Dynamics of density- and frequency-dependent selection

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The evolution of a multiallelic locus in a diploid monoecious population is investigated for weak selection in both discrete and continuous time. It is assumed that in the absence of selection the rate of change of population size N is a function only of N, and N converges exponentially to a unique, globally stable equilibrium, ^N. With arbitrary density- and frequency-dependent selection, after several generations have elapsed, N differs from ^N by terms of O(s) (i.e., bounded by a constant times s), where s is the selection intensity. The mean absolute fitness deviates from its equilibrium value (1 in discrete, 0 in continuous time) by only O(s2); its rate of change is O(s3). The evolution of the gene frequencies may be approximated with an error of O(s) by assuming that N = ^N and Hardy-Weinberg proportions obtain. When the frequency dependence is weaker than the selection intensity, the mean absolute fitness generally exceeds its equilibrium value by a quantity approximately proportional to the genic variance, the constant of proportionality being simply related to the rate of convergence of N to ^N in the absence of selection. Thus, after a short time, the population generally increases, and the rate of growth is proportional to the genic variance.

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