FOX: POPULATION SIMULATOR 



Wij 'J U U ^^^ Q 



3 n^,e-^Kil2i ^ j - 13 - tQ) 



Zij + uK 



x[l-e ^ '•/ '] (7) 



where ^j^ = 1, -3, 3, -1, respectively. With 

 the linear segmental growth option 



[l/Zy-l/(e^y-l)]} 



(8) 



Equations (7) and (8) are modified for monthly 

 calculations from Beverton and Holt's (1957) 

 equations 4.4 and 9.5 respectively. 



Maturation 



A maturity schedule of the sexes is necessary 

 to compute several relationships associated with 

 the reproductive sextor of GXPOPS. This is 

 accomplished in the simulator with two vectors 

 of age-specific values, one denoting the average 

 fraction of mature males in each year class dur- 

 ing the breeding season, 0^/, and the other 

 denoting the female fractions, (pfi. The mean 

 number of mature males, Nmi , and mean 

 number of mature females, Nf- , during month 

 j of the spawning season are 



A^ 



-Z;, 



•' 1 = 1 



(9) 



and 



N. 



-Z; 



>. = S ^fiNij (1 - e ^J)IZij ( 

 •' i= 1 



10) 



with 



A^: 



mjl\ 



(11) 



where s; is the mean monthly sex ratio. The 

 mean sex ratio for the breeding season is given 

 by 



i=[ .S sj]l(t's-ts+l) 



(12) 



where t^ and t'g are respectively the months 

 that breeding begins and ends. 



The maturation fractions, 0, are assumed 

 to be constants in the present version of 

 GXPOPS. It is possible that the may be 

 density-dependent in certain populations. One 

 can investigate the effect of differences in 

 at equilibrium with the current version of 

 GXPOPS, but it would be a simple matter to 

 reprogram GXPOPS to examine the conse- 

 quences of any functional hypotheses. 



Reproduction 



The major components of the reproduction 

 sector of GXPOPS are copulation, egg-carrying 

 (ovigerous period), and hatching. For the 

 common spawing characteristics of freely cast- 

 ing both spermatozoa and eggs, the copulation 

 and ovigerous period sector may be bypassed. 



The traditional concept of fertilization success 

 in fisheries population dynamics, excluding the 

 salmonids, is described by Beverton and Holt 

 (1957, p. 61) as: 



... if there is free liberation of gametes, with sperma- 

 tozoa greatly in excess of eggs, and especially if the 

 percentage of successful fertilisations is fairly high, then 

 the number of fertilised eggs would tend to be a con- 

 stant fraction of the numbers laid. In addition, a large 

 spawning population would tend to distribute eggs over 

 a rather wider area than a small one, so that the num- 

 ber of gametes per unit volume, which in such a situa- 

 tion determines the rate of fertilisation, would not be 

 expected to change much. 



Copulation is one mode of ensuring the fertiliza- 

 tion of eggs in sexual reproduction. The success 

 of fertilization, therefore, depends on the copu- 

 lation rate which is at least some function of the 

 sex ratio and density of the mature population. 

 Beverton and Holt recognized this may be the 

 case for a lobster population, in which copula- 

 tion occurs. 



Copulation is a part of reproduction in most 

 commercially exploited crustaceans, and some 

 fishes. With the exception of salmonid studies, 

 where a form of pseudocopulation occurs, popu- 

 lation dynamics studies of commercial fisheries 

 have ignored the effects of exploitation on the 

 rate of copulation. Conway (1969) presents the 

 most extensive quantitative treatment of repro- 

 duction in insects, whose copulation parallels 



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