FISHERY BULLETIN VOL 



'■-/■.SiV 



)-o. 



Thus the net biological gi'outh rate becomes r - f^ 

 and the condition for catastrophic behavior in 

 submodel B becomes 



aK > r 



fs- 



If we now consider effort E in the surface fishery 

 and mortality/', in the subsurface fishery as con- 

 trol variables (now assuming aK = constant), it is 

 clear that the surface of equilibrium A''-values has 

 the same nature as shown in Figure 6. Thus while 

 the surface fishery might be "subcatastrophic" in 

 the absence of any subsurface fishery, the de- 

 velopment of the latter might transform the sys- 

 tem into a catastrophic region. 



One further possibility is worth noting. As re- 

 marked earlier, the schooling behavior of tuna 

 may be influenced by environmental factors, par- 

 ticularly the depth of certain thermal isoclines. If 

 so, the system might switch randomly between 



catastrophic and noncatastrophic states. Under 

 these circumstances the fishery might exist for 

 sometime at a level of stable sustained yield, but 

 could suffer a catastrophic collapse induced by un- 

 usual, or unusually protracted environmental 

 conditions. 



The practical importance of these possibilities is 

 increased by the fact that CPUE is likely severely 

 to misrepresent the decline in abundance of the 

 tuna population. In the first simulation reported 

 below, for example (Figure 7), CPUE falls by only 

 2(y/r even though the tuna population declines by 

 over 997f . 



A SIMULATED CATASTROPHE 



Figures 7 and 8 show the outcome of two simula- 

 tions based on submodel B. (These simulations 

 employed the discrete-time version of the 

 population-dynamics submodel, as described in 

 Appendix A. Qualitatively the results are the 

 same as for the continuous-time model.) The fol- 

 lowing parameter values were utilized: 



K = 5,000 attractors 

 Xo = 0.5 



EFFORT (SDF ) 



FIGURE 7. — Simulation re.sults: model B. "catastrophic" case. EfTort (measured in 

 standardized days fishing iSDF'i is increased at years 1, 5. and 9. The final effort level 

 produces a catastrophic but gradual declme in the tuna population, which is not "picked 

 up" by thecatch-per-unit-effort iCPUEi index until the population has been essentially 

 eliminated, i Scales for the four curves are Imear but not related; see initial values 

 shown . I 



326 



