CLARK and MANGEL; AGGREGATION AND FISHERY DYNAMICS 



models of both population dynamics and the 

 school-formation process. We next discuss the be- 

 havior of our models in detail. 



Model A 



Figure 2(a) and (b) show the system of solution 

 trajectories (Nit), S(t)) for the Equation system 

 (17), for the two cases 



In these Figures, the effect of an increase in the 

 effort parameter E is to rotate the isocline S = in 

 a clockwise direction, thus decreasing both popu- 

 lation levels N^ andS^. The corresponding yield- 

 effort curves are shown in Figure 3(a) and (b) re- 

 spectively. 



The shape of these yield curves is easily 

 explained. Note from Equations (16) that the con- 

 stant 



oK < r and aK > r 



aK 



respectively. The system has a unique stable 

 equilibrium at the point {N^,Sj; the correspond- 

 ing sustained yield from the fishery is given by 



y = bx„ES^. 



N =0 



3 

 Q. 



o 



< 



Lj_ 

 IT 



3 



to 



(a) aK < r 



N = 



S = 



(b) aK > r 

 SUBSURFACE POPULATION (N) 



Figure 2. — Trtyectory diagram for model A: a stable equilib- 

 rium exists at the point IN^, S^). Case (a): intrinsic schooling 

 rate less than intrinsic growth rate; population cannot be de- 

 pleted below N by surface fishery. Case (b): intrinsic schooling 

 rate greater than intrinsic growth rate; population can theoreti- 

 cally be fished to arbitrarily low levels (see also Figure 3). 



represents the maximum net rate at which the 

 subsurface population A'^ aggregates to the sur- 

 face; this may be referred to as the "intrinsic 

 aggregation rate" (or "intrinsic schooling rate" in 

 the present model). If the intrinsic aggregation 

 rate p is less than the intrinsic growth rate r (see 

 Figures 2(a), 3(a)), then the population cannot be 

 exhausted by the surface fishery; in this caseN -> 

 TV > and y -►F > as effort £ -♦ -^-. (Figure 3(a) 

 shows yield increasing to a maximum level and 

 then declining as effort increases. This situation 

 arises if N < N/2, i.e., if p > r/2; otherwise, Y 

 simply increases to an asymptotic value Y .) 



(0 ) aK < r 



(b) aK > r 



EFFORT 



;e ) 



Figure 3. — Equilibrium yield-effort curves for model A. Case 

 la): intrinsic schooling rate less than intrinsic growth rate; yield 

 approaches a positive asymptotic value as effort approaches 

 infinity. Case (bl: intrinsic schooling rate greater than intrinsic 

 growth rate; yield approaches zero at finite effort level. 



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