ATKINSON: FISH POPULATION DYNAMICS 



interference effects may also be involved (Levine 

 1976; Vance 1978). Since my model deals only with 

 first-order effects, the components of the coefficient 

 vector a are defined as constants and assumed to 

 be related to the dominant competitive mechanisms 

 acting over the range of population densities ex- 

 pected in the simulation. 



The ft predation coefficient in Equation (4) corres- 

 ponds to the Lotka-Volterra term in the differen- 

 tial equation and implies unlimited attack capacity 

 per predator (May 1973). Relative values of these 

 vector components reflect the comparative attack 

 rates of the different predators in the model. The 

 effective ft coefficients perhaps should be reduced 

 when there are relatively few predators compared 

 with the size of population P { because of saturated 

 feeding. However, predation is probably a second- 

 ary factor under these conditions as competitive 

 limitations will tend to dominate. Based on first- 

 order arguments, constant ft components are as- 

 sumed to apply over a reasonable range of predator 

 densities. Leslie and Grower (1960) make a similar 

 assumption in the prey equation of their two- 

 component predator-prey model. Their predator 

 response equation, on the other hand, saturates at 

 high relative prey levels as in the present model. 



The prey form, represented in Equation (4), 

 reflects Ivlev's form (Equation (3)) and implies some 

 upper bound survival/growth rate under abundant 

 prey conditions. The present form also incorporates 

 a starvation mortality parameter, R, that describes 

 a worst case condition without prey. This param- 

 eter would typically equal one unless the M time step 

 is short or an alternative food source not explicitly 

 included in the modeling is available to sustain the 

 population. 



Component magnitudes of the prey coefficient 

 vector, y, relate differences in the relative efficien- 

 cy with which alternative prey are captured and 

 utilized for predator growth and/or survival. At 

 similar prey densities, a predator may utilize dif- 

 ferent capture methods and feed at higher or lower 

 rates depending on the size and behavioral charac- 

 teristics of a particular prey (Parsons and Takahashi 

 1973). Note, however, from the form of the expon- 

 ential prey term in Equation (4), that any one suffi- 

 ciently abundant prey population can satisfy the 

 predator feeding requirement. 



Finally, in comparing the present development 

 with traditional fishery models, note that Equation 

 (4) can be directly related to the single species 

 recruitment models of Ricker (1958) and Beverton 

 and Holt (1957) if the time step is defined as the 

 maturation time between spawning and recruit- 



ment. Also, a comparable fishing term can be broken 

 out of the survival/growth parameter as follows: 



S = 



S f S 



(5) 



where Sf is the fishing survival rate and S incor- 

 porates the remaining survival/growth effects. A 

 corresponding fishing mortality rate, /, can be 

 defined and related to fishing effort, Ej, as in the 

 Beverton and Holt (1957) model: 



/ = 



■\nS f 

 M 



e f E f 



(6) 



where E* is the fishing efficiency and Af is the fish- 

 ing area. The general compatability with traditional 

 fishery models is stressed. 



Spatial Redistributions 



A simplified picture of fish stock migratory pat- 

 terns during a typical life cycle is illustrated in 

 Figure 1. Adult fish move from the feeding grounds 

 to the spawning grounds and return; larval fish drift 

 from the spawning to the nursery ground; and 

 recruits join the adult stock on the feeding grounds. 

 The seasonal timing of these events is quite regular 

 as are the spatial regions to which the stock return 

 during the cycle (Cushing 1975). 



Large-scale spatial patterns will be represented 

 in the model by a number of "boxes" or compart- 

 ments, each with a defined size and each contain- 



FEEDING 

 GROUND 



SPAWNING 

 AREA 



NURSERY 

 AREA 



Figure 1.— Typical fish migratory pattern (from Cushing 1975). 



537 



