SILLIMAN; EXPERIMENTAL EXPLOITATION OF FISH POPULATIONS 



Equation (3) that the Hmit Pqo = -fo^xp (G), and by 

 substituting this in Equation (3), differentiating 

 and taking logs a growth term may be derived for 

 Equation (1): 



/(Pi) = AA-,(log,Pi^-log/'i). 



(4) 



For the competition term, Volterra (1928) used 

 ./'(P2 ) = f'l •^i-f*2 • Preliminary experimentation 

 showed that this term was unsatisfactory for the 

 guppy-swordtail experiments, since it was impos- 

 sible to obtain even a reasonably good fit using it. 



I also experimented with /(f 2) = ^1 (-^1 + -^2) ^^ 

 the theory that the sum of the populations, rather 

 than their product, might be controlling, but it was 

 equally unsatisfactory. The most suitable term 

 proved to be simply: 



/(P2) = C,P2 



(5) 



This term agrees with the reasonable idea that the 

 competitive effect on one population is propor- 

 tional to the size of the other. 

 For the fishing term I adopted from Fox (1970): 



6 8 10 12 



BROOD INTERVALS (3 weeks) 



16 



18 



f{X,) = q,X,P, = F,P,. 



(6) 



Substitution of Equations (4), (5), and (6) in 

 Equation (1) provides the model for the first 

 population: 



Figure 8. -Initial growth for the independent population of the 

 guppy, with fitted Gompertz curve. 



fitting the Fox (1970) model. The zero points plus 

 two other exploitation rates (relatively stable 

 periods considered to be equilibrium points: 

 guppy, weeks 316-334 and 355-373, Figure 2; 

 swordtail, 322-334 and 342-354, Figure 3) gave 

 three fitting points for each species (Figures 10, 



C?Pj/C?f = PjA-i(l0g,Pioo-l0g,Pi) -fiPg-^lA- C'') ^^1 Swordtoil- Pop Growth 



By exactly parallel derivation the model for the 

 second population is: 



dP^ldt = P,k 2(log,P2oo - log,^2) -C2P1- F,P,. (8) 



Thus the model for the competing populations 

 represents a modification of the Fox (1970) ex- 

 poential surplus-yield model, with the addition of a 

 term for competition. 



Determination of Constants 



Growth data were obtained from the indepen- 

 dent populations. Gompertz (1825) curves were 

 fitted to the initial growth period for both species 

 (Figures 8, 9), using the analog computer method 

 of Silliman (1967). Asymptotic levels were 38.7 g 

 for the guppy and 33.7 g for the swordtail. These 

 values were used for the zero exploitation levels in 



Asymptote 



2 4 6 8 10 12 14 



BROOD INTERVALS (4 weeks) 



Figure 9.— Initial growth for the independent population of the 

 swordtail, with fitted Gompertz curve. 



881 



