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12-YieldPerRec4

# 12-YieldPerRec4 - YIELD-PER-RECRUIT(continued The...

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YIELD-PER-RECRUIT (continued) The yield-per-recruit model applies to a cohort, but we saw in the Age Distributions lecture that the properties of a cohort do not apply in general to a collection of cohorts, which is what we usually encounter in practice. Now we will explore some consequences for the yield-per-recruit model of the simplifying assumption that the population is in equilibrium (constant survival, constant recruitment, constant growth). Yield-per-Recruit from Multiple Cohorts The Beverton and Holt model for yield-per-recruit is based on equations for mortality and growth of individuals within a single age-class. To derive the model we assumed that F and N(t e ) were both constant and we brought them from under the integral. Y t e t λ u F N t e ( 29 e Z - u t e - ( 29 W u ( ) d = This step is invalid unless F and N(t e ) are both constant through time. ==> Y F N t e ( 29 t e t λ u e Z - u t e - ( 29 W u ( ) d = If the fish population is in equilibrium (with constant annual recruitment, constant rates of natural and fishing mortality, and constant growth parameters), then the yield-per-recruit model we derived is also valid for the annual yield-per-recruit from the entire population. There is a proof of this in Beverton and Holt (1957) on pages 37-38. If the population is not in equilibrium, then the yield-per-recruit model is only an approximation. Recall the artificial age distributions that I constructed in a previous class. Year ==> Age 0 1 2 3 '95 100 '96 100 50 '97 100 52 26 '98 100 54 28 14 '99 100 56 30 15 If there are changes in annual survival (or recruitment), the age distribution changes from year to year. barb2se barb2se barb2se barb2se barb2se barb2se barb2se barb2se barb2se S = 50% 52% 54% 56% When there are changes in the rate of fishing mortality F or changes in the age-at-entry t e , the population will be out of equilibrium during a transition period, and the yield-per-recruit during the transition will not be the same as the equilibrium yield-per-recruit. Below is an artificial population that illustrates the problem. R' y denotes the number of fish entering the fishery in year y, the recruitment. Cohorts occur along each diagonal. The equations that describe survival for a cohort are reasonably simple because the recruitment term is a common factor for each age class. However, the pseudo-cohorts, which are the collection of cohorts present in any given year, have complicated survival equations because there is no common factor for recruitment. FW431/531 Copyright 2008 by David B. Sampson YieldPerRec4 - Page 69

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Year ==> Age 0 1 2 3 t e R' 0 R' 1 R' 2 R' 3 barb2se barb2se barb2se t e 1 + R' 0 e M - F 0 - R' 1 e M - F 1 - R' 2 e M - F 2 - barb2se barb2se t e 2 + R' 0 e 2 - M F 0 - F 1 - R' 1 e 2 - M F 1 - F 2 - barb2se t e 3 + R' 0 e 3 - M F 0 - F 1 - F 2 - ... ...
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12-YieldPerRec4 - YIELD-PER-RECRUIT(continued The...

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