FISHERY BULLETIN: VOL. 76, NO. 3 



unobserved such states may be), since the stock 

 will include simultaneously the young and the old, 

 the older having accumulated a probabilistic his- 

 tory of mortality, fecundity, and growth which 

 may differ considerably from the current schedule 

 that affects both. Various tactics for adjusting the 

 parametric mechanics of stock-production models 

 to such long-term, delayed influences are given by 

 Gulland (1969), Fox (1975), Walter,^ and others, 

 but in the case of the Pella-Tomlinson system the 

 difficulties have been compounded by artifacts of 

 the conventional analysis and by an instability 

 inherent to the mathematical indeterminacy of 

 the system itself. With the critical-point analysis, 

 most of those impediments will convert to tracta- 

 ble relationships or vanish altogether. We can 

 suppress the troublesome dual formulations as- 

 sociated with the conventional casting of the sys- 

 tem, we can uncouple the indeterminate exponent 

 and the coefficients of the governing equations, 

 and we can make explicit the relationships be- 

 tween parametric graph curvature and the man- 

 agement components of the system. 



THE REFORMULATED 

 GOVERNING EQUATIONS 



Stock-production models, as they are usually 

 defined, arise from the common premise that a fish 

 stock, when reduced by exploitation to a level 

 below some prior abundance, will always strive to 

 recover its former size in accord with some latent, 

 self-regulating mechanism of restoration. Irre- 

 spective of the compensatory details, any such re- 

 covery must accrue directly from the productivity 

 of the stock, and in the conventional representa- 

 tion of the Pella-Tomlinson system, the latent 

 capacity for biomass production in a stock of fishes 

 is given the dual formulation 



P(B) = ±aB" + b B. 



la) 

 lb) 



• 



P(B) being the production rate of the stock at stock 

 sizeB. Equation ( la) applies when exponents falls 

 on the range 0<n<l, and Equation (lb) applies 

 when n >1. In either case, all the critical compo- 

 nents of the system — maximum stock size, 

 maximum productivity, the stock level where 

 maximum productivity occurs — depend in some 



^Walter, G. G. 1975. Non-equilibrium regulation of fisheries. 

 Int. Comm. Northwest Atl. Fish. Res. Doc. 75/IX/131, 12 p. 



516 



way on the numerical value assigned to exponent 

 n. That is, root B^c is given by 



lll—n 



B 



the critical ordinate p (which corresponds to the 

 stock level where maximum productivity occurs) 

 is determined by 



P = 



1/1—?? 



while extremum coordinate m (which corresponds 

 to productivity P ma.x ' must be determined from 

 the formula 



m 



n \b ) 



the plus sign applying to Equation (la) and the 

 minus sign to Equation (lb). 



Although exponent n controls the graph curva- 

 tures of Equations ( la) and ( lb), the nonzero roots 

 and extrema are controlled hy By- and the coordi- 

 nate pair (p, m). As shown by Fletcher (1975), 

 coordinate m has no essential dependence on ex- 

 ponent n, and with the appropriate transforma- 

 tions the dual formulation (Equations ( la, b)) may 

 be suppressed. In consequence, either of the 

 parametric sets {m, p,Bx}or{m,/7.5x}will consti- 

 tute a complete set of independent governing 

 parameters for latent productivity in the Pella- 

 Tomlinson system, and the dual formulation 

 (Equations (la, b)) converts to the single differen- 

 tial equation for latent productivity 



P = ym 



it) -HI)"- <^» 



with y a purely numerical factor wholly prescribed 

 by n as 



,n/n — l 



y 



1  



(3) 



With the coefficients so cast, the sign reversals at 

 turning point /; = 1 become automatic, and the 

 consolidated interval of definition for n becomes 

 0<n <3c (the point n = 1 being a removable singu- 

 larity). With parameter m thus separated from n 

 in Equation (2), the undetermined exponent n 

 can be defined solely by the ratio p/B y- in the 

 relationship 



