FISHERY BULLETIN: VOL. 73, NO. 4 



z 

 o 



(- 

 > 



10 



20 



30 40 50 



3-WEEK PERIODS 



60 



70 80 



Figure 14.— Deviations from simulation modeL Broken lines 

 show lags for comparable portions of test and control popula- 

 tions. 



fluctuations to consider the models fitted under 

 independent and competing conditions. The 

 former were based on equilibrium population con- 

 ditions, whereas the latter recognized non- 

 equilibrium conditions and used nonlinear 

 differential equations capable of expressing con- 

 tinuous variation in population and yield under 

 stable-limit cyclic variation. Conclusions for 

 management may be different under the second 

 type of formulation, nevertheless I feel that the 

 conclusions drawn below are of value. 



Finally, it is pertinent to discuss what seems 

 likely to have been the start of an oscillatory fluc- 

 tuation in the independent population of the 

 swordtail. As mentioned under "Course of 

 Populations," number and biomass increased dur- 

 ing the final five brood intervals, contrary to what 

 might be expected as a result of the 16.7% exploi- 

 tation rate applied. The incipient oscillation may 

 have been triggered by the low level of biomass 

 reached just before it began, through overcom- 

 pensation of the population. This level was lower 

 than any that had been in effect since the initial 

 growth of the population, and it may have moved 



the population toward the high recruitment rates 

 .shown in Figure 7. 



CONCLUSIONS 



Extinction of the swordtail population in the 

 control pair (Figures 4, 12) as mentioned under 

 "Course of Populations," is compatible with the 

 theory of competitive exclusion first advanced by 

 Cause (1934). He stated that where two popula- 

 tions are fully competing, one will have a slight 

 advantage in growth or aggression and eventually 

 displace the other. This occurrence illustrates one 

 of the values of conducting population 

 experiments over a suflficient period for natural 

 phenomena to develop. The extinction of the 

 swordtail could hardly have been anticipated dur- 

 ing the first few months of the experiment, when 

 growth of the swordtail actually outstripped that 

 of the guppy. Cause's phenomenon of "mutual 

 depression" (Cause and Witt 1935) also was 

 exemplified in the experiments. Quantitative 

 measures of this were provided by the coefficients 

 of competition, r, and<:-2, determined (by succes- 

 sive trials) for Formulae (7) and (8). These were 

 0.071 and 0.120 for the guppy and swordtail, re- 

 spectively. These values show greater depression 

 for the swordtail than for the guppy and, 

 therefore, the superior competitive ability of the 

 guppy. Growth advantage for the guppy was in- 

 dicated by the values of k and /^ (Table 9), both of 

 which were greater for the guppy. 



My greatest interest in these experiments was 

 to discover what combination of exploitation rates 

 would produce the greatest sustainable yield for 

 the two populations. This problem can be 

 approached by calculating equilibrium yields for 

 pairs of population sizes P, and P.,- At equilibrium, 

 the left hand sizes of Equations (7) and (8) are 

 equal to zero; with the constants already deter- 

 mined, F] and i^gcan be calculated for any pair of 

 values P] and Pg- To obtain 3-wk yields, F^ and F., 

 were converted back to »?jand m.^^ by the formula 

 m = I - exp(-F). Then total 3-wk yields, com- 

 parable to the yields actually obtained in the 

 experiments (Table 8), represent the sum of /??jP, 

 (guppy) and m^2 (swordtail). Yields are directly 

 comparable for the guppy, but values in Table 8 

 must be multiplied by three fourths for the 

 swordtail. 



I expressed the total yields {m jPj -I- m^2^ ^^ ^^e 

 form of yield isopleths (Figure 15). Inspection of 



886 



