i 

I 

NOAA Technical Memorandum NMFS 

MAY 1982 

THE RELATIONSHIP BETWEEN CHANGES IN 

GROSS REPRODUCTIVE RATE AND THE 

CURRENT RATE OF INCREASE FOR 

SOME SIMPLE AGE STRUCTURED MODELS 

Tom Polacheck 

NOAA-TM-NMFS-SWFC- 19 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration 

National Marine Fisheries Service 

Southwest Fisheries Center 
NOAA Technical Memorandum NMFS 

The National Oceanic and Atmospheric Administration (NOAA), organized in 1970, has evolved 

into an agency which establishes national policies an manages and conserves our oceanic, coastal, and 

atmospheric resources. An organizational element within NOAA, the Office of Fisheries is responsible for fisheries policy and the direction of the National Marine Fisheries Service (NMFS). 

In addition to its formal publications, the NMFS uses the NOAA Technical Memorandum series to 

issue informal scientific and technical publications when complete formal review and editorial processing are not appropriate or feasible. Documents within this series, however, reflect sound professional 

work and may be referenced in the formal scientific and technical literature. 
NOAA Technical Memorandum NMFS 

This TM series is used lor documentation and timely communication 01 preliminary results. imterim reports. or 

special purpose information. and have not received complete lormal review, editorial control, or detailed editing 

MAY 1982 

THE RELATIONSHIP BETWEEN CHANGES IN 

GROSS REPRODUCTIVE RATE AND THE 

CURRENT RATE OF INCREASE FOR 

SOME SIMPLE AGE STRUCTURED MODELS 

Tom Polacheck 

Southwest Fisheries Center 

National Marine Fisheries Service, NOAA 

La Jolla, California 92038 

and 

Department of Biology 

University of Oregon 

Eugene, Oregon 92403 

NOAA-TM-NMFS-SWFC-19 

U.S. DEPARTMENT OF COMMERCE 

Malcolm Baldrige. Secretary 

National Oceanic and Atmospheric Administration 

John V. Byrne. Administrator 

National Marine Fisheries Service 

William G. Gordon, Assistant Administrator for Fisheries 
ThE RELATIOhSHIP BETWEEN ChAiiGES IN GROSS REPROUUCTIVt 

RATE ANU THE CURRENT RATE OF INCREASE FOR SOME 

SIMPLE AGE STRUCTURED M)DELS 

Tom Pol acheck 

Southwest Fisheries Center 

National Marine Fisheries Service, NOAA 

La Jolla, California 92038 

and 

Department of Biology 

University of Oregon 

Eugene, Oregon 92403 

The major criterion used for evaluating the status of dolphin stocks 

involved in the eastern tropical Pacific tuna purse seine fishery has been the 

ratip of current p pulation abundance to pre-exploitation abundance (NMFS, 

1976 ; Smith, 1979 ). Estimates of pre-exploitation abundance have been 

calculated using a population projection model which starts with an estimate 

of current abundance and adds animals killed while substracting net 

recruitment of animals in a sequential manner moving backwards in time. The 

net recruitment or net rate of increase has been estimated by taking the 

difference between estimates of the gross annual reproductive rate, G.A.K. 

(i.e., the pregnancy rate times the proportion of females that are mature 

times the proportion of females in the population), in exploited and 

unexpl oi ted stocks. Thi s procedure assumes 1) that survi Val rates are 

constant, over time and age, 2) that the net rate of increase is zero in 

unexploited stocks, and 3) that changes in G.A.R. can be directly translated 

into changes in the net rate of increase. Not considered in this procedure 

are the effects of the age structure of the population. If the age structure 

is unstable, then it is unlikely that changes in the G.A.K. would directly 

translate into changes in the net rate of increase and this procedure would 

produce an inaccurate estimate. A1 so, even when populations approach their 

stable age distributions, inaccurate estimates may result if survival rates 

are not constant with age. 

B 

This latter effect is examined in this report. 

One of the simplest age structured models is one in which there are two 

age classes, corresponding to mature and immature individuals. A simple 

'National Marine Fisheries Service. 1976. Report of the workshop on stock 

assessment of porpoi ses i nvol ved i n the eastern Pacific ye1 1 owf i n tuna f i shery 

(La dolla) July 27-31, 1976. Southwest Fisheries Center Administrative Report 

'Smith, T. (ea). 1575. Report of the status of porpoise stocks workshop 

(August 27-31, 1979, La Jolla, California). Southwest Fisheries Center 

Aaministrative Report No. Ld-79-41. 

NO. LJ-76-29. 
2 

Leslie matrix formlation of such a model is 

where 

F = the fecundity 

Sj = survival rate 

SM = survival rate 

Nj = the number of 

NM = the number of 

rate 

for immature individuals 

for mature i ndi vi dual s 

immature females 

mature females. 

The time units are the number of years to sexual maturity, and the net rate 

of increase, assuming a stable age distribution, is the eigen value x of the 

matrix -q, 

dSM2 + 4s. F =_I--- 'M + J 

2 

If the pregnancy rate equals the fecundity rate and a sex ratio is 50:50, 

the gross reproductive rate is 

(2 1 

For density dependent changes in F, it can be shown that X - G equals a 

constant if and only if S. = SM (see appendix). In other words, if adult 

fecundity rates are the oily vital rates changing in response to density, 
3 

observed changes in G.A.R. (AG) will be equivalent to changes in the current 

rate of increase (AX) if and only if adult and juvenile survival rates are 

equal. Some numerical examples of the differences between AX and AG are 

presented in Table 1. There are three points to note from this table: (1) the 

larger the differences between S. and SM, the areater the discrepancy between 

AA and AG, (2) if Sj <SM, then A% results in an overestimation of AX and vice 

versa, and (3) the magnitude of the discrepancy between AG and AX for fixed 

values of Sj and SA is areatest for small values of A. 

Unfortunately, if the age of maturity is not equal to one in the time 

units in which fecundity and survival have been measured, the conversion of 

fecundity rates into time units equal to the age of maturity is nontrivial 

even if one assumes fecundity rates are constant with aae. This is because 

the net fecundity rate over an extended time interval is a complex function of 

both the survival and fecundity rates during that interval (Keyfitz, 1972). 

Thus, the above model cannot easily be applied to dolphin populations as 

estimates for $he age of maturity range between 5 and 9 years (Perrin and 

Henderson, 1979 ; Perrin et al., 1976) while fecundity (pregna cy) rates have 

been estimated on an annual basis (e.g., Henderson et al., 1979 ). 8 

To obtain an idea of how large the discrepancy between AG and AX may be 

for the dolp$in estimates, a model was explored similar to equation 1 but in 

which Q and N were defined as 

Q 

3Perrin, W. F. and J. R. Henderson. 1979. Growth and reproduction rates in 

two populations of spinner dolphins, Stenella longirostris, with different 

histories of exploitation. Southwest Fisheries Center Administrative Report 

4Henderson, J. R., W. F. Perrin, and R. B. Miller. 1979. Gross annual 

production in dolphin populations (Stenella spp.) in the eastern tropical 

Pacific 1973-1976, working paper SOS//9/T , for status of porpoise stock 

work shop. 

NO. LJ-79-29. 
4 

Table 1. Comparisons of AX and AG for changes in F for model 1. Note 

AX and AG are calculated as the difference between adjacent rows. 

SM F X G AX AG AX/AG 

.9 .9 .25 1.1038 .2038 

.50 1.2578 .3577 

.75 1.3867 .4867 

.1540 .1539 1.000 

.1289 .1290 1.000 

.9 

.9 

.25 

.5 .25 1.0222 .2008 

.50 1.1226 .3459 

.75 1.2099 .4630 

1 .oo 1.2882 .5630 

.lo04 ,1451 .692 

.0873 .1171 .745 

.0783 .1000 .783 

.25 .25 .9648 .1986 

.50 1.0223 .3358 

.75 1.0745 .4417 

1 .oo 1.1227 .5289 

.057 5 .1372 .419 

.0522 .1059 .493 

.0482 .0872 .553 

.90 .25 .6155 .1778 

.50 .8074 .3086 

.75 .9560 .4203 

1 .oo 1.0519 .5197 

.1919 .1310 1.465 

.1486 .1115 1.333 

.1259 .0994 1.267 
5 

+- 

N = vector of the number of females in each age 

where 

1 to M and the M+l element equals the number of 

mature females , 

M = age of maturity 

SA = adult survival rate 

Si = age specific survival 

F = adult fecundity rate. 

By placing SA in the lower right hand corner of 0, survival rates are 

assumed to be constant past the age of maturity, M. Again assuming a 50:50 

sex ratio and that fecundity rates equal the estimated pregnancy rates, the 

gross reproductive rate for this model is 

NM+l G = _I_Ic. 

M+l 

Explicit expressions for X in terms of F and Si and for the relationship 

between AG and AX, given changes in F, are not possible. However, numerically 

it appears that AG = AX if and only if S- = SA. Similarly, the same general 

conclusions for the first, simpler, modej hold. Some numerical results are 

presented in Table 2, for an age of maturity equal to 7 years, and for 

estimates of vital rates which span current ranges for dolphin populations. 

The results indicate that with the generally accepted assumption for dolphins, 

of Sj < Shy AA could be equal to 60 to 100% of AG. This demonstrates the need 

for caution in assuming changes in G.A.R. are equal to net rates of 

increase. While the direction of the change is the same for both rates, there 

is no simple relationship between changes in their magnitudes, even when 

populations are tracking their stable age distribution. 

The above results all incorporate the assumption that only pregnancy 

rates have chanaed and that survival rates have remained constant. Changes in 

survival also result in changes in G.A.R. and the current rate of increase. 

However, the effect on G will tend to be less than on A . In some cases, it 

is possible to achieve rather large chanaes in X with only small changes in 

G. This is illustrated in Table 3 for the simple model (equation 1) given 

above. 

Positive changes in G.A.R. are evidence of positive changes in the net 

rate of increase (barring transitory age structure effects). A lack of 

significant change in G.A.R. does not necessarily mean that there has been no 

change in the current rate of increase. The relationship between changes in 

G.A.R. and changes in the current rate of increase is complex for 

age-structured models, but the results of chanqes in survival will tend to be 
6 

Table 2. Comparison of AA and AC for changes in F for model I1 when 

M = 7. Note AX and AG are calculated as the differences 

between adjacent rows. 

F x G AX AG AUAG 1 

SM 'j 

.95 .85 .15 1.0049 .09296 

.20 1.0179 .11189 

.25 1.0293 .12806 

.30 1.0396 .1422 

.0130 .0189 .684 

.0114 .0162 .705 

.0114 .0142 .806 

.95 .9 .10 1.00239 .06826 

.15 1.02055 .09041 

.20 1.03598 .lo878 

.25 1.04945 .12454 

.01817 .02215 .820 

.01543 .01837 .840 

.01347 .01576 .854 

, 

.95 .8 .10 .97966 .07186 

.15 .99141 .09571 

.20 1.00185 .11584 

.25 1.0113 .13205 

.30 1.01988 .14659 

.0118 .0239 .410 

.0104 .0201 .517 

.0094 .0162 .580 

.0086 .0145 .591 

.8 .95 .10 .9207 .0586 

.15 .9500 .0770 

.20 .973 1 .0925 

.25 .9924 .lo60 

.30 1.0090 .1180 

.0293 .0184 1.593 

.0231 .0155 1.494 

.0193 .0135 1.430 

.0166 .0120 1.383 

'Survival for ages 1 to M is constant and equals Sj 
7 

Table 3. Comparison of AX and AG for chanaes in survival rates for 

model 1. Note AX and AG are calculated as the differences 

between the first row of the table and each additional row. 

SM F x G AA AG AA/AG 

.90 .80 .3 1.1152 .2864 

.90 .90 .3 1.1373 .2874 

.95 .80 .3 1.1578 .2824 

.95 .90 .3 1.1790 .2891 

.0235 .00098 23.9 

.0426 .00184 23.5 

.0638 .0027 23.7 

underestimated by changes in G.A.R. while the results of changes in fecundity 

will tend to be overestimated by G.A.R. if juvenile surviva1,is less than that 

of adults. 
8 

LITERATURE CITED 

Keyfitz, N. 1972. Introduction to the mathematics of populations. AddisonWesley. Reading, Mass. 489 pp. 

Perrin, W. F., J. M. Coe, and J. R. Zweifel. 1976. Growth and reproduction 

of the spotted porpoise, Stenella attenuata, in the offshore eastern 

tropical Pacific. Fish. Bun., U.S. /4:229 -269. 
9 

AP PEIUD IX 

For model 1, to show that A minus G equals a constant for changes in F 

if and only if Sj equals Sh, note that from equation 3 

Thus, if - G equals a constant C then 

2 

-- A -c 

F+ 

Thus 

A= C + dC2 + 4C F 

2 

but from equation 2 

A= 

sM + JZ 

2 

Therefore, equation (6) to equal equat.ion (7), Sj must equal SF1 which must 

equal C. 
RECENT TECHNICAL MEMORANDUMS 

Copies of this and other NOAA Technical Memorandums are available from the National Technical 

Information Service, 5285 Port Royal Road, Springfield, VA 22167. Paper copies vary in price. 

Microfiche copies cost $3.50. Recent issues of NOAA Technical Memorandums from the NMFS 

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NOAA-TM-NM FS-SWFC- 9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

An annotated computerized bibliography of the use of karyotypic 

G. L. WORTHEN 

(January, 1981 ) 

Albacore trolling and longline exploration in eastern North Pacific waters 

R. M. LAURS, R. J. LYNN, R. NISHIMOT0,and R. C. DOTSON 

(March, 1981) 

Observations of albacore (Thunnus alalunga) fishing off California in 

relation to sea surface temperature isotherms as measured by an airborne infrared radiometer. 

analysis in the subspecific taxonomy of mammals. 

during mid-winter 1981. 

J. L. SQUIRE, JR. 

(May, 1981) 

Stock assessment activities within the National Marine Fisheries Service. 

(June, 1981) 

Planning double-tagging experiments. 

J. A. WETHERALL and M. Y. Y. YONG 

(May, 1981 ) 

Histological gonad analyses of late summer-early winter collections of 

bigeye tuna, Thunnus obesus, and yellowfin tuna, Thunnus albacares, 

from the Northwest Atlantic and the Gulf of Mexico. 

S. R. GOLDBERG and H. HERRING-DYAL 

(June, 1981) 

Status reports on world tuna and billfish stocks. 

STAFF, SWFC 

(July, 1981 

An evaluation of tagging, marking, and tattooing techniques for 

M. J. WHITE, JR., J. G. JENNINGS, W. F. GANDY and L. H. 

COR NE LL 

(November 1981 

Local stability in maximum net productivity levels for a simple model 

T. POLACHECK 

(April 1981) 

Environmental data contouring program EDMAP2. 

L. EBER 

(April 1982) 

small delphinids. 

of porpoise population sizes.