FISHERY BULLETIN: VOL. 71, NO. 4 



that of many crustaceans. He recognizes three 

 types of mathematical models for representing 

 reproduction: 1) derivations from the Lotka- 

 Volterra equations, 2) empirical models (essen- 

 tially multiple regressions from data sets with 

 a number of variables), and 3) structural models 

 of the causal mechanisms likely to be involved. 

 Derivations from the Lotka-Volterra equa- 

 tions assume that if mating is random, then the 

 rate of change of copulated females is propor- 

 tional to the expected rate of contact per fe- 

 male times the number of uncopulated females. 

 The rate of contact per female is assumed to be 

 proportional to the number of males, giving 



dNfldt = kcN^(Nf-Nf) 



(13) 



where * denotes copulated females and k^ 

 is the coefficient of copulation. The copulation 

 coefficient may be thought of as consisting of 

 two multipliers — the instantaneous coefficient 

 of males contacting females at random and the 

 fraction of these encounters that result in copu- 

 lation. Equation 13 has the solution 



-k^Nr 



or 



Nf=Nf(l-e ^ '"j 



-k N 

 p = {\-e ^ '^) 



(14a) 



(14b) 



where p is the fraction of mature females that, 

 at the end of the breeding season (scaled to be 

 of one unit length for convenience), have been 

 copulated. Equation (14b) may also be derived 

 probabilistically by assuming the number of 

 copulations is Poisson distributed such that /e^ 

 is the mean rate of contact resulting in copula- 

 tion per male, then 



P^ [x copulations] = e 



x\ 



with equation (14b), therefore, expressing the 

 probability that at least one copulation per fe- 

 male has occurred in one unit of time (Klomp, 

 Montfort, and Tammes, 1964 [cited in Conway, 

 1969]). This assumes that the population is 

 not aggregated but is randomly distributed over 

 the breeding grounds regardless of the size of 

 the population. Philip (1957) has also derived 



equation (14b) under slightly different circum- 

 stances. Writing equation (14a) in terms of the 

 sex ratio (males: females), s, and total mature 

 population, N^, we have 



^-iTh^N,[l-e-''ciTT-s)Na^_ (15) 



Therefore, as the total mature population de- 

 creases, or as the sex ratio increases, the 

 number of females being copulated with de- 

 crease under this model for reasonable values 

 of s in the virgin state. 



With the aid of equation (15) one can easily 

 follow the reasoning of Beverton's and Holt's 

 conclusions for large, long-lived fish popula- 

 tions with essentially a constant sex ratio. The 

 maximum sustainable average yield (MSAY) 

 in these populations is likely to occur at rela- 

 tively low rates of fishing, hence there would 

 be only a small reduction in population size 

 from the virgin state. If k^ were high, then 

 the reduction in Nf^ would be negligible. On 

 the other hand, if the population is short-lived, 

 the MSAY is likely to occur at a relatively 

 lower population level. If fishing also increases 

 the sex ratio, the deleterious effects on the size 

 of A^^ are compounded. 



Equation (13) may be extended to become 

 more realistic for a multiage population under 

 exploitation. Often the mortality during breed- 

 ing is neglected; however, exploitation may in- 

 crease the mortality significantly and the breed- 

 ing season may be protracted. Therefore, with 

 the additional assumption that males make no 

 distinction between year classes of females, for 

 any year class / during month j 



dNfildl = k^N^it) [Nfiit) - N^i it) ] - ZiN^i 



(16) 

 where 



/= 1 



^m(0= .^^'PmlNif' '•' (17) 



and 



n 



-Z;.t 



Nfi(l)= V 0^.^..^ ij (18) 



Rewriting equation (16) in terms of the ratio of 



1022 



