NOAA Technical Memorandum NMFS 

APRIL 1982 

LOCAL STABILITY AND MAXIMUM 

NET PRODUCTIVITY LEVELS FOR A SIMPLE 

MODEL OF PORPOISE POPULATION SIZES 

Tom Polacheck 

NOAA-TM-NMFS-SWFC- 17 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration 

National Marine Fisheries Service 

Southwest Fisheries Center 
NOAA Technical Memorandum NMFS 

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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 of preliminary results. imterim reports. or 

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

APRIL 1982 

LOCAL STABILITY AND MAXIMUM 

NET PRODUCTIVITY LEVELS FOR A SIMPLE 

MODEL OF PORPOISE POPULATION SIZES 

Tom Polacheck 

Southwest Fisheries Center 

National Marine Fisheries Service, NOAA 

La Jolla, California 92038 

and 

Department of Biology 

University of Oregon 

Eugene, Oregon 97403 

NOAA-TM-NMFS-SWFC-17 

U.S. DEPARTMENT OF COMMERCE 

Philip M. Klutznick, Secretary 

National Oceanic and Atmospheric Administration 

Richard A. Frank, Adminisirator 

National Marine Fisheries Service 

Terry L. Leitzell, Assistant Administrator for Fisheries 
CONTENTS 

Page 

................................................. I. Introduction 1 

11. Stability of the Projection Model.. .......................... 2 

I1 I. Calculating Maximum Net Productivity Level s.. ................ 5 

Literature Cited....... ............................................ 7 

Table 1. Critical combinations of r and z at which the 

stability behavior of equation 2 changes ............. 8 

Table 2. Comparison of M.N.P.L. as a proportion of N for 

for equations 12 and 13 for a range of valugs for 

the shape parameter (z) and maximum net reproductive 

rate (r) ............................................. 9 

Appendix 1. The relationship between the critical value of 

Nt/Np for stability and M.N.P.L. for a fixed 

kill occurring after reproduction .................... 10 

Appendix 2. An expression for the population level giving 

maximum net productivity for Equation l... ........... 13 
LOCAL STABILITY AND MAXIMUM NET PRODUCTIVITY LEVELS 

FOR A SIMPLE MODEL OF PORPOISE POPULATION SIZES 

bY 

Tom Pol acheck 

Southwest Fisheries Center 

National Marine Fisheries Service, NOAA 

La Jolla, California 92038 

I. INTRODUCTION 

A ratio of current population size to pre-exploitatIan population size of 

porpoise involved in the Eastern Tropical Pacifif tuna fishery has been used 

to evaluate the status of the stocks (NMFS, 1976 ; Smith, 1976’). The method 

of estimating pre-exploitation population size is complex and involves 

accounting for several variables. The most recent estimates are based on the 

foll owing discrete model of densi ty-dependent population growth: 

tZ Nt+l = (Nt - 0.5 Kt)[l+r (1 - - 11 - 0.5Kt , Nz P 

where 

Nt = the population size in year t 

Kt = the kill in year t 

r = the maximum net reproductive rate 

Np = the pre-expl oi tati on or equil i bri um popul ati on size 

z = a shape parameter for the density-dependent response. 

‘NMFS. 

involved in the eastern Pacific yellowfin tuna fishery. 

Jolla, California). -- SWFC Administrative Report No. LJ-76-29. 

‘Smith, T. (ed). 1979. Report of the status of porpoise stocks workshop 

(August 27-31, 1979, La Jolla, California). _I_ SWFC Administrative Report No. 

1976. Report of the workshop on stock assessment of porpoises 

(July 27-31, 1976, La 

LJ-79-41. 
2 

Two aspects of the model are considered here. These are 1) the stability 

properties of the model and 2) the calculation of the maximum net productivity 

level (M.N.P.L. ). 

11. STABILITY OF THE PROJECTION MODEL 

Discrete time models can exhibit unstable behavior especially when the 

reproductive rates are high and/or there is marked nonlinearity in the 

densi ty-dependent response. The local stabil i ty characteristics of a discrete 

time model or difference equation can be determined from its eigenvalue at 

equilibrium. If the model is of the simple one-dimensional form 

where Ne is of a single dimension, then the eigenvalue of F (xF) is 

the pariial de 'vative of F with respect to N evaluated at equilibrium value 

of N (i.e., The following conditions hold for the behavior of 

a one dimensional model near its equilibrium (N*). . If -1 <AF <O, then the 

model will have monotonic damping near N*. If 0 <hF <1, then the model will 

exhibit oscillatory damping near N*. , then the model will be 

sufficiently large values of A, chaotic behavior (i.e. there exist a finite 

number of periodic points plus an uncountable number of points whose 

trajectories are totally aperiodic but bounded). The above conclusions are 

taken directly from May and Oster (1976). 

"/8N I N*). 

repelling near N* and will exhibit limit cycles or, for 

These criteria can be used to evaluate the behavior of the projection 

model used in estimating porpoise abundance. The density-dependent model 

(ignoring the kills) is: 

(2) = F1(Nt) = Nt(l+r(l-( N t/N 

Nt+l P 

and the 

N -- a F1 - l+r-r(z+l) ( t/NpIZ 

aN t 

which reduces at the equilibrium (i .e. N=Np) to 

= (aF/aN I N*) = 1 -rz F1 

(3) 

Thus, the stability characteristics of the basic model are dependent on 

the product of rz and can be summarized as follows: 
3 

If 0 < rz < 1, then monotonic damping, 

if 1 < rz < 2, then oscillatory damping, 

if 2 < rz, then repel 1 i ng. 

Table 1 lists some combinations of the critical values of r and z 

corresponding to the above criteria. It can be seen there that for the values 

of r and z that have been c sidered reasonable for porpoise stocks (e.g. 

monotonic damKing as the population approaches equilibrium. 

~0.04 and z<20; Smith, 1979 gT population projections will exhibit smooth 

The question of the effect of a fixed quota on the above stability 

properties can be analyzed in a similar fashion. The case where the quota is 

taken after the annual net reproduction will be considered first. The case 

where the quota is taken before annual net reproduction is considerably more 

complicated. In the case where the quota is taken after net reproduction, the 

model is 

and 

where 

Q = a fixed annual quota 

N f = t/Np 

Note that the value of f at equilibrium is a function of the annual 

quota relative to the unexploited population size (N 1. The value of f can 

range anywhere between the ratio of M.N.P.L. to N If Q is greater 

than the potential yield at M.N.P.L., no positive $uilibrium value exists for 

the model. If one lets 

afd one. 

w = -(1 - fZ (l+z)) 

= fZ (l+z)-l 

(7) 

then the stability requirements of this model are the same as those for 
4 

the basic model (Fl), with W replacing z. Since, in general W<z (i.e. fZ <l), 

this fixed quota model will be more stable than the basic model. However, 

unlike z, W can be less than zero (if f is small enough) which would mean that 

hFz<O and the model would be unstable. For W to be less than zero, it would 

mean that 

The right-hand side of equation 8 is the same as the proportion of Np 

at which maximum net productivity occurs (see Part 111 below) and is the lower 

limit for f at which a positive equilibrium for this model can exist. 

Therefore, if the quotas are not so large as to drive the population to 

extinction, this fixed quota model will be more stable than the basic model. 

The situation in which the kills are taken prior to the net reproduction 

is more complex. In this situation, the model is 

and 

where 

and 

By letting 

Nz-1 

-- aF3 - l+r(l-(z+l)fZ+Qz -t- I 

NZ 

P 

aN t 

Y = -(l-(z+l-pz)fZ) 

= (z+l-pz )fZ-l 
5 

the stability requirements are the same as for the basic model F, but 

with Y replacing Z. Similar to the situation with F2, Y is less than Z since 

fZ <1 and P >O. Thus, this fixed quota model (F3) in general will be more 

stable than the basic model. 

Similar to the situation with F , Y can be less than zero but again only 

when f is below the proportion of N at which maximum net productivity 

occurs. #us, if the quotas are not so large as 

to drive the population to extinction, this model will also be more stable 

than the basic model. 

This is shown in Appendix I. 

The projection model that has been used for calculating porpoise 

abundances is actually a mixture of models F2 and F3. Since both of these 

models are more stable than the basic projection model Fl, the values in 

Table 1 should provide a guide to the combinations of critical values for r 

and z for which the models bifurcates and changes its stability behavior. For 

the range of values for r and z that have been used to project porpoise 

population sizes, the model exhibits monotonic damping as the population 

approaches an equilibrium. 

111. CALCULATING MAXIMUM NET PRODUCTIVITY LEVELS 

2 In Smith (1979 ), M.N.P.L. for various porpoise stocks was calculated by 

1 /z 

M.N.P.L. = (12) 

where N was estimated from calculations based on equation 1. The 

above eRpression for M.N.P.L. was derived by finding the maximum value for the 

net population growth curve for the basic density-dependent model 

(equation 2). This method of deriving M.N.P.L. fails to account for the 

timing of the kills relative to the timing of reproduction and results in a 

technically incorrect expression for M.N.P.L. for equation 1. Equation 12 is 

actually the correct expression for M.N.P.L. for the situation in which the 

kil Is occur after reproduction (equation 5). 

An explicit expression for M.N.P.L. for the model represented by equation 

1 is 

(rtztl) - J(r+z+l) 2 -2r-r 

M.N.P.L. = Np (- r 

(13) 

A derivation of this expression is provided in Appendix 2. 

equation 13, M.N.P.L. is not solely a function of the shape parameter z but is 

also a function of the maximum net reproductivity. This contrasts with many 

Note that in 
6 

common densi ty-dependent models in which harvesting is assumed (either 

explicitly or implicitly) to occur after reproduction. Table 2 provides 

calculations of M.N.P.L. as a proportion of N based on equation 13. Also 

presented in this tab1 e are the corresponding cal cul ati ons based on equati on 

12 for comparison. As illustrated in this table, equation 12 is a lower bound 

for the actual M.N.P.L. calculated by equation 13. Also, M.N.P.L. is still 

primarily dominated by the shape parameter z, at least for reasonably small 

values of r. The magnitude of the differences in M.N.P.L. as calculated by 

equations 12 and 13 increases with increases in r and with decreases in z. 

However, the actual differences tend to be small for values of r and z 

considered reasonable for porpoise population (i .e. w.10 and D1). As such, 

equation 12 represents a close approximation to the actual expression 

(equation 13) for M.N.P.L. for the projection model represented by equation 1. 

P 
7 

LITERATURE CITED 

May, R. M. and G. F. Oster. 1976. Bifurcation and dynamic complexity in 

simple ecological models. -- Am. Nat. 110:573-599. 
8 

Table 1. Critical combinations of r and z at which the stability 

behavior of equation 2 changes. 

Corresponding critical value of Z so that 

Value of r rz =1 rz =2 

.Ol 

.02 

.03 

.04 

.05 

.06 

.08 

.10 

.20 

.30 

.40 

.50 

.60 

.70 

.80 

.90 

1 .oo 

100.0 

50.0 

33.3 

25.0 

20.0 

16.7 

12.5 

10.0 

5.0 

3.3 

2.5 

2.0 

1.7 

1.4 

1.2 

1.1 

1 .o 

200.0 

100.0 

66.7 

50.0 

40.0 

33.3 

25.0 

20.0 

10.0 

6.6 

5.0 

4.0 

3.3 

2.9 

2.5 

2.2 

2 .o 
9 

Table 2. Comparison of M.N.P.L. as a proportion of Np for equations 12 

and 13 for a range of values for the shape parameter (2) and 

maximum net reproductive rate (r). 

Value of M.N.P.L. for 

Z Equation 12 

M.N.P.L. for 

Equation 13 

Value of r 

.02 .04 .06 .10 .20 .30 

1 .500 .501 .502 .504 .506 .512 .517 

2 .577 .579 .580 .581 .584 .590 .595 

3.5 .651 .652 .653 .654 .656 .661 .666 

6.0 .723 .724 .725 .726 .727 .731 .745 

11.5 .803 .803 .804 .805 .806 .808 .811 

20.0 .859 .859 .860 .860 .861 .862 .864 

J ’> 
10 

Appendix 1. The relationship between the critical value of Nt/Np for 

stability and M.N.P.L. for a fixed kill occurring after 

reproduction. 

M.N.P.L. for F3 

For F3 

where 

and at equilibrium 

NR 

1 +R 

Q =- 

M.N.P.L. will be achieved when 

0 = aQ/aN 

NR = c /1+,l/aN 

1 BNR NR aCl+RI = /1+R ] - - - 

aN (I+R)' aN 

Then 

where 

-- aNR - r - (z+l)hr 

aN 
11 

and 

-- - -zr N ‘-l/NPz . al+R 

aN 

Thus 

O= r- (z+l) hr --- NR (-z r N~-’/N~~) 

1 +R [l+RI2 

(r-rh) h (z+l)hr + - 

l+r-rh 

=r- 

= rh2 - (l+z+Zr) h + (l+r) , 

and 

(l+z+2r) & (l+z+zr) 2 -4r(l+r) h J 

2r 

Note only the negative square root term yields a value of h less than 1. 

Therefore at M.N.P.L., h must equal the negative root of equation 1. 

Value of h for which Y<O - 

From Equation 11 in Part I1 

where 

Thus 

Y = (z+l-P)fz -1 = (z+l-Pz)h - 1 

P = 0 /N 

- NR 1 - R --- - - - 

(1+R) N 1 +R 

Y= [,,l - “1 h-1 

= [z+l - -----1 rz - rhz 

l+R 

h - 1 

l+r-rh 
12 

For Y=O means 

rz - rhz ~- 1 0 = rz+1 - - 

L l+r-rh J 

= - rh2 + (l+z+2r)h -(I+?-) 

Thus 

Thus, for Y<O, h must be greater than the positive root of 2 or less 

than the negative root. Only the neaative root is less than 1 and is of 

interest (i.e., allows for a positive kill). Since equations 1 and 2 are 

identical, the critical value of h for which y<O equals the value of h at 

which M.N.P.L. is achieved. 
13 

If 

where 

Appendix 2: An Expression for the Population Level Giving 

Maximum Net Productivity for Equation 1. 

=(Nt-0.5K) (l+R) -0.5K t+l 

NZ R = r (1- t /N ‘1 P 

at equilibrium Nt+l = Nt, thus 

0 = (Nt-0.5K) (1+R)-0.5K-Nt 

0 = Nt R - K (1+0.5R) 

k = NtR/1+0.5R 

Dropping the subscript t, then 

- aK - a(NR) 1 - NR ao. 5~ - -- 

aN aN 1+0.5R (1+0.5RI2 aN 

Expressions for the two partial derivatives in the above equation are: 

= r-rtz+l)h 

where 

h = NZ /Npz 

and 

Nz P 
14 

Thus 

NR (-0.5rz NZ-l/N P-- K -- - r-r(z+l)h -- - - -- 

aN 1+0.5R ( 1+0.5R ) 

- r-r(z+l)h + O.5Rrzh 

2 1+0.5R (1+0.5R) 

For K to be a maximum, =O; therefore multiplying by (1+0.5R)/r yields aN 

O=( 1- (z+l)h ) (1+0.5R )+O. 5Rz h 

which can be reduced to 

0=( 1+0.5r -h(r+z+l )+h2(0. 5r) 

Thus 

2 

h= 

r 

Note only the negative square root term yields a value of h less than 1. 

Also, since h = (N/Np)Z then 

M.N.P.L. = Np (h)l/z I \ 
RECENT TECHNICAL MEMORANDUMS 

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An annotated computerized bibliography of the use of karyotypic 

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Observations of albacore (Thunnus alalunga) fishing off California in 

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analysis in the subspecific taxonomy of mammals. 

during mid-winter 1981. 

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(May, 1981) 

Stock assessment activities within the National Marine Fisheries Service. 

(June, 1981 

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

(May, 1981) 

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(June, 1981) 

Status reports on world tuna and billfish stocks. 

STAFF, SWFC 

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