the Ribes host. 

Next, the pine host target is developed. Only one computed variable (current 

tree height) and two input variables (table 3) are used here. This submodel provides 

the target area of an average tree each year by needle age and needle location (hole 

and branch). The output is expressed as square centimeters of infection surface. 

Note that current tree height is also used and so site quality is an additional 

variable. 

Individual pine tree infection is computed as the totai number of cankers on an 

average tree in the stand (ACPT), the Total number of Lethal branch Cankers (TLC) , 

and the Total number of Cankers derived from Stem-needles (TCOS) for each year of the 

simulation. Eight input variables (defined in table 4) interact with spore load 

(QBSAT) and Past Tree Height (PTH) to simulate the annual amount of new infection. 

The infection that develops annually on the average tree is expanded to a stand 

basis by the stand infection submodel. In all, 10 input variables (table 5) interact 

with the computed variables from the individual pine infection submodel, and the 

height function to simulate dead trees, damaged trees, recovered trees, clean trees, 

and healthy trees. Trees that recovered because of permanent canker inactivation 

feed back to the individual pine infection submodel. Trees that are resistant are 

removed from the system. Simulation output can be expressed as the level of ariy 

state variable over time or a disease progress curve for mortality, lethal branch 

infection, stem or needle-derived bole infections, or total infections. Note also 

that Total of Living Trees (TLT) per hectare is fed back to the Ribes density submodel. 

DIAGRAMMATIC AND WORD MODELS 

The simulation includes specification of varying levels and kinds of host 

resistance and other control measures, as well as specification of various epidemiological fitness traits (Nelson 1973) of the rust. The objective was to change the 

variables in order to observe their influences on the simulated development of rust 

damage on pine stands. Duration, seasonal frequency, and season of occurrence of 

infection periods can be specified also. Consequently, host-, fungus-, and environmentrelated parameters can be varied independently and their individual influences on the 

outcome of simulated epidemics studied. 

We are simulating infection on an average tree in a given stand. This average 

tree is characterized by an average target area, an average distance from basidiospore 

source to the tree, and, therefore, an average number of infections. The average 

infection level is then used to simulate infection of stand members on an individual 

basis. This simplification should work well since most white pine stands are evenaged. In this section, each flow will be defined in words, so that the biology of 

the model can be seen to be separate from the quantitative relationships. 

A submodel consists of a grouping of state variables (boxes) all defined by the 

same units such as bushes per hectare, spores per bush, infections per tree, or 

square centimeters per tree. Each state variable is delineated by the subscripted 

capital letter X, a number, and an acronym (for example, BClT). In the appendix 

description of the quantitative relationships, each flow will be quantified by an 

equation and each flow will be identified by a subscripted capital letter (J5 1)' 

, 

5 
RibeJ Density Submodel 

Generalized source of Ribes bushes (fig. 3) is partitioned each year into number 

of Ribes bushes per hectare for each of four Ribes species (5~1)(5~2) (5~3)(5~4). 

Rib~s bush populations are known to be influenced by amount of sunlight available. 

Some Ribes species are more responsive to sunlight than others; so the four principal 

Ribes species of northernrldaho are handled separately. Amount of sunlight reaching 

the forest floor depends largely on tree size and density. Second, Ribes bushes can 

be physically removed from the site or their populations reduced by other means. 

Thus, the number of bushes per hectare can be a function of management options. 

Third, the Ribes species are distributed according to their site requirements; a 

given site may not be suited for one or more of the species, requiring that density 

also be set independent of the sunlight relationship. All state variables and external 

variables relating to this submodel are defined in table 1 ~nd each flow path is 

described in table 2. 

Figure 3.--Box-and-arrow diagram of the ribes 

density simulation submodel (terms defined 

in table 1). 

6 

Ribes 

Density 

Function 

I 

I 

I 

, 

I 

I 

~ 

" , 

, 

, 

, 

\ 

\ 

\ 

I 

I 

/ 

\ 

RDl 

Bushes/ha 

RD2 

Bushes/ha 

RD3 

Bushes/ha 

RD4 

Bushes/ha 

2 

3 

4 
Table l.--Definition of variables used in Ribes density submodel 

Variable 

State variables: i 

X(l) RD. I 1 

X(2) RD. 2 1 

X(3) RD. 3 1 

X(4) RD. 4 1 

XeS) BUSHES 

User controlled variables: 

ERAD. 1 

i 

SITE 

TIME 

Computed variables: 

Flow path Flow name 

J(5,1) Ribes generation 

J (5,2) Ribes generation 

J Ribes generation (5,3) 

.J (5,4) Ribes generation 

Definition 

Ribes density, R. hudsonianum bushes per hectare 

Ribes density, R. inerme bushes per hectare 

Ribes density, R. lacustre bushes per hectare 

Ribes density, R. viscosissimum bushes per hectare 

Unspecified source of Ribes bushes 

If positive number = eradication level in bushes per 

hectare; 0 = Ribes species not present;-l = Ribes 

population under natural control. 

Index to Ribes species. I 

inerme, 3 = R. lacustre, 4 

R. hudsonianum, 2 

R. viscosissimum. 

Site index, total height at 50 years in feet. 

Current stand age, years 

See appendix I 

Table 2.--Ribes density submodel flow description 

Flow description Flow equation* 

0.05 + 2.15 PFS 16 . 58 ERAD(l) 

Annual production of RO(l) 0 ERAD (1) 

R. hudsonianum ERAD(l) ERAD(l) 

1.45 + 4.35 PFS l 1. 84 ERAD(2) 

Annual production of RO(2) 0 ERAD(2) 

R. inerme ERAD(2) ERAD(2) 

40.0 + 190.0 PFS IO . 96 ERAD(3) 

Annual production of RO(3) 0 ERAD~3~ R. lacustre ERAD(3) ERAD 3 

40.0 + 660.0 PFS 27 . 03 ERAD(4) 

Annual production of RD(4) 0 ERAD (4) 

R. viscosissimum ERAD(4) ERAD(4) 

R. 

< 0 

0 

> 0 

< 0 

0 

> 0 

< 0 

0 

> 0 

< 0 

0 

> 0 

*See Appendix I for discussion of flow equation derivation. 

7 
RibeJ Infection Submodel 

The Ribes infection submodel (fig. 4) is composed of four parts--one for each 

spore stage involved. The definitions of the variables used are given in table 3 and 

flows are described in table 4. Rust development and pine infection attributable to 

each of the four principal Ribes species of northern Idaho will be computed individually 

to create a composite spore load for each infection period by summation over all four 

Ribes species. Infection periods will be defined during the input of weather data. 

The model was made to respond to a variety of descriptions of infection period, which 

can be based on duration, date of occurrence, and average temperature. 

/ 

-:. .,..../ ./ 

./ 

I 

/ 

/ 

/ 

/ 

/ 

I 

I 

\ 

/ 

\ 

/ 

/ 

\ 

/'---

I 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

, 

Figure 4.--Box-and~arrow diagram of Cronartium ribicola infection on Ribes species 

(terms defined in table 3). 

8 
Table 3.--Definitions of variables used in the Ribes infection submodel 

Variable 

State variables: 

X(7S) 

X(76) 

X(77) 

X(7S) 

X(79) 

X(SO) 

X(Sl) 

X(S2) 

X(S3) 

X(S4) 

X(SS) 

X(S6) 

X (S7) 

X(SS) 

X(S9) 

X(90) 

SOURCE 

PASL 

AST 

ASGER 

ASTAL 

ASI 

USPOR 

VUSA 

USGER 

USTAL 

USI 

TMAX 

PGERT 

TGER 

QBAB. 1 

QBAB. 1 

i 

1 

2 

Definition 

Unspecified source of aeciospores. 

Potential aeciospore load, spores per square centimeter. 

Number of aeciospores per square centimeter trapped 

in a given infection period, spores per square 

centimeter of Ribes leaf (under surface). 

Number of aeciospores per square centimeter that 

are expected to germinate, spores per square centimeter. 

Aeciospores expected to produce germ tubes of 

adequate length to cause infection, spores per square 

centimeter. 

Aeciospores expected to cause infections during a 

given infection period, spores per square centimeter. 

Urediospores available for infection from both aecial 

and uredial infections, spores per square centimeter. 

Viable urediospores, spores per square centimeter. 

Urediospores germinated during a given infection 

period, spores per square centimeter. 

Urediospores expected to produce germ tubes of 

adequate length to cause infection, spores per 

square centimeter. 

Urediospores expected to cause infections during 

a given infection period, spores per square centimeter. 

Teliospores - maximum number available, spores 

per square centimeter. 

Probable number of teliospores available for 

germination, spores per square centimeter. 

Teliospores expected to germinate during a given 

infection period, spores per square centimeter. 

Quantity of basidiospores expected to be produced 

at Ribes hudsonianum bushes, spores per square 

centimeter. 

Quantity of basidiospores expected to be produced 

at Ribes inerme bushes, spores per square 

centimeter. 

(continued) 

9 
Table 3.--Definitions of variables used in the Ribes infection submodel (continued) 

Variable 

State variables: 

X(9l) 

X(92) 

X (93) 

QBAB. 1 

QBAB. 1 

QBSAT 

i 

3 

4 

User controlled variables: 

AGMAX 

AIYR. 1 

AMHL 

BMASS. 

1 

BSYR. 1 

DAAI 

DAP 

DIP(DAAl) 

GPF. 1 

i 

TARGET. 

1 

TCW. 

1 

Definition 

Quantity of basidiospores expected to be produced 

at Ribes lacustre bushes, spores per square 

centimeter. 

Quantity of basidiospores expected to be produced 

at Ribes viscosissimum bushes, spores per square 

centimeter. 

Quantity of basidiospores expected at trees, 

spores per square centimeter. 

Temperature where aeciospore germination is maximum, 

degrees C. 

Aeciospore infection yield ratio expressed as infections per germinated aeciospore and indexed by Ribes 

species. 

Average maximum hourly load of aeciospores, spores 

per square centimeter per hour. 

Biomass of urediospores/teliospores produced by a 

single infection and indexed by Ribes species, m3 . 

Potential basidiospore yield ratio indexed by Ribes 

species, basidiospores/teliospore. 

Days after April 1 to midpoint of an infection 

period indexed to infection periods (0 to many/yr). 

Date of aeciospore appearance expressed as days 

after April 1 (l/yr). 

Duration of infection period, indexed by infection 

period, hours. 

Growth point factor, indexed by Ribes species and 

expressed as Ribes bushes/clump. 

Index to Ribes species. 1 = R. hudsonianum, 2 = 

R. inerme, 3 = R. lacustre, 4 = R. viscosissimum. 

Ribes target, indexed by species, expressed as 

cm2 /bush. 

Average telial column width, indexed by species, 

expressed in microns. 

(continued) 

10 
Table 3.--Definitions of variables used in the Ribes infection submodel (continued) 

Variable 

User controlled variables: 

TIP (DAAl) 

TSAR. 1 

TSD 

TSL 

TSUR. 1 

UGMAX 

UIYR. 1 

USD 

USL 

Computed variables: 

Definition 

Average temperature of an infection period, indexed 

by infection period, expressed as degrees C. 

Teliospores from aeciospores infection ratio, indexed 

by Ribes species expressed as proportion of spore 

biomass arising from aesiospore infections that is 

devoted to production of teliospores. 

Average teliospore diameter, microns. 

Average teliospore length, microns. 

Teliospores from urediospores infection ratio, 

indexed by Ribes species expressed as proportion of 

spore biomass arising from aeciospore infections 

that is devoted to production of teliospores. 

Temperature where urediospore germination is maximum, degrees C. 

Urediospore infection yield ratio, indexed by Ribes 

species and expressed as infections per germinated 

spore. 

Average urediospore diameter, microns. 

Average urediospore length, microns. 

See appendix I 

11 
f-L 

N 

Flow path 

J(75-+76) 

J (76-+77) 

J(77-+78) 

J (78-+79) 

J (79-+80) 

J(80-+8l) 

Flow name 

Aeciospore generation 

Aeciospore availability 

Aeciospore germination 

Aeciospore germ tube 

growth 

Aeciospore infection 

establishment 

Urediospores from 

aeciospores and 

uredispores 

Table 4.--Ribes infection submodel flow descriptions 

Flow description 

Computation of potential aeciospore 

load expressed in aeciospores per square 

centimeter per infection period 

Computation of number of aeciospores 

available per square centimeter of 

Ribes leaf surface during a 

specified infection period 

Calculation of number of trapped 

aeciospores expected to germinate 

during a given infection period per 

square centimeter of leaf surface 

Computation of number of germinated 

aeciospores per square centimeter of 

leaf surface expected to produce 

germ tubes of sufficient length for 

penetration of Ribes 

Computation of number of expected 

aeciospore germ tube penetrations 

that establish infection per square 

centimeter of lower leaf surface 

Computation of urediospore 

production by infections caused by 

aeciospores 

PASL. 1 

AST. 1 

ASGER. 1 

ASTAL. 1 

A

1.9

A

ASI~ = AST

1

USPOR. 1 U
f-' 

(,N 

Flow path 

J(8l-+82) 

J(82-+83) 

JC83-+S4) 

J(84-+85) 

JCS5-+8l) 

JCSO-+86) 

J(85-+86) 

Table 4.--Ribes infection submodel flow descriptions (continued) 

Flow name 

Urediospore viability 

Urediospore germination 

Urediospore germ tube growth 

Urediospore infection 

Urediospore infection from 

urediospores 

Teliospore generation 

Flow description 

Computation of numb~r of germinable 

urediospores per square centimeter 

in a given infection period 

Computation of number of 

urediospores that germinate 

during a given infection period 

Calculation of number of germinated 

urediospores expected to produce germ 

tubes of sufficient length to penetrate 

Computation of number of urediospore infections per square centimeter of Ribes 

leaf caused by urediospore infections 

Computation of the number of 

urediospores produced per square 

centimeter of Ribes leaf by 

urediospore infections 

Computation of number of teliospores 

expected to arise from aeciospore 

and urediospore derived infections 

VUSA. I 

USGER. 1 

USTAL. 

US I. 

1 

I 

lJSPOR. I 

TMAX-. I 

2

US

T

+ T
I--' 

oj:::. 

Table 4.--Ribes infection submodel flow descriptions (continued) 

Flow path 

JC86+87) 

JC87+88) 

JC88+89) 

JC89+93 

;90+93 

;91+93 

;92+93) 

Flow name 

Teliospore viability 

Teliospore germination 

Basidiospore generation 

Basidiospore distribution 

*See appendix I for derivation of flow equations 

Flow description 

Computation of number of teliospores 

viable during a given infection 

period per square centimeter of 

leaf surface 

Computation of number of teliospores 

expected to germinate during a 

given infection period 

Computation of number of basidiospores 

produced for each square centimeter 

of underside leaf surface per 

infection period 

Computation of number of basidiospores 

per square centimeter of flat surface 

at the white pine tree 

PGERT. 1 

TGER. 1 

QBAB. 1 

QBSAT 

TMAXi·

PGERT .. P

1 

TGER .. BS

4 

l: 

i=l 

1 

[

QBA

282

Q

+ 
The infection period defines that period of favorable environmental circumstances 

during which spores are released, transported to an infection court, germinate, and 

establish a new thallus. 

The two criteria used to define Duration of Infection Period (DIP) were (1) S 

consecutive hours of assumed 100 percent relative humidity as determined from hourly 

precipitation records where each hourly record was required to have at least 0.01 inches 

of rain, and (2) moisture in the woods was assumed to remain at or near 100 percent 

relative humidity for 2 hours after the 5-hour wet period if the period ended during 

darkness. Thus, the minimum infection period was 5 hours. Next, date of occurrence 

of the infection period was expressed as number of Days After April 1 to the midpoint 

of the period (DAAl). Average Temperature of an Infection Period (TIP) was obtained 

by summation of daily maximum and minimum temperatures divided by the number of 

records summed. The infection periods can be redefined so that one could investigate 

the influence of this definition on the simulated outcome or use data from a specified 

location. 

The infection process will be divided into five major areas: trapping, spore 

viability, spore germination, germ tube growth rate, and probability of penetration 

(infection yield ratio). Details of trapping will be discussed for aeciospores, 

urediospores, and basidiospores. Viability is assumed to be controlled by a timedependent function. Germination is divided into germination lag period (the time 

between onset of conditions for germination and beginning of germination); germination period (the time from beginning to end of germination); and germination 

percentage. All three of these facets of germination are easily measured, as is germ 

tube growth rate. The last process, probability of penetration, will be considered 

as a catchall. We will assume that only Infection Period Duration (DIP) and Infection 

Period Temperature (TIP) will influence the germination process. But all kinds of 

factors from rust and pine genotype to chemicals applied to control the rust will be 

assumed to influence probability of penetration. This probability can be measured by 

supplying optimum temperature and duration conditions, then varying all other factors. 

We have accordingly measured Basidiospore Infection Yield (BIY) and Aeciospore Infection 

yield (AIY) and have incorporated a place for Urediospore Infection Yield ratio 

(UIY), but have not measured it. It is expected that the model will be very sensItIve 

to each of these values. Thus, they can all be used to make many kinds of adjustments 

to simulate the influence of control chemicals, weather, Ri.bes and pine resistance 

genes, and rust genetic variation. Each specific IY will be discussed in the appropriate place. 

Within the commercial range of western white pine in Idaho, the four principal 

species of a total of 15 Ribes species are R. hudsonianum (R. petiolare), R. viscosissimum, R. lacustre, and R. inerme (Mielke and others 1937). Intensification 

occurs at different rates on R. hudsonianum, R. inerme, and R. viscosissimum (Mielke 

and others 1937), but little if any intensification occurs on R. lacustre (Mielke and 

others 1937; Buchanan and Kimmey 1938). Consequently, calculation of basidiospore 

loads must take into account Ribes species, in addition to Ribes population densities, 

aeciospore loads, and weather. Thus, each state variable is actually one of a set of 

four that will be indexed by i (Index to Ribes Species). This submodel will be 

described for R. hudsonianum (i=l). 

The first problem was to develop a model describing the amount of Ribes leaf 

surface to be expected for each of the species. In general, the four Ribes species 

under consideration complete their growth by the first of July in northern Idaho 

(Mielke and others 1937). We assumed that the amount of Ribes material available on 

July 1 will be available while aeciospores are flying, which is generally about midMay in northern Idaho. Huch of the available data is on the basis of individual 

bushes. Mielke and others (1937) supply data on the number of leaves per bush for 

the four species and Lachmund (1934) supplies data on the average size of the 

leaves for the four species (table 5). 

15 
Table 5.--Average number of leaves per bush and average si ze of leaves in square 

centimeters for four Ribes species 

Number of Leaf Number of Leaf surface/ 

Species i leaves size 1 leaves/bush2 bush 

cm2 cm2 

R. hudsonianum 1 385 35.45 470.6 16.7 x 10 3 

R. inerme 2 325 14.06 578.9 8.1 x 10 3 

R. lacustre 3 342 6.68 907.3 6.1 x 10 3 

R. viscosissimum 4 362 16.20 240.2 3.9 x 10 3 

lCalculated from Lachmund (1934, table 2). Includes both open and shade forms. 

2Calculated from figure 3 and table 5, Mielke and others (1937) and includes open, 

shade, and partial-shade forms. 

We decided to model Ribes infection from the viewpoint of individual infections 

(spores) because of some interesting epidemiological perspectives that came out of 

the study of single aeciospore isolates of C. ribicola on detached R. hudsonianum 

leaves. These studies were designed to investigate the genetics of C. ribicola, but 

the following observed features of the life cycle of the rust were pertinent to our 

modeling effort: 

1. Rarely does an aeciospore infection produce teliospores directly. 

2. Infections started from urediospores produce a mixture of teliospores and 

urediospores. 

3. The relative proportion of urediospores and teliospores seems to be influenced 

by leaf physiology because urediospore populations derived from a single aeciospore 

(rust genetics controlled) and placed on several single leaves from one Ribes clone 

(Ribes genetics controlled), each in a separate Petri dish and all held at 21°C, 

will produce a variable ratio dish to dish. 

4. Temperature seems to be involved in some way because urediospore infections 

held at 13°C produce mostly telia, whereas those kept at 21°C produce mixes. 

5. The average partitioning seems to be about a 1:1 ratio (urediospores: 

teliospores) of the spore yield capacity of each infection on a biomass basis at 

21°C. 

6. A given infection seems to have a definite life expectancy, thus constant 

spore production requires continued reinfection. 

7. The stages in the life of an individual aeciospore infection seem to require 

14 days incubation, urediospore production peaks at 25 days, trailing to nontransferable 

(nonviable) urediospores at about 35 days in sealed Petri dishes at 100 percent 

relative humidity. 

16 
8. The stages in the life of an individual urediospore infection are 14 days 

incubation followed by urediospore and teliospore appearance at 14 days and then 

peak urediospore and teliospore production between 25 and 28 days. 

9. Teliospores are produced in the enclosed Petri dishes and germinate as soon 

as mature, whether they are held at a constant 21°C, kept at 13°C nights and 21°C 

'days, or held at a constant 13°C. 

These observations lead to a description of the life cycle of C. ribicola that 

is somewhat different from current belief (Van Arsdel and others 1956). The current 

thought is that a 13°-16°C night temperature is required to trigger formation of 

significant numbers of teliospores and that about 30 days are required from inoculation 

with aeciospores to formation of teliospores. Also, the teliospores typically show 

up about 2 weeks after the 13°C night temperatures are initiated. Based on our 

observations, we believe that the "trigger" is a temperature shock to the urediospores 

produced by the initial aeciospore infection and a temperature drop that causes the 

formation of dew needed for the urediospores to reinfect. Then the lower temperatures 

cause the new uredial infections to produce mostly teliospores. The net result is a 

new cycle of infections from urediospores, which in turn produce teliospores. In 

our experience with detached leaf culture of C. ribicola, we have tried repeatedly 

to obtain teliospores by using aecial infections and a temperature trigger. Poor 

yields were obtained until we began making urediospore inoculations. Then, teliospores 

were produced in as little as 14 days after urediospore inoculation, even under a 

constant 21°C. 

Our work with detached leaf culture has also led to our strong feeling that a 

given infection produces spores and then dies. Thus, continuation of the rust over 

the summer requires the proper matching of periods favorable for infection and of 

viable inoculum. More on this later. 

With that view of the life cycle of C. ribicola in mind, we went to the literature 

in search of more information. In 1918, L. H. Pennington and W. H. Snell studied 

uredial generations in the Adirondack region of New York State (reported by Spaulding 

1922). They observed seven periods of urediospore production. The first generation 

started on May 28 and reached a peak on June 12, 14 days later. The second generation 

peaked about June 27, and the third peaked about July 12. In order to assess the 

timing accurately, one would have to have the timing of the infection periods. But, 

the appearance and general features of Pennington and Snell's periods seem to match 

well with stages of our hypothesized life cycle. Also, the appearance of telia in 

the above case matched our observations. Telia appeared first on June 28 with the 

second generation of urediospores and then appeared with all subsequent generations. 

If a specific niching of viable urediospore populations and infection periods is 

required for the successful increase of C. ribicola, then several factors about the 

epidemics on Ribes become evident. First, each individual group of aecial infections 

has a set of future infection periods with which it must match for greatest buildup. 

Thus, the more cohorts started, the higher the chance of a matching weather sequence. 

With this thought in mind, some data from 1931 through 1936 (Stillingerl) were analyzed. Occurrence of wet periods and the level of Ribes infection were given for each 

of the 6 years. There appeared to be a high degree of correspondence between the 

length of dry period between wet periods and the level of infection. A 2l-day cycle 

of wet periods throughout the summer seemed to be related to highest levels of 

teliospore production. 

lStillinger, C. R. 1936. Infection studies at Newman Lake, Washington 1936, 

p. 128-137. In 1936 annual report, Spokane Office of Blister Rust Control (Mimeographed). U.S. Dep. Agric. Bur. Plant Quar. 

17 
If the above is true, then the number and temporal distribution of periods of 

infection by aeciospores becomes important. Knowledge about the relative abilities 

of aeciospores and urediospores to spread the rust also becomes important. During 

the course of our investigations into single-spore cultures of the rust, we have 

transferred thousands of single aeciospores and hundreds of single urediospores. A 

fine glass needle about the same diameter as the spores being transferred was used. 

The aeciospores and urediospores exhibit strikingly different behavior patterns 

toward electrostatic forces and water. Aeciospores appear to have an active electrostatic behavior. These spores are either attracted to or repelled by the glass 

needle. Sometimes it is nearly impossible to remove them from the glass needle and 

at other times it is nearly impossible to pick them up. In regard to water, aeciospores 

seem to be repelled by water. It is very difficult to make an aeciospore adhere to 

the surface of a large (100~ diameter) water drop. They are most easily placed on a 

leaf surface covered with droplets about 10-lS~ in diameter. They, of course, 

require free water for germination; but their electrostatic' and water behavior 

patterns fit the theory of wind transport. Also, aeciospores seem to come from the 

aeciurn in a dehydrated state (they look like deflated footballs). They then hydrate 

(inflate) quickly when placed on a wet leaf. 

On the other hand, urediospores show the opposite behavior. These spores are 

easy to pick up with the glass needle -- they appear to be much less active electrostatically. When placed on the surface of a water drop, they quickly leave the 

needle and slowly drift to the junction between the drop surface and the leaf surface. 

Also, urediospores seem to be turgid when in association with the sorus, but they 

quickly dry out with a complete loss of viability when placed in a dry environment. 

We have noted much better inoculation success when urediospores have been transferred 

en masse to water drops rather than to the wet surface of a leaf. 

The water and electrostatic behavior differences between the two kinds of 

spores lead us to believe that aeciospores are a dispersion spore-type influenced 

greatly by the wind and that urediospores probably are moved very little by the wind, 

but are readily transported by insects and water. Thus, urediospore spread should be 

limited largely to bushes that were the site of an aeciospore infection. A search 

of the literature shows that the two spore stages function as expected. Viable 

aeciospores have been known to travel up to 400 miles (Mielke 1943). Urediospores 

have been trapped up to 1,000 yards from a known source (Spaulding 1922), but their 

effectiveness in causing disease at this distance is unknown. Snell (1920) reported 

that most urediospore spread was limited to the bush infected by aeciospores. It 

seems likely to us that spread is usually limited to within the bush and we will 

model accordingly. 

Variation in incubation periods, ,spore yields, ratio of urediospores to teliospores in both aecial and uredial infections, spore echinulation patterns, and 

telial column length and width all could modify the epidemiological fitness (Nelson 

and McKenzie 1973; Nelson 1973) of the Ribes portion of the life cycle of C. ribicola. 

Changes here cause changes in the sporidial load faced by the pine host. Many 

opportunities may exist to modify and to monitor these epidemiological fitness 

traits. Although these aspects are not included, such knowledge should eventually be 

integrated into the rust management plans to fine tune the model. 

Next, we wanted to develop a model of the level of infection on Ribes leaves or, 

more specifically, the level of basidiospore inoculum to expect in differing seasons 

and localities for each infection period. The level measured by percentage of leaf 

surface bearing telia could not be used. This measure varies a great deal, depending 

on the interaction of weather and site (Mielke and others 1937), and on the susceptibility of the Ribes leaves to aeciospore infection. Leaf susceptibility is related 

somewhat to leaf age and varies from species to species and from open to shade form 

leaves (Lachmund 1934). The model should be able to process Ribes species distribution 

patterns and environmental input data into predicted basidiospore loads season by 

season and locality by locality. 

18 
~ 

': 

6 

First, we assume that all the aecial infections from a single infection period 

form a single cohort. Then, each cohort can be thought of as developing according to 

the scheme shown in figure 5. Study of the figure shows the importance of pattern of 

infection period occurrence. Temporal variation of aeciospore, urediospore, and 

teliospore viability and occurrence, that could not be incorporated into the figure, 

are illustrated in figures 6 and 7. We assume that the urediospore and teliospore 

viability-occurrence curves are -sharp. The assumption is based on data obtained 

from Spaulding (1922), Spaulding and Rathbun-Gravatt (1925b), and our experience with 

single aeciospore cultures. If our assumptions are correct, it is obvious that 

shifts of a few days one way or the other in the infection period peak can cause 

major differences in the amount of multiplication and the ultimate level of basidiospore 

I (, 

k 

!: 

I 

I 

< 

:': 

I 

I 

i 

I~ I 

I 

I 

m I 

I 

; .. " I 

.~ UREDIOSPORES I 

'.' ... 

I TELIOSPORES 

I 

I 

I 

I 

I 

~~",~j~~:t;(.'rJ. • lb'~~IT~·~~ . ~ AECIOSPORES ~ 

'I 

I 

I 

I 

I 

I 

I 

I 

I 

UREDlOSPORES 

I 

I 

I 

I 

I 

I 

I ., I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

'.b I 

~'o 'I 'I 

~'o 'I 

~'o 'I ~Io 'I ~Io 'I ~Io 'I J 'I ;1 'I " 'I " 'I :1 " , 'I ! 'I " '\ 'I 'I 'I 'I 

30 90 00 110 120 130 140 150 160 170 180 190 

Days after April 

Figure 5.--Theoretical distribution of individual cohorts of Cronartium ribicola 

infection on Ribes sp. Assumptions: (1) aeciospore presence is Apri lIto May 31; 

(2) aeciospore infections produce only urediospores; (3) urediospore infection 

produces 1:1 ratio of urediospores to teliospores; and (4) urediospore viability of 

one cohort set at 45 days and teliospore viability set at 70 days. Occurrence of wet 

periods as in northern Idaho(1934)and according to Paine (1947) low pine infection 

occurred. 

19 

! 200 
inoculum. Figure 5 also shows why 1934 was a poor pine infection year (Paine 1947). 

A good early buildup was followed by an absence of wet periods when the teliospores 

were present. If instead, an infection period had fallen at about day 110 (late 

July) a massive buildup would have resulted. Then, an infection period of sufficient 

duration in mid-August would have resulted in high pine infection. The Ribes infection 

submodel (fig. 4) was developed to operate on the basis of infection periods that 

can be defined in different ways as already discussed. The current form makes use 

of days after April to midpoint of infection period (DAAl) , Duration of Infection 

Period in hours (DIP), and average of maximum and minimum Temperatures for the 

Infection Period (TIP). 

1.0 

:E 

:::J 0.8 

:E 

x 

~ 0.6 LI0 

Z 

0 

I- 0.4 a:::: 

0 

0... 

0 

a:::: O. 2 0... 

o 4 8 12 16 20 24 28 32 36 40 

DAYS AFTER AT START OF PRODUCTION WAPA) 

Figure 6.--Assumed distribution of Cronartium ribicola aeciospores as based on 

.description of occurrence by Mielke and Kimmey (1935). Curve produced by Weibull 

function (equation 2 where: a = 0, b = 22.0, c = 3.6, and x = DAPA). 

1.0 ,-, 

~ 0.8 

U rediospores I \ 

::> I \ 

:E , \ 

x , \ 

~ 0.6 , \ 

, \ 

u... \ 0 , 

z , \ 

0 0.4 f \ l- I \ a:::: \ 0 I 0... , \ 0 0.2 \ a:::: , \ 0... I \ 

I " 

0 8 16 24 32 40 48 56 64 '72 80 88 

DAYS AFTER INOCULATION 

Figure 7.--Assumed distribution of urediospore and teliospore availability based on 

data and descriptions of Spaulding (1922) and Spaulding and Rathbun-Gravatt (1925b). 

(proportion urediospores = equation 2 where a = 12, b 20, c 2.5, w = 2.477, and 

x DAYS; proportion teliospores = equation 2 where a = 16, b = 27.5, c = 2.25, 

w = 2.416, and x = DAYS). 

20 
The number of infections expected from an aeciospore load expressed as spores 

per square centimeter will be calculated for each species. If the amount of infection 

from each cohort of aeciospore infections is calculated separately, then the entire 

season can be simulated for each species. R. hudson~anum generally inhabits creek 

bottoms and small openings in the forest. Lloyd (1961) showed that dew is found in 

canyon bottoms more often than it is in upland areas and that once formed it lasts 

longer, Thus, we assumed that this species would be subject to longer infection 

periods (3 hours longer) than the other three, which are generally upland species. 

No infections will occur after an infection period of less duration than the time 

needed for spores to germinate and penetrate Ribes leaves. 

The box-and-arrow diagram will now be described flow by flow in words. But, 

this submodel will cycle by infection periods and Ribes species whereas the others 

cycle once annually. 

AECIOSPORE INFECTION 

computation of potential aeciospore load (75+76, fig. 4). Prediction of average 

number of infections per bush requires knowledge of aeciospore load, but this value 

is not presently available because it has not been measured. Aeciospore loads 

should be monitored for several years at a range of sites. In the absence of data, 

we will assume an Aeciospore average Maximum Hourly Load (AMHL) that sets maximum 

Potential Aeciospore Load (PASL) for an area. 

Computation of number of aeciospores trapped per cm2 of Ribes leaf surface 

during a specified infection period (76+77). This relationship was developed from 

the results of Mielke and Kimmey (1935), who listed aeciospore appearance dates, 

date of peak cast, and end of cast at seven sites over a number of years. We adjusted 

all their observations to a common blister-appearance time and calculated the average 

number of days to beginning of aeciospore cast (16 days), number of days to peak 

cast from first appearance (21 days), and duration of the cast (49 days). These 

features were used to develop the assumed availability curve shown in figure 6. 

Calculation of proportion of trapped aeciospores expected to germinate during a 

given infection period (77+78). To incorporate infection-period temperature as well 

as duration into the model, aeciospore and urediospore germination tests were conducted. 

Fresh aeciospores or urediospores were mixed with distilled water and sprayed on 

water agar in Petri dishes. Fifteen inoculated plates were placed in incubators set 

at 5-degree intervals from 0° to 30°C so that one observation per temperature per 

hour from 2 hours to 16 hours and one observation at 24 hours after start of germination could be made. Each plate was discarded after percent germination (based on 

100 spores) and germ tube length (based on 20 spores) were recorded. The results 

are presented as temperature-dependent plots in figures 8, 9, 10, 11, 12, 13, and 

14. Curves in figures 7, 8, 11, 13, and 14 were developed using the Weibull function 

(eq. 2 appendix I) (Bailey and Dell 1973). 

21 
1.0 

8 0.8 Ioe:( 

z 

:E 

~ 0.6 (,!) 

z 

o 

I- 0.4 0::: 

o 

a.. 

o 

~ 0.2 

~--. 

/. " 

I 

I 

16°C Optimum,' 

, 

, 

, 

" 

, 

, 

" . .............• aIIIII::...-'-.... .....&.....L.....L_~"--..I.....,I;;,I1IIIIo 4 8 12 16 20 24 28 

12 

10 

8 

6 I4 

2 

o 4 

24 

20 

16 

12 

I8 

4 

o 

TEMPERATURE (OC) 

I ... ------Limits -------t~ ... 1 

8 12 16 20 

TEMPERATURE ( DC) 

24 28 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

14 ... __ ------Limits------~ 

I 

Urediospores A 

Aec iospores • 

TEMPERATURE (oC) 

22 

32 36 

Figure 8.--Proportion germination of Cronartium ribicola aeciospores as a 

function of temperature. 

Solid line shows optimum 

at 20°C according to 

author's unpublished data 

and dotted line shows optimum at 16°C according 

to Van Arsdel and others 

(1956). Optimum is variable to simulate a variable 

epidemiological fitness 

trait, (proportion = equation 2 where: a = 3.0, 

b WBA, c = WCA, w = 10.3, 

x = temperature). 

Figure 9.--Relationship 

between temperature and 

period of germination lag 

of Cronartium ribicola 

aeciospores. Data points 

from author's unpublished 

data; curve fit manually. 

Figure 10.--Relationship 

between temperature and 

period of germination of 

Cronartium ribicola aeciospores and urediospores. 

Data points from author's 

unpublished data; curve 

fit manually. 
Figure ll.--Growth 

rate of Cronartium 

ribicola aeciospore 

germ tubes. Data 

points from author's 

unpublished data 

and curve produced 

by the Weibull 

function (equation 

2 where a = 3.5, 

b = 15.0, c = 3.5, 

w = 126.37, and 

x temperature). 

Figure l2.--Relationship between temperature and period 

of germination lag 

of Cronartium ribicola urediospores. 

Data points from 

author's unpublished 

data; curve fit 

manually. 

Figure l3.--Growth 

rate of Cronartium 

ribicola urediospore germ tubes. 

Data points from 

author's unpublished 

data and curve produced by Weibull 

function (equation 

2, where a = 3.5, 

b ; 15.0, c = 3.5, 

w • 136.23, and 

x temperature). 

80 

60 

~ 40 

:::t. 

20 

o 

TEMPERATU RE ( °C) 

12 I 

1 

Limits 1 

,. "I 10 

8 

~ LLJ 6 :E 

I4 

2 

0 4 8 12 16 20 24 28 32 36 

TEMPERATURE (oC) 

80 

60 

~ 40 

::L 

20 

o 

TEMPERATU RE ( °C) 

23 
l.0 

8 

I- 0.8 « 

z 

~ 

85 0.6 ~ 

z 

0 

I- 0.4 c::: 

0 

Cl.. 

0 

c::: 

Cl.. 0.2 

, ..... I ' , \ 

I 

I 

ISoC Optimum,' 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

4 • 12 16 20 24 

TEMPERATURE (OC) 

28 32 36 40 

Figure l4.--Germination of Cronartium ribicola urediospores 

as a function of temperature. 

Dotted line shows optimum at 

l5°e according to author's unpublished data and solid line 

shows optimum at 20 0e, according to Van Arsdel and others 

(1956). Optimum is variable 

to simulate a variable epidemiological fitness trait 

(curves simulated by Weibull 

function, equation 2, where 

a = 7, b = WBU, c = weu, w = 

10.3, x = temperature). 

Three features of germination were utilized: first, the proportion germinated 

as a function of temperature (fig. 8); second, the period of germination (fig. 9); 

and third, the amo~nt of time a population of spores required to complete germination 

as a function of temperature (fig. 10). 

computation of number of germinated aeciospores expected to produce germ tubes 

of sufficient length for penetration (78~79). It has been observed that germ tubes 

grow flat on the surface of Ribes leaves and that penetration occurs through stomata 

and then only when the germ tube passes directly over the stomatal opening (personal 

communication, J. Y. Woo). The germ tubes seem to grow to a maximum length of 

around 500 to 600~ (authors' unpublished data). Also, germ tubes of less than 100~ 

or more than 300~ in length are seldom seen to penetrate (personal communication, J. 

Y. Woo). Given a temperature dependent GERm Tube growth Rate of Aeciospores(GERTRA) 

(fig. 11), probability of penetration by a germinated spore should be a function of 

germ tube length and relative densities of stomata and aeciospores. 

computation of expected aeciospore germ tube penetrations that establish infection 

(79~80). We have determined Aeciospore Infection Yield Ratio (AIYR(ISP)) on detached 

Ribes hudsonanium leaves, but Lachmund's (1934) study of aeciospore infection provided 

us with some additional information. His data were presented as bar graphs of infection 

spots per square inch obtained from artificial inoculation of leaves in 10 age groups 

from four Ribes species. Values were read from the bar graphs, averaged over all 

leaf ages and leaf types (open form and shade form), and converted to infections per 

square centimeter (table 6). A second study (Pierson and Buchanan 1938a) provides a 

good comparison of aeciospore and urediospore infection levels. In the spring of 

1976, we made 7,500 single spore transfers and obtained 69 takes (authors' unpublished 

data). If we assume a similar success ratio in the wild, we can expect 0.009 infections for every aeciospore trapped by the lower surface of R. hudsonianum leaves. 

We felt that this single determination of infection yield would be insufficient; 

so some infection experiments were conducted on detached R. hudsonianum leaves. All 

the tests were based on two Ribes clones and aeciospores from a single canker. This 

canker was on a Pinus flexilis seedling that was inoculated in one of our large-scale 

progeny tests. Ten leaves were collected from each of two Ribes clones growing in a 

greenhouse. Each leaf was placed on a wet filter paper in a Petri dish. One rubbercement-coated cover slip was placed next to the leaf and inside the Petri dish. A 

water suspension of aeciospores was sprayed onto the leaf and cover slip with an 

aerosol-powered chromatography sprayer. The Petri dish was then sealed and placed in 

a growth chamber held at a constant l7°e and l6-hour photoperiod of about 600 fc. 

24 
Germination of the spores was recorded after 24 hours and infection after 16 days as 

number of pustules per square centimeter of leaf surface. We assumed one pustule per 

penetration and the results are shown in table 7. Evidently there is some sort of 

specific genetic interaction between Ribes and C. ribicola, since we obtained good 

infection on clone 1 and no infection on clone 2. We do not know the extent of the 

genetic interaction or its influence on the epidemic. 

Table 6.--Relative levels of Cronartium ribicola aeciospore and urediospore infection 

on the four principal Ribes species of northern Idaho 

AeciosEore infections I Urediospore infections 

Per cm2 Per cm2 

Species ISP leaf Relative leaf Per cm2 leaf 

R. hudsonianum 1 16.89 1. 00 2 3.8 2 5.7 

R. inerme 2 14.57 0.86 

R. lacustre 3 7.13 .42 

R. vi scosissimum 4 4.65 .27 

lCalculated from Lachmund (1934). 

2Calculated from Pierson and Buchanan (1938a). 

Table 7.--Infection ratio of Cronartium ribicola aeciospores on detached leaves of 

Ribes hudsonianum obtained from two clones 

Clone 1 Clone 2 

Percent Number Infections/ Spores/ Percent Number Infections/ 

Leaf germinated germi- cm2 infec- germinated germi- cm2 

number nated/cm2 ion nated/cm2 

1 38 250 21 11. 9 43 295 0.0 

2 53 420 20 21. 0 55 160 0.0 

3 57 415 90 4.6 54 135 0.0 

4 70 605 9 67.2 56 145 0.0 

5 .60 175 13 13.5 60 135 0.0 

6 62 225 34 6.6 70 190 0.0 

7 61 215 0 0.0 66 135 0.0 

8 72 245 10 24.5 51 155 0.0 

9 62 255 14 18.2 45 75 0.0 

10 73 225 1 225.0 44 140 0.0 

Totals 3030 212 1565 0.0 

25 
computation of urediospore production by infections caused by aeciospores (80~8l). 

Our goal was to simulate curves of teliospore availability such as those published 

by Kimmey (1945). These curves are not available for our species and conditions. 

We also lack information on the role of leaf drop and relative yields of urediospores 

and teliospores by an infection developing from an aeciospore as discussed. Regarding 

leaf drop, we will assume for the present that our viability-occurrence curves (fig. 

7) will be sufficient. Regarding urediospore-teliospore partitioning,we will include 

a variable ratio of urediospores to teliospore production for both uredial and 

aecial infections (TSAR. and TSUR.) (see table 3 for definitions). 1 1 

We relied on our single-aeciospore-culturing experience for many of the details 

of partitioning spore production. A single aeciospore infection on a detached R. 

hudsonianum leaf can produce about 12,000 urediospores, according to our visual 

estimates. We also estimate that a single urediospore infection under similar 

conditions can produce about 10 telial columns, if no urediospores are produced. If 

urediospores average l5w x 32w (Colley 1918), then 12,000 spores = 68xl0 6w3 of 

spore volume. Also, we find that teliospores average about lOw x 40w (authors' 

unpublished data) and that infections on R. hudsonianum produce telial coiumns that 

average 100w in diameter and 900w in length. Thus, 10 telial columns = a yield of 

spore volume of 70xl06 w3 • The two estimated values are in close agreement. 

The size of the telial columns produced by c. ribicola on the different Ribes 

species varies. Mielke and others (1937) describe the size differences in qualitative 

terms. R. hudsonianum produced large and closely spaced columns. R. inerme produced 

somewhat more spar5ely spaced, but longer columns. R. viscosissimum produced short 

stout columns more sparsely spaced than those of R. inerme. R. lacustre produced 

sparsely spaced columns that were longer than the others, but thinner. These qualitative descriptions were used to construct some estimated quantitative dimensions in 

conjunction with two pieces of quantitative data that were available. Colley (1918) 

states that telial columns average about 100W in diameter and may be up to 2,000w in 

length although both length and diameter vary greatly. Taylor (1922) estimates that 

1 in2 of fully infected R. lacustre leaf surface will produce 281,280 sporidia at 

6,000 sporidia per column. These values lead to about seven columns per square 

centimeter (calculated by the authors). 

Mielke and others (1937) describe telial production on the four Ribes species on 

a scale based on relative mass of telial columns per unit area. With R. lacustre as 

1, R. viscosissimum was 2, R. inerme was 12, and R. hudsonianum was 20. 

Teliospores are packed rather tightly into the columns and are of a somewhat 

irregular shape. If we measure the total length of a chain of teliospores and divide 

by the number in the chain, however, we find that teliospore length averages about 

40w. If we assume that spore size is invariate over Ribes species, then we can make 

some estimates about column dimensions based on our current information and estimate 

the number of teliospores per column for each Ribes species. The estimated dimensions 

are: Ribes hudsonianum 100w x 900w; R. lacustre 50w x 1,200w; R. viscosissimum 100w 

x 450w: and R. inerme 100w x 1,200w. 

If the R. hudsonianum values are taken as 1, the relative mass values for the 

other species become (1) R. lacustre 0.05, (2) R. viscosissimum 0.1, and (3) R. 

inerme 0.6. Let us assume that these mass values represent a measure of the relative 

production capabilities of the different Ribes species. The rust should be studied 

on each Ribes species to determine the relative influence of Ribes species and 

environment on telial column parameters and spore biomass production. Some understanding of within-species variation and variation within different combinations of 

rust and Ribes is also needed. We will assume that all species produce the same size 

teliospores and urediospores, but that the productivity and telial column size and 

diameter varies among species. If R. hudsonianum can produce 70xl06 w3 of spore 

26 
volume per infection, then R. lacustre can produce 3.5xl06~3 (70xlO~·O.05), R. 

viscosissimum 7xl06~3, and R. inerme 42xl06~3. These values complete those necessary 

to model the rust on Ribes. 

If each aeciospore infection is assumed to produce a given amount of biological 

material (BMASS.), as discussed, this yield can be in the form of urediospores or 

teliospores. OUr evidence indicates that infections resulting from aeciospores 

behave differently than infections arising from urediospores in regard to the relative 

proportions of urediospores and teliospores. We will use the input variables TelioSpores from Aeciospores Ratio (TSAR.) and TelioSpores from Urediospores Ratio (TSUR.) 1 1 

to control the partitioning of the infection production potential for the two kinds 

of infections on the four Ribes species. Germination lag and germ tube growth are 

used in the same way to model urediospore infection as to model aeciospore infection, 

but the functions are different as illustrated in figures 12 and 13. 

computation of number of germinable urediospores in a given infection period 

(8l~82). Out of all the urediospores produced by a certain cohort, the proportion 

viable (UV) is determined by the spore-age relationship shown in figure 7. 

computation of number of urediospores that germinate during a given infection 

period per square centimeter of Ribes target (82~83). The proportion of viable 

urediospores per square centimeter of leaf surface that will be expected to germinate 

in any given infection period of unlimited duration will be provided by the function 

illustrated in figure 14. The interaction of rust genes and temperature is assumed 

to control this function. 

computation of number of germinated urediospores expected to provide germ tubes 

of sufficient length for penetration (83~84). The rationale is the same as for flow 

(78~79). 

Computation of number of infections caused by urediospores during a given 

infection period (84~85). An infection yield ratio is used to control this flow as 

it was for flow (79~80). 

Computation of number of urediospores ar~s~ng from urediospore infection (85~8l). 

Urediospores are produced by infections arising from urediospores as well as aeciospores. So, this portion of the model is recycled through this feedback flow. 

PRODUCTION OF TELIOSPORES 

Computation of number of teliospores expected to arise from aeciospore (80~86) 

and urediospore (85~86) derived infections. The number of potential telispores 

(TMAX) is calculated from a summation of spore biomass left after removal of urediospores. Remember that teliospores can come from infections that resulted from both 

aeciospore and urediospore infection; thus, the two flows are considered in tandom. 

BASIDIOSPORE PRODUCTION 

Computation of number of teliospores viable in a given infection period (86~87). 

Teliospore viability varies with age (fig. 7) and temperature during formation (Van 

Arsdel and others 1956); thus a viability function is necessary. In northern Idaho, 

the temperature seldom exceeds the limit critical to formation viability (35°C); so, 

we incorporated only the teliospore-age variable at this time. 

27 
computation of number of teliospores expected to germinate in a given infection 

period (87+88). Teliospores germinate to produce basidiospores. Assuming that the 

number of teliospores per square centimeter of Ribes leaf is known for each Ribes 

species, what quantity of basidiospores can be expected? A model of teliospore 

germination was developed that yields percent of teliospores germinated (PCAST) as a 

function of infection period duration, teliospore column diameter, and teliospore 

diameter. The detailed construction of this model is given in appendix III. 

computation of number of basidiospores produced for each square centimeter of 

underside leaf surface of Ribes hudsonianum per infection period (88+89). Each 

germinating teliospore can produce a maximum of four basidiospores, but under natural 

conditions we know very little about how many are actually produced per teliospore. 

So, we elected to reserve a spot for an input variable called BasidioSpore Yield 

Ratio (BSYR.), which is expressed as basidiospores per teliospore. 1 

computation of number of basidiospores produced for each square centimeter of 

underside leaf surface of R. inerme per infection period (88+90). This flow is a 

repeat of flow (88+89) with a repeat of all flows between 77 and 88 with changed 

parameters to match R. inerme. 

computation of number of basidiospores produced for each square centimeter of 

underside leaf surface of R. lacustre per infection period (88+91). Same explanation 

as previous flow. 

computation of number of basidiospores produced for each square centimeter of 

underside leaf surface of R. viscossisimum per infection period (88+92). Same 

explanation as previous flow. 

DISTRIBUTION OF BASIDIOSPORES TO PINE TREES 

computation of number of basidiospores per square centimeter of flat surface at 

the white pine tree (89, 90, 91, 92+93). As already indicated, the model includes a 

repeat of the flows 75 through 89 for the remaining three major Ribes species. In 

several places, details differ like aeciospore and urediospore infection yield ratios, 

urediospore versus teliospore partitioning, telial column parameters, and duration of 

infection periods. After the expected number of basidiospores per hectare is computed 

for each of the four Ribes species, simulated spore load is summed over Ribes species 

and is distributed to pine trees on the basis of Average Ribes-growth-point (Bush) to 

Tree Distance (ABTD.) and the number of bushes per Growth-Point or Growth-Point 

Factor (GPF.). The 1 separate spore loads (spores/bush) for each infection period are 

used to cal~ulate spores per hectare by multiplying by the number of bushes per 

hectare (state variables 1, 2, 3, and 4, Ribes density submodel). ABTD. and GPF. are 

then used to calculate the average spore load in spores per square centimeter at 1 the 

trees. 

Pine Host Target Submodel 

This submodel (fig. 15) computes the target of an average tree as a function of 

site index, stand age, host genotype, and weather. The computed target is partitioned 

into current, l-year-old, and 2-year-old branch needles and current and l-year-old 

stem needles. Variable definitions for the target submodel are given in table 8 and 

flow descriptions are given in table 9. 

Six characteristics of the pine target were incorporated into this model: 

number of needles, placement of needles (branch or stem), number of rows of stomata 

per needle, needle length, needle age, and pine genotype (resistance). These characteristics are fully discussed under the pine infection model. 

28 
PISDS 

cm2/tree 

RAISS 

cm2 /tree 

25 

, 

" , 

\ 

26 

'" ...... 

" 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

PISB 

cm2/tree 

32 , \ 

, \ 

QISSI \ 

cm2/tree \ 

\ 

27 , \ 

J 

./ \ 

-

QISS2 

cm2/tree 

28 \ 

I 

, 

, 

" .......... ..... -

\ 

\ 

\ 

\ 

, 

" " 

QISB3 

cm2/tree 

-

30 

Stomatal 

Function 

Needle 

Length 

Function 

QISB2 

cm2/tree 

31 

RAISB 

cm2/tree 

QISBI 

cm2/tree 

" ", ..... -

Figure lS.--Box-and-arrow diagram of pine-host target. Variable definitions are in 

table 8. 

29 

33 
Table 8.--Definition of variables used in the pine-host-target submodel 

Variable 

State variables: 

X(lO) SINK 

X(2S) PISDS 

X(26) RAISS 

X(27) QISSI 

X(28) QISS2 

X(29) SOURCE 

X(30) QISB3 

X(3l) QISB2 

X(32) PISB 

X(33) RAISB 

X(34) QISBI 

User controlled variables: 

RSPN 

QNEEDL 

Computed variables: 

Definition 

Sink that accepts dead needles. 

Potential infective surface on direct stern needles, 

square centimeters per tree. 

Stomatal row adjusted i~fective surface stern needles, 

square centimeters per tree. 

Quantity of infective surface on stern needles in 

their first year, square centimeters per tree. 

Quantity of infective surface on stern needles in 

their second year, square centimeters per tree. 

Unspecified source of infective surface. 

Quantity of infective surface on branch needles 

in their third year, square centimeters per tree. 

Quantity of infective surface on branch needles 

in their second year, square centimeters per tree. 

Potential infective surface bran~h needles, 

square centimeters per tree. 

Row adjusted infective surface on branch needles, 

square centimeters per tree. 

Quantity of infective surface on branch needles in 

their first year, square centimeters per tree. 

Rows of stomata per needle, number rows/needle. 

Quantitative needle length factor, proportion of 

maximum needle length. 

See appendix I. 

30 
toN 

J->. 

Flow path Flow name 

J(29-+32) Branch needle generation 

J (32-+33) Stomatal row adjustment 

J(33-+34) Needle length adjustment 

J(34-+3l) Aging infective surface 

J(3l-+30) Aging infective surface 

JC30-+l0) Needle death 

JC29-+2S) Stem needle generation 

JC2S-+26) Stomatal row adjustment 

J(26-+27) Needle length adjustment 

JC27-+28) Aging infective surface 

JC28-+l0) Needle death 

Table 9,--Pine-target-submodel flow description 

Flow description 

Computation of potential infective 

surface on current year branch 

needles 

Potential infective surface in cm2 jtree 

on current year branch needles is 

adjusted for number of stomatal rows 

per needle 

Row adjusted IS on current year branch 

needles is adjusted for needle length 

Branch-needle IS on needles in their 

current year ages one year to give rise 

to IS on second-year branch needles 

Branch-needle IS on needles in their 

second year ages one year to give rise 

to IS on third-year branch needles 

Branch needles lost at end of third 

growing season due to natural death 

of needles 

Computation of potential infective 

surface on current year stem needles 

Adjustment of potential IS on current 

year stem needles for number of stomatal 

rows 

Stomatal-row-adjusted stem-needle IS is 

adjusted further for needle length by 

the stem-needle factor QSNEDL 

Computation of first-year aging infective 

surface on stem needles in their current 

year 

Stem needles lost at end of second growing 

season due to natural death 

*See appendix I for derivation of flow equations. 

PISB 

RAISB 

QISBI 

QISB2 

QISB3 

SINK = 

PISDS 

RAISS 

QISSI = 

QISS2 = 

SINK = 
The concept of infective surface was developed to facilitate the modeling of the 

target area. The investigations of Clinton and ~cCormick (1919) and Chapman2 

indicated that infection by basidiospores occurred through the stomates. This was 

confirmed by Patton and Johnson (1967). The basidiospores land on a needle, germinate, 

and the germ tube grows in a random direction (Hansen 1972). When the tip of the 

germ tube happens to penetrate a stomate, it grows between the guard cells into the 

substomatal chamber, where it produces a substomatal vescicle, and then an infection 

hypha (fig. 16). The infection hypha penetrates a mesophyll cell and the infection 

is established. If one assumes that this entire process must be completed by utilization of energy stored within the basidiospore, then growth of germ tubes on pine 

needles should provide a clue to their growth potential. What is the average distance 

a spore can be away from a stomate and still have reserves to establish an infection? 

Germ tubes averaged about lOOw in length when spores were germinated on eastern white 

pine (Hansen 1972) or on water agar at pH 6.8 and 16°C (Bega 1957). 

~------100~------------~ 

Figure l6.--Diagramatic representation of a Cronartium ribicola basidiospore 

infecting a pine needle. 

2Chapman, C. H. 1934. A study of the movement of mycelium of Cronartium 

ribicola Fischer during the incubation period in Pinus monticola Dougl. In Blister 

rust work in the far West, p. 293-298. (Mimeographed) U.S. Dep. Agric., Div. Plant 

Dis. Control. 

32 
The distance from the top of the subsidiary cells to the bottom of the substomatal 

cavity averaged about SSw on a few sections of western white pine needles (authors' 

unpublished data). This leaves about SOw of growth potential. How can we envision 

the utilization of this potential? Since stomates are arranged in rows and the 

distance from the end of one stomate to the beginning of the next is about the same 

as the length of a stomate (20w, authors' unpublished data), SOw perimeters around 

succeeding stomata in a row overlap. Therefore, the important measure becomes the 

"infective width" of the stomatal row. We have assumed infective width to be 100w 

(fig. 17). Infective surface is taken as the square centimeters of needle area within 

a tree that a spore can land on and have a chance to establish an infection. Factors 

influencing this parameter that must be considered in calculating target are needle 

length and number of rows of stomata. 

Figure 17.--The concept of infective surface is illustrated by this diagramatic 

representation of germinating basidiospores on the surface of a pine needle (infective 

surface in square centimeters per tree = average number of rows of stomata/needle x 

number of needles per tree x average length of stomatal row x 0.01 cm). 

, 

computation of potential Infective Surface (IS) on current year branch needles 

(29~32). State variable 29 is a source of infective surface that is partitioned out 

each year to branch and stem needles according to the branch-needle and stem-needle 

growth functions. Potential Infective Surface Branch (PISB) is based on number of 

current year branch needles. 

Potential branch-needle infective surface adjusted for stomatal-row-nu~er 

(32~33). In this formulation, we have provision for a given value of Rows of Stomata 

Per Needle (RSPN) to be used for calculation of IS. This variable is important 

because it can apparently vary a great deal under the influence of both environmental 

and genetic factors. 

Stomatal-row-adjusted branch-needle IS adjusted by a needle-length factor 

(33~34). Environment, genes, and tree age or size probably all influence needle 

length. This flow provides a place for such sources of variation. 

33 
Branch-needle IS on needles in their current year ages 1 year to give rise to IS 

on second-year branch needles (34~3l). According to available data (Buchanan 1936), 

there are no losses of branch-needles between their first and second years, so the 

entire contents of QISBI must be transferred to QISB2 (see table 8 for definitions). 

Branch needle IS on needles in their second year ages 1 year (3l~30). The 

equation mediating this flow was derived from Buchanan's (1936) needle-removal table. 

His sample showed that 64.2 percent of the branch needles of a particular cohort are 

retained for their third year. 

Branch needle IS lost at the end of the third growing season due to natural 

death of needles (30~lO). In western white pine, about 4 percent of the needles in 

the entire crown survive 4 years and in the upper crown 4-year survival is even lower 

(Buchanan 1936). Consequently, we assumed that all needles die and are cast at the 

end of their third season. All the contents of QISB3 are moved to the sink. Thus, 

new infections that occur on needles in their third season are assumed to be of no 

consequence. 

computation of potential stem needle IS (29~25). Needles are also located on 

the bole. These stem needles are of particular importance because infections originating here reach canker status directly on the bole. There is no canker mortality due 

to dying branches and, since the needles are located on the first year wood, boles 

are of small diameter so damage occurs quickly. For these reasons, stem needles are 

treated separately from branch needles. Infective surface flows from IS source (X29) 

to PISDS (Potential Infective Surface Direct-Stem). 

Adjustment of potential IS for stem-needle (PISDS) (25~26). The use of RSPN was 

discussed under flow (32~33). 

Stomatal-row adjusted stem-needle IS is adjusted for needle length by the stemneedle factor (QSNEDL) (26~27). QNEEDL, defined in table 8, is used to compute 

QSNEDL. This change is necessary because stem needles tend to be longer than branch 

needles. 

computation of i-year aging of infective surface on stem needles in their 

current year (27~28). The assumption was made that none of the current year stem 

needles die before their second season. Thus, all of the IS in X(27) will be transferred to X(28) for the next cycle. 

Infective surface moves to the sink after stem-needle death (28~lO). Our field 

observations lead us to believe that stem needles generally die and are cast at the 

end of their second season. Thus, new infections on stem needles in their second 

season are cast before they can become cankers. This means that only first-year stem 

needles need be considered as targets. 

We now have calculated infective surface values of first-year stem needles 

(QISSl), first-year branch needles (QISBl), and second-year branch needles (QISB2). 

These variables are transferred now to the pine infection submodel and combined with 

basidiospore load from the Ribes infection submodel to create a simulated amount of 

infection on an average tree. 

Pine Infection Submodel 

Pine infection is outlined in figure 18. This process is complex but the 

general aspects of pine infection delineated by Hirt (1935 and 1942) and Spaulding 

and Rathbun-Gravatt (1925a) have been preserved. Infections that occur on needles 

attached directly to the bole are of special significance as already discussed. For 

34 
\ , 

" 

,,-

1/'" U ,..--_....JO..-_...., 

P' TNISI 

I 

I 

BSTAL 

spores/cm2 

96 

----- ( Pine Target) 

BSGER 

spores/cm2 

95 

----

TNIBI 

BSTIS 

spores/cm2 

94 

TI2BI 

" \ 

Infections/tree t-~/t-~~ Infections/tree I-----l'l-----::~ 

12 

Infections/tree t---.. --~ 

/ 

I 

I 

\ 

\ 

/ 

\ 

7 

TI2S1 

Infections/tree t--v----3t 

8 

I 

/ / 

TI3S1 

/ 

/~--~ 

/ 

/ 

/ 

Infections/tree t---:it---::~ / 

/ 

/ / 

9 

-- /------ ./ 

/ 

I 

/ 

/ 

/ 

/ 

/ 

I 

TCOS 

Infections/tree 

II 

ACPT ANCPT 

Infections/tree ~-"""---llnfections/tree 

99 98 

\ / 

\ I 

\ 

\ 

/ 

\ 

\ 

I 

I 

I 

I 

I 

I 

I 

I 

I 

, 

--

CIIBI 

Infections/tree 

13 

/ ~---

\ 

I 

I 

I 

I 

I 

I 

I 

\ 

TLBC 

Infections/tree 

14 

--- - _-::::.. ....... - --

\ 

I 

I 

A 

, 

/ 

" " 

/ '" 

15 

CI2BI 

Infections/tree 

16 

\ 

\ 

\ 

"- ----,,"" " ........ -

CI1B2 

Infections/tree 

20 

CI2B2 

Infections/tree 

22 

\ '-- .--------- ---

\ 

\ 

'\ , ~-:::::.-==------- - -_.-:::'_- -- - -- ........ -----

" " 

",. 

/ 

"- , 

.-

" 

---

QBSAT 

spores/cm2 

93 

TI3BI 

Infections/tree 

17 

CI3BI 

Infections/tree 

18 

CI3B2 

Infections/tree 

24 

TNIB2 Tl2B2 TI3B2 

'----------------------------~ Infections/tree I----.--~ Infections/tree t---~--~ Infections/tree 

19 24 23 

I \ / ,_ ..... 

Figure 18.--Box-and-arrow diagram of Cronartium ribicola infection of the pine host. 

35 
review, they can begin the girdling process immediately and their numbers cannot be 

reduced by the death of branches; consequently, the silvicultural control treatment 

of pruning is not applicable to infections from this source. The probability of 

infection on stem needles becomes an important variable when estimating probable 

damage and working out control strategies. The definitions of the state variables 

found in this submodel are given in table 10 and descriptions of flows are in table 

11. 

Table 10.--Definitions of variables used in the pine infection submodel 

Variable 

State variables: 

X(lO) SOURCE 

X(ll) TN1Sl 

X(12) T12Sl 

X(13) T13Sl 

X(14) TCOS 

X(lS) TNIBI 

X(16) CIlBl 

X(17) TLBC 

X(18) T12Bl 

X(19) C12Bl 

X(20) T13Bl 

X(21) CI381 

Definition 

Unspecified source of pine infections. 

Total number of new needle infections in current 

year on stem needles infected in their first season. 

Total number of needle infections in current year 

that are in their second season and are in needles 

infected in their first season. 

Total number of needle infections in current year 

that are in their third season and are in needles 

infected in their first season. 

Total number of cankers that appear on the stem in 

the current year from infections in stem needles. 

Total number of new needle infections in current 

year on branch needles in their first season. 

Number of canker infections from l-year-old branch 

needle infections that were infected in their first 

year. 

Total number of potentially lethal branch cankers 

that appear in current year. 

Total number of needle infections in current year 

that are in their second year and growing in branch 

needles that were infected in their first year. 

Number of canker infections from 2-year-old branch 

needle infections in needles that became infected 

in their first year. 

Total number of needle infections in current year 

that are 3 years old and growing in branch needles 

that were infected during their first year. 

Number of canker infections from branch needle 

infections in their third year and growing in needles 

that became infected during their first year. 

36 
Table IO.--Definitions of variables used in the pine infection submodel (continued) 

Variable 

X(22) TNIB2 

X(23) CIIB2 

X(24) T12B2 

X(2S) CI2B2 

X(26) TI3B2 

X(27) CI3B2 

X(93) QBSAT 

X(94) BSTIS 

X(9S) BSGER 

X(96) BSTAL 

X(97) BSI 

X(98) ANCPT 

X(99) ACPT 

User controlled variables: 

BIY 

DIP(DAAI) 

Definition 

Total number of new needle infections in current 

year on branch needles in their second year. 

Number of canker infections from l-year-old branch 

needle infections growing in needles that became 

infected during their second year. 

Total number of needle infections in current year 

that are 2 years old an~ growing in branch needles 

that became infected during their first year. 

Number of canker infections from 2-year-old branch 

needles that became infected in their second year. 

Total needle infections in their third season and 

growing in branch needles that became infected 

during their second growing season. 

Number of canker infections from 3-year-old branch 

needle infections growing in branch needles that 

became infected during their second growing season. 

Quantity of basidiospores at an average tree, 

basidiospores per square centimeter. 

Number of basidiospores trapped per unit infective 

surface, basidiospores per square centimeter. 

Number of basidiospores expected to germinate after 

being trapped, basidiospores per square centimeter. 

Number of basidiospores expected to produce germ 

tubes of adequate length for penetration, basidiospores per square centimeter. 

Number of basidiospores expected to penetrate and 

establish infections, basidiospores per square 

centimeter. 

Average number of new cankers per tree each year. 

Accumulated number of cankers per tree inbred by 

infection period. 

Ratio of spots to trapped and germinating basidiospores, spots per square centimeter:basidiospores 

per square centimber IS. 

Duration of infection period, hours. 

(continued) 

37 
Table lO.--Definitions of variables used in the pine infection submodel (continued) 

Variable 

FER 

NRF 

QNLR 

RGRN 

SNAIF 

TEF 

TIP(DAAl) 

Computed variables: 

Definition 

Fungus encirclement rate, meters/year. 

Needle retention factor, year of branch needle 

retention. 

Needle lesion reduction resistance factor. 

Rust growth rate down the needle, mm/month. 

Secondary needle age infection factor, proportionality constant. 

Trapping efficiency factor of branch and stem 

needles. 

Average temperature of an infection period, indexed 

by infection period, expressed as degrees C. 

See appendix I. 

38 
Vl 

\.0 

Flow path 

J(93-+94) 

J(94-+9S) 

J(9S-+96) 

J(96-+97) 

P 

I J (6-+7) 1 

J(6-+l2) 

Table ll.--Pine infection submodel flow descriptions 

Flow name 

Basidiospore trapping 

Basidiospore germination 

Basidiospore germ tube growth 

Basidiospore infection 

establishment 

Basidiospore infections summed 

over infection periods 

Basidiospore infections summed 

over infection periods 

Flow description 

Computation of number of 

basidiospores trapped by 

a square centimeter of 

pine infective surface 

Computation of number of 

trapped basidiospores 

expected to germinate 

during a given infection 

period 

Computation of number of 

germinated basidiospores 

expected to produce germ 

tubes of sufficient length 

to penetrate 

Computation of number of 

basidiospores penetrating 

to establish infection 

Calculation of number of 

new infections per tree 

for entire season on 

current-year stem needles 

Calculation of number of 

new infections per tree 

on current-year branch 

needles 

Flow equ

BSTIS 

BSGER 

BSTAL 

BSI BS

TNISI 

TNIBI 
+::-

o 

Flow path 

J (6-d9) 

J(7-+11) 

J (7-+8) 

J(8-+11) 

J (8-+9) 

J (9-+11) 

Table ll.--Pine infection submodel flow descriptions (continued) 

Flow name 

Basidiospore infections summed 

over infection periods 

Canker appearance 

Needle infections aged 

Canker appearance 

Needle infections aged 

Canker appearance 

Flow description 

Computation of number of 

new iJfections pet tree on 

second-year branch needles 

Computation of number of 

incipient cankers from 

stem needles infected in 

their first year 

Computation of number of 

stem-needle infections 

that failed to reach canker 

status in 1 year 

Computation of number of 

2-year-old stem needle 

infections that reached 

canker status in current 

year 

Computation of number of 

2-year-old stem needle 

infection that failed to 

reach canker status in 

current year 

Computation of number of 

3-year-old stem needle 

infections that reach 

canker status in current 

year and summation of all 

stem-needle cankers to 

appear in current year 

Flow equatio

P 

TNIB2 L(B

1 

TCSI = TNIS

TI2S1 TNI

TCS2 TI2S

TI3Sl TI2S

Tcas TCSI 
.j:::.. 

I-' 

Flow path 

J (12~13) 

J (12~lS) 

J (1S~ 16) 

J (1S~17) 

J(17~18) 

Table 110 - -pine infection submode.l flow descriptions (continued) 

Flow name 

Canker appearance 

Needle infections aged 

Canker appearance 

Needle infections aged 

Canker appearance 

Flow description Flow eq

Computation of number of CIlBl 

l-year-old needle infections 

in branch needles infected in 

their first year that reach 

canker status after 1 year 

Computation of l-year-old TI2Bl 

branch needles infected in 

their first year that failed 

to reach canker status in 

1 year 

Computation of number of \ 

2-year-old branch needle 

infections from needles 

infected in their first 

year that reach canker 

status in the current year 

Computation of number of 

2-year-old branch needle 

infections on needles 

infected in their first 

year that failed to reach 

canker status in current 

year 

Computation of number of 

3-year-old branch needle 

infections that reached 

canker status in current 

year 

CI2Bl 

TI3Bl 

CI3Bl 
Table ll.--Pine infection submodel flow descriptions (continued) 

Flow path Flow name Flow description Flow equation* 

CIlBl' (O

J(13-+l4) Lethal-canker generation Calculation of number of CLlBl 

new lethal cankers from 

last year's incipient 

cankers from branch needle 

infections on needles 

in their first year 

J (16-+14) Lethal-canker generation Calculation of number of CL2Bl CI2Bl.(O,

new lethal cankers from 

last year's incipient 

cankers from 2-year-old 

branch needle infections 

on needles infected in 

their first year 

.j;:;;. 

J (18-+14) Lethal-canker generation Calculation of number of CL3Bl CI3Bl.(0.

N 

new lethal cankers from 

last year's incipient 

cankers from 3-year-old 

branch needle infections 

J (19-+20) Canker appearance Calculation of number of CIlB2 TNIB2'b- 2

incipient cankers from 

l-year-old infections on 

branch needles infected 

in their second year 

J (19-+21) Needle infections aged Calculation of number of TI2B2 TINB2-CIl

l-year-old infections on 

branch needles that failed 

to reach canker status 
Table ll.--Pine infection submodel flow descriptions (continued) 

Flow path Flow name Flow description Flow equation* 

J(2l-+22) Canker appearance Calculation of number of CI2B2 TI2B2·b- 2O 

incipient cankers from 

2-year-old infections on 

branch needles infected in 

their second year 

J (21-+23) Needle infections aged Calculation of number of TI3B2 TI2B2-CI2B2 

2-year-old infections on 

branch needles that failed 

to reach canker status 

J(23-+24) Canker appearance Calculation of number of CI3B2 TI3B2 

incipient cankers from 

3-year-old infections on 

branch needles from I-year-

+:- old needle infections on VI needles infected in their 

second year 

CIlB2(0.24'

J(20-+l4) Lethal canker generation Calculation of number of CLlB2 

new lethal cankers from 

last year's incipient 

cankers from 2-year-old 

branch needle infections 

on needles infected in 

their second year 

CI2B2(0.24

J (22-+14) Lethal canker generation Calculation of number of CL2B2 

new lethal cankers from 

last year's incipient cankers 

from 2-year-old branch needle 

infections on needles infected 

in their second year 
+:>-

+:>-

Table ll.--Pine infection submodel flow descriptions (continued) 

Flow path Flow name 

J (24-+14) Lethal canker generation 

J(6-+98) Total new cankers 

J(98-+99) Canker accumulation 

*See appendix I for derivation of flow equations. 

Flow description 

Calculation of number of 

new lethal cankers from 

last year's incipient 

cankers from 3-year-old 

branch needle infections 

on needles infected in 

their second year 

Calculation of number of 

new cankers that appear 

each year 

Calculation of accumulated 

number of cankers to appear 

on an average tree after a 

given period of exposure 

Flow equation* 

CL3B2 CI3B2(0.24

ANCPT TCaS + CIlB

+ CI2B

ACPT ACPT + ANCP
computation of basidiospores trapped by a square centimeter of pine infective 

surface (93~94). After the basidiospores are produced, they must escape the Ribes 

canopy and penetrate the pine canopy. Both of these operations are dependent on the 

relationships among spore size, spore density, windspeed, and leaf size and shape. 

One way to gain an understanding of these relationships is to look at needles and 

leaves as spore traps and calculate the trapping efficiencies under different conditions 

(Gregory 1952). A paper by Edmonds (1972) provides the methodology for such calculations. Trapping efficiencies of pine needles and Ribes leaves are given in table 12. 

The movement of spores through the pine canopy is envisioned as follows: Any slight 

air movement will cause the spores to be moved around the needles rather than onto 

them because of the very low trapping efficiency of the needle. A few basidiospores 

would be trapped at 450 em/sec (10 mi/h) where the efficiency reaches about 5 percent. 

At 900 em/sec (20 mi/h) the efficiency reaches 25 percent. Thus wind speeds of 45 to 

225 em/sec (1 to 5 mi/h) , which may be best for dispersal, may be worst for deposit jon. 

This leads to the conclusion that dispersal and deposition 'can be viewed as two 

completely different processes and that most basidiospores are probably deposited on 

the needles rather than trapped by impaction phenomenon. So long as the wind blows, 

the basidiospores remain in suspension, but during quiet periods the basidiospores 

s'ettle out onto the pine needles. Another possibility is that rain will wash basidiospores out of the air even though the wind is blowing (Schrodter 1960). This subject 

brings up the question of the importance of variation of wind velocity in the total 

infection process/of pines by C. ribicola basidiospores. 

Table 12.--Impaction efficiencies in percent for basidiospores with a diameter of lO~ 

and a specific gravity of 1 

Trap Trap width 

cm 

Pine needle 0.1 

Ribes leaf 7.0 

Calculated from Edmonds 1972. 

1 mi/h 

(45 em/sec) 

<l. 0 

<l. 0 

Horizontal windspeed 

2 mi/h 4 mi/h 

(90 em/sec) (180 em/sec) 

<l. 0 2.5 

<1.0 <1. 0 

20 mi/h 

(900 em/sec) 

25.0 

<1.0 

For example, wind during a wet period could prevent the spores from being 

deposited before they lose their viability and thus prevent disease development. A 

high wind, for example, 1 125 em/sec (25 mi/h) , could increase the efficiency of 

trapping by impaction in the tops of tall trees; consequently, wind profile could 

influence the amount of infection in short and in tall trees and in densely stocked 

and in thinly stocked .stands. 

Since basidiospores behave as a penetrator, not an impactor, the most meaningful 

measure of infection potential should be number of spores deposited on slides or 

stationary glass rods rather than some other type of spore sampler. One would want a 

trapping system that was inefficient in slow winds, but efficient under calm conditions. 

The role of rain could be assessed by using the methodology that has been 

developed for trapping spores from rain (Rowell and Romig 1966; Roelfs and others 

1970). The whole thing could be tied together by knowing the concentration of 

basidiospores in the air. This could be determined with a volumetric spore trap. 

It is conceivable that the character of the wind during a favorable infection period 

with respect to moisture and temperature could determine the effectiveness of the 

infection period. 

45 
All of our current available information on the infection yield ratio (spots per 

spore cast) has been obtained in the calm interior of infection tents with the 

basidiospores settling directly onto the pine foliage and glass-slide traps. Since 

most basidiospores of C. ribicola are probably deposited this way in nature, our 

simulation will assume a spore deposition model rather than an impaction model. 

computation of number of trapped basidiospores expected to germinate during a 

given infection period (94~95). ~1any factors can influence the likelihood that a 

spore will cause an infection even after it is trapped. The first consideration is 

that the basidiospore land where the germ tube can physically reach a stomata and 

cause an infection. We have attempted to solve this problem by our concept of 

infective surface, by considering only the needle surface from which a spore can 

reach a stomata. So, our next problem is basidiospore germination. 

If the assumed relationships between germination and germ tube growth rate are 

the same for basidiospores as for aeciospores and urediospores, then probability of 

infection by basidiospores after they are trapped can be modeled in a similar fashion. 

When sporidia were germinated on eastern white pine needles (Hansen 1972) under 

favorable temperature and moisture conditions, at least 95 percent germinated. 

Spores deposited on P. lambertiana in a saturated atmosphere and temperatures from 

13°C to 22°C germinated at nearly the 95 percent rate over the range (Bega 1960). 

Since Bega (1960) found 100 percent germination of basidiospores on P. monticola and 

the above determination was also near 100 percent, we have assumed 100 percent 

germination between the temperatures 1°C and 24°C. Germination started at about 2 h 

after casting (Hansen 1972) and at 15°C and pH 3 to 4 was completed at 3 h (Bega 

1960). These findings were incorporated into functions for germination lag and 

germination period (fig. 19 and 20). Hansen (1972) reported a germ tube growth rate 

of 3.S ~/h for the first 30 h, and a decreasing rate for another 42 h on artificial 

membranes. Bega (1960) measured germ tube length on artificial media maintained at 

a pH of 4 at various temperatures. Length was measured 24 h after spore deposition. 

The hourly rate was determined by the authors and plotted (fig. 21). First, we 

needed the value for lag time between the onset of the infection period and completion 

of basidiospore germination. During this period, teliospores must germinate and the 

disseminated basidiospores must germinate and penetrate. It apparently takes about 

6 to 7 hours for teliospores to begin to cast basidiospores, another 2 to 8 hours for 

the basidiospores to commence germination (fig. 19), and an additional 2 to 24 hours 

for the population of basidiospores to complete their germination (fig. 20). Then an 

additional 10 to SO hours would be required for the germ tubes to grow and penetrate. 

10 

8 

6 

4 

2 

o 

14I ... .... ------Limit-------.. ~~1 

I 

TEMPERATURE (OC) 

46 

Figure 19.--Assumed 

relationship between 

temperature and 

period of germination 

lag of Cronartium 

ribicola basidiospores. Data point 

at minimum is from 

Bega (1960); and 

shape of curve is 

assumed to be the 

same as for C. 

ribicola basidiospores. 
Figure 20.--Assumed relationship between temperature 

and period of germination 

of Cronartium ribicola 

basidiospores. Shape of 

curve estimated from author's unpublished data for 

aeciospores and urediospores and data from Bega 

(1960). 

Figure 2l.--Growth rate 

of Cronartium ribicola basidiospore 

germ tubes. Data 

points from Bega 

(1960) and curve from 

Weibull function 

where a = 0.02, 

B 16.8, c = 2.8, 

W 12.634, and 

X temperature. 

8 

6 

4 

2 

o 

o 4 8 12 16 20 24 28 32 36 

TEMPERATURE (OC) 

computation of number of germinated basidiospores expected to produce germ tubes 

of sufficient length to penetrate (95~96). If probability of penetration is dependent 

on length of germ tube under ideal conditions, then the number of trapped and germinated 

BasidioSpores expected to produce germ Tubes of Adequate Length (BSTAL) to penetrate 

should be a function of infection period duration and temperature as both will influence 

germ tube length. 

computation of number of basidiospores penetrating to establish infection 

(96~97). It seems reasonable to assume that all the basidiospores that produce germ 

tubes of adequate length and are located on the infective surface will not reach the 

state of an established BasidioSpore Infection (BSI). What is the probability of a 

successful penetration? A measure of this probability is the ratio of spots found to 

spores trapped (Basidiospore Infection Yield ratio or BIY) under optimum conditions 

of temperature, moisture, and duration. BIY has been measured in the case of C. 

ribicola basidiospores on primary needles of P. strobus and on current year's secondary 

needles of P. monticola. 

Hansen (1972) found that when he applied 100 spores per square millimeter to nine 

P. strobus seedlings, the ratio of spots to basidiospores trapped varied from 0.0 to 

0.14 with an average of 0.004 (calculated by the authors on the assumption that 

Hansen inoculated 40 square millimeters of needle surface per seedling (Hansen 1972, 

p. 84 and 94). These nine seedlings were inoculated and then immediately placed in a 

47 
dew chamber. Ten more seedlings were inoculated, covered with moist plastic bags, 

and placed in the dew chamber. These trees yielded a ratio of 0.0005 to 0.011 and 

averaged BIY = 0.0043 spots per spore trapped (inoculated and ratio calculated as 

above). Thus, for primary needles of Pinus strobus the expectation is that the ratio 

of spots formed to basidiospores cast should be about 0.004 x average trapping 

efficiency, which equals 0.004 x 0.08 = 0.00033. 

The ratio is stated in terms of spots per spores cast because all the available 

western white pine data are on this basis. In western white pine, one of our current 

studies at Moscow was inoculated in September 1973 by our standardized method (Bingham 

1972a) except that rubber-cement-coated slides were stationed on a 2-foot grid 

throughout the inoculation. The slides were removed at the end of the inoculation. 

A lacto-phenol analin-blue fixing staining solution was placed on the slides and a 

cover slip added. Spores were then counted in a random sample of 10 fields of 0.175 

mm2 each on each slide. In June of 1974, spot types were tallied (see McDonald and 

Hoff 1975), and a sample of 20 slide stations was taken in order to obtain an estimate 

of the spot per basidiospore cast ratio. All trees within 1 foot of each of the 20 

slides were studied. There were 20 planting spots within the prescribed distance 

from each slide, but some trees were clean (covered as controls) and some had died. 

In all, spots on 201 trees were used. All the spotted trees within 1 foot of each 

slide were used to calculate the ratio of spots to spores trapped on that slide. The 

average was 6.1 trees per slide. The average infection yield ratio (spots per square 

centimeter:spores cast per square centimeter) for each slide was used to calculate a 

mean BIY (i = 0.000087 ± 0.00001) of current year needles on 3-year-old trees. We 

will use the value of 0.000087 spots per basidiospore cast for current year western 

white pine needles. 

There is apparently a resistance mechanism that causes a reduction in the rate 

of needle lesion frequency (Hoff and McDonald 1980). The difference between the 

average ratio of the four lowest-spotting families and the four highest-spotting 

families out of 120 families studied was 2.15 to 18.4. Thus spots were 8.56 times 

more frequent on the high spotters than on the low. More data on this important 

aspect of rust-resistance biology were collected for inclusion in this paper by 

analyzing records from the 1970 progeny test designed to screen resistant-tree seed 

orchard materials. Vaseline-coated slides were used to trap basidiospores during the 

artificial inoculation. Several hundred seedlings were included in each family. 

Number of basidiospores per square centimeter was calculated from the spore traps and 

number of spots per square centimeter of infective surface was determined on a sample 

of plants from each family (table 13). The ratios of spots to spores cast mayor may 

not be comparable to the ratios already discussed because of the difference in spore 

trapping techniques; but the evidence clearly indicates that some families are high 

spotters and others are low spotters. The tenfold difference is in agreement with 

our earlier findings discussed above. The reduced-needle-lesion-frequency resistance 

mechanism was worked into the model as a BIY reduction factor, Quantity of Needle 

Lesion Reduction (QNLR). Thus, if we wanted to simulate the development of the rust 

on a population of trees characterized by the needle-lesion resistance factor, we 

would set QNLR as equal to 0.10. 

Resistance mechanisms in the pine host are not the only possible factors that 

could reduce the BIY. As pointed out earlier, control chemicals, weather, rust 

genes, and other conditions could influence the ratio. These items will be introduced 

through the user-controlled variable BIY as they were in the case of aeciospore 

infection (AIYR(ISP)) and urediospores (UIYR(ISP)). 

Calculation of number of new infections per tree on current-year stem needles 

(6+7). Since BSI was computed on the basis of current year secondary needles only, 

one more adjustment is needed to compute expected infection on stem needles. We do 

not know whether spores can be trapped only on upward surfaces or if all surfaces are 

equally effective traps. Our assumption is that only upward surfaces will trap. On 

48 
Table l3.--Relationship between spore cast and frequency of spot formation for a 

group of full-sib families 

Degree of 

spotting 

Low spotters 

High spotters 

Families 

19x17 

208x207 

20x2l 

69x185 

86x166 

84x155 

l76x2l5 

l59x78 

l17x123 

237x202 

353x182 

Spore cast 

(spores/ cm2 ) 

2,660 

2,500 

3,000 

2,660 

2,000 

2,000 

3,160 

2,410 

2,250 

1,750 

2,330 

Lesion frequency 

(lesions/ cm2 

infective surface) 

0.16 

.11 

.14 

.20 

.1S' 

.12 

1. 47 

1. 72 

1. 79 

1. 06 

2.31 

Lesions/ 

trapped 

spore 

0.00006 

.00004 

.00005 

.00008 

.00009 

.00006 

0.0005 

.0007 

.0008 

.0006 

.0010 

this basis, needle orientation on stem and branches was subjectively studied. 

The conclusion was that for stem needles 0.50 of the stomatal bearing surface~ were 

oriented upward and that 0.25 of the branch needle surfaces were likewise oriented. 

Thus, respective reduction will be made in amount of infAction on each needle type. 

computation of number of new infections per tree on current-year branch needles 

(6~12). This flow is calculated the same as flow (6~7), except that the needle 

orientation factor is 0.25 instead of 0.5. 

computation of number of new infections on second-year branch needles (6~19). 

Primary foliage is known to be much more susceptible than secondary foliage (Patton 

1967; authors' unpublished data). Also, data from Pierson and Buchanan (1938b) 

indicated that current-year needles of western white pine become spotted at a frequency 

14.5 times less than 1- and 2-year-old needles. If this is so, then year-old secondary 

needles should have a BIY of 0.0013 (0.000087 x 14.5). This value is larger than 

Hansen's, but is in agreement with the findings over the years that P. strobus is 

more resistant than P. monticola (Bingham 1972b). Other results (Lachmund 1933) 

indicate that the current year's needles are much less susceptible to infection than 

older needles (table 14). Lachmund's finding was based on the detailed analysis of 

the pattern of appearance of young cankers after a particularly severe infection 

year. In all, 5,879 cankers were inspected and recorded as to the year of wood on 

which they occurred relative to the supposed infection year. The trees were described 

as "thrifty trees ranging from approximately 10 to 30 years of age." Lachmund's 

(1933) results show the following pattern of canker occurrence: current year wood, 

0.10; l-year-old wood, 0:53; 2-year-old wood, 0.31; and 3-year-old wood, 0.05. In 

order to interpret these findings, we must know the distribution of needles by 

needle-age class. The relative target contribution by each needle-age class was 

estimated from table 19 (see pine target submodel). The proportion of cankers on the 

wood that matched with current needles was divided by the proportion of target in 

that class to give cankers per unit of target as shown in column C of table 14. This 

analysis requires the assumption that needles are held to the end of their fourth 

growing season and that all the infections occurred in one season. 

49 
:; 

Table l4.--Relative susceptibility of western white pine needles of different ages to 

Cronartium ribicola as measured by appearance of young cankers and needle spots 

A B C 

Cankers 1 Needle 2 

Needle age 2 years distribution Cankers/ 

at time of after 10 to unit target 

infection infection 30 years 

Current 0.1 0.43 

1 year .53 .3S 

2 years .31 .20 

IData obtained from Lachmund (1933). 

2Data obtained from table 4, this paper. 

A/B 

0.23 

l. 39 

1. 55 

D 

Relative 3 

difference 

16.04 

6.74 

3Value obtained by l-year-old/current and 2-year-old/current. 

4Data obtained from Pierson and Buchanan (193Sb). 

Spots 4 / Relative 

100 cm2 difference 

l.3 

lS.9 14.54 

lS.S 14.46 

Pierson and Buchanan (193Sb) reported on needle spotting on current and older 

needles. They potted seedlings 5 to 7 years old growing in the absence of the rust 

and placed them under heavily infected Ribes bushes during September and October. 

The trees were then placed in a greenhouse to allow for development of the rust. 

After 66 days, the number of spots was determined. The results were recorded as the 

number of spots per 100 square centimeters of needle surface. 

In Lachmund's (1933) study, 1- and 2-year-old needles were six times more susceptible than current needles and in Pierson and Buchanan's (193Sb) report 1- and 2-yearold needles were 14.5 times more susceptible. 

Thus, we will use an input variable to account for the difference in susceptibility 

between current and older needles. Secondary Needle Age Infection Factor (SNAIF) will 

currently be set equal to the average difference (10.25) as shown by the data of 

Pierson and Buchanan (193Sb) and Lachmund (1933). 

computation of number of incipient cankers (11) from stem needles infected in 

their first year (7~11). What will be the pattern of occurrence of canker appearance 

from any given year's infection? Little concrete information is available. Slipp 3 

artificially inoculated branches of western white pine seedlings and saplings and 

followed the appearance of new cankers over time. The number to appear by the fourth 

year was taken as 100 percent and the distribution of the preceding year's cankers 

was calculated. At the end of I year, 7.5 percent were visible; 2 years, SO.4 percent; 

and 3 years, 9S.S percent. 

Other results (author's unpublished) show a similar pattern with the added 

feature of genetic variation of incubation period. Two-year-old seedlings from 40 

full-sib families were artificially inoculated in September. The following September 

the families varied in percent of their seedlings showing stem symptoms out of the 

3Slipp, A. E. 1949. Canker development following artificial inoculation. Univ. 

Idaho, Moscow, 137 p. (Mimeographed) 

50 
total per family that showed stem symptoms by the fourth year after inoculation. The 

low family showed only 7 percent of the stem with stem symptoms and the highest 

family showed 60 percent. These data were used to construct curves of percentage 

cankered on years-after-inoculation. The relationship is shown in figure 22. 

100 b = 25 b = 12 b = 5 

80 

VI 

c::: 

UJ 

~ 60 

z 

« 

u 

..... 

z 

UJ 40 

u 

c::: 

UJ 

a.. 

20 

Figure 22.--Assumed rate of appearance of 

Cronartium ribicola cankers for three 0 1 2 3 4 

different rates of rust growth in needles 

of Pinus monticola. YEARS AFTER INOCULATION 

computation of number of stem-needle infections that failed to reach canker 

status in 1 year (7~8). This flow is self-explanatory. The needle infections that 

did not produce cankers are simply aged 1 year. 

Computation of 2-year-old stem-needle infections that reached canker status in 

current year (8~11). The same relationships are used as in flow (7~11), but the 

variables have changed because of the additonal year. 

Computation of number of 2-year-old stem-needle infections that failed to reach 

canker status in current year (8~9). This flow is the difference between needle 

infections in their second year (TI2Sl) and the cankers that appeared from these 

infections (TCS2). 

Computation of number of 3-year-old stem-needle infections (9) that reach 

canker status in current year (9~11). This flow is composed of all remaining infections. 

Computation of number of l-year-old needle infections in branch needles infected 

in their first year (12~13) that reach canker status after 1 year (13). This model 

uses the same relationship as flow (7~11). 

Computation of l-year-old branch-needle infections on needles infected in their 

first year that failed to reach canker status in 1 year (12~15). This flow is computed 

by obtaining the difference between year-old needle infections and their associated 

incipient cankers (CIlBl). 

Computation of number of 2-year-old branch-needle infections from branch needles 

infected in their first year that reached canker status in the current year (15~16). 

This relationship is the same as in flow (8~11). 

51 
Computation of number of 2-year-old branch-needle infections on needles infected 

in their first year (15) that failed to reach canker status in the current year 

(15~17). Computation of the difference is again called for. 

Computation of number of 3-year-old branch.needle infections (17) that reached 

canker status in the current year (17~18). The entire contents of X(17) flow to 

X(18). 

Calculation of number of new lethal cankers (14) from last year's incipient 

cankers from branch-needle infections on needles in their first year (13~14). There 

is a relationship (fig. 23) between the probability that a branch canker will cause 

a bole canker (lethal canker) and the distance from canker to bole (Slipp 1953). 

The relationship has also been established, in the form of published curves (Childs 

and Kimmey 1938), between proportion of lethal cankers and tree height at time of 

infection. We constructed a composite curve (fig. 24) from the published data and 

one can readily see that proportion of cankers that are lethal is related to tree 

height at time of infection. 

Since there is a definite relationship between tree height and branch length, 

there should be a relationship between a certain size of "crown cap" and proportion 

of cankers expected to be lethal. That is, there should be some length of crown 

from the top down that would be associated with the proportion of all cankers on a 

tree that eventually would reach the bole. This idea was tested by calculating the 

proportion of target (needles) in a crown cap 1 m in length. The result (fig. 24) 

was a close approximation of the composite lethal canker curve. 

Calculation of number of new lethal cankers from last year's incipient cankers 

from 2-year-old branch-needle infections on needles infected in their first year 

(16~14). The explanation of this flow is analogous to flow (13~14). 

Calculation of number of new lethal cankers (14) from last year's incipient 

cankers from 3-year-old branch-needle infections on needles infected in their first 

year (18~14). The explanation of this flow is analogous to flow (13~14). 

Calculation of number of incipient cankers (20) from l-year-old infections on 

branch needles infectea in their second year (19~20). This flow is analogous to 

flow (12-+13). 

Calculation of number of l-year-old infections on branch needles that failed to 

reach cal~er status (19~20). The explanation of this flow is analogous to flow 

(12--+ 15) . 

Calculation of number of incipient cankers (22) from 2-year-old infections on 

branch needles infected in their second year (2l~22). This flow is analogous to 

flow (12~13). 

Calculation of number of 2-year-old infections on branch needles that failed to 

reach canker status (2l~23). This flow is analogous to flow (l2~15). 

Calculation of number of incipient cankers (24) from 3-year-old infections on 

branch needles infected in their second year (23~24). This flow is analogous to 

flow (17~18). 

Calculation of number of new lethal cankers (14) from last year's incipient 

cankers from l-year-old branch-needle infections on needles infected in their second 

year (20~14). This flow is analogous to flow (13~14). 

Calculation of number of new lethal cankers (14) from last year's incipient 

cankers from 2-year-old branch-needle infections on needles infected in their second 

year (22~14). This flow is analogous to flow (13~14). 

52 
1. 00 

\ 

\ 

....J \ 

\ « O. 80 \ :J: \ I:i \ ....J \ (!) \ 

Z \ 

\ 

L.&.J \ !Xl 0.60 \ 

c::: \ 

\ L.&.J \ :::.t:::: \ Z \ « \ u \ 

u.. \ 

0 0.40 \ 

\ >- I- \ /\ Assumed at l normal rust growth rate ....J 

\ I !Xl 

« \ 

!Xl \ 

0 0.20 \ c::: \ 

c- \ 

\ 

\ 

" .... ........ 

--- 0 20 40 60 80 100 120 140 

D I STANCE FROM BOLE TO BRANCH CANKER (CM) 

Figure 23.--Probability that Cronartium ribicola branch canker will cause a bole 

(lethal) canker on Pinus monticola as expected from Slipp's (1953) data and an 

assumed probability if normal rust growth rate were reduced by one-half. 

L.&.J 

....J 1.00 0 

!Xl 

:J: 

U 

i:5 

c::: 

....J 0.80 

== 

~ 

c::: 

L.&.J 

:::.t:::: 

z 

« 0.60 u 

:J: 

U 

Z 

« 

c::: 

!Xl 0.40 I-« 

:J: 

I-

~ 

....J O. 20 !Xl 

« 

!Xl 

0 

c::: 

cO 

I 

I 

I 

I 

I 

I 

I 

I 

I 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ , 

2 

\ , 

, 

" " , 

' .... ........ 

4 

------------------- ---- Simulated 

~-------- Calculated 

6 8 10 12 14 

TREE HEIGHT, TIME OF INFECTION (M) 

Figure 24.--Calculated (data from Childs and Kimmey, 1938) and target-area-simulated 

probability of a Cronartium ribicola branch-canker reaching the bole of a Pinus 

monticola. 

53 
Calculation of number of new lethal cankers (14) from last year's incipient 

cankers from 3-year-old branch-needle infections on needles infected in their second 

year (24~14). This flow is analogous to flow (13~14). After all flows have been 

calculated, the number of bole and lethal-branch cankers can be determined by 

summation. 

Also, if needle retention could be reduced to 2 years instead of 3 on the 

branches and 1 year instead of 2 on the stem, there would be a great reduction in 

effective target. If the 3 years' branch needles are partitioned equally on the 

tree (table 19) and second-year needles are 10 times more susceptible than currentyear needles, then elimination of second-year needles as a target would reduce 

effective target by 91 percent and there would be no effective target on the stem. 

Thus, we have included provisions for taking all stem needles and the second-year 

branch needles out of the model through the Needle Retention Factor (NRF) that can 

stop flows (6~7) and (6~12). 

Calculation of number of new cankers to appear on average tree (98). This is a 

bookkeeping sort of flow. State variables 11, 13, 16, 18, 20, 22, and 24 are summed 

each year and placed in X(98). 

Calculation of accumulated number of cankers to appear on an average tree 

(98~99). This flow serves to keep a record of all infections over all simulated 

years. The value in X(98) is added each year to the previous year's total. 

Stand Infection Submodel 

The preceding parts of the simulator all dealt with computation of the amount 

of rust infection on an average tree. The objective of the stand infection submodel 

is to expand the average infection to a stand and simulate disease progress curves 

for the stand. The factors assumed to influen~e the curves are host-growth rate, 

fungus-growth rate, host genotype, stand age, rust genotypes, natural canker inactivation 

(Kimmey 1969; Hungerford 1977), and the interaction of all the above and inoculum 

load over years. The box-and-arrow diagram (fig. 25) for stand infection illustrates 

the general concept, and the variables are defined in table 15 and flow descriptions 

are given in table 16. 

This submodel is complex because of relatively large numbers of interconnected 

flows. The order of discussion will follow the flows beginning with (35~43) and 

ending at state variable (50), which is the part of the model dealing with seedlings. 

The next route will be (35~36) and pertains to stem cankers originating from stem 

needles. The branch (37~38~39) leads to death by cankers that remain always active 

and branch (37~45~46~47) leads to death by cankers that are temporarily inactive. 

The (35~44) route develops the branch canker relationships, and also contains both 

always active and temporarily inactive branch cankers. 

Computation of number of infected seedlings (35~43). Branches on trees up to 

1 m tall are so short that all cankers are considered lethal. So no distinction by 

type of canker is necessary and all cankered trees follow the same path. 

54 
",------ -- ---- ----- ----- - - --- - --...... ~-, , / 

I 

I 

I 

, 

\ 

\ 

/"'- -- ....... 

r--~----....... ~ \ 

, 

, 

\ 

eSOR 

Trees/ha 

50 

........ --;::---

I 

, 

/' 

se2T 

Trees/ha 

37 

" 

seGS 

Trees/ha 

38 

seTD 

Trees/ha 

39 

/ 

\ 

\ 

\ 

'\ 

, 

...... 

" \ 

I 

I 

/ 

/ 

I 

I 

\ 

\ 

, 

I 

I 

I 

, 

" ..... 

eS3Y 

Trees/ha 

48 

selT 

Trees/ha 

36 

SelRS 

Trees/ha 

45 

...... ...... ..... ..... 

'" \ 

selGS 

Trees/ha 

46 

TORose 

Trees/ha 

47 

\ 

\ 

\ 

'\ , 

---

'\ 

, 

...... ..... 

eS2Y 

Trees/ha 

42 

eLNT 

Trees/ha 

35 

ORRBe 

Trees/ha 

56 

I 

/ 

/ ./ ,. 

" " 

I 

\ 

\ 

eS1Y 

Trees/ha 

43 

BelT 

Trees/ha 

44 

' ..... 

'\ 

\ 

'\ 

, , 

I / .; 

..... -

BelAS 

Trees/ha 

51 

BelSG 

Trees/ha 

55 

' ..... _-- _____________ c _______ _ 

--_ .... / ,. 

I 

I 

l 

I 

I 

I 

I 

/' 

I 

ANRT 

Trees/ha 

74 

Trees/ha 

49 

I 

I 

./ 

I 

I 

'" 

BeSG 

Trees/ha 

60 

ORBC 

Trees/ha 

61 

Figure 25.--Box~and-arrow diagram of Cronartium ribicola infection ln Pinus monticola 

stands. 

55 

I 

I 

I 

I 

I 
Table IS.--Definition of variables used in stand infection submodel of the white pine 

blister rust simulation 

Variable 

State variables: 

X(3S) CLNT 

X(36) SCIT 

X(37) SC2T 

X(38) SCGS 

X(39) SCDT 

X(42) CS2Y 

X(43) CSIY 

X(44) BCIT 

X(4S) SCIAS 

X(46) SCIGS 

X(47) TDRDCC 

X(48) CS3Y 

X(49) BC2T 

X(SO) CSDR 

X(SI) BCIAS 

X(SS) BCISG 

X(S6) DRRBC 

X(60) BCSG 

Definition 

Number of clean trees in stand, trees per hectare. 

Number of stem cankered first year trees, trees 

per hectare. 

Number of stem cankered 'second year trees, trees 

per hectare. 

Number of trees in stem canker girdle storage, 

trees per hectare. 

Number of stem canker direct dead trees, trees 

per hectare. 

Number of cankered seedlings with second year 

cankers, trees per hectare. 

Number of cankered seedlings with first year canker, 

trees per hectare. 

Number of branch cankered first year trees, trees 

per hectare. 

Number of trees in stem canker inactive storage, 

trees per hectare. 

Number of trees with stem canker and in girdle 

storage, trees per hectare. 

Number of direct stem cankers, trees per hectare. 

Number of cankered seedlings third year cankers, 

trees per hectare. 

Number of lethal branch cankered 2-year trees, 

trees per hectare. 

Number of cankered seedlings dead from rust, trees 

per hectare. 

Number of trees with lethal branch cankers inactive 

and in storage, trees per hectare. 

Number of trees with reactivated lethal branch 

cankers in storage for girdling, trees per hectare. 

Number of trees dead due to rust from reactivated 

lethal branch cankers, trees per hectare. 

Number of trees with lethal branch cankers in storage 

for girdling, trees per hectare. 

56 (continued) 
Table l5.--Definition of variables used in stand infection submodel of the white pine 

blister rust simulation (continued) 

Variable 

X(6l) DRBC 

X(74) ANRT 

User controlled variables: 

BCIL 

BCPIL 

FER 

ICIAS 

SCIL 

SCPIL 

SORL 

SRRL 

STAND 

Computed variables: 

Definition 

Number of trees dead due to rust from lethal branch 

cankers, trees per hectare. 

Accumulated number of rust resistant trees, trees 

per hectare. 

Branch canker inactivation level, percent. 

Branch canker permanent inactivation level, percent. 

Fungus encirclement rate, meters/year. 

Inactive period for cankers inactive, years. 

Stem cankers inactivation level, percent. 

Stem canker permanent inactivation level, percent. 

Spots only resistance level, percent. 

Stem reaction resistance level, percent. 

Initial stand in trees per hectare. 

See text 

57 
Ul 

00 

Flow path 

J (35-+43) 

J(43-+74) 

J(43-+42) 

J(42-+74) 

J (42-+48) 

J(48-+50) 

J(35-+36) 

Table l6.--Stand infection submodel flow descriptions 

Flow name Flow description Flow equation

Seedling infection 

Spots-only resistance 

Infected seedlings age 

Stem-reaction resistance 

Infected seedlings age 

Infected seedlings age 

Stem infections 

Computation of number of 

infected seedlings per 

hectare 

Computation of number of 

seedlings that possess 

spots-only resistance 

Computation of number of 

infected seedlings that age 

one season 

Computation of number of 

seedlings that possess 

stem-reaction resistance 

CSlY 

SORT 

CS2Y 

SRRT 

Seedlings supporting 2-year- CS3Y 

old cankers age 1 year 

Computation of number of CSDR 

seedlings 1 meter and under 

in height at time of canker 

appearance that die in the 

current year 

Computation of number of SCIT 

trees that have incipient 

stem needle cankers appear 

from a single season's 

infection 

CLNT(l

CSlY'SO

CSlY-SO

CS2Y'SR

CS2Y-S

CS3Y + 

CLNT(l
In 

~ 

Flow path 

JC36-+74) 

JC36-+37) 

JC37-+74) 

JC37-+3S) 

JC37-+4S) 

JC37-+38) 

Table 16.--Stand infection submodel flow descriptions (continued) 

Flow name 

Spots-only resistance 

Infected trees age 

Stem-reaction resistance 

Permanent inactivation 

Temporary inactivation 

Girdle storage 

Flow description 

Computation of number of 

direct-stem infected trees 

that have spots-only 

resistance 

Compilation of number of 

direct-stern-infected trees 

with cankers that will age 

1 more year 

Flow equation

SSORT SCIT·

SC2T = SCIT-S

Computation of number of SSRRT SC2T·

infected trees over 1 meter 

in height that exhibit stemreaction resistance from 

stem-needle cankers 

Computation of number of 

trees returned to CLNT due 

to permanent inactivation 

of stem cankers arising 

from stem needles 

Computation of number of 

trees infected with stem 

cankers from stem needles on 

which all cankers will be 

temporarily inactivated 

for the period ICIAS 

Computation of number of 

trees supporting stem 

cankers that came from 

stem needles and are not 

resistant or do not have 

all cankers inactivated 

TSCPI CSC2

TSCTI = -CSC2

SCGS SC2T-S
0\ 

o 

Flow path 

J(38-+39) 

J(45-+46) 

J(46-+47) 

J(35-+44) 

J(44-+74) 

J(44-+49) 

J(49-+74) 

Table 16.--stand infection submodel flow descriptions (continued) 

Flow name 

Girdle process 

Reactivation process 

Girdle process of reactivated 

cankers 

Branch infection 

Spots-only resistance 

Branch infections aged 

Stem reaction resistance 

Flow description Flow equation

Computation of number of 

trees in SCGS that will die 

in a given year 

SCTD 

Computation of number of SCIGS 

trees that have at least one 

direct-stern-needle inactivatedcanker reactivated 

Computation of number of TDRDSC 

trees that have had inactive 

cankers reactivated that will 

die in a given year 

SCGT 

RSCIA

SCIT

Computation of number of 

trees to have new lethal 

branch cankers appear in 

the current simulated year 

BClT 

(CLNT

Computation of number of 

branch-cankered trees that 

have spots-only resistance 

BSORT BClT

Computation of number of 

branch-cankered trees that 

age 1 year 

BC2T 

Calculation of stem-reaction BSRRT 

resistance trees in the 

branch cankered population 

BSIT-B

BC2T
0"-

~ 

Table 16.--Stand infection submodel flow descriptions (continued) 

Flow path Flow name 

J(49-+3S) Permanent inactivation 

J(49-+S1) Temporary inactivation storage 

J(49-+60) Girdle storage 

J(60-+61) Girdle process 

J(SI-+SS) Reactivation process 

J(SS-+56) Girdle process 

*See appendix I for derivations 

Flow description 

computation of number of 

BC2T trees that have all 

branch cankers permanently 

inactivated 

Compilation of number of 

trees that will have all 

branch cankers temporarily 

inactivated for the period 

ICIAS 

Computation of number of 

trees that will go into 

storage for time it takes 

the rust to grow down the 

branch, girdle the stem and 

kill the tree 

computation of number of 

trees in BCSG that will die 

in a given year 

Computation of number of 

trees with all branch 

cankers that will have at 

least one canker reactivate 

Computation of number of 

trees that will be damaged 

by reactivated cankers in 

TGIR02 years after cankers 

have been inactive for 

ICIAS years 

Flow equation

TBCPI (BC2T 

TBCTI (BC2T

BCSG BC2T-B

DRBC BCTD 

BCISG RBCIA

DRRBC BCITO 
The proportion of the stand infected in anyone year is determined by a regression of proportion infected on average number of cankers per tree (fig. 26). The 

data for this regression were obtained from anywhere we could find mention of or 

could calculate percentage of the stand infected versus average number of cankers in 

the stand (Childs and Bedwell 1948; Childs and Kimmey 1938; Mielke 1943; Buchanan 

1938; Macleod4

; and authors' unpublished data). In all, 167 data points were found. 

All influencing factors such as tree size, number of years exposed, and site were 

ignored. Our curve was nearly the same as Childs and Kimmey's (1938) curve for 

infection by 5-foot height classes and was very close to Van der Plank's (1975) 

theoretical curve. We assume that the number of years of exposure has no bearing on 

this curve. That is to say, it makes no difference if a stand is exposed to 5 years 

of inoculum level L (total inoculum load = 5L) or to a single large inoculum load of 

5L, the percentage of the stand infected would be the same in each case. Many of 

the data points used in the regression (assuming that 1 needle lesion = 1 canker) 

were from one application of heavy inoculum load on seedlings in the nursery under 

artificial conditions (authors' unpublished data). These points seem to match up 

with others that were obtained from field data on much larger trees exposed to many 

years of natural inoculum. We also arranged the data into age (size) classes, but 

no constant relationship was found. It seems that age and number of years of 

exposure are not as important as infection level. This regression is a very sensitive 

function in the model. Any predictive value the model may have would depend a great 

deal on the accuracy of the infection probability function. 

We make the additional assumption as did Childs and Kimmey (1938) that the same 

general relationship must exist for lethal cankers as exists for all cankers. 

o 

UJ 

IU 

UJ 

u.. 

1.0 

Z 0.8 

VI 

0::: 

I.J.J 

co 

::E 

UJ 

::E 0.6 o 

Z 

;:: 

VI 

u.. 

o 0.4 Z 

o 

I0::: 

o 

a.. 

~ 0.2 a.. 

o 

/ 

/ " 

", 

.",. 

" 

.., .., 

-------------------- ----- ----- ~ ---------------- -------- ----- ------ --- -- -- I /' I ,... 

I ./"'" 

I ./ 

I / 

I / 

I / 

I / 

I / 

I / 

I / 

I / 

I / 

I / 

I I 

I I 

I / 11 

I 

2 4 6 8 10 12 14 16 18 

AVERAGE NUMBER OF CANKERS PER TREE 

Figure 26.--Regression of proportion of Pinus monticola stand infected on average 

number of Cronartium ribicola cankers per tree. Dotted lines are 95 percent confidence 

limits. 

4 Macleod, R. L. 1937. Pine disease survey and scouting for blister rust in the 

inland empire, 1937, p. 112-116. In 1937 annual report Spokane office blister rust 

control. (Mimeographed) U.S. Dep. Agric., Bur. Plant Quar. 

62 
computation of number of seedlings that possess spots-only resistance (43~74). 

The proportion of the stand exhibiting spots-only resistance (McDonald and Hoff 

1970 and 1971) is given by the externally supplied variable--Spots Only Resistance 

Level (SORL). This resistance is actually expressed during the movement of the rust 

down the needles, but since the infection portion of the model simulates infection 

on an average tree, this needle resistance cannot be accounted for on an individual 

tree basis until this point in the simulation. Hence, the calculation is made and 

trees resistant by reason of the spots-only mechanisms are removed from X(43) before 

additional flows are calculated. 

Computation of one season's aging of seedlings with l-year-old cankers (43~42). 

The seedlings remaining after removal of resistant individuals are aged I year. 

Computation of number of seedlings that possess stem resistance (42~74). The 

externally supplied variable, Stem Reaction Resistance Level (SRRL), is used to calculate 

this flow. SRRL is the proportion of the stand that can express a Stem-Reaction 

Resistance Level (see Hoff and others 1973). The flow is placed at this point in 

the simulator because it takes 2 years of canker development for the stem resistance 

to show up. 

Seedlings supporting 2-year-old cankers age 1 year (42~48). The resistant 

trees are removed, and the remainder advanced 1 year. An important point to keep in 

mind is that only the first cankers on a tree count, since this is the canker that 

will be first to do damage. 

Computation of the number of seedlings 1 m and under in height at time of 

canker appearance that die in the current year (48~SO). Our experience shows that 

seedlings generally die within 3 years of canker appearance. The entire contents of 

X(48), trees that have been cankered 3 years, are flowed to X(SO). In turn, X(SO) 

is not emptied so that the number of seedlings simulated to have died from C. 

ribicola can be accumulated. 

Compilation of number of trees when average stand height is over 1 m to have 

incipient stem-needle cankers appear from a single season's infection (3S~36). If 

plants are over I m tall then a proportion of the uninfected stand (CLNT) is transferred to the first stem-needle cankered stage state variable X(36). 

Computation of the number of direct-stem-infected trees that have spots-only 

resistance (36~74). Resistant individuals are moved to the resistant tree accumulation state variable (74). 

Compilation of the number of direct-stem-infected trees with cankers that will 

age 1 more year (36~37). Needle-spats-only resistance removes trees from the population and the remaining infected trees develop for one additional year. 

Computation of number of infected trees over 1 m in height that exhibit stemreaction resistance from stem-needle cankers (37~74). The only difference between 

this flow and flow (42~74) is that the trees are assumed to be over I m in height 

for the current flow. 

Computation of number of trees returned to CLNT due to permanent inactivation 

of stem cankers arising from stem needles (37~3S). This flow deals with the problem 

of canker inactivation described by Kimmey (1969) and Hungerford (1977). Certain 

cankers are permanently inactivated while others remain inactive for a time then 

reactivate. This is treated as a probability. If the amount of permanent inactivation, or the amount of reactivation, the average number of direct stem cankers per 

infected tree, and the amount of inactivation are given, we can calculate the probability of a tree exhibiting only inactive cankers. The inactivation and reactivation 

63 
levels can be given as external variables and the average number of cankers per 

infected tree can be given by the calculations discussed above. Theoretically, if a 

tree has one direct stem canker and the probability of inactivation is 0.4, then the 

probability of that tree having only an inactive canker is 0.4 or if 10 trees have 

one canker each then four of those trees should have no active cankers. To digress 

for a moment, there are three things that can happen to a tree supporting cankers: 

(1) some cankers continue to grow; therefore disease development follows a normal 

course; (2) all the cankers are permanently inactivated and the infected tree would 

be classified as clean; (3) all the cankers are inactivated for a period of time and 

then one or more continue development. 

Thus, the flows from state variable 37 to state variables 74, 35, and 45 are 

controlled by the probability of an infected tree having stem resistance, that all 

cankers have been permanently inactivated, or that all cankers have been temporarily 

inactivated. 

computation of number of trees infected with stem cankers from stem needles on 

which all cankers will be temporarily inactivated (37~5). The logic for calculating 

this flow is the same as outlined above. 

computation of number of trees supporting stem cankers that came from stem 

needles and are not resistant or have not been inactivated (37+38). This flow is 

computed as the remainder after removal of resistant trees and trees with inactive 

cankers. 

computation of number of trees in X(38) that will remain in X(38) for the 

period required to complete the girdling process (38+39). Period required for bark 

cankers established from stem-needle infections to girdle the bole is determined by 

a function called GIRDLE (appendix IV). TGIRDI computes the storage period for 

stem-needle infections and TGIRD2 computes the period for branch-needle infections. 

Thus, the time required for girdling is calculated separately for cankers originating 

from stem and from branch needles. Computation of the storage period for stemneedle cankers is based on current tree height, past tree height, rust encirclement 

rate, tree circumference growth rate, and the length of time that elapses before 

death occurs after the girdle is complete. 

computation of number of trees with all direct stem-needle cankers temporarily 

inactivated and placed into storage for the duration of the inactivation period 

(45+46). So little is known about the inactivation process that we have assigned an 

external variable (ICIAS) to define the inactive period. 

computation of the number of trees that have had inactive cankers for the 

period ICIAS and now have reactivated cankers that will be stored until the girdling 

process is complete (46+47). The flow simply holds the infected trees with no 

change for the period ICIAS as set by the user. 

computation of the number of trees to have new lethal branch cankers appear in 

the current simulated year (35+44). After trees supporting stem-needle cankers have 

been removed from CLNT, we must next account for trees supporting lethal branch 

cankers. 

computation of number of branch-cankered trees that have spots-only resistance 

(44+74). The rationale for this flow is the same as flow (43+44). 

computation of number of branch-cankered trees that age 1 year (44+49). After 

the resistant trees are removed, the infected trees are transferred to X(49). 

Calculation of stem-reaction resistance trees in the branch-cankered population 

(49+74). The rationale for this flow was discussed under (42+74). 

64 
computation of number of BC2T trees with permanently inactivated branch cankers. 

Such trees have returned to the uninfected state for reinfection (49~35). An 

externally supplied variable sets the level of permanent branch canker inactivation. 

Compilation of number of trees that will have all branch cankers temporarily 

inactivated for the period ICIAS (49~51). Most of the r~tionale has been discussed. 

An external variable sets the level of temporary branch canker inactivation and 

another external variable sets the length of the inactive period for both branchneedle and stem-needle cankers. 

Computation of number of trees that will go into storage for the time it takes 

the rust to grow down the branch, girdle the stem, and kill or top-kill the tree 

(49~60). This flow simply sets up the value that is to be stored. The next flow 

executes the storage process. 

Computation of number of trees in BCSG that ~lill die in a given year (60~61). 

The rationale for this flow is discussed under flow (38~39). 

Computation of number of trees on which all branch cankers have been inactivated 

and on each of which at least one canker will be reactivated (Sl~5S). The period is 

defined by the external variable IeIAS. 

Computation of number of trees that will be damaged by reactivated cankers in 

TGIRD2 years after cankers have been inactive for ICIAS years (55+56). This flow 

adds the inactive period IeIAS to the INDX time already computed in order to simulate 

the tree growth that occurred during IeIAS. The circumference of the point of entry 

is now larger; therefore, girdling will take longer. 

DISCUSSION 

Verification 

You have seen the step-by-step process of model formulation. The next step 

will be programing the model for computer operation. Our programing objective will 

be to allow for the annual status of any state variable to be plotted and to incorporate various operational modes from complete random selection of infection weather to 

control of infection year and stand age when infection occurs so that various "what 

if" games can be played. After programing, the simulation will have considerable 

research value. But, before being used to directly assist in management decisions, 

it must be verified. The steps that should be taken range from calibration-prediction 

tests to verification of individual components through experimentation. 

The simulator is nothing more than a hypothesis about the real world and, as 

such, it can never be complete. But, it should be able to assist us in our reading 

of real situations and predicting alternate outcomes. Obviously, we cannot predict 

weather, but we can read weather and other input data and compare actual and predicted 

results. Many such predictions could be made and the simulator could be modified as 

data are accumulated. At the conclusion of the general verification procedure, the 

range of the simulator's applicability and capability should be well understood. 

Verification should include mapping Ribes and pine populations, monitoring 

aeciospore production and rust buildup on Ribes (bush by bush), and recording weather 

data such as days after April I to midpoint of an infection period, infection period 

duration, average temperature of infection period, and pattern of infection period 

occurrence. Occurrence of incipient cankers should then be recorded each year for a 

65 
period of years. The input data (weather; pine host parameters, such as stocking 

age, needle length, and number of stomatal rows; and Ribes bush species and density) 

would be used to predict the levels of the various applicable state variables. Some 

outputs such as basidiospore load need not be measured directly, because the objective 

would be to predict pine damage from easily measured parameters. Aeciospore load 

does need to be measured directly, because we do not have a value of aeciospore load 

that is a required input parameter. Ribes, pine, rust, and weather variables would 

be entered for a given year over a number of different sites that would vary in 

aspect, elevation, site quality, pine age, Ribes density, and Ribes species composition. In 5 to 10 years, the simulator could be verified, calibrated, and ready for 

use by forest managers. 

Additional Research 

The model formulation phase of the blister rust simulator has highlighted many 

areas in need of additional research. Areas of greatest need will be discussed in 

order of their appearance in the model formulation. The first major function 

HEIGHT (appendix II) was modeled on the basis of a very small amount of data. The 

mensurational literature contains little on annual height increment of western white 

pine. In fact, the whole area of the relationship between tree growth and target 

area needs additional work. For example, there is one report relating some crown 

parameters to number of needles (Buchanan 1936) in white pine; but more data are 

needed in order to address the effects of site quality, aspect, elevation, slope, 

and competing vegetation. In addition, little is known of the influences of environment on needle length and number of rows of stomata, both of which appear to have a 

considerable influence on effective target. 

The relationships shown in figures 27 through 30 are crude empirical functions 

of the influence of sunlight on Ribes density. These important relationships could 

be refined experimentally for many different Ribes species and sites. These are 

important relationships to understand because with understanding there is an opportunity 

for us to manage pine stocking density, which can have a direct effect on Ribes 

density. Conversely, stands opened by the death of trees susceptible to blister 

rust may result in increased Ribes density. Some work has been done on these ecological 

relationships (Wellner 1946), but much more is needed to fully integrate these 

possibilities into a rust management plan and to make this model more realistic. 

Greater breadth of data is needed to round out our understanding of the four 

important physical aspects of pine target--needle number, needle length, number of 

stomatal rows, and period of needle retention. We need to know more about the 

relative contributions of both environment and genetics to variation of these traits 

because there may be some opportunities for breeding or stand manipulation to shift 

stands to an unfavorable condition for the rust. It might be possible to regulate 

needle length and number of stomatal rows by stocking density and/or species mixes, 

as well as by breeding. Aspect and nutrient status of the site could also influence 

these characters. We should acquire enough understanding so that we do not inadvertently make one or more of the factors more favorable to the rust. Period of needle 

retention is of special importance and will be discussed more fully here. 

More knowledge is needed concerning the Ribes target as well. How do various 

site conditions influence number of leaves per bush, size of leaves, and stomatal 

frequency? How does Ribes genetic susceptibility vary from site to site and species 

to species? For example, in our detached leaf inoculations, hundreds of urediospore 

transfers were made to four clones of Ribes hudsonianum. One clone became infected 

at about the 85 percent rate; another was positive about 50 percent of the time; but 

the two remaining clones remained disease-free. Research is needed on the hypothesis 

that this situation could be explained by a group of specific races of the rust, 

because the two clones that will not accept our restricted rust population were 

lightly infected in the field. 

66 
3 

c::x:: 

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:::::> 

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c 

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c 

0 

V) 

"C 

:::J 

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0::1 

u. 

0 

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:::::> 

z 

20 30 40 50 60 70 80 90 100 

PROPORTION, FULL SUN 

Figure 27.--Number of Ribes hudsonianum bushes per hectare as a function of proportionfull-sun. 

6 

5 

c::x:: 

:::t: 

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:::::> 

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:::::> 

z 

20 30 40 50 60 70 80 90 100 

PROPORTION, FULL SUN 

Figure 28.--Number of Ribes inerme bushes per hectare as a function of proportionfull-sun. 

67 
« 

:1 ::I: 

tr. 

u..J 

I 160 c.n 

=> 

CC! 

II 120 J 

a::: I 

w.... 

0 

a:::: 

I.I.J 

cc 80 

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=> 

z 

40 

20 30 40 50 60 70 80 90 100 

PROPORTION, FULL SUN 

Figure 29.--Number of Ribes lacustre bushes per hectare as a function of proportionfull-sun. 

7001 

I 

600 j 

I 

I c.( 

~ 500.1 VI . I oJ..J 

:J: 

I Vl 

::J 

co 

E 400 ::J 

E 

';:;; 

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0 

u 

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100 

20 30 40 50 60 70 80 90 100 

PROPORTION, FULL SUN 

Figure 30.--Number of Ribes viscosissimum bushes per hectare as a function of proportionfull-sun. 

68 
In addition to the above, we performed a survey of C. ribicola infections on R. 

viscosissimum at the West Fork of Merry Creek outplanting site (Bingham and others 

1973) on August 5, 1977. We randomly selected 92 bushes from a strip across the 

center of the outplanting. Of the 92 bushes inspected, 46 were disease free; 16 

supported very sparse infections, and 20 were heavily infected. All three kinds of 

bushes appear to be located at random. Heavily infected bushes were always surrounded 

by disease-free or lightly infected bushes (see Hahn 1928). 

These two observations and Hahn's (1928) findings under greenhouse conditions 

indicate that rust resistance probably exists in Ribes populations. How widespread 

is it, what are the genetic and environmental components of its variation, and can 

we manage stands of white pine so as to increase the resistance in Ribes or at least 

not decrease it? 

I 

Very little is known about the patterns of aeciospore dispersal and viability. 

Aeciospore inoculum density should be measured at a number of sites and those levels 

related to nearby aeciospore sources. We need more information about the duration 

of aeciospore viability and period of release, and little is known about the aeciospore 

infection process. Can we expect to exercise any control over aeciospore loads or 

do we want to try? 

This modeling effort has uncovered a series of questions about both aeciospore 

and urediospore infection processes. A model of infection is presented simply to 

suggest a framework to facilitate additional studies. The ideas about individual 

infection life cycles and productivity should be subjected to experimentation. We 

should learn how these aspects of C. ribicola are controlled and how they contribute 

to the epidemic. Are there opportunities for management or pitfalls to be avoided? 

Some aspects needing research are influence of temperature and moisture on 

aeciospore and urediospore germination percentage, period of germination lag, period 

of germination, and germ tube growth rate on both water agar and leaves. Additional 

knowledge is needed about the relative patterns of spore attachment and stomatal 

location. How are electrostatic forces involved in the infection process? How 

important are the various methods of spore dispersion (wind, water, and animals) to 

natural inoculation with urediospores? 

In spite of all the past work on and the importance of teliospore production, 

it still remains an uncontrollable variable. One cannot routinely produce teliospores 

in a laboratory setting. The factors that influence teliospore production in the 

field are not really known. Can aecial infections produce teliospores or are 

urediospore infections required? What triggers the initial formation of teliospores? 

Once an individual infection begins producing teliospores, can it subsequently 

produce more urediospores? 

In addition to knowing the trigger(s) that change urediospore production to 

teliospore production, we need more data on variation in telial column length, 

diameter, and growth density. There appears to be a strong Ribes species component 

to all these factors, and an environmental component probably also contributes. 

Knowledge about these rust traits is important because column size and density 

influence rate of release of basidiospores which have a direct bearing on the number 

of basidiospores released per infection period. For example, if the same number of 

teliospores were "packaged" in many long, slender columns as opposed to a few long 

thick columns, the long slender columns would enjoy a considerable advantage in 

causing pine infections during short infection periods because of a higher basidiospore 

release rate. Ribes lacustre tends to produce long slender columns and also appears 

to cause more damage than would be expected on the basis of telial column density. 

Perhaps the rust on R. lacustre can make more efficient use of short infection 

periods. How many other subtle differences in the rust might exist both within and 

69 
between Ribes species? How much of this variation could be used to our advantage, 

and how much constitutes pitfalls that can hinder our efforts to develop a workable 

strategy of rust management? 

Another area of teliospore biology that is in need of further research is 

germination and basidiospore production ratio. What conditions influence germination 

of teliospores and the number of basidiospores produced per teliospore germinated? 

This information is needed for incorporation into the model and may be of value in 

developing alternative control measures. Also a better understanding is needed to 

facilitate screening of pine populations for resistance by way of more efficient use 

of available inoculum. Many of the traits concerned with teliospore production and 

germination may be amenable to some sort of manipulation procedure, either genetic 

or environmental. 

The basidiospore dispersion portion of this model was based more on simple 

assumptions than available knowledge about the movement of fungal spores in the air 

could lead to. But, specific empirical knowledge about Cronartium ribicola basidiospore 

dispersion was used to make the simplified assumptions. Since a more sophisticated 

approach would require a large additional effort, the simplified model should be 

tested before further work is done on basidiospore dispersal. 

Despite many reports (Patton and Johnson 1967; Hansen 1972; Hansen and Patton 

1977; Hirt 1942) the infection of pine by basidiospores is poorly understood. 

Results of artificial inoculations are extremely variable. According to the theoretical treatment by Van der Plank (1975), there should be a tight linear relationship 

between the amount of inoculum and the amount of disease. Regarding blister rust on 

western white pine, this relationship appears to be very loose (authors' unpublished 

results), as shown by the fact that the value defined as basidiospore infection 

yield ratio (BIY), which is the slope of the regression of infections on inoculum 

load, is highly variable. 

Many factors can influence this ratio, and most have already been discussed. 

The primary research needs are: (1) to measure the influence of temperature and 

moisture on percentage of basidiospore germination, germination lag period, germination period, germ tube growth rate, and germ tube morphology; (2) to measure the 

influence of pine genotype and Ribes species on the above rust traits; and (3) to 

determine the role of needle surface waxes and electrical charges on spore trapping 

in different environmental conditions and on various host genotypes. 

Another area needing attention is the transport of the spore to the near vlclnlty 

of the needle. What role does wind play? We have noticed considerable variation in 

basidiospore size. Does this variation of basidiospore size interact with atmospheric 

conditions to cause variations in amounts of disease in such a way that management 

options can reduce losses? Does basidiospore size variation relate in some way to 

racial variation of C. ribicola? How does needle orientation influence amounts of 

infection? What are the genetic and environmental components of the variation in 

needle orientation? 

The relative susceptibility of secondary needles of different ages is a feature 

of the blister rust system that is of primary concern, because the literature reports 

that current-year needles are up to 14 times less susceptible than needles in their 

second or third year. If this is true, then a very appealing control option would 

be to defoliate trees to rid them of 2- and 3-year-old needles after a heavy infection 

season. Differential defoliation chemicals such as Ethephon (Griffing and Ursic 

1977) need to be tested. Breeding western white pine for 2-year branch needle and 

I-year stem needle retention is also possible. If one were to develop this idea, we 

believe the first task should be to investigate differential susceptibility more 

fully. Some knowledge about the effects of differential defoliation on growth is 

already available (Linzon 1958). 

70 
Available knowledge about the rate of appearance of cankers from needle infections 

and growth of cankers down branches is limited to the original white pine population. 

Natural selection by the rust over the past 40 years for increased resistance in the 

pine could have caused large changes in these relationships. So, current wild pine 

populations may behave differently than expected. These relationships should also 

be considered in artificial breeding programs. Currently, they are not. Delayed 

appearance of the rust in the stem could lead to greater losses of infections when 

needles are naturally shed. A factor that might work in opposition is that infected 

needles in the absence of needle-shed resistance may be selectively retained by the 

tree. 

The portion of the simulator that partitions branch infections into harmless 

branch cankers and lethal branch cankers is based on limited data. This portion 

should be studied further if sensitivity analysis demonstrates that it is an important 

relationship. We need a better understanding of how natural inactivation, biological 

agents, resistance, and environment influence a canker's capacity to grow from the 

branch into the bole. These relationships have an important bearing on the management 

options one might select. Breeding for a rate of reduced rust growth or development 

of genetic and cultural means to increase tree growth rate could greatly influence 

amount of rust damage. 

The key function in the stand infection submodel was derived from the regression 

of proportion infected on average number of trees infected. It appears that this is 

another highly sensitive relationship in the model. A much larger data base is 

needed for the regression of stand infection on average number of cankers per tree. 

Even though the curve illustrating this function is close to the theoretical curve 

given by Van der Plank (1975) for diseases in general and by Childs and Kimmey 

(1938) for random blister rust infection in a homogeneous stand, we still need to 

know how the environment, tree size, tree age, inoculum load, pine genotype, and 

pine species influence this function. 

Knowledge about natural inactivation is very limited; also, a minimum amount of 

effort was expended on this portion of the simulation. There should be, however, a 

good deal more effective way to represent even the amount of information that is 

available. Since we do know that inactivation, both permanent and temporary, occurs 

(Hungerford 1977), sensitivity analysis should be used to determine the importance 

of canker inactivation before additional effort is expended on recasting the model 

or acquiring new knowledge. Management options ranging from breeding to utilization 

of biological control agents might apply here. 

The representation of girdling of boles could use a more complete treatment of 

canker placement in the crown and a better calculation of period of growth down the 

branches to the stem. Again, due to lack of knowledge, a simplistic representation 

was developed to provide a place for this aspect of the disease. If additional 

modeling efforts are deemed necessary in the future, a more complete model can be 

substituted. 

A very important piece of information that is related to many aspects of the 

whole model is growth rate of the fungus. We have made the assumption that growth 

rate of the fungus is independent of growth rate of the host. Since we have no 

information on the influence of site or other factors on fungus growth rate, the 

parameter (FER) is a user-controlled variable. Because of the potential importance 

of this variable, it should receive high priority in any future modification or use 

of this simulator. 

The current form of the blister-rust simulator is an initial and most incomplete 

representation of the real system. But, it still provides a partial definition of 

the boundaries of the blister-rust system. Definition should lead to a clearer 

71 
concept of how a change (application of a management option) in one part of the 

system is connected to a change in another part. Such understanding can greatly 

facilitate our ability to formulate experiments, to gain knowledge about the system, 

and to develop management prescriptions that will reduce the impact of the rust on 

ecosystems containing white pines. 

72 
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957. 

Spaulding, P., and A. Rathbun-Gravatt. 

U.S. Dep. Agric. Bull. 

1925a. Conditions antecedent to the infection of white pines by Cronartium ribicola 

in the northeastern United States. Phytopathology 15:573-583. 

Spaulding, P., and A. Rathbun-Gravatt. 

1925b. Longevity of the teliospores and accompanying urediospores of Cronartium 

ribicola Fischer in 1923. J. Agric. Res. 31:901-916. 

Strand, M. A., and L. F. Roth. 

1976. Simulation model for spread and intensification of western dwarf mistletoe 

in thinned stands of ponderosa pine saplings. Phytopathology 66:888-895. 

Taylor, ~1. W. 

1922. Potential sporidia production per unit in Cronartium ribicola. Phytopathology 

12:298-300. 

76 
Van Arsdel, E. P., A. J. Riker, and R. F. Patton. 

1956. The effects of temperature and moisture on the spread of white pine blister 

rust. Phytopathology 46:307-318. 

Van der Plank, J. E. 

1975. Principles of plant infection. 216 p. Academic Press: New York and 

London. 

Waggoner, P. E. 

1974. Simulation of epidemics, p. 137-160. In Epidemics of plant disease: 

mathematical analysis and modeling. J. Kranz, ed. Springer-Verlag, New York. 

Waggoner, P. E., and J. G. Horsfall. 

1969. EPIDEM: a simulation of plant disease written for a computer. Conn. 

Agric. Exp. Stn. Bull. 698. 

Wellner, C. A. 

1946. Recent trends in silvicultural practice on National Forests in the western 

white pine type. J. For. 44:942-944. 

Wellner, C. A. 

1948. Light intensity related to stand density in mature stands of the western 

white pine type. J. For. 46:16-19. 

Zadoks, J. C. 

1971. Systems analysis and the dynamics of epidemics. Phytopathology 61:600-

610. 

Zadoks, J. C. 

1972. Methodology of epidemiological research. Annu. Rev. Phytopathol. 10:253-

276. 

77 

APPENDIXES 

79 

APPENDIX I 

Quantitative Models 

The derivation of relationships among state variables will be detailed in this 

section. Forcing functions, donor-controlled or donor-recipient controlled interactions, are developed for each flow within a given submodel. Each flow will be 

discussed under a subheading composed of the flow name. In referring to the boxand-arrow diagrams, remember that the dotted lines show where various functions 

apply when it is possible to have one derivation apply to the relationship between 

more than one set of state variables. Only the flow equations and their derivation 

are discussed in this section. 

RIRFS DENSITY SUBMODEL 

Ri/J(,J Generation 

The principal derivation in this submodel was the Ribes density function for 

each of the four major Ribes species; but before we begin that derivation we must 

look at the computaton of Proportion Full Sun (PFS) , which depends on tree height 

and density. Tree height is computed by function HEIGHT from site and age data as 

discussed in appendix II. Stocking density and tree height are used to model stand 

basal area in square meters per hectare. Basal area was defined as the total cross 

sectional area of stems 1 inch and larger at breast height per unit of land area, 

where 

Cross sectional area nd.b.h. 2/4, 

0.786 d.b.h. 2. 

We assumed that the height calculated by function height is the average height for 

all trees in the stand being simulated (Current Tree I-leight (CTH]), and that the 

d.b.h. of a tree of height CTH is the average d.b.h. for the stand. D.b.h. as a 

function of tree height was obtained from a regression of total tree height on 

d.b.h, (Deitschman, Glenn H., unpublished, this Station): 

D.b.h. in meters CTH - 1. 14 

75.59 

Current Basal Area per hectare (CBA) is estimated by multiplying stems per hectare 

by the average cross sectional area as follows: 

CBA = TLT-0.786 ([CTH-l.14]/75.59)2 

where 

TLT = Total number of Living Trees per hectare (computed variable). 

An empirical function that relates PFS (Proportion Full Sunlight) to basal area 

(Wellner 1948) was used as follows: 

Light intensity (PFS) e- (0.02-CBA). 

81 
The blister rust eradication program accumulated extensive records on the 

numbers and species of Ribes plants removed during the first working. These records 

are summarized in table 17 for the Kootenai, Cabinet, Clearwater, St. Joe, Coeur 

d'Alene, and Kaniksu National Forests of northern Idaho and western Montana (Swanson 5). 

The data were recorded by stand type (open reproduction, dense reproduction, 

open pole, dense pole, open mature, dense mature, and burn), without definitions. 

In order to associate a light level with Ribes density, a few assumptions were made 

using white pine stand tables (Haig 1932). We assumed reproduction to be under 20 

years old, pole-sized material to average 50 years of age, and mature trees to 

average 140 years of age. Open reproduction was assumed to be stocked at 200 trees 

per acre, open pole, 100 trees per acre, and open mature, 25 trees per acre. Armed 

with these assumptions, we calculated the levels of light intensity for each type of 

stand (table 17). This method assumed a constant relationship between shading and 

basal area, which we know is not strictly true, but present~y we have no better 

information. 

Table l7.--Number of Ribes bushes eradicated per hectare during the years 1929 to 1937 

in northern Idaho National Forests 

Stand Proportion R. R. R. R. 

type full sun 1 lacustre viscosissimum petiolare inerme 

Open 

reproduction 0.99 217 613 1.0 6.3 

Dense 

reproduction .82 77 69 .4 3.0 

Open pole .96 103 93 .6 3.4 

Dense pole .43 52 31 .05 1.3 

Open mature .91 160 82 .9 1.4 

Dense mature .22 29 8 .05 1.6 

Burn 1. 00 169 670 2.2 4.6 

, lRelative amount of sunlight reaching the forest floor. See text for method of 

determination. 

5 

Swanson, H. E. 1937. Ribes eradication, Inland Empire, 1937~ p. 23-25. In 1937 

annual report Spokane office blister rust control. (Mimeographed) U.S. Dep. Agric., 

Bur. Plant. Quar. 

82 
Finally, we fit a power function to the data in table 17 to derive the relationship 

giving bushes per hectare as a function of proportion of full sunlight (fig. 27, 28, 

29. and 30). Each species follows the same general trend,but the specific curve for 

each is different. The general function (Ribes density function) that seems to best 

fit the available data is: 

wi1ere 

estimated parameters; 

for 

R. hudsonianum i 1, aOl 2.15, a21 16.58, 

R. inerme i 2, aO 2 1.45, a12 4.35, a22 11.84, 

R. lacustre i 3, aO 3 40.0, a13 190.0, a23 10.96, 

R. viscosissimum i 4, a04 40.0, a14 660.0, a24 27.03. 

These relationships are illustrated in figures 27, 28, 29, and 30. If a Ribes 

eradication program had been applied in the case of Ribes viscosissimum the input 

variable would be ERAD4 (eradication level of R. viscosissimum in bushes per hectare) 

instead of being computed by the density function. This value is less than 0 when 

no eradication program is underway, 0 when this species is absent, and some value 

greater than 0.0 that is equal to the number of bushes per hectare left after·the 

eradication has been performed. If eradication was not practiced or the bush 

density was not set, then the empirical function just derived that relates Ribes 

density to Proportion of Full Sunlight (PFS) comes into play. The ribes density 

submodel and the various management options are summarized in equation 1: 

J (5-+ 1 , 2, 3, or 4) 

RIRFS INFECTION SUBMODEL 

Aeciospore Infection 

a2' aoi + aliPFS 1; for ERAD. < 0 1 

0 for ERAD. 0 1 

ERAD. for ERAD. > O. 1 1 

RD. 1 

(1) 

Aeciospore generation.--The derivation of this flow equation is simple. The assumed 

or measured value of Aeciospore Maximum Hourly Load (AMHL) is multiplied by the 

duration of the infection period (DIP) in hours to give the potential spore load in 

aeciospores per square centimeter for the designated infection period. Thus: 

J ( 75-+76) AfvlH L DIP i 

PASL. 1 

where 

PASL Potential Aecispore Load 

i index to Ribes species. 

83 
Aeciospore availability.--We felt that a normal curve best fitted the pattern of 

occurrence of aeciospores over time (fig. 6), and that the Weibull function (Bailey 

and Dell 1973) adequately described this distribution: 

where 

c-l -((x-a)/b)c y = e(c/b) ((x-a)/b) -e (2) 

a,b,c = variable parameters 

e = base natural log 

x = known parameter 

y unknown parameter. 

In order to use the above approach, we needed to develop some way to determine 

date of appearance of aeciospores. In the spring of 1976, we collected aeciospores 

in our nursery beds located at Moscow, Idaho; in a natural stand near Clarkia, 

Idaho; and in a natural stand at the Priest River Experimental Forest. Blisters 

appeared in early April at Moscow and in early May at Clarkia and Priest River. 

These times of appearance were matched with the summation of average daily temperature 

(maximum and minimum averaged). In each case, blisters appeared when the average 

temperature summed to 80°C degree days in no more than 10 days after a 24-hour 

average of less than 4°C. These criteria can be used to establish an expected 

appearance date from published weather records. Our experience indicated that about 

7 days after appearance, the first blisters break. Hence, Date of first Aeciospore 

aPpearance (DAP) will be recorded as date of blister appearance + 7 days and will be 

expressed as days after April 1 (also see Hirt 1964, p. 14). The aeciospore availability 

function (fig. 6) is used to compute proportion of the peak spore load that is 

available on a specified date. 

The number of AecioSpores Trapped per square centimeter during a given infection 

period (AST) is calculated as a function of days after appearance of the aeciospores 

as shown in figure 6, using the parameters listed in fig. 6 and equation (2): 

where 

J(76,77) W(A1 2. 6)_e(-A1

3

.

6

) -PASL 

AST. 1 

Al DAYINF/22.0, 

W 1.954 

c/b from equation 2 times a constant to scale function to maximum value of 

1;, this parameter will also be used in other versions of the Weibull function, 

DAYINF = DAAI - DAP. 

Aeciospore germination.--The first problem was to model germination of the available 

aeciospores as a function of temperature and assumed rust genotype. Rust genetics 

is introduced by assuming that optimum germination temperature is strongly inherited 

(fig. 8) and is controlled by input variable AGMAX as shown below: 

where 

_(A2)WCA PAGER = 10.295 (WCA/WBA)-A2-(WCA-l.0) e 

PAGER = Proportion of available Aeciospores on the underside of Ribes leaves 

that are expected to GERminate, given an infection period of unlimited duration and 

a given average temperature (TIP), 

84 
WCA and WBA 

from the rust), 

parameters that control position of the peak (the genetic response 

A2 = (TIP - 3.)/WBA, 

TIP = average-temperature of the infection period. 

WCA and WBA are parameters that control the shape of the Weibull function. We 

derived equations that give parameters WCA and WBA as a function of population mode: 

WBA = 7.095 + 0.19242*GMA + 0.0253l4*GMA2 

and 

WCA 0.857046 + 0.1726l8*GMA + 0.002759*GMA2 

where 

GMA Mode 

AGMAX-3. 

The constant, 3, is subtracted from AGMAX to adjust the mode as given by the 

Weibull function to the x-axis. As illustrated in figure 8, optimum germination 

temperature of rust aeciospores can be varied. 

Next a model of germination as a function of infection period duration was 

developed. Ultimately rust genetic variation could be incorporated here, too, but 

at present it is not. Proportion of GERMinated Aeciospores (GERMA) is assumed to be 

controlled by both temperature and duration of the infection period (DIP). According 

to our data, the GERmination Lag period of Aeciospores (GERLA) varies with temperature 

as shown in figure 9. A parabola provides a good empirical first approximation of 

this relationship. GERmination Period of Aeciospores (GERPA), that is the time 

required to complete germination once it has started, also varies with temperature 

(fig. 10), and appears to fit the parabolic function: 

where 

GERLA 

GERPA 

0.04*(TIp2) - 1.67*TIP + 19.43 

0.1*(TIp2) - 3.57*TIP + 35.86 

TIP average temperature of infection period in degrees C, 

GERLA time required for germination to begin in hours, 

GERPA time required to complete germination after it has begun in hours. 

(3) 

(4) 

We can model expected germination as a function of DIP, if we assume that any 

given population will germinate according to the standard normal distribution with a 

range equal to GERPA encompassing +3 S.D. and a mean of GERPA/2.0. Then proportion 

germinated of the population will be a function of duration and temperature of the 

infection period: 

GERMA CNORMP(A,B,C)*PAGER 

where 

GERMA = proportion germinated, 

CNORMP = normal distribution function obtainable on most computers, 

85 
A DIP - GERLA, 

B GERPA/6.0, 

C GERPA/2.0. 

The flow is computed by multiplying the number of aeciospores trapped (AST) by the 

proportion expected to germinate (GERMA): 

J(77,78) AST.*GERMA 1 

ASGER. 1 

Aeciospore germ tube growth.--If the length of the germ tube is a function of GERm 

Tube growth RAte (GERTRA) and duration of germ tube growth (DIP-GERLA), then GERm 

Tube Length of Aeciospores (GERTLA) becomes: 

GERTLA = GERTRA-(DIP-GERLA). (5) 

Since length of tubes should be distributed normally, the function CNORMP was used 

to compute proportion of GERminated Aeciospores with germ tube of proper length to 

Penetrate (PGERAP): 

PGERAP CNORMP(GERTLA, 33.3, 200.0), 

where 

GERTLA = average length attained by germ tube when infection period ends, 

33.3 = microns per standard deviation for the range 100w to 300w, 

200.0 = average of the distribution. 

Then number of AecioSpores germ Tubes of Adequate Length (ASTAL) is computed by: 

J(78,79) PGERAP*ASGER. 1 

ASTAL. 1 

Aeciospore infection establishment.--The average aeciospore infection yield ratio for R. 

hudsonianum (AIYR.) used in the model is the average of the experimentally determined 

values (table 7) klready discussed: 

where 

AIYR. 0.05 1 

i = R. hudsonianum index = 1, 

0.05 = constant obtained by dividing 11 infections per square centimeter by 

230 spores/cm2. 

This means that we should expect one aeciospore out of every 20 germinated 

aeciospores on the leaf surface of R. hudsonianum to penetrate and establish an 

infection. Many factors such as leaf age, moisture and temperature conditions, rust 

genotype, Ribes genotype, and Ribes species should influence this value. Consequently, 

much more study should be devoted to the determination and use of this infection 

yield ratio. Using the R. hudsonianum ratio we obtained and the information in 

86 
table 6, we computed the following values: for R. inerme, (AIYR2 = 0.037); for R. 

lacustre, (AIYR3 = 0.018); and for R. viscosissimum, (AIYR4 = 0.012). The number of 

AecioSpore Infections (ASI) derived per square centimeter of leaf surface on R. 

hudsonianum is: 

J(79,80) ASTAL. -AIYR. 1 1 

AS!. . 1 

Urediospore Infection 

Urediospores from aeciospores and urediospores.--The simulation of the number of 

urediospores produced by each aeciospore infection of a given cohort that will be 

available for causing new infections during a given infection period is determined 

by: 

UMASS. 1 ASI. -BMASS.-(1.0-TSAR.) 1 1 1 

where 

ASI. = number of aeciospore infections per square centimeter of leaf underside 

surface 6f a given species (i=l to 4) established in each infection period from 12 

to 60 days previous to the current period, 

BMASS. = production capability of the rust on a given Ribes species in cubic 

microns (f6r R. hudsonianum, 70xl06~3), 

TSAR. = proportion of BMASS. to be devoted to te1iospores, 1 1 

UMASS. = total mass of urediospores present in current infection period. 1 

Then the calculation of UredioSPORe production from aecial infection on a given Ribes 

species (USPOR.) is: 1 

where 

and 

J(80,8l) UMASS./(TI-USL-USD2 ) + BMASS. -(l-TSUR.) 1 1 1 

USPOR. 1 

TI 3.141, 

USL UredioSpore Length in microns, 

USD UredioSpore Diameter in microns, 

B~~SS. = spore production capability of given infection, 1 

TSUR. 1 proportion of BMASS~ produced by uredia1 infection that is devoted to 

te1iospore production, 

B~~SS. - (1-TSUR. ) 1 1 movement of USPOR. from X(85) and represents urediospores 

produced by uredil1 infections (see J(85,87)). 

87 
Urediospore viability.--This flow is based on equation 2 and the parameters given in 

figure 7 for urediospore viability (UV); so that: 

where 

and 

UV = 2.477(A4)1,Se-(A4)2.S 

A4 (DAYS-12.0)/20.0, 

DAYS = DAAl(M) - DAAl(L), 

DAAIM 

DAAIL 

current infection period date in days after April 1, 

last infection period date in days after April 1. 

Also, this flow must be programed so that all cohorts of urediospores are computed 

separately, because each cohort will have a different UV. But all viable urediospores 

will contribute to the next round of infection (see fig. 5). In simplified form: 

J(8l,82) UV 

VUSA .. 

1 

USPOR. 1 (6) 

Urediospore germination.--This flow is calculated by multiplying state variable 82 

by proportion of viable spores expected to germinate during a given infection period. 

The computation begins: 

where 

weu 

PUSGER = 9.577 (WeU/WBU)AS(WeU-l.O)e-(AS) 

PUSGER = Proportion of viable UredioSpores produced by one cohort of aecial or 

uredial infections per square centimeter that will GERminate in a given infection 

period of unlimited length and temperature of TIP. 

weu and WBU = curve shape parameters set by UG~~X (a user controlled variable 

from table 3) using same equations as for weA and WBA, except GMU = UGt-lAX-7, 

AS = (TIP-7.0)/WBU~ 

The remainder of this flow is calculated as in flow (77+78). But the constants 

of the equation for urediospore germination lag (GERLU) change (fig. 12): 

GERLU = O.03(TIP)2 - 1.19(TIP) + 12.8 (7) 

The constants of the equation for urediospore germination period (GERPU) remain the 

same as for GERPA (equation 4). Thus: 

GERMU = eNORMP(A,B,C)·PUSGER, 

A DIP - GERLU, 

B GERPA/6.0, 

C GERPA/2.0. 

88 
Finally: 

J(S2,S3) VUSA. ·GERMU 1 

USGER .. 

1 

Urediospore germ tube growth.--The equations are the same as flow (7S~79) except 

that some constants change. The GERm Tube Growth Rate (GERTRU) equation for urediospores 

(fig. 13) is: 

GERTRU = l36.23(A3)2.Se -(A3)3.S (S) 

The GERm Tube Length attained by Urediospore germ tubes (GERTLU) 

growth rate (GERTRU) times growth period (DIP-GERLU): 

is the product of 

GERTLU = GERTRU*(DIP-GERLU) 

Then Proportion of GERminated Urediospores with germ tubes of proper length to 

Penetrate (PGERUP) is: 

PGERUP = CNORMP(GERTRU, 33.3,200.0) 

and number of UredioSpores with germ Tubes of Adequate Length (USTAL.) is: 1 

J(S3,S4) PGERAP • USGER. 1 

USTAL .. 

1 

Urediospore infection--We have assumed that the Urediospore Infection Yield Ratio, 

UIYR., will have a range of values different from AIYR.; although we have no observation§. The number of infections expected from any one1cohort of aecial infections 

will be provided by: 

J(S4,SS) USTAL. • UIYR. 

USI .. 

1 

1 1 

Urediospores infection from urediospores.--Since infections caused by all cohorts of 

aecial infections must be accounted for, the calculations from X(Sl) to X(8S) must 

be repeated for all viable aecial cohorts (see fig. S). Then, since the urediospores 

arlslng from urediospore infections are assumed to be identical to urediospores 

arising from aeciospore-caused infections, all the calculations from X(Sl) to X(8S) 

must be repeated again. Since the different uredial cohorts will be of different 

ages, and urediospore viability is assumed to be a function of cohort age (fig. 7), 

each repeat will have 'a different urediospore viability value (UV in equation 6). 

The same is true for the aeciospore-derived-infection repeats of the last flow. All 

these infections are accumulated in state variable X(8S), UredioSpore Infections 

(USI). Also, each repeat of urediospore-derived infection can have a new partitioning 

of the spore biomass. The variable TSUR is used as follows: 

J(SS,Sl) BMASS. • (l-TSUR.) 1 1 

USPOR. 

1 

S9 
Teliospore Production 

Teliospore generation.--These flows are computed by calculating the sum of the 

teliospore biomass produced for a given cohort and then dividing by the expected 

size of a teliospore to yield the number of teliospores Thus: 

TSAR . • ASI. . BMASS . 

J(80,86) 1 1 1 

1T • TSL • TS02 

J(85,86) TSUR. . US I. . BMASS. 1 1 1 

1T • TSL . TS02 

TMAX. J(80,85) + J(85,86) 1 

where 

1T 3.141, 

TSL teliospore length in microns, 

TSO teliospore diameter in microns. 

Teliospore viability.--The proportion of available teliospores in any given infection 

period TMAX is multiplied by the teliospore viability function to give the number of 

Potential GERminated Teliosores (PGERT): 

where 

J (86,87) A9 l . 25 e-(A9)2.25 TMAX . • 2. 416 • 1 

PGERT. 1 

A9 = (OAYS-16)/27.5. 

Teliospore germination.--Number of Teliospores expected to GERminate (TGER) is 

calculated by: 

where PCAST 

J(87,88) PGERT. • PCAST 1 

TGER. 1 

(see fig. 7) 

proportion of viable teliospores that germinate in a given infection 

period (see appendix III for derivation). 

Basidiospore generation.--Thus, the Quantity of Basidiospores produced At Ribes 

Bushes (QBAB.) is computed: 1 

J (88,89; 88,90; 88,91; 88,92) TGER. • BSYR. 

1 1 

QBAB. 1 

where 

i 1 to 4 to index the four major Ribes species. 

90 

QBAB. 1 
Basidiospore distribution.--The basidiospores produced by the rust on the Ribes 

plants are windborne to the pine hosts. As is characteristic of all wind-disseminated 

pathogens, the number of spores reaching a given target is primarily dependent on 

the distance from the target to the spore source (Schrtldter 1960). Many other 

factors can influence the spore load-distance relationship like number of spores 

released, size and shape of spores, horizontal wind velocity, turbulence, and the 

density of point sources over a given area. We are not in a position to consider 

any atmospheric influences in this paper; so we are limited to spore number, source 

density, and physical spore characteristics. A theoretical treatment of the interaction 

of spore traits and flight pattern is also beyond the scope of this paper; however, 

these problems can be simplified through empirical use of Buchanan and Kimmey's 

(1938) plot of number of cankers per unit of needles exposed as a function of distance 

from a point source of basidiospores. 

Their empirical relationship was used to develop the dlstribution function. 

Since we use the concept of infection on an average tree, the number of basidiospores 

deposited on the average tree should be a function of dilution due to distance from 

spore source to tree in a given area. The criteria imposed for calculation of the 

average distance were regular spacing of the pine host and of the point basidiospore 

sources (Ribes bushes). This restriction would not be necessary if data were available 

on the relative spatial relationships of both hosts. The average distance was 

graphically determined (fig. 31) for a set of different stocking densities of pine 

and point Ribes sources and was found to be largely independent of pine stocking at 

the densities tested and to be very dependent on point source stocking density 

(table 18). Thus, average tree to point source distance was determined for a range 

of Ribes stockings from 5.9 bushes/ha to 2,500 bushes/ha and a pine stocking of 

2,500/ha. A plot of number of Ribes growing points per hectare yielded a curvilinear 

relationship (fig. 32). Growing points per hectare were used in place of bushes 

because Ribes bushes tend to grow in clusters and the concept of points gives more 

modeling flexibility as will be seen later. The average Ribes to tree distance was 

expressed as follows: 

ABTD = e- (.503 . 1n(RGP) + 3.65) (9) 

where 

ABTD average distance in meters, 

e base of natural logarithm, 

In(RGP) natural log of the number of Ribes Growing Points per hectare, 

RD .. 

1 

Table l8.--sample of graphically determined average distance from pine tree to Ribes 

growth point for four combinations of stocking 

Pine trees/ Ribes bushes/ Average tree to 

hectare hectare bush distance 

Number Number Meters 

2,500 39 6.08 

2,500 6.25 15.29 

1,111 44 6.25 

1,111 5.90 15.51 

91 
0· 

o 

G 

~~F---Ribes growth point 

I 

I 

I 

I 

I 

J--e----0---e---

I 

I 

I 

I 

I 

~- - -0- - - -G--1 Q 

Figure 31.--Illustration of method used to calculate average spore source to target 

distance. 

20 

:E 

L.J..J 16 L.J..J 

a::: 

I0 

I- 12 :::r::: 

V') 

=> 

co 

L.J..J 8 

u 

z 

« 

lV') 4 0 

IX 

0 20 40 60 80 100 120 140 160 180 

NUMBER OF R I BES G ROW I NG PO I NTS IHA 

Figure 32.--Plot of average distance from pine trees to ribes growing points per 

hectare. Average distances determined graphically as explained in the text. 

92 
If the average distance from the target and the number of spores produced by the 

point source are known, how can this information be transformed into an expected 

spore load? 

If one assumed that the ratio of cankers formed to spores trapped is independent 

of distance from spore source, then calculation of the percentage of total infection~ 

in each ring of Buchanan and Kimmey's (1938) experiment should reflect the percentage 

distribution of basidiospores in the rings. Such a calculation was made (fig. 33) 

and the resulting relationship was used to estimate basidiospore load as a function 

of average distance on the basis of concentric rings 3 m wide (fig. 34). In order 

to convert to absolute values, one must calculate the area within each ring. This 

area is found to be as follows: 

2 

A Al (2n-l) n 

where 

A area in the nth ring, n 

Al area in the center ring, 

n number of the nth ring. 

0.5 

0 

UJ 

u 

:::::> 

0 0.4 

0 

0:::: 

a.. 

Vl 

UJ 

0:::: 

0 

a.. 0.3 Vl 

0 

0 

Vl 

« 

co 0.2 LL0 

Z 

0 

to:::: 

0 0.1 a.. 

0 

0:::: 

a.. 

o 2 4 6 8 10 12 14 

CONCENTRIC RING NUMBER (RING WIDTH, 3 M) 

16 

Figure 33.--Distribution of Cronartium ribicola basidiospore load from a point source 

into concentric rings 3 meters wide. Calculation based on count of cankers on 

Pinus monticola located in concentric rings around a point source (Buchanan and 

Kimmey 1938). 

93 
Figure 34.--Illustration of 

concentric rings used to calculate simulated Cronartium 

ribicola basidiospore loads 

on basis of average bush to 

tree distance and ring area. 

We have simulated the quantity of basidiospores produced, calculated the area in a 

ring, determined the percent of the spores to be expected in a given ring, and can 

identify the ring by the average point source to tree distance. So, the basidiospore 

load expected from that point source can be computed. But, the background basidiospore 

load from other point sources in the vicinity must be considered. It has been 

estimated that 99.9 percent of the spores from a point source near the ground would 

land within 100 m of the source (Gregory 1952). This seems to be true for c. 

ribicola as indicated by Buchanan and Kimmey (1938) and Kimmey and Wagener (1961), 

if we assume that infections reflect basidiospore load. Thus, we calculated background 

as follows: The percentage of the basidiospores expected to fall within the boundary 

defined by the nth ring was subtracted from 1 to give the percentage of basidiospores 

beyond the average distance from source to tree. Background basidiospore load was a 

function of basidiospore load, average tree to source distance (ring number), and 

density of growing points. Basidiospore load expressed as Quantity of Basidiospores 

At Trees (QBAT) was calculated for the four Ribes species as follows: 

F = (89,93) , (90,93) , (91,93) , (92,93) 

QBAB.ePBIRX 111 QBAB. eGPF.e (l-PCT) 

(282,743.34(2RX-l) + 100,000,000 

where 

e TARGET. e RD. 

1 1 

= QBAT. 1 

QBAT. = basidiospore load resulting from Ribes sp. at average tree in spores 1 per square centimeters, 

QBAB. Quantity of Basidiospores expected per square centimeter of leaf surface 

At Ribes §p. Bushes, 

94 
PBIRX Proportion of QBAB. Basidiospores expected In Ring x, 1 

-0.8RX 

e + 0.007, 

RX = number of ring x, 

282,743.34 = area in the center ring in square centimeters, 

GPF. = number of Ribes sp. growth points per hectare, 1 

RD. bushes of Ribes sp. per hectare (from Ribes density submodel), 1 

(10) 

PCT accumulated proportion of basidiospores expecte~ in rings 1 through x, 

TARGET. = Ribes sp. target = square centimeters leaf area (one side) per bush 

(from table 1 5), 

Then: 

100,000,000 = square centimeters in 1 hectare. 

J(89,93) 

J(90,93) 

J(9l,93) 

J(92,93) 

QBSAT 

QBAT 1 , 

QBAT2, 

QBAT 3, 

QBAT4· 

J(89,93) + J(90,93) + J(9l,93) + J(92,93) 

4 

= L: QBAT. 

i=l 1 

PINF TARGET SUBMODEL 

Branch Targel 

Branch needle generation.--An empirical function was derived that relates number of 

new needles to tree height (fig. 35). A bit of discussion of the derivation of this 

relationship is in order. The first step consisted of determining the total number 

of needles in crowns on trees of varying height. Buchanan (1936) measured the crown 

height and width and estimated the number of needles on 6,809 western white pines 

ranging in height from 0.03 to 9 m. He then developed an empirical formula that 

gives the number of needles as a function of crown height and width: 

Number of needles = l7,680-crown length-(2-crown radius)1.16 

where 

Crown length and radius are in meters. 

95 
Vl 

:::> 

e 

::c 

!::: 

Vl 

I..U 

....J 

o 

I..U 

I..U 

Z 

We 

a:: 

I..U 

CXl 

400 

300 

200 

~ 100 z 

o 4 8 12 16 20 

TREE HEIGHT (M) 

24 

Figure 35.--Number of new 

pinus monticola needles 

I as a function of total 

height plotted from new 

needle column of table 19. 

This formula was coupled with Buchanan's (1938) published crown width and length 

values fOT a group of trees ranging from 1.5 to 48 m in height to develop a curve of 

number of needles as a function of tree height. Since adequate data to form the 

lower end of the curve were not available, we collected, dissected, and counted the 

number of needles on 28 trees ranging from 0.1 to 2.71 m in height. The completed 

curve is shown in figure 36. This curve and the needle counts were used to develop a 

life table for number of needles as a function of tree height (table 19). The number of 

needles from the fiTst year column was then plotted (fig. 35) to produce the curve of 

number of new needles as a function of tree height. 

800 

700 

600 

Vi 

:::> 

0 

:r: 

t::: 500 

~ 

§ 

z 

w... 

0 400 0::: 

I..LJ 

co 

2: 

:::> 

z 300 

200 

100 

o 4 12 16 20 

TREE HE IGHT (M) 

24 

96 

Figure 36.--Total number of 

branch needles in crown of 

Pinus monticola as a function of total height (calculated from Buchanan, 1936 

and 1938). 
Table 19.--Life table of branch needle populations on a western white pine modeled 

after Buchanan's empirical formula and crown size data 

Tree 

age 

Years 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

Tree 

height 1 

Meters 

0.2 

.26 

.38 

.59 

.99 

1. 20 

1. 50 

2.00 

2.50 

3.00 

3.40 

3.90 

4.30 

4.80 

5.30 

5.90 

6.40 

7.00 

7.60 

8.20 

8.80 

9.40 

10.10 

10. 70 

11.40 

12.00 

12.70 

13.40 

14.10 

14.80 

15.50 

16.20 

16.90 

17.70 

18.40 

19.10 

19.80 

20.50 

21.30 

22.00 

22.70 

23.40 

24.10 

24.90 

25.60 

26.30 

27.00 

Total branch 

need1es 2 

85 

321 

1,024 

2,592 

5,252 

8,628 

15,175 

22,473 

30,476 

39,087 

46,370 

55,920 

63,892 

74,244 

84,997 

98,396 

109,951 

124,258 

139,020 

154,213 

169,818 

185,817 

204,958 

221,755 

241,791 

259,327 

280,193 

301,483 

323,184 

345,281 

367,763 

390,619 

413,838 

440,807 

,464,774 

489,076 

513,705 

538,653 

567,549 

593,161 

619,071 

645,275 

671,767 

702,389 

729,479 

756,842 

784,471 

New 3 

needles 

60 

245 

740 

1,695 

3,085 

4,455 

8,739 

10,874 

13,992 

18,114 

19,273 

25,018 

26,501 

31,681 

36,302 

41,755 

44,890 

52,561 

57,640 

62,829 

69,984 

75,497 

84,531 

88,755 

98,767 

103,579 

113,206 

121,779 

128,727 

138,372 

146,748 

155,036 

164,590 

176,684 

182,423 

183,222 

203,367 

211,237 

225,750 

231,797 

242,342 

254,119 

262,064 

277,181 

284,053 

294,839 

307,270 

1-year-old 

need1es 4 

Number 

25 

60 

245 

740 

1,695 

3,085 

4,455 

8,739 

10,874 

13,992 

18,114 

19,273 

25,018 

26,501 

31,681 

36,302 

41,755 

44,890 

52,561 

57,640 

62,829 

69,984 

75,497 

84,531 

88,755 

98,767 

103,579 

113,206 

121,779 

128,727 

138,372 

146,748 

155,036 

164,590 

176,684 

182,423 

193,222 

203,367 

211,237 

225,750 

231,797 

242,432 

254,119 

262,064 

277,181 

284,053 

294,839 

2-year-old 

needlesS 

o 

16 

39 

157 

475 

1,088 

1,981 

2,860 

5,610 

6,981 

8,983 

11,629 

12,373 

16,062 

17,014 

20,339 

23,306 

26,807 

28,819 

33,744 

37,005 

40,336 

44,930 

48,469 

54,269 

56,981 

63,408 

66,498 

72,678 

78,182 

82,643 

88,835 

94,212 

99,533 

105,667 

113,431 

117,116 

124,049 

130,562 

135,614 

144,932 

148,814 

155,584 

163,144 

168,245 

177,950 

182,362 

lHeight calculated with Brickell's (1970) site quality formula for ages 7 through 

48; a~es 2 through 6, actual height. 1 16 

Buchanan's empirical formula 1.36LW· = thousands of needles (1936) and 

Buchanan's crown measurements (1938) were used to plot total needles, ages 7 

throu~h 48. Fig. 35 values were read from plot. 

Actual counts of new needles, ages 2 through 6 and calculated other ages new 

needles = total needles -1-year-01d needles -2-year-01d needles. 

4Assume no needle mortality from current to 1-year-01d class so previous year's 

current needles become current year's 1-year-01d needles. 

SAssume 35.8 mortality from 1-year-01d class to 2-year-01d class 

97 
These curves and the life table are very similar to those of Douglas-fir (Mit~ell 

1974). Thus, 

ANNN = 3490_CTH1. 365 (11 ) 

where 

ANNN = number of new branch needles as a function of CTH (fig. 35). 

A relationship that predicts Potential Infective Surface on branch needles was 

developed as follows: rows of stomata were counted (fall 1973) on a sample of 1972 

and 1973 needles taken from 6-year-old nursery grown western white pine (author's 

unpublished data). In all, eight needles from each of 32 trees were examined. Onehalf the needles were of 1972 origin and one-half of 1973 origin. Number of rows 

varied from 4.17 to 7.1 from one year to the next within the, same seedling and from 

4.67 to 8.37 for one seedling to another within 1 year. The above findings only 

indicate some of the variation one might expect. The kind of variation indicated in 

this small sample could lead to doublings or halvings of target areas. In order to 

accommodate this variation, we computed the average number of rows and standard 

deviation on the 256 needles (x = 6.34, s = 0.89). If we assume a normal distribution 

and accept a limit of +4 standard deviations, then the maximum number of stomatal 

rows expected is in the vicinity of 10 (0.89 x 4 + 6.34 = 9.8). 

Maximum needle length is assumed to be 10 cm. This assumption is based on an emplrlcal relationship between needle length and tree height (fig. 37) obtained from table 20. 

In the word model discussion of infective surface, maximum width of the infective 

strip in which each stomatal row is centered was identified as 0.01 cm (100~). 

Thus, each needle has approximately 1 cm2 maximum infective surface and 

J(29,32) 

o 

3490 CTH l . 365 

PISB. 

4 8 12 16 20 24 28 

TREE HEIGHT (M) 

Figure 37.--Length of Pinus monticola branch needles as a function of tree height. 

98 
Stomatal row adjustment.--The computation of this flow is performed by dividing 

maximum number of rows, 10, by RSPN (Rows of Stomata Per Needle) to give the reduction 

factor RFACT. Then: 

where 

J(32,33) 0.1 PISB • RSPN 

RAISB 

RAISB = Row Adjusted Infective Surface on Branch needles. 

Needle length adjustment.--In order to accommodate some sources of variation for 

needle length, we have included an empirical equation (fig. 37 and table 11) that 

can account for tree size and a user-controlled factor QNEEDL to be set to mimic 

other sources of variation. When using the equation, the ~alculation is 

Branch needle length = 

8.6 

1 + 0.ge-0.75CTH 

and 

Branch needle length QNEEDL = 10 

or we can set QNEEDL to provide the desired needle length. Thus, the flow is: 

J(33,34) RAISB • QNEEDL 

QISBI. 

(12) 

Aging infective surface.--Since QISB2 is not needed in the current year the equation is: 

J(34,31) QISBI 

QISB2. 

Aging infective surface.--Buchanan (1936) showed that 64.2 percent of the branch 

needles of a particular cohort are retained for their third growing season. So: 

J(30,31) 0.642·QISB2. 

QISB3. 

Needle death.--Buchanan (1936) showed that most branch needles die at the end of 

their third growing season; so, all needles in state variable 30 will be moved to 

state variable 10. Thus, 

J(30,10) QISB3 

+ SINK. 

99 
Stem Needle Target 

Stem needle generation.--The stem-needle growth function computes the amount of new 

stem needle infective surface by the following equation: 

J(29,2S) = 10_12_67S(1_e-l.lSCTH)_0.01 

where 

10 maximum number of stomatal rows, 

12 maximum needle length multiplier where the value is 12.0 instead of 10 

because stem needles average 1.2 times branch needle length (table 20), 

67S_(1_e-l.lSCTH) = new direct stem-needle function (fig. 38), 

0.01 = infective strip width in centimeters. 

(13) 

Equation 13 resulted from counting the number of new stem needles on trees varying 

in height from 0.1 m to 2.71 m (same trees as discussed under branch needle generation). 

An equation was found that fit the empirical curve (fig. 38). This relationship was 

developed from a minimum of data, but like so many other empirical functions in this 

model, it could easily be made more representative, either by statistical fit or 

preferably by biological understanding. Finally, the above equation reduces to: 

J(29,2S) = 8l0(1.0_e-l.lSCTH) 

PISDS. 

Stomatal row adjustment.--The derivation and use of RSPN was discussed under flow 

( 3 2 , 3 3) . Th us: 

where 

J(2S,26) O.l-PISDS-RSPN 

RAISS 

RAISS = Row Adjusted Infective Surface on direct-Stem needles. 

11 b 1 e 20. - -,1verage lengths of needles obtained from the stems and branches of western 

I.vhi te pine trees of varying ages 

Tree 

age 

Years 

Tree 

height) 

:Jeters 

0.26 

.3S 

.59 

6 .99 

7 1.20 

1. 50 

9 2.(10 

10 2.50 

50 27.45 

/\verages 

Trees 

sampled 

No. 

18 

Length of stem 

needles 2 

X C.\'. 

Length of branch 

needles 

x C.V. 

- - - - - - - ."Ii 11 imeters 

54 + 9.5 O.lS 50 + 3.9 O.OS 

61 - 7.9 .13 S5 + '+ 6.3 .11 

70 '+10.3 .15 56 -

+ 3.1 .06 - - (19 + 5.3 .OS 5S + 4.4 .OS 

72 '+13.01 .1Sl S9 -

+ S.1 .09 - 71 (12 

S7 (16 

S9 72 

39 + 14.7 .17 

0.14 O.OS 

Branch/ 

stem3 

1. OS 

1.11 

1. 25 

1. 19 

1. 22 

1. 15 

1. 32 

1. 24 

1. 20 

lilelnhts calculated by Brickell's (1970) formula. 

2Sev~ral hundred needles per tree ",ere measured, but no statistical significance 

should be placed in these data. They are used for illustrative purposes. 

3Ratio of length of branch needles to length of stem needles. 

100 
V"! 

UJ 

.....J 

0 

UJ 

UJ 

z 

:2: 

UJ 

lV"! 

I 

IU 

UJ 

e::: 

0 

u.. 

0 

e::: 

UJ 

C£l 

:2: 

:::::l 

Z 

700 

600 

500 

400 

300 

200 

100 

o 4 8 12 16 

TREE HEIGHT (M) 

20 24 

Figure 38.--Number of new 

Pinus monticola needles 

located directly on the 

stem as a function of 

tree height. 

Needle length adjustment.--QSNEDL is obtained by dividing the needle length equation 

Ceq. 12) by 12 instead of 10 because stem needles are longer than branch needles: 

Thus, 

QSNEDL = C 8·~0.75CTH ) /12 . 1+0.ge 

J C26 ,27) RAISS-QSNEDL 

QISSI. 

Aging infective surface.--Since no first year stem needles die, the entire contents 

of state variable 27 flow to state variable 28. 

J C27 ,28) QISSl 

QISS2. 

Needle death.--The assumption was made that all the stem needles in their second 

year die. So: 

J C28 ,10) QISS2 

SINK. 

101 
PINE INFECTION SUBMODEL 

Basidiospore Infection 

Basidiospore trapping.--Basidiospores of C. ribicola were cast onto primary needles 

of P. strobus inside a 'settling tower and the ratio of spores trapped to spores cast 

(spores per square millimeter) was determined (Hansen 1972). Four measurements 

reported were 0.01, 0.06, 0.08, and 0.18, x = 0.08 and will be called "average 

trapping efficiency." Settling tower experiments similar to Hansen's should be done 

to get a large number of trapping efficiency ratios for needles from different 

species. different needle ages, different tree ages, and from trees showing different 

kinds of resistance because variation from this source could be an important contribution 

to variation of IY. 

Thus, BSTIS (number of BasidioSpores Trapped on Infective Surface) was computed 

as follows: 

J(93,94) 

where 

QBSAT-TEF, 

BSTIS, 

TEF Trapping Efficiency Factor of branch and stem needles (user controlled 

variable). 

Basidiospore germination.--The mathematical relationship for describing this flow was 

constructed in much the same way as aeciospore germination was (flow 77,78). Equations 

14 and 15 are based on figures 19 and 20, respectively. Thus: 

BGL = 0.03(TIP)2 - 0.95TIP + 9.52 (14) 

where 

BGL Basidiospore Germination Lag time in hours, 

TIP average Temperature of the Infection Period in °c, 

and 

BGP 0.1(TIP)2 - 3.l7TIP + 27.12 (15) 

wl1ere 

SGP = basidioSpor~ Germination Period in hours. 

The germination is assumed to be normally distributed over time with a range equal to 

SGP and a starting point equal to BGL = 6.0, where 

6 = time in hours required for teliospores to germinate and produce basidiospores. 

Therefore: 

GERMB CNORMP(A, B, C) 

102 
where 

then: 

where 

A DIP - (BGL + 6.0), 

B SGP/6.0, 

C SGP/2.0, 

J(94,95) BSTIS-GERMB 

BSGER, 

BSGER = number of trapped BasidioSpores expected to GERminate during a given 

infection period per square centimeter IS. 

Basidiospore germ tube growth.--The number of BasidioSpores per square centimeter of 

leaf surface expected to produce germ tubes of Adequate Length for penetration (BSTAL) 

is calculated by: 

where 

PGERBP 

J(95,96) 

CNORMP(BGTL,S.3,75.0) 

PGERBP-BSGER 

BSTAL 

BGTL BTGR-A = Basidiospore Germ Tube Length in microns, 

BTGR Basidiospore germ Tube Growth Rate 

l2.634(AS)1,Se-(AS)2.S 

A = DIP-(BGL+6.0), 

AS = (TIP-0.2)16.S, 

S.3 = range of penetration length = (100~-50~)/6 S.D., 

75.0 = germ tube length when 50 percent have penetrated. 

(16) 

Basidiospore infection establishment.--The number of species having germ tubes of 

adequate length for penetration are reduced by the basidiospore infection yield 

ratios (BIY) and a needle resistance factor QNLR. The flow is calculated: 

J(96,97) 

where 

BSTAL-BIY-QNLR, 

BSI, 

BIY = spots/spore trapped on a square centimeter of infective surface of current 

year needles, 

QNLR = factor to adjust for presence of reduced-needle-lesion-frequency resistance. 

103 
Basidiospore infections summed over infection periods.--This flow is computed by 

combining basidiospore infections per square centimeter of IS, target in IS per tree, 

and reducing the number of infections by the stern-needle orientation factor. Thus: 

p p 

L J(6,7) = L (BSI-0.S-QISS1) 

1 1 

TNISl 

where 

p 

tree. 

L summation of infections from each infection period to give a season total, 

1 

P number of infection periods, 

I 

BSI = basidiospore infections per square centimeter of current year needles, 

QISSl = infective surface of current year stern needles, square centimeters per 

And since needle infections do not yield cankers for at least a year, the summed flow 

need not be changed. 

Basidiospore infection summed over infection periods.--This flow is the same as the 

last one except that the needle orientation factor changes: 

where 

QISBl 

P 

L J(6,12) 1 

P 

L (BSI·O.2S-QISB1), 

1 

TNIBl 

current year branch needle IS, square centimeters per tree. 

Basidiospore infections summed over infection periods.--This is like the last except 

a new factor was added to account for the increased susceptibility of needles in 

their second season. This factor is SNAIF (Secondary Needle Age Infection Factor), 

So: 

P 

L J(6,19) 1 

P 

L (BSI-0.2S-SNAIF-QISB2) 

1 

TNIB2. 

Needle Infection to Cankers 

Canker appearance.--The fundamental relationship of this flow was depicted in figure 22. 

An empirical equation that describes canker appearance is: 

where 

b 

below), 

_ 20 (b -x) % cankered = b (17) 

parameter that is related to growth rate of the rust down the needle (see 

x = years after inoculation with range of 1 to 4 years YR. 

104 
The b parameter was related to host resistance through the assumption that a resistance 

mechanism can decrease growth rate of the fungus in the needles. If a b value of 24 

gives a curve similar to the most susceptible families and the growth rate of the 

rust down the needle is 10 mm per month (Chapman2 ), then this point defines the 

susceptible interaction (fig. 39). By the same reasoning, if the curve for resistance 

families matches the empirical curve with b equal to 6 and if we assume that fungus 

growth rate was cut in half, then the straight line between the two points should 

provide the relationship between rust growth rate down the needle and the b parameter. 

An input variable RGRN (Rust Growth Rate in the Needle) is supplied to simulate 

resistance to the rust expressed as a reduction of rust growth in the needle. This 

variable allows us to simulate the rate of canker appearance following needle infection. 

Figure 39.--Relationships between b parameter 

of equation 17 and growth rate of Cronartium 

ribicola down Pinus monticola needles. 

24 

20 

16 

.0 8 

4 

o 2 4 6 8 

GROWTH RATE OF RUST (MMI Mo) 

If no resistance is present (RGRN = 10 mm/mo), then we can expect 60 percent of 

needle infections in X(ll) to flow to X(14). But if RGRN 5 mm/mo, then we would 

expect only 6 percent of the infections to flow. Thus: 

J(7,11) 

-1. 5 

TNISleb- 20 (b ) 

TCSl, 

where 

b = 3.6 RGRN - 12.0, 

RGRN rust growth rate in the needle, millimeters per month, 

TCSI cankers appearing after 1 year from stem needles infected in their first 

year. 

Since Tcas is a sum of all cankers appearing in a given year from stem-needle 

infections, its calculation will be deferred. 

Needle infections aged.--The needle infections that did not develop to canker status 

are aged 1 year. Cankers are removed from the needle infections: 

J (7 ,8) TNISI-TCSl, 

TI2Sl. 

105 

10 
Canker appearance.--In this flow needle infections in their second year reach canker 

status and are summed with cankers appearing from J(7,11)' Thus: 

where 

J (8,11), 

b = 3.6-RGRN - 12.0, 

-3.0 

TI2Sl_b- 20 (b ) 

TCS2, 

TCS2 = cankers appearing after 2 years from stem needles infected in their first 

year. 

Needle infections aged, 

The flow is self-explanatory. 

J(8,9) I 

TI2Sl-TCS2, 

TI3S1. 

Canker appearance.--All the remaining infections from stem needles infected during 

their first growing season become cankers. All the cankers to appear from stemneedle infections in a given year can now be summed: 

J(9,11) TI3Sl, 

Tcas = TCSI + TCS2 + TI3Sl. 

Canker appearance.--The same logic is applied to the calculation of number of incipient 

branch cankers as was used to calculate the number of new stem-needle cankers. Thus: 

-1. 5 

TNIBleb- 20 (b ) J(12,13) 

CIlBl. 

Needle infection aged.--Thus: 

J(l2,lS) TNIBI - CIlBl 

TI2Bl. 

Canker appearance.--This equation has been explained; thus: 

-3.0 

TI2Bleb- 20 (b ) J(lS,16) 

CI2Bl 

Needle infections aged.--So: 

J(lS,17) 

Canker appearance.--So: 

J(17,18) 

TI2Bl 

TI331. 

TI3Bl 

CI3Bl. 

CI2Bl 

106 
Lethal Canker Portion 

Lethal canker generation.--If needles are assumed to be uniformly distributed throughout 

the crown, the expected number of lethal cankers can be derived as follows: 

where 

Then: 

or: 

LC NLC 

TC TN 

LC number of lethal cankers, 

TC number of cankers total, 

NLC = number of needles in lethal cap, 

TN = number of needles total. 

Proportion lethal NLC 

TN 

NLC = TN-(proportion lethal). 

(18) 

NLC was tabulated for a range of given tree heights (TH) and is presented in table 21. 

The next problem was to find the length of the crown cap that is proportional to 

the lethal canker curve. 

Since, 

Total needles 8,725 (tree height)1.365 

from figure 36, then: 

__ (Total needles)1/1.365 Tree height 8,725 

and if the relationship between Tree Height (TH) and Total Needles (TN) holds for any 

length of TH and number of TN then: 

where 

LCCL = ( NLC )1/1.365 

8,725 

LCCL = length of lethal crown cap, 

NLC = number of needles in lethal crown cap. 

Lethal Crown Cap Length (LCCL) was computed and the result is given in table 21, The 

next step was to plot the crown cap lengths as a function of tree heights. The 

relationship was linear for total heights of 1 m and greater (fig. 40). Since trees 

under 1.5 m in height are labeled as seedlings where all cankers are lethal, only the 

linear portion of the curve is needed. Thus: 

LCCL = 0.23-TH + 0.6 (19) 

when expressed as a function of tree height. 

107 
§ 

0.... 

« 

U 

I 

z 

s: 

0 

~ 

U 

I 

-I 

« 

:c 

ILLI 

-I 

u... 

0 

:c 

I-

(.!) 

Z 

LLI 

-I 

Table 21.--Calculation of length of lethal-crown-cap in Pinus monticola under attack by 

C. ribicola 

Tree 

height 

Meters 

0.5 

.75 

l. 00 

1. 25 

1. 50 

2.00 

3.00 

4.00 

5.00 

6.00 

7.00 

8.00 

10.00 

12.00 

14.00 

16.00 

Total 

need1es 1 

Number 

3,387 

5,891 

8,72S 

11,832 

15,175 

22,474 

39,087 

57,887 

78,498 

100,680 

124,258 

149,102 

202,193 

259,327 

320,059 

384,051 

Proportion 

of cankers 

letha1 2 

1. 00 

1. 00 

.80 

.67 

.54 

.42 

.31 

.26 

.24 

.22 

.21 

.20 

.19 

.18 

.17 

.17 

ITotal number of needles = 8,725(tree height)1.365. 

Needles in Lethal 

lethal cap 

cap3 height 4 

Number l!eters 

3,387 0.5 

5,891 .75 

6,980 .85 

7,927 .93 

8,195 .96 

9,439 1. 06 

12,117 1. 29 

15,,051 1.49 

18,840 1. 76 

22,150 1. 98 

26,094 2.23 

29,820 2.46 

38,417 2.96 

46,679 2.96 

54,410 3.82 

65,289 4.37 

2Proportion cankers lethal = (0.65/tree height at infection l .

25 ) + 0.15 (empirical 

curve figure 24). 

3Number of needles in lethal cap = proportion of cankers lethal times total number . 

needles. 

4Lethal cap height = (number of needles in cap/8,72S)1/1.365. 

4.0 

3.0 

2.0 

1.0 

0 2 4 6 8 10 12 14 16 

TREE HEIGHT AT TIME OF INFECTION (M) 

Figure 40.--Postulated relationshi.p between length of lethal crown cap in Pinus 

monticola and tree height at time of infecti.on by Cronartium ribicola. 

108 
Another question that we investigated was the influence of canker growth rate on 

the development of the disease within individual trees. Of specific interest was the 

effect of fungus growth rate on proportion of branch cankers that are lethal. If we 

assume that reduction of fungus encirclement rate (FER) would bring about a proportional 

reduction in distance a branch canker could traverse to become a bole canker (fig. 23), 

then there should be a similar influence on the slope of the crown-cap-height function. 

Then, the slope of the length-of-lethal-cap equation (0.23 in equation 19) can be 

stated as a function of FER and a proportionality constant (x): 

Slope = xFER 

and 

x = Slope/FER 0.24 yr/m 

where 

FER 0.958 m/yr, 

Slope = 0.23. 

Thus, 

LCCL (0.24FER)ePTH(I)+0.6 (20) 

where 

FER = fungus encirclement rate in meters per year, 

PTH(I) = height of tree at time of infection, 

I = index to height, which is number of years from needle infection to appearance 

of cankers. 

Since proportion of lethal cankers is equal to number of needles in cap divided by 

number of needles total (equation 18) then: 

and 

Then: 

NLC = (8,725(0.24 eFER)ePTH(I) + 0.6)1.365 

TN 8,725(PTH(I))1.365. 

Proportion of lethal cankers (8,725(0.24 eFER)ePTH(I)+0.6)1.365 

8,725(PTH(I))1.365 

which leads to: 

J(13,14) 

e (( 0 . 24 FER) PTH (I) + 0.6 ) 1 .365 

CIlBl PTH(I) 

CLlBl, 

where 

PTH(I) = height of tree in meters at time of infection. 

I number of years from needle infection to canker appearance 

TIME-l in this case. 

109 
Lethal canker generation.--These flows are analogous to flow (16~17). 

J(16,14) 

where 

I = TIME - 2. 

° 25 FERoPTH(I) + 0.6)1.365 CI2Bl(--·--------~~--~ PTH (I) 

CL2Bl 

Lethal canker generation.--

J(l8,14) 

CI3Bl(0.24oFERoPTH(I) + 0.6 1.365 

PTH(I) ) 

CL3Bl, 

where 

I TIME - 3. 

Canker appearance.--

J(19,20) 

-1. 5 

TNIB2 ob- 20 (b ) 

CI1B2. 

Needle infections aged.--The needle infections that have not appeared as cankers are 

aged 1 year. The quantity is determined by subtraction: 

J(19,2l) TINB2-CI1B2 

TI2B2. 

Canker appearance.--Rationale has been discussed: 

J(2l,22) 

Needle infections aged.--

J(2l,23) 

TI2B2ob-20(b3.0) 

CI2B2. 

TI2B2-CI2B2 

TI3B2. 

Canker appearance.--All remaining needle infections appear as cankers: 

J(23,24) TI3B2 

CI3B2. 

Lethal canker generation.--

J(20,14) 

where 

I TIME - 1. 

CI1B2(0.24oFERoPTH(I) + 0.6) 1.365 

PTH(I) 

CL1B2 

110 
Lethal canker generation.--

J(22,14) 

I = TU1E - 2. 

CI2B2(0.24eFERePTH(I) + 0.6)1.365 

PTH(I) 

CL2B2 

Lethal canker generation.--

J(24,14) 

where 

I = TIME - 3. 

CI3B2(0.24eFERePTH(I) + 0.6) 1.365 

PTH(I) 

CL3B2 

So, Total Lethal Branch Cankers (TLBC) to appear in a given year is: 

TLBC = CLlBl + CL2Bl + CL3Bl + CLlB2 + CL2B2 + CL3B2. 

Total new cankers.--This flow provides the Average number of New Cankers Per Tree 

(ANCPT). 

J(6,98) Tcas + CIlBl + CI2Bl + CI3Bl + CIlB2 + CI2B2 + CI3B2 

ANCPT. 

Canker accumulation: This flow Accumulates all Cankers Per Tree over all years 

(ACPT). 

J(98,99) ACPT + ANCPT 

ACPT. 

5T AND INFECTION SUBMODEL 

Seedlings 

Seedling infection.--The equation controlling this flow is based on the regression 

given in figure 26. 

where 

Proportion Infected in Year Total cankers 

1.0 

1. 228 

1. ° + ANCPT 

PIYT 

ANCPT = Average number of New Cankers Per Tree in year Y. 

(21) 

CLNT will be set equal to STAND, which is the initial number of clean trees per 

hectare, and is a user-controlled variable. Thus: 

J(35,43) CLNTepIYT 

CSlY. 

1" 
" ,-. 

Spots-only resistance.--A user-controlled variable that fixes the level of spots-only 

resistance (SORL) controls this flow. 

J(43,74) 

where 

eSlY·SORL 

SORT, 

SORT = number of trees expressing Spots Only ResisTance. 

Infected seedlings age.--The spots-only-resistant trees are removed from X(43) by 

subtraction. 

J(43,42) eS1Y - SORT 

eS2Y. 

Stem-reaction resistance.--This flow is self-explanatory. 

J(42,74) 

where 

eS2Y·SRRL 

SRRT 

SRRT = number of trees expressing Stem-Reaction ResisTance. 

Infected seedlings age.--The resistant trees are removed and the remaining trees aged 

1 year. 

J(42,48) eS2Y - SRRT 

eS3Y. 

Infected seedlings age.--Oead seedlings are accumulated 3 years after cankers appear 

or 5 years after infection occurs. This state variable is used in the calculation of 

disease progress curves and is calculated: 

J(48,50) 

Stenl Needle Infect ion 

eS3Y + eSOR 

eSOR. 

Stem infections.--The proportion transferred is computed by the infection-probability 

function which has already been discussed (eq. 21). But, the variable TeOS is used 

in place of ANePT: 

l.0 Proportion Infected in Year with Stem cankers 

1.0 l. 228 + --- TeOS 

Then: 

J(35,36) 

112 
Spots-only resistance.--The use of SORL has been explained. 

where 

J(36,74) SClT·SORL 

SSORT 

SSORT = number. of Stem-needle infected trees showing Spots-Only ResisTance. 

Infected trees age.--The mechanics of this flow have been explained. 

J(36,37) = SCIT - SSORT 

= SC2T. 

Stem reaction resistance.--

where 

J(37,74) SC2T·SRRL 

SSRRT 

SSRRT = number of Stem-needle infected trees showing Stem Reaction ResisTance. 

Permanent inactivation.--Calculation of number of trees having all cankers either 

permanently or temporarily inactivated was treated as a probability problem. For 

example, assume the following: 

Number of trees in X(37) = 10; 

Level of permanent inactivation 0.2; 

Average number of cankers = 2.3. 

In the instance of individual trees, the probability would be equal to the product of 

the individual probabilities. The probability of all cankers being permanently 

inactivated on a tree with two cankers would be 0.2 x 0.2 or 0.04. Although we know 

it not to be strictly accurate, we will assume that the same reasoning will apply to 

the average number of cankers per infected tree as follows: if the infected trees 

average 2.3 cankers per t2e~ the probability of all the cankers on a tree permanently 

inactivating would be 0.2' or 0.025. If X(37) contained 10 infected trees then 

0.25 trees would flow to X(35). We have used the same logic in deriving the temporary 

inactivation function. Thus the proportion returned to X(35) is calculated: 

CLNT = SCPILSCPIT.SC2T 

where 

SCPIL Stem Canker Permanent Inactivation Level, 

SCPIT average number of Stem Cankers Per Infected Tree from cankers that 

appear in current year (assume that cankers inactivate when they appear). 

113 
SCPIT is equal to the total number of stem needle cankers that appear during the 

current year divided by the number of stem-infected trees for the same year. The 

number of new stem infections is equal to TCaS (number per average tree) times the 

number of stems in the stand X(35), and the number of stem infections is equal to 

X(35)-(1.O(1.O+(1.228/TCaS))). Thus, 

SCPIT TCaS-X(35) 

which reduces to: 

Then: 

where 

SCPIT 

J(37,35) 

Tcas + 1.228. 

(SC2T - SSRRT)-SCPILSCPIT 

TSCPI 

TSCPI = number of Trees with Stem Cankers Permanently Inactive. 

(22) 

Temporary ina~tivation.--The derivation of the inactivation function has been discussed. 

where 

J(37,45) (SC2T - SSRRT - TSCPI)SCILSCPIT 

TSCTI 

TSCTI = number of Trees with Stem Cankers Temporarily Inactivated. 

Girdle storage.--The number of trees supporting at least one active stem canker that 

resulted from a direct stem infection that will be girdled by the rust. 

J(37,38) SC2T - SSRRT - TSCPI - TSCTI 

SCGS. 

Girdle process.--This flow is computed by subroutine GIRDLE (appendix IV), according 

to the following arguments and is the time required for the rust to complete the 

girdling process: 

SCGT flow out of GIRDLE (Stem Canker Girdled Trees), 

SCGS number of trees flowed in, 

I = flag for TGIRDI equation (appendix IV), 

TEM37B = number of trees moved into SCGS, 

3 = index for PTH(INDX) determines canker location on bole, 

o flag that indicates Branch Growth Period (BGP, see eq. 23) is not needed. 

114 
The number of Stem Canker Trees Dead (SCTD) in a given year is: 

J(38,39) 

SCTD 

SCGT 

SCGT + SCGT. 

Reactivation process.--Flow of trees that have had inactive stem cankers reactivated. 

The argument definitions for the GIRDLE function in this case are: 

RSCIAS = flow out of the storage array = Reactivated Stem Cankers after InActivation 

Storage. 

CIAS = identification of the storage array = Canker InActivation Storage, 

3 = flag that indicates TGIRD = ICIAS (Time to GIRDle '= Inactive Canker InActivation 

Storage period), 

and 

TEM37A = number of trees to be moved into the array CIAS, 

o no INDX value because infection point not computed, 

o flag that indicates BGP is not computed, 

J(45,46) RSCIAS 

SCIGS. 

Girdle process of reactivated cankers.--Flow of trees supporting reactivated stem 

cankers while girdling is completed. The arguments for subroutine GIRDLE in this 

case are: 

SCITD quantity of trees to flow from X(46) to X(47), 

SCIGS 

Storage, 

identification of the array used = Stem Canker Inactivation Girdle 

1 = flag to identify the TGIRD formula used, 

CIAS quantity of trees flowed into array SCIGS to reside for TGIRD years, 

INDX flag used to identify PTH(INDX) used in computation of circumference that 

canker must encircle, 

So: 

o = flag indicating that BGP is not to be computed. 

J(46,47) = SCIGS 

TDRDSC. 

115 
Branch Needle Infection 

Branch infection.--Since we are still in the same time cycle, the amounts removed or 

added from CLNT must again be subtracted or added because the SIMCOMP compiler does 

not complete the flows until the end of the cycle. We compute: 

where 

J(35,44) = (CLNT-SCIT + TSCPI) 1.0 

BClT 

SCIT = flow X(35)+X(36) 

TSCPI = flow X(37)+X(35) 

TLBC = value of X(17). 

1.0 

1. 228 

+ --- TLBC 

Spots-onlY resistance.--To obtain this flow: 

where 

BSORT 

J(44,74) BCIT-SORL 

BSORT 

number of trees with Branch canker Spots-Only Resistance. 

Branch-infected trees aged I year.--

J (44.49) BSIT - BSORT 

BC2T. 

Stem-reaction resistance.--

where 

J(49,74) BC2T·SRRL 

BSRRT 

BSRRT = number of trees with Branch canker Stem Reaction ResisTance. 

Permanent inactivation of branch cankers.--AII variables and rationale have been 

discussed. Computation of number of lethal Branch Cankers Per Lethally infected Tree 

(BCPLT) uses equation 22 with the substitution of TLBC for TCOS and BCPIL is an 

externally supplied variable. So, 

where 

TBCPI 

J (49,35) (BC2T - BSRRT)BCPILBCPLT 

TBCPI 

number of Trees with Branch Cankers Permanently Inactivated 

(BS2T -BSRR-BSCPI)BCILBCPLT 

116 
Temporary inactivation storage.--

where 

J(49,Sl) =(BC2T - BSRRT - TBCPI)BCILBCPLT 

= TBCTI 

TBCTI = number of Trees with Branch Cankers Temporarily Inactivated. 

Girdle storage.--This flow computes the number of trees with active branch cankers 

that will await completion of the girdling process. 

where 

J (49,60) BC2T - BSRRT - TBCPI - TBCTI 

BCSG 

TSBCG = number of Trees to be stored in Branch canker Girdle. 

Girdle process.--The equation for this flow was developed from the word model and 

empirical equations based on Buchanan's (1938) data (fig. 41,42). The derivation 

is: 

where 

Years to reach bole = 

0.S-0.66(0.24-FER-PTH(I)+0.6)-0.3S/pTH(I) 

0.SFER(O.44PTH(I)+0.3) 

0.5 = division of branch length by one-half, 

0.66 = division of lethal-crown-cap height by two-thirds, 

(0.24-FER-PTH(I)+O.6) = height of lethal-crown-cap (from eq. 20), 

0.35 /PTH(I) = crown radius at base (figure 41), 

0.5 FER = fungus growth rate down the branch given as growth rate of the fungus 

around the bole, (FER)-O.S, 

(O.44PTH(I)+O.3) = total height of the crown (figure 42). 

8 

6 

4 

2 

o 2 4 6 8 10 12 14 16 18 

TREE HEIGHT (M) 

Figure 41.--Plot of Pinus monticola crown radius on total height (data from 

Buchanan, 1938). 

117 
12 

10 

8 

~ 

:::r: 

le.!) 

z 

LLJ 6 ....J 

Z 

s: 

0 

c:::: 

u 4 

2 

o 2 4 6 8 10 12 14 16 18 

TREE HE I GHT (M) 

Figure 42.--Plot of Pinus monticola crown height on total height (data from 

Buchanan, ] 938) . 

The above expression reduces to: 

Years to reach bole /PTH(I) (O.03 o FER-PTH(I)+O.069) 

FER(O.22PTH(I)+O.l5) 

INDX = BGP .• 

+ 2 

(23) 

INDX gives the period required for growth of the rust down the branch. This value is 

the period from appearance of an incipient canker until this canker reaches the stern. 

The 2 is added because it took the canker 2 years to reach state variable X(49). 

INDX is used in two places. First, it is used to compute the expected distance from 

the top of the tree at time of infection to the point where the canker is simulated 

to enter the bole. The development of this idea was treated earlier. Also, in the 

case of the current flow, the additional 2 years are needed so BGP is set equal to 

INDX. Second, INDX is used in the computation of circumference to arrive at a 

girdle period, in which case the additional 2 years are not needed; so 2 was subtracted 

from the constant, 5, in TGIRDI and the constant becomes 3 in TGIRD2. The arguments 

for TGIRD2 are CTH, PTH(INDX), CLF, FER, TCGR, and BGP. All have been discussed and 

defined. Thus, 

GIRDLE(BCTD,BCGS,2,TSBCG,INDX,BGP) 

computes storage period, where: 

BCTD quantity of trees flowed out, 

BCGS array identity, 

2 = flags TGIRD2, 

118 
SBCG quantity of trees flowed in (trees Stored for Branch Canker Girdle), 

INDX time from appearance of incipient canker to stem entry for height 

calculations, 

and 

BGP = period for growth down branch, 

J(60,6l) BCTD 

DRBC. 

Reactivation process. --The derivation of GIRDLE has been di,scussed. 

RBCIAS = quantity of trees flowed out of the array, 

BCIAS = branch canker inactivation storage the array identity, 

3 = flag that TGIRD (storage period) is equal to ICIAS, 

TBCTI = number of trees flowed into array BCIAS (number of Trees with Branch 

Cankers Temporarily Inactivated), 

and 

o flag that INDX is not used, 

o flag that BGP is not used, 

J(Sl,SS) RBCIAS 

BCISG. 

Girdle process.--The time required for reactivated branch cankers to girdle the bole 

is given by subroutine GIRDLE and the following arguments: 

BCITD number of trees flowed out of the array, 

BCIGS branch canker girdle and branch-growth period storage array, 

2 = flag that TGIRD2 is the equation used, 

TICR = number of trees flowed into the array BCIGS (number of trees with inactive 

cankers reactivated), 

INDX = flag used to identify PTH(INDX) used in computation of circumference that 

canker must encircle, 

BGP = length of canker growth period down the branch in years, 

and 

J(SS,S6) 

BCITD 

DRRBC. 

119 

APPENDIX II 

Function Height 

One of the principal forcing variables for the blister rust model is tree 

height. The following paragraphs explain the construction of a model that produces 

height increment as a function of stand age and site index. 

HTI is a function of stand age (TIME), an internal or external variable, and 

site index (SITE), an external variable. The value of HTI is used to update average 

Current-Tree Height (CTH), a computed variable. Successive'annual estimates of CTH 

will be retained for up to 20 years in an array (PTH(I)), which is indexed by the 

number of years from current simulated time. 

The interaction between C. ribicola, P. monticola, and Ribes spp. is related to 

stand height through its influence on light penetration and other variables. Also, 

the quantity and quality of susceptible pine host target area is assumed to be 

directly linked to CTH and values of past average stand heights are used to estimate 

average bole circumference at particular points of infection. These estimates are 

combi~ed with estimates of fungal growth rate to project the length of time required 

for the rust to girdle the bole. The computation of Ribes density, used for predicting 

the quantity of inoculum available, is derived from an index of tree crown height 

and number of living trees in the stand (TLT). 

Function "height" predicts HTI through a two-component model: 

1. An equation to predict the expected value of the increment. 

2. A mechanism designed to draw a random value of HTI from a population 

(assumed to be normally distributed) described by the expected value 

and an estimate of the variance about the expected value. 

These components were derived as follows: Brickell (1970) used a Richard's function 

as a model for a western white pine site index equation: 

5 = blH(1_b2e-b3A)-b4 

where 

e = base of the natural logarithm, 

5 site index (feet at 50 years b.h. age), 

H average height of dominant and codominant trees, 

A age of the oldest dominant tree in the even-aged stand, 

This equation was solved for H, 

121 
and then differentiated with respect to age to obtain an expression for instantaneous 

rate of increment in the average height of dominant and codominant members of the stand 

(h) : 

(24) 

where 

Figure 43, derived by substituting Brickell's (1970) parameter estimates, shows 

that equation 1 has sufficient flexibility to mimic the expected pattern of change 

in tree height with respect to time. In order to derive parameter estimates appropriate 

to our model needs and modeling assumptions, we required observations of annual 

increment from a sample of trees, drawn without respect to crown class, from a wide 

range of site indices. Appropriate sources of data were records of two other studies. 

1,000 

800 

~ 

~ 

IZ 

LLJ 600 

:2: 

LLJ 

c::: 

U 

Z 

I- 400 :r:: 

~ LLJ 

:r:: 

200 

o 20 40 60 80 100 120 

AGE OF OLDEST DOMINANT IN EVEN- AGED STAND (YRS) 

140 

Site index 

90 

70 

50 

160 

Figure 43.--Expected annual increment in the average height of dominant and codominant 

memhers of a western white pine stand as a function of age and site index (adapted 

from Brickel]~ 1970). 

The first of these was a set of 124 estimates of annual increment resulting 

from measuring internode lengths on 28 young white pine that were growing on an 

excellent site (estimated site index 112). These trees were dissected and all 

branch and stem internodes were measured and needles per internode counted to supply 

more data on needle distribution and number for the pine target simulation. The 

trees were collected near Elk River, Idaho, and ranged in age from 3 to 9 years. 

The second set of data was information recorded on trees selected for progeny 

testing as part of the blister rust resistance breeding program (Bingham and others 

1971). Data included height, age, and site index for all candidate trees. In 

addition, 29 of the candidate trees, distributed among five site index classes, were 

photographed in 1952 and 1953. The bole axis and film plane were nearly parallel 

and the view of the bole and crown from the photo point was unobstructed. These 

122 
photographs were mounted on an Addo-X increment-core measuring device. Total height 

and all internode lengths that could be accurately associated with a tree age were 

measured. Appropriate scale factors translated these measurements so that an additional 

629 observations of annual increment were developed. The distribution of observations 

among site indexes and age classes is shown in table 22. 

Table 22.--Distribution of western white pine height increment data among age and 

site classes 

Age 

class 

0-5 

6-10 

11-15 

16-20 

21-25 

26-30 

31-35 

36-40 

41-45 

46-50 

51-55 

Total 

Site index 

53 63 75 104 112 

- - - - - - - - - Number of increment measurements 

4 

7 

15 

15 

15 

15 

12 

2 

85 

5 

15 

15 

15 

15 

4 

69 

8 

14 

15 

22 

22 

20 

17 

15 

7 

4 

144 

1 

17 

46 

60 

59 

45 

25 

1 

254 

109 

22 

19 

25 

20 

6 

201 

Total 

110 

47 

84 

119 

123 

103 

75 

37 

30 

19 

6 

753 

The distribution shows paucity of data at young ages in the low site classes 

and older ages in the high site classes. At ages where annual height increment is 

near maximum (20 to 40 years), all classes, nevertheless, are fairly well represented. 

Parameters, estimated from these data using Marquardt's (1966) nonlinear least 

squares algorithm, are as follows: 

bI 0.653617 

b2 1.024494 

b3 0.024202 

b4 2.071822 

These parameters were substituted into equation (24) and the expression was simplified 

to the form given in equation (25). 

h = 0.033577ese(1-1.0244gee -0.024202 eA)1.07l82 ee -0.024202 e

A 

where 

h average annual height increment on a stand basis (meters) 

variable HTI, 

computed 

(25 ) 

S estimated site index (feet at 50 years b.h. age) external variable SITE, 

A average stand age = computed variable TIME. 

123 
Equation (25) is used to compute the expected value of the annual increment in 

average stand height. 

The theory describing the estimation of the variance about the regression 

surface for nonlinear models is not well developed. ~1arquardt's algorithm, however, 

minimizes an objective function, which has been transformed by a Taylor series 

expansion to an expression which is linear with respect to the estimated parameters. 

Assuming that parameter estimates are stable, the confidence bands about the regression 

surface can be approximated in accordance with linear theory. This extension of 

linear theory was deemed adequate to meet our modeling needs. Figure 44 illustrates 

the relationship between annual increment in average stand height and stand age for 

site index 75, and displays the 95 percent confidence bands computed from equation 

(26). Equation (26) was derived from the output of the fitting procedure and incorporated 

into the simulation program. 

~ 1,000 ~ 

I-

~ 

~ 

UJ 

~ 800 0 

z 

« 

IVl 

UJ 

<.!) 600 --------- « 

a::: 

> 

UJ 

« ..... ............ --- -- ..... ..... ~ 400 ..... ..... -- ..... I- ..... ........... 1, z 

UJ ..... ..... ::E 

UJ -... a::: 

c...> 200 - z 

-I 

« 

:::::> 

z 

z 

« 0 10 20 30 40 50 60 70 80 

STAND AGE (YRS) 

Figure 44.--Expected annual increment in average stand ~eight (h) versus stand age for 

site index 75 western white pine. Bands surrounding h represent approximate 95 percent 

confidence bands. 

4 4 

{L: L: 

i=l j=l 

ptp .. -(P.-p".)-(P.-P".)}- PHT ~J ~ ~ ] ] (26) 

where 

ptp .. 

procedur~-? 

the elements of the inverse moment matrix resulting from the fitting 

P.,P. = the partial derivatives of equation (24) with respect to parameters i 

and j,~evJluated at the appropriate values of sand A. 

(PI = d (eq. 1) I A A A A ) 

dbl {S,A,bl,b2,b3,b4} 

p.,p. = the average values of the partial derivatives of equation (24) with 

respect t6 parameters i and j resulting from the fitting procedure, 

124 
PHT the mean squared residual. 

Prediction of HTI is a three-step procedure: 

1. Equation (25) is evaluated to determine expected value, 

2. Equation (26) is evaluated to estimate the variance about the expected 

value, 

3. A number is drawn from· the normal distribution described by the estimated 

variance and the expected value. 

Function "height" yields total height of an average tree in the stand at an age 

equal to TIME. 

125 

APPENDIX III 

Subroutine Spores 

A method was needed that would accept input information about the nature of 

infection periods and Ribes species and calculate the "expected" germination percentage 

of teliospores. The following model was constructed to make this calculation. 

If the rate of basidiospore cast, or its integral, cumulative basidiospore 

cast, can be expressed as a function of telial column size parameters, then the 

number of basidiospores produced in an infection period should be estimable from 

teliospore numbers, telial column size parameters, and the duration of the infection 

period. 

Bega (1959) conducted an experiment wherein individual telial columns were 

placed in ideal environments and allowed to germinate to exhaustion. He reported 

cumulative basidiospore cast by I-hour time intervals for four telial columns of 

varying lengths and diameters. To our knowledge, these data are the only published 

quantitative information on the relationship between telial column size and basidiospore production for C. ribicola. 

We converted these data to show cumulative basidiospore production as a proportion 

of total production for each telial column (fig. 45). These data show a 5- to 8-

hour delay between the onset of suitable environment and the beginning of basidiospore 

cast and an inflection at 13 to 17 hours. Following this inflection, cumulative 

basidiospore cast generally increases at a decreasing rate until the column is 

exhausted. Bega (1959) concluded that the magnitude of the decreasing rate appears 

to be related to column size. 

lV'! 

< u 

V'! 

LLJ 

e:::: 

0 

a.. 

V'! 

~ 

0 

V'! 

< co 

u... 

0 

Z 

0 

~ e:::: 

0 

a.. 

0 

e:::: 

a.. 

LLJ 

> 

~ 

:5 

=> 

2 

=> 

u 

1.0 

0.8 

0.6 

0.4 

0.2 

0 

1.0 

0.8 

0.6 

0.4 

0.2 

0 

Telial Column No.1 

Size: 322 )J x 55 )J 

10 20 

Telial Column No.2 

5 ize : 770 J..I x 74)J 

10 

1.0 

0.8 

0.6 

0.4 

0.2 

30 0 

30 40 

Telial Column No.4 

Size: 1330 JJ x 105 )J 

10 

1.0 

0.8 

0.6 

0.4 

0.2 

o 

20 30 

Telial Column No.3 

5 ize : 378)J x 74lJ 

10 20 

HOURS OF ENVIRONMENT SUITABLE FOR GERMINATION 

40 50 60 

30 40 

Figure 45.--Cumulative percent of Cronartium ribicola basidiospores cast from telial 

column of various lengths and diameters (calculated from Bega, 1959). 

127 
We next graphed the data to show the proportional hourly rate of basidiospore 

cast (fig. 46). Several patterns were observed: 

z 

o 

IU 

::::J 

0.20 

g 0.10 a::: 

Cl.. 

u.J 

a::: 

o 

Cl.. 

Vl 

o 

Cl 

Vl 

<x:: 

co 

u. 

o 

u.J 

Io 

~ 0.20 --l 

<x:: 

z 

o 

Ia::: 

~ 0.10 o 

a::: 

Cl.. 

o 

Column No.1 

Size: 378lJ x 55 lJ 

10. 20 

Column No.2 

Size: 770 lJ x 74lJ 

10 20 

30 0 

30 

Column No.4 

Size: 1330 lJ X 105 lJ 

10 

o 

Column NO.3 

Size: 378lJ x 74lJ 

10 20 

HOlJRS OF ENVIRONMENT SUITABLE FOR GERMINATION 

60 

30 

Figure 46.--Proportional rate of basidiospore production observed from four Cronartium 

ribicola telial columns of various lengths and diameters (calculation based on data 

from Bega 1959'). 

1. Each rate curve was positively skewed with a maximum in the vicinity of 14 

to 20 hours. 

2. The height of the maximum was inversely related to column diameter. 

3. Each maximum was succeeded by a series of local maxima, which were spaced 

at 4- to 8-hour intervals and decreased in magnitude as time of exposure to germination 

conditions increased. 

4. The length of time required to exhaust a column was directly related to 

column diameter. 

These patterns strongly suggest that the rate of basidiospore cast is related to the 

arrangement of teliospores within a column. 

We examined several sections of telial columns to see if there was some apparent 

geometric arrangement of teliospores that could be expressed as a simple model. In 

longitudinal section, teliospores are arranged as chains of bulging cylinders with 

diameters slightly larger in the middle than at each end. Adjacent chains are 

offset so that the center bulge of a teliospore in one chain closely conforms to the 

restricted area at the joining of two teliospores in the adjacent chains (see 

Colley 1918). Spore arrangements observed in horizontal sections reflected this 

offset as teliospore diameters appeared to be quite variable. The uniformity of 

spore size exhibited in vertical sections, however, suggests that this variation is 

128 '* u.s. GOVERNMENT PBINTING OFFICE: 1980-777-()28/59 Region NO.8 
an artifact of the vertical arrangement. It would seem reasonable to consider the 

individual teliospore as a cylinder with a diameter equal to the average of the 

teliospore diameters observed in a column cross section. 

We further observed that teliospores, when examined in a cross section, were 

arrayed in rings about a single center spore or a center cluster of four spores. 

These patterns can be translated into simple geometric models (fig. 47) for which 

the number of spores in a ring (N.) can be approximated from the proximity of the 

ring to the center of the column tfor the center cluster, i=l). For columns with a 

single center spore, 

N. ~ {

I, if i=l 

6 e (i-l) if i>l, 

and for columns with a center cluster of four spores, 

N. = 6i-2 ~ 

The structure of the innermost ring can be determined from the "diameter" of the 

telial column when measured in units of average teliospore diameter. An even column 

diameter implies a center cluster of four spores; and an odd column diameter implies 

a single center spore. 

Key to germination dynamics 

0-6 hrs • 18-24 hrs 

• 6-12 hrs • 24-30 hrs 

• 12-18 hrs 0 30-36 hrs 

Figure 47.--Geometric model of the arrangement of teliospores within a Cronartium 

ribicola telial column. The section at the left shows the arrangement of spores if 

the inner-most ring is composed of a single spore. The timing of basidiospore 

production relative to the start of the infection period is illustrated by the 

shaded teliospores. Both sections represent columns composed of four rings and 

would theoretically completely germinate in 36 hours. 

129 
This simple geometric model of column structure is well suited to interpretation 

of the proportional hourly basidiospore cast rate data. Colley (1918) observed that 

teliospores appear to germinate in order of proxiMity to the exterior of the column. 

Assuming that teliospore viability is relatively constant throughout a column, the 

maximum rate of basidiospore cast should then correspond to the germination of the 

outer ring, resulting in the observed positive skew in the production rate curves. 

If we further assume that basidiospore cast from a specific ring is normally distributed 

over a fixed time interval, then the sequential germination of successively inward 

rings should result in a pattern of regularly spaced local maxima, decreasing in 

magnitude as time of exposure to germination conditions increases. Since these rate 

curves never approach zero between local maxima, an overlap between the germination 

periods of adjacent rings would seem likely. 

Finally, if the length of time required to exhaust a ring is fixed, columns 

wi th more rings, that is, greater diameters, wi 11 necessari l'y require more time to 

completely exhaust. Columns with relatively large diameters, however, will have 

proportionately fewer spores in the outside ring. Therefore, the greater the 

diameter of the column, the smaller the magnitude of the maximum in the proportional 

cast rate curve. 

We used this interpretation as the framework for a model to describe the rate 

of basidiospore cast. We assume a 6-hour delay between the start of an infection 

period and the beginning of basidiospore cast. We further assume that the cast from 

an individual ring will be normally distributed over a l2-hour period with a standard 

deviation of 2 hours. Overlap between the germination periods of adjacent rings is 

approximated as 6 hours. These time parameters were estimated from the pattern of 

occurrence of maxima in the cast rate curves for Bega's (1959) columns. Simulations 

based on the size characteristics of these columns show a close correspondence to 

observed cast rates (fig. 48). We wish to emphasize that the model nevertheless is 

based on a rather loose, but plausible, interpretation of an extremely small sample 

and is quite tentative. 

§ SIMULATED COLUMN NO.4 oa 

...-I 

X 1,000 z - Ob serve d C oum I n No 4 - Odd No. of Rings Even No. of Rings 

~ I- - - U 

:::::> 

c 

0 800 a::: - I- - Q.. 

LLJ 

a::: - - 0 

Q.. 

Vl 

~ 600 - - c 

Vl - - c:( 

co 

u.. 

0 400 - - LLJ 

I-

~ - ..J 

~ 0 200 - -

~ 0 

Q.. 

0 

a::: 

Q.. 0 J ih I I I I 

10 30 50 70 20 40 60 20 40 60 

HOURS SINCE START OF INFECTION PER 100 

Figure 48.--Comparison of observed and simulated proportional rates of basidiospore 

production from a Cronartium ribicola telial column l330lJ long and 105lJ in diameter 

(observed data from Bega, 1959). 

130 
The information required to drive this model of teliospore germination is 

Telial Column diameter (Width) (TCW), average TelioSpore Diameter (TSD), and the 

expected Duration of the Infection Period (DIP). Using the above described time 

parameters, these arguments are employed to compute the proportion of total potential 

basidiospores that will be cast (PCAST). The distribution of cast rate for an 

individual ring is incorporated as an array (CNORMZ) with elements corresponding to 

the integrals of the standardized normal density function evaluated for 1/20 increments: 

-2,5 -x2/2 1 CNORMZ(l) --f e dx 0,0062 

/2n _00 

+2,5 -x2/2 

CNORMZ(ll)= _l_f e dx = 0,9938 

/2n _00 

CNORMZ is then indexed by the number of hours that a particular ring has been 

casting basidiospores, The sequence of model calculations begins with the estimation 

of cross-sectional dimensions of the telial column: telial column diameter, 

NTSD = TCW(ISP)/TSD + 0,5, 

and the number of spore rings in the column, 

NR NTSD/2,O + 0,51. 

Next, the infection period duration is adjusted to reflect the 6-hour lag, 

DIP = D - 6,0, 

and the length of time required to completely exhaust the column is estimated, 

ETIME = 6.0*(NR + 1,0). 

If DIP is greater than ETIME, all available teliospores are assumed to germinate and 

PCAST is returned to the calling program with a value of 1,0, Otherwise, the proportion 

of total basidiospores cast must be computed from the estimated column dimensions. 

To facilitate estimation of the proportion of available basidiospores cast when 

the column does not exhaust, the proportion of teliospores in each ring RATIO(I) 

must be computed, For convenience of summation, RATIO is indexed from the outside 

of the column in, corresponding to the theoretical order of germination. The value 

of RATIO for a particular ring can be derived from the number of rings in the column 

and the number of teliospores in a Cross SECTion of the column (CSECT). For columns 

with a single center spore, 

CSECT = 3,O*NR*(NR-l.0)+l,O 

and 

{ (6.0*(J-l.O))/CSECT for I<NR 

RATIO (I) 

l,O/CSECT for I=NR 

where 

J NR - I + I, 

131 
For columns with a center cluster of four spores, 

CSECT = 4.0 * NR + 3.0 * NR * (NR-l.O) 

and 

RATIO(I) = ((6.0*J) - 2.0)/CSECT. 

Next, the Number of Complete Rings Germinating is computed: 

NCRG = (DIP - 6.0)/6.0+0.05 

and PCAST is set equal to the sum of the first NCRG values for RATIO(I). 

With the assumed time parameters, one or two rings may 'not completely germinate. 

The value of PCAST must be adjusted to include the proportions of basidiospores cast 

from these partially germinated rings. The first of these partially germinated 

rings will have an index: 

I = NCRG+l. 

The proportion of available basidiospores cast by the Ith ring is determined by the 

appropriate reference to the array CNORMZ: 

INDX = DIP - 6.0*NCRG, 

and PCAST is adjusted accordingly: 

PCAST = PCAST + RATIO(I) * CNORMZ(INDX). 

Finally, INDX is decremented by 6, I is incremented by 1, and the adjustment to 

PCAST is repeated to reflect the contribution to basidiospore cast from the (NCRG+2) 

ring. 

132 
APPENDIX IV 

Subroutine Girdle 

A way was needed to hold some infections at a steady state while the simulator 

cycled tree growth and infections on other trees. We accomplished the holding 

action by feeding simulated infected trees into an array at a specified point and 

then advancing the array 1 year at a time. These "infected" trees would then come 

"out" of the array at the desired time to be flowed to the next state variable. The 

following derivation mainly concerns the method of calculating the time a given kind 

of "infected" tree will remain in storage. The first derivation pertains to the 

storage time required for girdling of the stem by a canker that reaches the stem 

through a stem-needle. Thus: 

TGIRDl(A,B,C,D) ((O.042-(A-B)-(A-l.l4))/(A-l.40))/(C-D)+S 

where 

A CTH, 

B PTH(INDX) PTH(3) , 

C FER, 

o TCGR = tree circumference growth rate, 

5 additional time in years it takes after girdling for death to occur. 

CTH is read from function HEIGHT. INDX = 3 in this case because the infection 

would be on wood 3 years old if it reached the bole in 2 years from an infection on 

a needle infected in its first year. Since second-year stem needles are assumed to 

drop immediately after infection, only first-year needles can have infection and 

most infection from these needles should reach the bole in 2 years, the value of 3 

was selected. This number is used to calculate the circumference of the bole so 

that girdle time can be calculated. The infection is considered to be a point on 

the bole when the infection is 2 years old. We know this is not completely accurate 

but we have no data with which to model otherwise. 

The second general formulation pertains to cankers that formed on branches. So: 

TGIRD2(A,B,C,D,E,F) = 

((O.042-(A-B+C)-A-I.14))/A-I.40))/D-E) +3 +F 

where 

A CTH 

B PTH(INDX) 

C CLF 

o FER 

E TCGR 

F BGP 

133 
The circumference is determined by the following scheme for both stem-needle 

and branch-needle cankers, where 

1. Bole is a perfect cone. 

2. D.b.h. is measured at 1.4 m. 

3. D. b.h. CTH - 1. 14 

75.59 (Deitschman, Glenn H. unpublished). 

4. Circumference = 2nr. 

5. Solve for similar right triangles (fig. 49). 

a 

~ 

PTH(x) 

Figure 49.--Family of right triangles 

solved to simulate circumference of 

a Pinus monticola stem at the point 

of entry of a Cronartium ribicola 

bole canker. 

a c b b d 

where 

a = CTH-PTH+CLF, 

ad 

c 

b radius in meters 

c = CTH - 1. 4, 

-

at 

d CHT 1. 14 radius * 75.59 = 2 

6. CLF = Crown length factor. 

point of rust 

at b.h. 

entry, 

(27) 

c 

CTH 

CLF is defined as the average distance from top of tree at time of infection to 

point where branch canker and stem needle canker will enter the bole and is a function 

of the height at time of infection and growth rate of the rust. We will continue 

the discussion of CLF, but please keep in mind that this value is used to compute 

TGIRD2, not TGIRD1. This point, defined as the distance down the bole from the tip 

of the crown, is determined as follows. The basic target we are concerned with is 

the lethal crown cap as will be discussed. This is the portion of the crown where 

134 
the stem needles are located and is where the branches are short enough that branch 

cankers can reach the bole. Equation (20) provides the length of the lethal crown 

cap as a function of tree height and fungus growth rate. We will assume, based on a 

subjective evaluation of the distribution of needles within the cap, that the most 

likely point of bole penetration is two-thirds of the way down the bole in the 

lethal crown cap. Thus, 

CLF = 0.66((0.24*FER)*PTH(x) + 0.6) 

and 

Circumference at point of entry 

(CTH-l. 14 

2 TI (CTH-PTH(x)+CLF) 151.2 

CTH-l.40 

which reduces to: 

Cl"rcumference t "t f t 0.042(CTH-PTH(x)+CLF) (CTH-l.14) a pOln 0 en ry = CTH-l.40 

where the variables have been previously named. The equation in the case of stemneedle infections is as above minus CLF. The net fungus encirclement rate determines 

the girdling period. Thus, 

Years to girdle Circumference of bole at entry point 

Tree circumference growth rate - Fungus encirclement rate 

and the equation is: 

where 

TGIRDl 

(0. 042 (CTH-PTH) (I))(CTH-l.14) 

CTH-l.40 

FER-(0.00032s:SITE-0.0012s) 

I = INDX which was previously discussed, 

SITE = index of site quality, 

FER = fungus encirclement rate, an external variable, 

0.0032s*SITE-0.0012s = TCGR = Tree Circumference Growth Rate. 

The equation for TCGR was derived from a regression of tree height on diameter 

(Deitschman, Glenn H.~unpublished, this Station). Since the rate of diameter growth 

is about the same at different heights in a tree (Chapman and Meyer 1949), we assumed 

that the rate of increase in circumference at breast height was equal to the rate of 

increase anywhere else on the bole. That the rate of circumference increases as a 

function of SITE was derived by multiplying d.b.h. at SO years as determined by 

equation (27) by TI and dividing the result by SO years for SITE = SO, 70, and 90. 

When plotted, the result conformed to the linear equation: 

TCGR = 0.0032s*SITE-0.0012s 

where 

TCGR tree circumference growth rate in meters per year, 

SITE height in feet at SO years of age. 

In the case of flow (38~39), the length of time that the tree remains in X(38) 

is set by the computation of TGIRD1, which in this flow, is the time required to 

complete the girdling process. But, infected western white pine live an average of 

135 
' .... , 

5 additional years after girdling before death occurs (Hungerford, R. D."unpublished, 

this Station). Thus, trees are retained in X(38) for girdle time + 5 years. This 

I~storage" function is accomplished by indexing the number of trees flowing from 

X(37), labeled (TEM37B), into an array at point TGIRDI. Each element of the array 

is then stepped up I year each cycle. At the end of TGIRD1 years, the number of 

trees in the top slot is advanced to X(39) or the trees "die." 

136 
McDonald, G. I., R. J. Hoff, and W. R. Wykoff. 

1981. Computer simulation of white pine blister rust epidemics. I. Model formulation. USDA For. Servo Res. 

Pap. INT-258, 136 p. Intermt. For. and Range EXp. Stn., 

Ogden, Utah 84401. 

A simulation of white pine blister rust is described in 

both word and mathematical models. The objective of this 

first generation simulation was to organize and analyze the 

available epidemiological knowledge to produce a fo~ndation 

for integrated management of this destructive rust of 5-need1e 

pines. Verification procedures and additional research needs 

are also discussed. 

KEYWORDS: Computer simulation of epidemics, integrated pest 

management, Cronartium ribicola, Pinus monticola, 

deployment of resistance 

McDonald, G. I., R. J. Hoff, and W. R. Wykoff. 

1981. Computer simulation of white pine blister rust epidemics. I. Model formulation. USDA For. Servo Res. 

Pap. INT-258, 136 p. Intermt. For. and Range EXp. Stn., 

Ogden, Utah 84401. 

A simulation of white pine blister rust is described in 

both word and mathematical models. The objective of this 

first generation simulation was to organize and analyze the 

available epidemiological knowledge to produce a foundation 

for integrated management of this destructive rust of 5-needle 

pines. Verification procedures and additional research needs 

are also discussed. 

KEYWORDS: Computer simulation of epidemics, integrated pest 

management, Cronartium ribicola, Pinus monticola, 

deployment of resistance 
The Intermountain Station, headquartered in Ogden, 

Utah, is one of eight regional experiment stations charged 

with providing scientific knowledge to help resource 

managers meet human needs and protect forest and range 

ecosystems. 

The Intermountain Station includes the States of 

Montana, Idaho, Utah, Nevada, and western Wyoming. 

About 231 million acres, or 85 percent, of the land area in the 

Station territory are classified as forest and rangeland. These 

lands include grasslands, deserts, shrublands, alpine areas, 

and well-stocked forests. They supply fiber for forest industries; minerals for energy and industrial development; and 

water for domestic and industrial consumption. They also 

provide recreation opportunities for millions of visitors each 

year. 

Field programs and research work units of the Station 

are maintained in: 

Boise, Idaho 

Bozeman, Montana (in cooperation with Montana 

State University) 

Logan, Utah (in cooperation with Utah State 

University) 

Missoula, Montana (in cooperation with the 

University of Montana) 

Moscow, Idaho (in cooperation with the University of Idaho) 

Provo, Utah (in cooperation with Brigham Young 

University) 

Reno, Nevada (in cooperation with the University 

of Nevada)