17-EquilYield1

17-EquilYield1 - TOTAL EQUILIBRIUM YIELD Recall that the...

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TOTAL EQUILIBRIUM YIELD Recall that the yield-per-recruit model enabled us to examine the problem of growth overfishing but, because we assumed that recruitment was constant, the model did not tell us whether a given level of fishing might result in recruitment overfishing . Now we will combine Beverton and Holt's model for yield-per-recruit with their model for the stock-recruit relationship and build a model for long-term equilibrium yield, which will let us examine the problem of recruitment overfishing. Although this model was originally developed and described in Beverton and Holt (1957), it has received relatively little attention from fisheries scientists. Combining Yield-per-Recruit with a Stock-Recruit Model For many fish species it is at least approximately true that the number of eggs laid per mature female is directly proportional to the weight of the female. ( No. Eggs Laid Per Female ) = k · ( Female Body Weight ) If this relationship is valid, then the total number of eggs laid during the lifetime of a cohort of fish is given by E k t R u N fem u ( ) W fem u ( ) d = where E is the total number of eggs laid; k is the number of eggs laid per female body weight; t R is the age at first reproduction; N fem is the number of females at age; W fem is the female weight at age. If the amount of eggs as a proportion of bocy mass (k) is not constant with age then parameter k cannot be brought from under the integral sign. The formulation above assumes that egg production is a continuous process, but for many fish species spawning is highly seasonal. We will ignore this complication. Also, we will assume that the sex ratio is constant and that both sexes have the same weight at age so that we do not need separate integrals for males versus females. Given these assumptions it follows that E k' t R u N u ( ) W u ( ) d = k' t R u B u ( ) d = This equation uses k' rather than k because both sexes are included. The cohort's egg production is proportional to its spawning stock biomass (SSB). Our simplified problem is to model the spawning stock biomass and then relate that to the subsequent production of offspring and recruits to the adult population. For simplicity let's assume that the age at recruitment t r is less than the age at first spawning t R , and that the maximum exploitable age t λ is infinite. We must consider two cases: FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 107
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------- M ------ ------------- M+F ------------- (a) t e t R —|——————|—————|——————— Age, t t r t e t R ------------- M ------------ ------- M+F ------- (b) t R t e —|—————|—————|———————— Age, t t r t R t e For t e t R , the spawning stock biomass is SSB N t R ( 29 t R u e Z - u t R - ( 29 W u ( ) d = SSB R e M - t e t r - ( 29 e Z - t R t e - ( 29 t R u
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17-EquilYield1 - TOTAL EQUILIBRIUM YIELD Recall that the...

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