FISHERY BULLETIN: VOL. 73, NO. 4 



approach seems to offer a means of predicting the 

 Hmits of stability of trophic webs against pertur- 

 bation. 



(B) Reproductive Time Lag 



A limited study of the effects of the length of 

 reproductive time lag was made using a represen- 

 tative individual model of the simplest trophic 

 web, PI 1000 (a food base and the fish predator, 

 species E). The food base was of the exponential 

 growth type and the predator employed an Ivlev 

 feeding function. Reproductive lags of 0, 2.50, and 

 6.25 yr were tried with a model that was otherwise 

 basically the same. These three alternatives 

 correspond respectively to the assumptions: 1) that 

 offspring are mature when spawned, 2) that they 

 take 2.50 yr to reach the "representative" stage, 3) 

 that they take 6.25 yr to reach this stage. The 

 second assumption is reasonable for species E. 



The system was initially perturbed by starting 

 with species E at 20% above its standard 

 equilibrium population. The results for the 

 biomass of species E are shown in Figure 7. It is 

 clear that with increasing reproductive lag, the 



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Figure 7.-Effects of reproduc-tive time lag on the response of a 

 single species to an initial perturbation in its population. The 

 three cases illustrated have reproductive time lags of 0.00, 2.50, 

 and 6.25 yr, respectively. 



regulation of the system about its standard 

 equilibrium becomes weaker; i.e., the biomass of 

 species E, and other variables (not shown), reach 

 more extreme oscillatory amplitudes. Larger 

 amplitudes always incur greater risk of disaster. 

 For example, in the runs shown here, a different 

 sexual maturity criterion was used-knife-edge 

 maturity at 80% of the standard equilibrium body 

 weight. In the 6.25-yr lag run, this weight was 

 reached at about 59 yr into the simulation, and 

 after the 6.25-yr lag, it so reduced recruitment and 

 the species E population that the system became 

 unstable. This instability, which did not occur in 

 the other runs, was due to the long lag in recruit- 

 ment response to change in fecundity with 

 changing food availability. 



Except where otherwise stated, all the results 

 presented here are for representative individual 

 models having 2.50-yr lag and age class models 

 having 2.00-yr lag. These are reasonable for the 

 species involved. They are mutually consistent 

 because in the age class model, reproductive 

 products are summed over a full year and produce 

 recruits 2.00 yr after the end of the year. Thus the 

 average lag is about 2V2 yr for the age class model 

 also. 



(C) Age Class Effects 



A number of simulations were run with an 

 explicit 4-age class model. Some results involving 

 starvation have been shown above. Other exercises 

 investigated the capabilities of this more accurate 

 type of population model to regulate in the normal 

 manner. Figure 8 illustrates the response of a 

 simple food base-fish predator PllOOO system with 

 Ivlev feeding function to an initial perturbation of 

 the fish predator population. The "mean total 

 population" is a variable obtained by summing the 



4 

 populations of all the age classes, ^^ N^, during 



each computational increment of the year and 

 taking the arithmetic average of these values. 

 "Population mean annual biomass" is obtained by 

 similarly summing and averaging the biomass 



values, ^ ^i^i  These variables are shown in 



i = 1 



Figure 8 as percents of their standard equilibrium 



708 



