TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 

 FOR STOCK-PRODUCTION MODELS 



R. Ian Fletcher' 



ABSTRACT 



The time-dependent formulations of the Graham-Schaefer and Pella-Tomlinson systems are re- 

 structured so as to accommodate directly the critical-point parameters of their respective governing 

 graphs; the resulting parametric system accounts for the behavior of either model wholly in terms of its 

 management components. The indeterminate exponent and the coefficients of the Pella-Tomlinson 

 equations are uncoupled and the dual formulations associated with the conventional casting of the 

 system are eliminated; the governing equations and corresponding solutions are cast into composite 

 forms and the sign changes of coefficients become automatic. The previously obscure relationships 

 between management parameters and variable graph curvature in the Pella-Tomlinson model are 

 expressly formulated; maximum sustainable yield is shown to be independent of the indeterminacy of 

 the system. Time-delay estimators for both systems are formulated. 



We analyze here, in a deterministic setting, cer- 

 tain of the transient, nonlinear mechanisms 

 employed in the modelling of stock and yield dur- 

 ing periods of imbalance between fishing removals 

 and stock productivity. The general method of 

 analysis, which appeals primarily to the direct 

 parameterization of critical points, will apply to 

 any nonlinear scheme of exploitation and gross 

 production, but it applies in particular to the 

 Graham-Schaefer hypothesis (Graham 1935; 

 Schaefer 1954) and to the "generalized" model of 

 Pella and Tomlinson ( 1969). Since control of either 

 system rests ultimately with the control of critical 

 points, we restructure the parametric definitions 

 accordingly and the governing equations for both 

 systems are then controlled directly by parame- 

 ters of management significance. 



Typically, either system reflects the determinis- 

 tic premise that a stock of fishes, otherwise held by 

 exploitation at levels below a prior abundance, 

 will constantly strive to recover its numbers in 

 accord with some innate, self-regulating, and re- 

 peatable mechanism of restoration. Any such res- 

 toration must accrue from the productivity of the 

 stock, and by Graham's hypothesis, the inherent 

 or latent capacity for productivity in a stock of 

 fishes depends jointly on the current size of the 

 stock (in numbers or biomass) and the difference 

 between the current and potentially maximum 



'Center for Quantitative Science in Forestry, Fisheries, and 

 Wildlife, University of Washington, Seattle, WA 98195. 



sizes. Whence, in terms of time-dependent 

 biomass B, and with the proportionality 

 coefficient defined as the ratio of "intrinsic" 

 growth rate k and 6^ , Graham's formula for latent 

 productivity P takes on the familiar form 



P(B) = kB 



B 



'-B\ 



(1) 



Manuscript accepted September 1977. 

 FISHERY BULLETIN: VOL. 76. NO. 2. 1978. 



Of the two expanded terms, the first governs the 

 intrinsic, exponential capacity for growth of the 

 population's biomass, while the negative, non- 

 linear term provides the damping that ultimately 

 slows growth as B(t) approaches its asymptotic 

 maximum B^. The two terms, in their algebraic 

 sum, govern the latent productivity of the stock at 

 any stock size between zero and JS^^- Parameter k, 

 as we shall see, is coupled analytically and 

 phenomenologically to parameter By:, but the de- 

 pendence of k on root Bx in Equation ( 1) can be 

 supressed in favor of the direct parameterization 

 of maximum productivity (which, in the complete 

 exploitation model, we identify with maximum 

 yield rate I. 



In the Pella-Tomlinson model, the parametric 

 controls for latent productivity exceed by one the 

 total number of such parameters in Graham's 

 formulation, an increase in freedom that comes at 

 considerable cost to tractability, both analytical 

 and statistical. The differential equation that gov- 

 erns latent productivity in the Pella-Tomlinson 

 system has the indeterminate form 



377 



