26-Surplus4 - SURPLUS PRODUCTION REVISITED Extensions to...

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B dB/dt a b SURPLUS PRODUCTION REVISITED Extensions to the Graham-Schaefer Production Model The standard Graham-Schaefer surplus production model is based on various simplifying assumptions, including the following: Fishing mortality is proportional to fishing effort, F = q·f ; the response in biomass to any changes is immediate; and the system is entirely deterministic with r and K constant. Now we will relax each of these assumptions in turn to examine how they influence the behavior of a fishery whose dynamics are governed by logistic growth. The Assumption that F = q·f Instead of assuming that the rate of fishing mortality F is directly proportional to the amount of fishing effort f, suppose there is a handling time for each unit of biomass caught and that dY dt q τ f 1 q h B + B = We saw this model in Catch Process 3. Parameter τ (tau) is the fraction of a trip spent fishing and handling; h is the handling time per unit catch. The presence of a nonzero handling time can fundamentally alter the equilibrium characteristics of the exploited fish stock. In the Graham-Schaefer model there is never more than one nonzero equilibrium value for B(t). With the handling time model, however, there can be two nonzero equilibria provided the slope of dY/dt versus B at the origin is greater than r and provided h is large enough so that the horizontal asymptote for dY/dt is sufficiently small. (The slope of dY/dt versus B at the origin is q· τ ·f; the asymptote is τ ·f/h.) B <= Stable Stable / <= Unstable In the Graham-Schaefer model (the graph on the left) the equilibria are always stable . After any small perturbation to B, the system will return to the original equilibrium (provided the parameters remain constant) because fishery removals exceed natural growth when biomass is greater than the equilibrium level, which causes the biomass to decrease, and vice versa when biomass is less than the equilibrium level. In the model with handling time (the graph on the right), when there are FW431/531 Copyright 2008 by David B. Sampson Surplus4 - Page 169
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two nonzero equilibria (as in the higher yield curve), the one closer to the origin is unstable. Any small perturbation to B will move the system either to the stable equilibrium at the origin, or to the larger equilibrium biomass. Condrey (1984), on the Supplemental Reading list, discusses this model and its properties. Even with a yield model that is linear with B we can get a system that has two nonzero equilibria,
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26-Surplus4 - SURPLUS PRODUCTION REVISITED Extensions to...

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