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simpler. Combining our study of mortality and that of individual growth takes us in interesting directions. Suppose that survival to age t is given by the exponential distribution e~mt, where the mortality rate is fixed and that if the organism matures at age t, when length is L(t), then lifetime reproductive output isfL(t)b, where f and b are parameters.

For many fish species, the allometric parameter b is about 3 (Gunderson 1997); for other organisms one can consult Calder (1984) or Peters (1983). The parameterf relates size to offspring number (much as we did in the study of egg size in Atlantic salmon). We now define fitness as expected lifetime reproductive success, the product of surviving to age t and the reproductive success associated with age t. That is F(t) = e~mfL(t)b. Since survival decreases with age and size asymptotes with age, fitness will have a peak at an intermediate age (Figure 2.2b). It is a standard application in calculus to find the optimal age at maturity.

Show that the optimal age at maturity, tm, is given by tm = - log k m + bk

In Figure 2.3,1 show optimal age at maturity as a function of k for three values of m. We can view these curves in two ways. First, let's fix

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