FISHERY BULLETIN: VOL. 73, NO. 4 



evidence (Simpson 1951; Bagenal 1957, 1967; 

 Beverton 1962; Pitt 1964; LeCren 1965; Bagenal 

 and Braum 1971) that in most fish with adequate 

 food supply above metabolic demands, fecundity is 

 strongly dependent upon body weight. Regres- 

 sions on weight usually fit better than regressions 

 on length or age (Bagenal 1957, 1967; Nikolskii 

 1962). It seems reasonable to represent the rate of 

 production of reproductive material, S, (or ac- 

 cumulation of body stores for that purpose) as a 

 simple function of weight. Although more general 

 functions have been proposed (Bagenal and 

 Braum 1971), apparently most regressions so far 

 fitted using data from specimens have been quite 

 close to the linear expression. 



S = uW, 



(4) 



where u is a constant. In the present simulations, u 

 = 0.1 in all cases, based on average values for 

 several species and both sexes (Bagenal 1957, 1967; 

 LeCren 1958, 1962; Mann 1965; Norden 1967; 

 Phillips 1969). The linear function is truncated 

 near its lower end at a weight corresponding to 

 sexual maturity. This is consistent with the 

 general observation that the onset of sexual ma- 

 turity in fish appears to be a function of size rather 

 than age (Beverton and Holt 1959; LeCren 1965). 

 Exceptions for individual species are noted in 

 Bagenal (1957). 



Trophic factors regulate the animal's fecundity 

 through their effect on body weight. Also, when 

 food intake becomes sufficiently low, there must 

 not be enough energy above metabolic demands 

 for normal fecundity. The scanty field data 

 available suggest that usually fish sacrifice growth 

 for reproduction, so that as food intake decreases, 

 fecundity stays at or near normal (with decreased 

 growth) until the net energy above metabolic ex- 

 penditures is less than the normal fecundity 

 requirements, after which fecundity decreases 

 (Mackay and Mann 1969). The model operates in 

 this way. 



The ration, C, under any instantaneous set of 

 conditions, is obtained from the maximum ration, 

 Cn,ax . and the current abundance of the prey which 

 constitutes the food supply. C^^^ is dependent on 

 body weight, and its current value can be deter- 

 mined from Equation (1) if the maximum growth 

 rate, Gn,^^ , is known. Since Gma.x is a function of the 

 current size of the individual, this function is 

 required. Data from appropriate ad libitum feed- 

 ing experiments with a particular species of 



interest could be fitted to the appropriate function 

 to give continuous values of G^^^ . The von Ber- 

 talanffy growth function is a convenient one to 

 which growth data from a large number of fish 

 species have been fitted (e.g., Beverton and Holt 

 1959; Ursin 1967). In its differential form it 

 expresses growth 



GyB-'^iWoo'^m^^'-W), 



(5) 



where k and Woo are numerical fitting parameters 

 (k corresponds to the ^k of Ursin 1967, and to 3 

 times the K of Beverton and Holt 1959). Woo 

 corresponds to a theoretical maximum weight, 

 asymptotically approached. Values of k and \\^ 

 for the present simulations are taken for certain 

 illustrative species from Beverton and Holt (1959) 

 and Ursin (1967:2421-2423). Equation (5) is 

 employed in the model with a constant coefficient 

 of 4.0 as an arbitrary standard adjustment to 

 represent the highest feeding conditions. This 

 gives a relationship between values over the full 

 feeding range (e.g., zero, maintenance, and 

 maximum ration) consistent with those observed 

 in long-term feeding and growth experiments. 

 With the C^ax term thus expressed, the Q^^^ term 

 is simply Equation (2) with a = a^^^ and using the 

 current weight, W. The S^^^ term comes from 

 Equation (4). Thus 



'^max 



k 



max 



(6) 



There is a considerable and developing body of 

 theory, for which evidence continues to ac- 

 cumulate, that where environmental conditions 

 are fairly stable, a predator's ration may be 

 expressed as a fraction, r, of its maximum ration, r 

 being a simple function of the abundance of prey, 

 P. This approach is taken as a useful long-term 

 ecological approximation, in which short-term 

 behavioral factors, factors affecting the acces- 

 sibility of the prey, etc. are smoothed out. Several 

 expressions for this relationship have been 

 proposed. 



Three alternative expressions for simple preda- 

 tion with no explicit competitive effect between 

 predator individuals were used in the model in 

 different simulation runs. These are: 



Linear: 

 Ivlev: 



Holling: r = 



l-e-^''(Ivlev 1961b) 

 P 



(7A) 

 (7B) 



m. 



(Holling 1959) (7C) 



698 



