Chaos from Linear Frequency-Dependent Selection

Lee Altenberg

The American Naturalist 138 (1): 51-68. 1991.

Abstract
The simplest diploid model of frequency-dependent selection can generate periodic and chaotic trajectories for the allele frequency. The model is of a randomly mating, infinite diploid population with non-overlapping generations, segregating for two alleles under frequency-dependent viability selection. The fitnesses of each of the three genotypes is a linear function of the frequencies of the three genotypes. The region in the space of the coefficients that yields cycles and chaos is explored analytically and numerically. The model follows the period-doubling route to chaos as seen with logistic growth models, but includes additional phenomena such as the simultaneous stability of cycling and chaos. The general condition for cycling or chaos is that the heterozygote deleteriously effect all genotypes. The kinds of ecological interactions that could give rise to these fitness regimes producing cycling and chaos include cannibalism, predator attraction, habitat degradation, and disease transmission.