DISCRETE-TIME DIFFERENCE MODEL FOR 

 SIMULATING INTERACTING FISH POPULATION DYNAMICS 



C. Allen Atkinson 1 



ABSTRACT 



The dynamics of interacting fish populations are modeled using a coupled set of discrete-time difference 

 equations. The basic equations describe predator-prey and competitive relationships analagous to the 

 first-order expressions used in standard differential equation models. Population births and aging are 

 represented using a modified Leslie matrix. A spatial representation is also incorporated and consists 

 of a number of separate compartments, each containing interacting population groups which can be inter- 

 changed between compartments during a given time period. The potential applicability of the discrete- 

 time formulation is demonstrated via a simulation of the multispecies fish populations within the Califor- 

 nia Current during the sardine population collapse of 1930-60. 



Numerous mathematical models of interacting multi- 

 species fish populations are found in the literature 

 (Riffenburgh 1969; Saila and Parrish 1972; May et 

 al. 1979; Steele 1979). Depending on the nature of 

 a particular ecosystem and the desired resolution 

 level for its components and processes, these models 

 can become extremely complex (Parrish 1975; 

 Anderson and Ursin 1977; Laevastu and Favorite 

 1978). The major limitation in practical fisheries ap- 

 plications is the lack of sufficient field data to ade- 

 quately estimate many of the model parameters, 

 particularly the population interaction terms in com- 

 plex multispecies models (Goodall 1972). 



The two objectives in the present multispecies 

 model development are 1) to establish a general 

 mathematical form applicable to a variety of prac- 

 tical fisheries problems and 2) to provide an efficient 

 computational tool for simulating complex multi- 

 species systems. The latter feature has implications 

 for dealing with the problem of model parameter 

 uncertainty via specialized Monte Carlo and non- 

 linear programming procedures as discussed by 

 Atkinson (1985). 



The proposed formulation consists of a unique set 

 of discrete-time difference equations that describe 

 first-order dynamic processes affecting some ar- 

 bitrary number of interacting fish populations at one 

 or more trophic levels. The discrete equations are 

 particularly well suited for computer implementa- 

 tion. There are no requirements for sophisticated 

 integration routines (e.g., Runge-Kutta, Adams- 

 Moulton), and the equations have inherent numerical 



'System Science Applications, Inc., 121 Via Pasqual, Redondo 

 Beach, CA 90277. 



Manuscript accepted January 1986. 

 FISHERY BULLETIN: VOL. 84, NO. 3, 1986. 



stability. Difference equations are also compatible 

 with fisheries data sets (e.g., eggs and larvae sur- 

 veys) which are usually sampled seasonally. 



The essential biological processes represented in 

 the model are spawning, growth, mortalities, age 

 class structure, nonuniform spatial distributions, 

 and migrations. Certain of these features, such as 

 spawning, sexual maturation, and migrations, are 

 often most conveniently described in a discrete form 

 as assumed in the model. Seasonal time steps are 

 natural increments for consideration as the values 

 of appropriate model parameters can then be easily 

 changed to relate seasonal fish behavior. 



The mathematical details of the discrete-time 

 difference model are developed below. The special 

 problem of estimating model parameters in practical 

 applications is also briefly discussed. The dynamics 

 of the California Current fish populations are then 

 modeled and simulation runs performed correspond- 

 ing to the period of the sardine collapse in 1930-60. 

 Comparisons are made between the simulation 

 results and the actual (estimated) population 

 responses. 



DEVELOPMENT OF THE DISCRETE-TIME 

 DIFFERENCE EQUATIONS 



The dominant first-order ecological processes af- 

 fecting fish populations are modeled by discrete-time 

 difference equations. For convenience in the mathe- 

 matical development, these processes are assumed 

 to occur in the following sequence during a given 

 time period: 1) individual growth and mortalities; 

 2) spatial redistributions of the surviving members; 

 and 3) births and age class changes of the surviving, 



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