plus recruits and minus catch and deaths) to be 

 compared with the actual counts made. Some of 

 the observed discrepancies were no doubt due to 

 unobserved deaths (dead fish eaten by others 

 before seen) or to errors in counting. That some 

 errors occurred is not surprising. Each expected- 

 actual comparison involved as many as 17 separate 

 counts. During each main count while exploiting 

 the stocks the counter had to keep in mind the total 

 number, the number caught, the state of maturity 

 of each fish, and the sex of each mature fish. 



The distribution of discrepancies for selected (to 

 provide representative data) periods (Table 7) 



Table 7.-Count discrepancies for selected periods: swordtail, 

 April 1970 to June 1972; guppy, March 1970 to June 1972. Values 

 represent "expected" number subtracted from the actual count. 



Guppy 



1 

 1 



1 

 2 

 1 

 1 

 1 



2 

 1 

 3 

 3 

 2 

 3 

 1 

 1 

 1 

 3 

 1 

 2 



31 



Total 



23 



shows that negative discrepancies (actual less 

 than expected) exceed the positive for both 

 swordtails and guppies. This no doubt arose from 

 the unrecorded natural mortalities mentioned 

 above. The two positive discrepancies for the 

 swordtail probably represent counting errors. For 

 the guppy, however, the fairly large proportion of 

 positives exceeding three fish suggests that 

 unrecorded recruitment occurred. Apparently 

 some of the guppy "fry" escaped detection, even 

 though a thorough search of the tanks was made. 

 This phenomenon is in keeping with the observed 

 greater hardiness of the guppy, and suggests that 

 the superiority in recruitment for the guppy was 

 even greater than indicated in the stock-recruit- 

 ment relations reported above. 



FISHERY BULLETIN: VOL. 73. NO. 4 



SIMULATION MODEL 



Mathematical Derivation 



Data of population weight reflect growth of in- 

 dividual fish as well as recruitment and mortality, 

 and all of the analyses below will be in terms of 

 weight. Development of the formulae requires a 

 fairly extensive list of symbols, which are defined 

 below. 



P = Total population weight in grams. 



t — Time from start, in 3-wk periods. 



X = Fishing effort in arbitrary units. 



q = Catchability coefficient. 



F = Instantaneous rate of fishing mortality 



{ = qX). 

 m = Three-week rate of fishing mortality. 

 G = Constant of the Gompertz growth curve. 

 k = Constant of the Gompertz growth curve 



and of the Fox (1970) population model. 

 . J — Empirical constants. 



cji 



Adding a term for the effect of fishing to the 

 formulae of Volterra (1928) gives a pair of 

 differential equations: 



rfPj/f/f =,/'(Pi)-/(P2)-./'(^i), 

 dP^ldt = f{P,) -f{P^) -f{X^). 



(1) 

 (2) 



In these equations, the first term of the right 

 hand side is for population growth; the second, for 

 competition; and the third, for the effect of 

 fishing. The development is exactly parallel for the 

 two equations, and only that for Equation (1) will 

 be outlined below. 



For the growth term Volterra (1928) used/(Pj ) 

 = JiPj-Z^jPf. This is the logistic growth curve, 

 which requires symmetrical population growth. 

 Growth for the guppy under fishing (equilibrium 

 yield) was shown by Silliman and Gutsell (1958) 

 and Silliman (1968) to be distinctly asymmetrical. 

 The Gompertz (1825) curve, introduced as a 

 population yield model by Fox (1970), is suitable 

 for asymmetrical growth and will be shown in the 

 section on determination of constants to be suit- 

 able for initial population growth in both the 

 guppy and the swordtail. This is expressed: 



P, = Poexp[G-Gexp(-A-OJ. (3) 



It can be shown by mathematical analysis of 



880 



