nSHERY BULLETIN VOL 77. NO 2 



tion will be modeled below.) Setting dS/d? = 0, we 

 obtain the following "catch equations": 



bXpaKEM 



(Model A) 



y = 



(11) 



) bxoaKQ*EN 



{oiN+bxoQ''E (Model B) . 



These equations appear not to be of a standard 

 form, as encountered either in ecology (where YIN 

 would be termed the "functional response," see 

 Fujii et al. ), or in economics (where Y would be 

 termed the "production function" of the fishery, 

 see Clark 1976, sec. 7.6), or in the fisheries litera- 

 ture (Paloheimo and Dickie 1964; Rothschild 

 1977 ). This unfamiliarity is perhaps to be expected 

 since, as far as we know, the peculiar "skimming" 

 process of the purse seine fishery has not previ- 

 ously been modeled. Equations (10) are however 

 closely analogous to the Michaelis-Menten equa- 

 tion of enzyme kinetics (White et al. 1973) as 

 might be expected from the observation that the 

 attractors serve to "catalyze" the purse seine 

 fishery, see Appendix B. 



Regarding the catch Equations (11), let us ob- 

 serve that both submodels exhibit a saturation 

 effect with respect to fishing effort £, whereas only 

 submodel B exhibits a saturation effect with re- 

 spect to tuna abundance N . For a fixed background 

 population level N, the catch rate Y bears an 

 asymptotic relationship with fishing effort £. For 

 small E we have, from Equations (11): 



[bXoaNK 



pxoQ* 



KE 



(Model A) 

 (Model B) 



(12) 



Since Q* = aA^//3 in Model A, these expressions are 

 in fact the same for the two submodels, and concur 

 with the standard Schaefer fishery production 

 function. For large E we have 



lim y 



oNK 



(13) 



»Fuju, K., P. M. Mace, and C. S. Holling. 1978. A simple 

 generalized model of attack by predator. Unpubl. manuscr., 39 

 p University of British Columbia, Institute of Animal Re- 

 source Ecology, Vtmcouver, B.C., Canada V6T 1W5. 



322 



for both submodels. For submodel B we also have 

 (for fixed £) 



lim y = bxoKQ*E (Model B). (14) 

 AT-^oo 



FISHERY DYNAMICS 



As our submodel of population dynamics of the 

 subsurface tuna population, we adopt the familiar 

 Schaefer logistic model (Schaefer 1957): 



■^ = rN{l-N/N)-i 

 at 



(15) 



where r = intrinsic growth rate 



N = environmental carrying capacity 

 e = net rate of transfer to the surface pop- 

 ulation. 



The net rate of transfer, 6, is obtained from Equa- 

 tions (2) and (5): 



{aNK - I3S (Model A) 



6= { (16) 



[oNKa -SIS*) (Model B). 



Our dynamic models of the surface tuna fishery 

 then consist of the simultaneous system of Equa- J 

 tions(10)and (15). For convenience we rewrite the ' 

 two systems as follows: 



dS 



= aiiw — ps — oxo^.^ I 



(17) 



Model A: ^= aKN-fiS-bxoES 

 at 



dN 

 dl 



dS 



G(N)-{aKN-iiS)} 



Model B: -^ = aKN{l- S/S* ) - bxoES | 



dN 

 dl 



G(N)-aKN(l-S/S*) 



where G{N) = rA?(l -N/N). 



(18) 



(19) 



Although the difference between these two 

 models may appear minor, their qualitative be- 

 havior turns out to be quite dissimilar. Their be- 

 havior is also quite different from the standard 

 Schaefer model (Schaefer 1957). As indicated by 

 results discussed in the appendices, however, the 

 qualitative behavior of the above models seems to 

 be characteristic of a wide variety of alternative 



