ATKINSON: FISH POPULATION DYNAMICS 

 N t M 



PU t +1) = I I F%JLt) gf n (t) Sf n (t) e-J*V 



'J v ' n = l m=\ J 



x g-Wtt) [l _ fl£(f) r r,/"(0] 



x Pfi(0 (9) 



where m is summed over all spatial compartments 

 M; n is summed over all population subgroups N{, 

 and F\jjf) is defined by 



A(t);j = l,n>2 

 b m (t);j ^ l,n = j 

 a in {t);j ^ 2,n = j -1 

 ; otherwise. 



n„ (o = 



(10) 



The model parameters in Equations (9) and (10) 

 consist of maximum survival/growth rates (S), star- 

 vation mortality rates (R), transport terms (g), fecun- 

 dity factors (/), age class changes (a and b), and 

 population interaction coefficients (a, ft, and y). Time 

 dependency is indicated for all parameters except 

 the interaction terms. Space dependency is assumed 

 to apply to all but age class changes and interaction 

 terms. If the parameters are described by probabi- 

 listic functions, the model becomes a stochastic 

 representation. 



The above difference model represents a com- 

 prehensive description of coupled fish population 

 dynamics and is proposed for general application. 

 The form of Equation (9) is particularly well suited 

 for computer implementation; it provides an effi- 

 cient time-step simulation capability without requir- 

 ing a numerical integration scheme. The model can 

 be conveniently programmed on a mini-computer 

 system and used to simulate complex multispecies 

 population dynamics. 



MODEL PARAMETER ESTIMATION IN 

 PRACTICAL APPLICATIONS 



The predictive power of the difference model in 

 practical applications is obviously dependent on the 

 knowledge of the ecosystem processes and the abil- 

 ity to estimate the associated parameters used in 

 the modeling. This situation is true for any eco- 

 system model whether it consists of difference 

 equations, differential equations, or any other for- 

 mulation. In fact, I (1980) showed that difference 

 equations representing multispecies populations can 



be used to approximate the complex response modes 

 of differential equations by relating parameters and 

 choosing suitably small differencing time steps. I 

 also showed that the difference model suffers from 

 a similar sensitivity to the parameter estimates; the 

 problem becomes more severe with increasing eco- 

 system complexity. 



Certain parameters in either difference or dif- 

 ferential equation models can be roughly estimated 

 from field and/or laboratory studies. Examples in- 

 clude fecundity and growth rates of individual fish 

 which can be observed directly. Population-level 

 parameters, such as interaction and transport 

 terms, are more difficult to estimate given the 

 dynamic, wide-ranging nature of fish behavior. Even 

 with extensive field sampling and the use of multi- 

 variate statistical techniques to sort out stochastic 

 environmental features (Reid and Mackay 1968; 

 Mobley 1973; Poole 1976), these parameter esti- 

 mates will typically have a large degree of 

 uncertainty. 



The potential advantage of difference models in 

 dealing with parameter uncertainty is related to 

 their computational efficiency. When parameter 

 uncertainty is represented in a probabilistic frame- 

 work, Monte Carlo procedures can be applied to 

 statistically describe population response character- 

 istics based on large numbers of simulation runs. 

 Probabilistic descriptions of parameter uncertain- 

 ty can express both the inherent stochastic nature 

 of the ecosystem and the parameter estimation er- 

 ror. One problem is that the stochastic ecosystem 

 features, which are of primary interest, will typically 

 be masked in the statistics by the large parameter 

 estimation errors if realistic values for the latter are 

 included. 



I (1980, in press) used nonlinear programming 

 (NLP) techniques to treat parameter uncertainty in 

 dynamics models for a general class of ecosystem 

 problem. My approach is summarized below; it has 

 been used for resolving parameter estimates in the 

 difference model application discussed in the section 

 that follows. 



An NLP problem can be stated in the following 

 general form: 



where x is the variable vector with upper and lower 

 bounds of x and x m , respectively; f(x) is the so- 

 called objective function; and g{x) is a vector func- 

 tion of implicit constraints. 



539 



