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# APPENDIX BDERIVATION OF HAMILTON’S EQUATION FOR THE FORCE OF NATURAL SELECTION ON MORTALITY

We start with a derivation of the Euler–Lotka equation, which was developed for describing population growth (Lotka, 1907, 1913). Crow and Kimura (1970, pp. 3–20) defined four deterministic models of population growth: discrete nonoverlapping generations (model 1); continuous random births and deaths (model 2); overlapping generations, discrete time intervals (model 3); and overlapping generations, continuous change (model 4). The Euler–Lotka equation pertains to model 4, in which, unlike in model 3, the process of population change occurs in a continuum of time and, unlike in model 2, individuals are not regarded as equally likely to die or reproduce at all times. We consider how the Malthusian parameter m relates to a measure of fitness applicable to the simple model of population growth for discrete nonoverlapping generations or model 1 (Crow and Kimura, 1970, pp. 5–6). Let Nt be the number of individuals in the population at time t, measured in generations, and w or the Darwinian fitness be the average number of progeny per individual. The population number in generation t can be expressed in terms of the number in the previous generation, t − 1, by Nt = wNt − 1. The relation between Nt −1 and Nt − 2 is the same as that between Nt and Nt − 1. If w remains constant, we can write Nt = w(wNt − 2) = w2Nt − 2. Continuing this process, we obtain

(B.1)

where N0 is the population size in ...

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