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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)- per-unit-effort 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 Non-Equilibrium 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 non-equilibrium 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 time-averaged 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 time-averaged 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 yield-per-recruit, but in reverse, to estimate time-averaged biomass from estimates of biomass at...
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- Fall '09