CLARK and MANGEL AGGREGATION AND FISHERY DYNAMICS 



K = -aiKT+bExo{^^ C,} +|3iCi +gK{K). 



Steady states of the system are obtained if we set 

 the left-hand sides in Equation i B22) equal to zero. 

 We then find that the steady states are determined 

 bv: 



C 





(n>2) (B23I 



gK(K) = 



J=i 



Equations (B22l and (B23) seem to represent a 

 fairly realistic model of the fishery dynamics. A 

 full analysis of these equations would be quite 

 illuminating. However, as it is, the analysis of this 

 model quickly becomes intractable. In order to 

 illustrate the behavior of this model, we will 

 analyze the case /; = 2 (for arbitrary y, , y.^); 



Pi 



C■^ +7,r . - Co 



P2 



bE\a 

 Cj ^ A' + harvest of 7 J tons (B24) 



bE\n 

 C2 "A + harvest of 71 +73 tons. 



The results of the analysis of three-(or higherJstep 

 mechanisms should be similar to the analysis of 

 the two-step mechanism. 



The multistep model provides a picture of the 

 tuna-porpoise bond which appears to be relatively 

 realistic. For example, we may imagine that the 

 first Vi schools are bound strongly to the complex 

 (a, large, /^, small) and that the next y.^ schools are 

 bound less strongly (a.^ < a,. /32 ^" /^i '■ Sharp's 

 (19781 discussion of the effect of the thermocline 

 on the tuna-porpoise association supports this 

 model. In particular, it seems likely that the a, and 

 13, depend upon the location of the thermocline. 



The kinetic equations corresponding to the mul- 

 tistep model are 



f = gr -OiT' ' A-ttaT'^ Ci + /3i Cj 



+ /^2C2 + 7il3iCi +72(^2C2 'B25) 



^ = ^h- -ai r" A + (JjCi + 6£xo(Ci + Cg) 



(B26i 



+ 1^202 - bE\oC\ (B27I 



C2 = ttaCiT'^ -^2C2-bE\QC2  (B28i 



In the steady state, we have 

 azCiT'' 



Co 



(^2 + bExo  



(B29) 



Adding the steady-state version of Equations 

 (B25)-iB28) gives 



f^K = 



(B30) 



which we assume has the solution K = A,,  0. The 

 steady-state version of Equation (B27), using 

 Equation (B29), is: 



= ttiAT'' -l^iCi -asCiT'-' - bExoCi 



+a2^2T'"C^I(^2 + bE\o) 



which can be solved to give the steady-state level 

 ofC, complexes; 



<"l = 



aiKT^' 



i3i ^a2T'' + bExo -a2p2T^ ' IW2 + ^^Xo) 



(B31) 

 The instantaneous harvest rate is given by: 

 Y = 6£Xo(7iCi +(72 + 7i)C2)So (B32) 



bExoSoaiKT'- 

 |3i +a2T"- + 6£:xo -ot2ii2T" IW2 + ''^Xo) 



X (7i +72a2r"7(|32 + 6£\o))- (B33) 



335 



