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21-Surplus3

# 21-Surplus3 - SURPLUS PRODUCTION(continued Transition to a...

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SURPLUS PRODUCTION (continued) Transition to a New Equilibrium The following materials are adapted from Fletcher (1978), on the Recommended Reading list. Because B(t) approaches the new equilibrium value asymptotically, it takes an infinite amount of time to actually reach the new equilibrium. However, we can determine how long it will take to get to within any fixed proportion of the new equilibrium following a sudden change in the rate of fishing mortality F. If the initial rate of fishing mortality F 0 is less than the new rate F 1 , then the old equilibrium biomass was greater than the new B e and we want to determine how long it takes for B(t) to reach the value B e + ε ·B e . We want to find t lag such that B t lag ( 29 B e ε B e + = B e 1 ε + ( 29 = ==> B t lag ( 29 B e 1 ε + = ==> 1 1 C exp r F 1 - ( 29 - t lag + 1 ε + = where C new_B e old_B e - old_B e = and r 4 MSY K = Transition After an Increase in F. t B 0 t lag Here is a picture of what we are doing. B e · ε { B e Solve the following equation for t lag 1 1 ε + ( 29 1 C exp r F 1 - ( 29 - t lag + = 1 1 ε + 1 - C exp r F 1 - ( 29 - t lag = ==> 1 1 ε + ( 29 - 1 ε + 1 C exp r F 1 - ( 29 - t lag = ln ε - 1 ε + 1 C r F 1 - ( 29 - t lag = ==> t lag ln 1 ε + ε - new_B e old_B e - old_B e r F 1 - = FW431/531 Copyright 2008 by David B. Sampson Surplus3 - Page 134

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Transition After a Decrease in F. t B 0 t lag Suppose F 1 is less than F 0 (i.e., the new B e is greater than the old B e ). In this case the approach to the new equilibrium is from below and B t lag ( 29 B e ε B e - = B e 1 ε - ( 29 = ==> B t lag ( 29 B e 1 ε - = The solution for t lag is t lag ln 1 ε - ε new_B e old_B e - old_B e r F 1 - = B e B e · ε { In either case, the yield that accumulates during the transition period (0,t lag ) is Y 0 t lag t F B t ( ) d = 0 t lag t F B e 1 C exp r F - ( ) - t [ ] + d = Y F B e 0 t lag t 1 1 C exp r F - ( ) - t [ ] + d = The integral is of the form u 1 1 a exp b - u ( ) + d u exp b u ( ) exp b u ( ) a + d = ln exp b u ( ) a + ( ) b Arb + = 0 X u 1 1 a exp b - u ( ) + d ln exp b X ( ) a + ( ) b ln 1 a + ( ) b - = 1 b ln exp b X ( ) a + 1 a + = FW431/531 Copyright 2008 by David B. Sampson Surplus3 - Page 135
We can use this result with b = r - F and a = C and write the equation for Y as Y F B e r F - ln exp r F - ( ) t lag

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21-Surplus3 - SURPLUS PRODUCTION(continued Transition to a...

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