FISHERY BULLETIN: VOL, 77. NO, 2 



APPENDIX A 



The purpose of these appendices is to test the 

 robustness of our models, by introducing alterna- 

 tive submodels for population dynamics (Appen- 

 dix A) and for the schooling process (Appendix B). 

 In this appendix we replace the continuous-time 

 Schaefer model by a discrete-time stock- 

 recruitment model. We postulate a fishing season 

 of given length, during which the stock is fished 

 down, followed by an interim season which results 

 in the replenishment of the stock. The purpose of 

 this exercise is not particularly to provide a more 

 realistic model of tuna population dynamics, but 

 simply to enquire whether our main results are 

 independent of the type of model employed. (See 

 Clark 1976, chapter 7, for a general discussion of 

 models of the sort considered here, i 



Our alternative Model A is governed by the 

 equations 



dt 

 dN_ 

 dt 



= -(aKN-j]S) 

 /V(0) = R,S{0) = aKR/li 



0<t<T 



(AD 



(A2) 



where R denotes recruitment prior to the fishing 

 season, and T denotes the length of the fishing 

 season. In Equation (A2) we also assume, for 

 simplicity, that the surface population S reaches 

 equilibrium with ^V before the fishing season be- 

 gins. 



LetP = NiT) denote escapement at the close of 

 the fishing season. From the linearity of Equa- 

 tions (Al) it follows that P is a linear function of 

 R = .V(0): 



C, .« 



iC, 



constant <1 1. 



(A3) 



Clearly C,.; is a decreasing function ol'E; it is easily 

 seen that 



•a -* 



^.C/.- = expi-aKT). 



(A4i 



If G(P) now denotes the stock-recruitment func- 

 tion, the coupling between successive years is 

 given by the equations 







(A5) 



If, for example, GiP) is quadratic: 

 G(P) = ePil -PIP). 



(A6) 



the equilibrium escapement level P  is easily cal 

 culated: 



P = 



pa - VgCt:) 

 





> 1 

 s 1, 



Exhaustion of the stock by the surface fishery is 

 thus possible if and only if 



/ 



/ 



a. 



/Yco 



3 



(b) aKT > o- 

 ESCAPEIVIENT ( P ) 



FiC.URE 10. — Fishery dynamics for the discrete-time modeLs; 

 schooling model B G = net population growth curve; Y - catch 

 curve; Y, = limiting position of Y; P' = population equilibrium 

 forgiven Y. Case (a): intrinsic schooling rate less than intrinsic 

 growth rate; escapement population cannot be reduced below 

 level P by .surface fishery. Case (bl: intrinsic schooling rate 

 greater than intrinsic growth rate; population can be fished to 

 arbitrarily low levels; P* denotes an unstable equilibrium. (The 

 corresponding yield-effort curve is similar to Figure 5(bl.l 



3.30 



