FLETCHER: RESTRUCTURING OF PELLATOMLINSON SYSTEM 



two. Figure 2 illustrates the individual effects of 

 set {m, p, B^} on the graph of Equation (11). 



In equilibrium model (11), biomass level B^:; var- 

 ies parametrically as a function of equilibrium 

 fishing mortality F. In terms of parameters m, n, 

 Boc, the relationship becomes 



B. 





(12) 



Qco' Uneocploit&d. stack. Lewi 



p : I hfz tricrfrujLSi lai^ol for MS Y 



Figure 2.— The graph of Equation (111, equilibrium yield vs. 

 equilibrium stock size in the Pella-Tomlinson system, as control- 

 led by independent parameters m, p, Bj.. 



When exponent n of the system takes any value on 

 0</i <1 then y<0, in which case we can see by Equa- 

 tion (12) that B:,, >0 no matter how great the value 

 of F. That is, when 0<n<l there exist equilibrium 

 adjustment levels of stock biomass for all mag- 

 nitudes of fishing mortality large and small; such 

 a stock defies annihilation. Should exponent n >1, 

 however, the corresponding stock can have non- 

 zero adjustment levels B^ only whenF<ym/Bx. 

 That is, when n >1 and when fishing mortality ex- 

 ceeds the critical value ym/Bx, the "adjustment" 

 level corresponds to extinction and Equation (12) 

 does not apply. 



Upon the substitution of Equation (12) into 

 Equation (11), the direct relationship between 

 equilibrium yield and equilibrium fishing mortal- 

 ity becomes 



1 

 T V ym J 



B. 



(13) 



and the fishing mortality that maximizes Equa- 

 tion ( 13) is given by Equation (9). That is, with the 

 substitution of F^gy into Equation (13) then 

 Y*/t = m. 



Under the equilibrium conditions, the conven- 

 tional quantity U (which signifies accumulated 

 catch per unit of fishing effort as a function of 

 fishing intensity f/r) can be cast into the restruc- 

 tured form 



\ °° jm T I 



(14) 



which eliminates the explicit appearance of catch- 

 ability coefficient g, permitting instead the direct 

 quantification of maximum sustainable yield m. 

 Quantities U and J7x have the customary mean- 

 ings 



Y 

 U = —f' ^Y* being the yield accumulated over 



time interval r as a consequence of ef- 

 fort f). 

 U^ ^ qB^ (q being the individual probability of 

 capture per unit of fishing effort f). 



Should the accumulation interval t be a year, the 

 variable U becomes annual CPUE (catch per unit 

 of effort) and the variable [It becomes effort per 

 annum. With exponents >1 in the Pella-Tomlinson 

 system, no steady-state CPUE exists for a fishing 

 intensity in excess of critical value ym /U^.. But if n 



519 



