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Unformatted text preview: 1 2 3 4 5 time, t Biomass & Yield SURPLUS PRODUCTION (continued) We just saw that at equilibrium there is a linear relationship between catch (weight) peruniteffort and effort for a population whose growth follows the logistic model. In practice, fishing effort and CPUE are almost never constant over time. As a consequence, it is almost never appropriate to estimate the parameters K, r, and q from a linear regression of CPUE against effort! Schaefer's Method for Analyzing NonEquilibrium Data Schaefer's main contribution to fisheries science was not the parabolic surplus production model that we usually associate with his name. His main contribution was introducing a method for analyzing catch and effort data arising from nonequilibrium conditions. Consider the catch in weight that accumulates over a one unit time interval (0,t) , during which the instantaneous rate of fishing mortality F is constant. avB(t) is timeaveraged biomass and F has units [1/t]. Y t ( ) F avB t ( ) = Assume that the rate of fishing mortality can vary between time intervals but is constant during any given interval. The yield during interval i is Y i F i avB i = B(1) B(4) B(3) B(2) Y = F avB Y 1 = F 1 avB 1 Y 3 = F 3 avB 3 Y 4 = F 4 avB 4 Y 2 = F 2 avB 2 If the population is truly at equilibrium, then B(t) = B(t+ t) for all values of t. Let us consider the population biomass just at the endpoints of each interval. If it happens that B(t+1) is equal to B(t), then removals by the fishery during the interval (t,t+1) must have been just equal to the biological production during the interval. If the harvest was in excess of the natural population growth, then FW431/531 Copyright 2008 by David B. Sampson Surplus2  Page 126 time Biomass & Yield i i 1 + B(t+1) would be less than B(t). If the growth was in excess of the harvest, then B(t+1) would be greater than B(t). Schaefer's method is based on the idea that for any level of biomass there exists a rate of yield which, if it is taken, will keep the population biomass at a constant level. Let Ye denote such an equilibrium yield. It consists of the actual yield removed from the population Y plus a factor to adjust for any change in the biomass B. Ye i Y i B i + = where B i B i 1 + ( ) B i ( ) = B(i) B i B(i+1) Y 1 In this illustration the actual yield during the (i)th interval was so large that the biomass declined. As a consequence, B for the interval is negative. To calculate values for the equilibrium yield we first need to estimate the B(i). We can derive approximate values from the timeaveraged biomass. B i 1 + ( ) 2245 0.5 avB i avB i 1 + + ( 29 B i ( ) 2245 0.5 avB i 1 avB i + ( 29 B i B i 1 + ( ) B i ( ) = 2245 0.5 avB i 1 + avB i 1 ( 29 By the way, we used this form of approximation in Ricker's generalized model for equilibrium yieldperrecruit, but in reverse, to estimate timeaveraged biomass from estimates of biomass at...
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 Fall '09

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