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B
dB/dt
a
b
SURPLUS PRODUCTION REVISITED
Extensions to the GrahamSchaefer Production Model
The standard GrahamSchaefer 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 GrahamSchaefer 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 GrahamSchaefer 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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View Full Document 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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This document was uploaded on 12/06/2011.
 Fall '09

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