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Cumulant Dynamics in a Finite Population: Linkage Equilibrium Theory

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  • Magnus Rattray
  • Jonathan L. Shapiro
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    Abstract

    The evolution of a finite population at linkage equilibrium is described in terms of the dynamics of phenotype distribution cumulants. This provides a powerful method for describing evolutionary transients and we elucidate the relationship between the cumulant dynamics and the diffusion approximation. A separation of time-scales between the first and higher cumulants for low mutation rates is demonstrated in the diffusion limit and provides a significant simplification of the dynamical system. However, the diffusion limit may not be appropriate for strong selection as the standard Fisher-Wright model of genetic drift can break down in this case. Two novel examples of this effect are considered: we show that the dynamics may depend on the number of loci under strong directional selection and that environmental variance results in a reduced effective population size. We also consider a simple model of a changing environment which cannot be described by a diffusion equation and we derive the optimal mutation rate for this case.

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    Bibliographic Info

    Paper provided by Santa Fe Institute in its series Working Papers with number 99-07-055.

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    Date of creation: Jul 1999
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    Handle: RePEc:wop:safiwp:99-07-055

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    Related research

    Keywords: Evolutionary dynamics; mutation-selection dynamics; theoretical population biology; diffusion approximation;

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