Chaos from Linear Frequency-dependent Selection

—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. The possibilities for complex dynamical behavior from even the simplest models of population growth regulation (May, 1974, 1976) have led to the examination of conditions that produce cycling and chaos in a wide variety of models in population biology. Most studies of chaos in population dynamics have focused on the dynamics of population size. Models in which cycling or chaos is produced by natural selection acting on genetic variation have received less attention. Asmussen (1979, 1983) and Felsenstein (1979) have examined density-dependent selection in populations exhibiting chaos, but in their models, the chaotic behavior results not from the presence of genetic variability, but from the form of density regulation that is acting. May (1979, 1983) examined a symmetric, two allele model of frequency-dependent selection in which the heterozygote fitness is the geometric mean of the homozygote fitnesses; he showed that when the homozygote’s fitness decreases monotonically with its frequency, there could be at most a 2-point limit cycle, but no chaotic dynamics. The few models of frequency-dependent selection that have been found to produce chaos either involve fitnesses that are complex functions of the genotype frequencies, with steep peaks or non-analytic points, or post hoc choices of fitness functions in order to produce chaos. The latter includes the model of Cressman (1988a), in which the quadratic logistic curve is simply grafted into the recursion, and the model of May (1979, 1983) where the fitness functions are solved to produce a system equivalent to 1