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Cumulant Dynamics in a Finite Population: Linkage Equilibrium Theory

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  • Magnus Rattray
  • Jonathan L. Shapiro

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.

Suggested Citation

  • Magnus Rattray & Jonathan L. Shapiro, 1999. "Cumulant Dynamics in a Finite Population: Linkage Equilibrium Theory," Working Papers 99-07-055, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:99-07-055
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    Keywords

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

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