SURPLUS PRODUCTION REVISITED (continued)
The Assumption of Constant r and K
Suppose the parameters r and K of the GrahamSchaefer surplus production model are not
constant, but instead are subject to random shocks.
One qualitative approach for studying the
perturbed system is to examine the "characteristic return time", which is a measure of how rapidly
the system will respond to perturbations or changed conditions.
Recall that the solution to the
differential equation underlying the GrahamSchaefer model is
B t
( )
B
e
1
C exp
r
F

(
)

t
⋅
[
]
⋅
+
=
where
B
e
K
1
F
r

⋅
=
and
C
B
e
B
0

B
0
=
The characteristic return time for the GrahamSchaefer model is
T
R
1
r
F

=
.
When r  F is near zero,
the term exp[ ( r  F )·t ] changes slowly and B(t) approaches the
equilibrium B
e
very sluggishly.
When r  F is large, B(t) rapidly approaches B
e
.
Below (on the left)
is the graph of T
R
versus F.
Fishing Mortality, F
Return Time, TR
1
r
r
Fishing Mortality, F
TR(F) / TR(0)
1
r
The vertical intercept is equal to the inverse of the intrinsic growth rate 1/r .
We can standardize
the curve by dividing T
R
(F) by T
R
(0).
The standardized curve, which is shown in the graph on the
right, has a vertical intercept equal to one.
With the Schaefer model the characteristic return time increases steadily as F increases, indicating
that B(t) becomes increasingly unresponsive with exploitation. Fishing reduces a stock's ability to
respond to changing conditions.
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 Fall '09
 Variance, Population Ecology, Probability theory

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