11-YieldPerRec3 - YIELD-PER-RECRUIT(continued Grow...

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YIELD-PER-RECRUIT (continued) Grow overfishing results in sub-optimal stock sizes and wasted economic resources (too much fishing effort and the fuel and labor required to produce the fishing effort). Allen's Method is a simple technique for determining whether growth overfishing is occurring. Allen's Method for Determining the "Best" Minimum Age This is a method for determining whether fishing is eumetric. Given a fixed value for the rate of fishing mortality F, the problem is to determine whether the age-at-entry t e will produce the maximum yield from a cohort over its fishable lifespan. An approximate solution to this problem was first given in Allen (1953), "A method for computing the optimum size limit for a fishery;" Nature 172:210. Allen's Rule is to choose t e so that W t e ( 29 av W ( ) F Z = where W(t e ) is the weight of a fish when it attains the age-at-entry t e and av(W) is the average weight of a fish in the total catch from the cohort. To apply this rule simply measure W(t e ), av(W), F, and Z. If W(t e ) < ( F / Z )·av(W) , then W(t e ) and t e are too small. If W(t e ) > ( F / Z )·av(W) , then W(t e ) and t e are too large. This method assumes that W(t) is an increasing function of age t, and that all fish in the exploitable portion of the cohort are equally susceptible to both sources of mortality F and M. The following proof of Allen's Rule is from Fletcher (1987), on the Supplemental Reading list. Start with the equation for yield from a cohort over its fishable lifespan. Y t e t λ u F N u ( ) W u ( ) d = and for N substitute the following survival equation . N t ( ) N t e ( 29 e Z - t t e - ( 29 = N t r ( 29 e M - t e t r - ( 29 e Z - t e Z t e = = N t r ( 29 e M - t e e M t r e M F + ( ) t e e Z - t N t r ( 29 e M t r e F t e e Z - t =
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