Dynamics of density- and frequency-dependent selection.
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(s(2)); its rate of change is O(s(3)). 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.
Department of Biophysics and Theoretical Biology, The University of Chicago, 920 East 58th Street, Chicago, Illinois 60637.
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Nagylaki