CLARK and MANGEL AGGREGATION AND FISHERY DYNAMICS 



POPULATION ( N ) 



(a) aK < r 



LU 



>- 



( b) aK > r 

 EFFORT ( F ) 



Figure 5. — Equilibrium yield-effort curves for model B. Case 

 (a); intnnsic schooling rate less than intrinsic growth rate; yield 

 approaches a positive asymptotic value as effort approaches 

 infinity. Case lb); intnnsic schooling rate greater than intrinsic 

 growth rate; yield undergoes a catastrophic transition when 

 effort exceeds critical level E,.. 



in order to return the system to the original stable 

 equilibrium. 



The behavior of our model (submodel B) can be 

 described in terms of Figure 6, in which the hori- 

 zontal plane represents the "control space," with 

 efforts as the basic control and intrinsic schooling 

 rate aK as a parameter (which in some cases 

 might also be subject to manipulation, or to 

 stochastic variation). The vertical axis represents 

 subsurface stock size A^. The surface S is the locus 

 of equilibrium solutions for our model. 



Two possible paths for the development of the 

 fishery are also shown in Figure 6. (Simulated 

 versions of these paths will be presented below.) 

 Path I, corresponding to Figure 5(a), occurs if «A' 

 < r; here there is a steady decline in the equilib- 

 rium population level N = N as the effort parame- 

 ter increases. (If £ varies rapidly over time, then 

 equilibrium conditions will not prevail, and the 

 actual development path will diverge from Path I 

 lying on i.. Figure 6 is still useful for understand- 

 ing the dynamics in this case, however.) 



Path II, with aK > r, behaves similarly to Path I 



INTRINSIC 



SCHOOLING RATE (aK ) 



Figure 6. — Catastrophic surface il) corresponding to model 

 B; This surface describes the eauilibrium population level (N) 

 as a function of effort I E) and intrinsic schooling rate faK) Path I 

 represents the development of the fishery, as effort increases, in 

 thecase that aK < r, while Path II corresponds to the case oK > 

 r In the latter case the fishery experiences a catastrophic col- 

 lapse at point P. 



for small levels of effort, but then suddenly falls 

 over the "edge" of the catastrophe surface 1, at 

 point P. I Notice that for aK - r the surface 5i folds 

 under itself, the upper sheet N = N and the lower 

 sheet N = being stable equilibria, while the 

 middle sheet N = N' is unstable. This surface 

 shape is the typical "cusp" catastrophe of Thorn 

 1975.) 



The management implications of the theory will 

 be discussed later; the question of robustness of 

 the models will be taken up in the appendices. 



Figure 6 stresses the significance of the parame- 

 ter p = aK for the interactive dynamics of aggre- 

 gation and fishing. For tuna, p may be age- 

 dependent, as suggested by the differences in age 

 distribution between longline and purse seine 

 catches. Also, as noted previously, p may vary over 

 time and space as a result of environmental gra- 

 dients. The theoretical consequences of such com- 

 plexities have yet to be investigated (Mangel see 

 footnote 6). 



A "cusp" catastrophe surface similar to that de- 

 picted in Figure 6 can also be used to describe the 

 response of the tuna fishery to simultaneous 

 exploitation of the surface schools and the subsur- 

 face (background) population. If a given level of 

 fishing mortality f\ is applied to the subsurface 

 population, the effect will be to replace our 

 dynamic Equation (15) by 



325 



