14-Recruitment2

14-Recruitment2 - MODELS FOR VARIABLE RECRUITMENT...

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MODELS FOR VARIABLE RECRUITMENT (continued) The other model commonly used to relate recruitment strength with the size of the parental spawning population is a model developed by Beverton and Holt (1957, Section 6), which is on the Recommended Reading list. We will use the same notation as before. E S k E 0 is the total number of eggs, which then become larvae, then juveniles, and then adults. are the number of spawning adults, the parents of E. is the average number of eggs laid per spawning adult. is the total number of eggs laid. E 0 = k·S The Beverton and Holt stock-recruit model comes from the solution to the following simple differential equation. dE dt M E - E = where M E M α E + = α ·E is the density-dependent rate of natural mortality. M is the density-independent rate of natural mortality. Here the density dependent term is proportional to the cohort abundance, not to the parental abundance (as in the Ricker model). This form of mortality could arise from processes such as competition for food or other scarce resources. This differential equation is much more complicated to solve than the one that led to the Ricker model. As before, we start by separating the variables. dE dt M α E + ( 29 - E = dE M α E + ( 29 E dt - = But, now what do we do? We can solve this equation by separating the left hand side, which is the product of two fractions, into an equivalent sum of two fractions. This is sometimes described as the method of partial fractions . The following example illustrates the general method. We can integrate the sum on the right. The integral of a sum is the sun of the integrals. c X a - ( ) X b - ( ) A X a - B X b - + = We need to find values for A and B that satisfy the above equation. To do so, multiply together the two terms on the right hand side and then eliminate the denominator. c X a - ( ) X b - ( ) A X b - ( ) B X a - ( ) + X a - ( ) X b - ( ) = c A X b - ( ) B X a - ( ) + = A X B X + A b - B a - = The left hand side of this equation contains no terms in X, which implies that FW431/531 Copyright 2008 by David B. Sampson Recruitment2 - Page 82
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0 A X B X + = and c A - b B a - = ==> B A - = and c A - b A a + = A a b - ( ) = ==> A c a
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14-Recruitment2 - MODELS FOR VARIABLE RECRUITMENT...

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