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27-Surplus5

# 27-Surplus5 - SURPLUS PRODUCTION REVISITED(continued The...

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SURPLUS PRODUCTION REVISITED (continued) The Assumption of Constant r and K Suppose the parameters r and K of the Graham-Schaefer 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 Graham-Schaefer 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 Graham-Schaefer 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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27-Surplus5 - SURPLUS PRODUCTION REVISITED(continued The...

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