Stable cycling in discrete-time genetic models.

Abstract

Examples of stable cycling are discussed for twolocus, two-allele, deterministic, discrete-time models with constant fitnesses. The cases that cycle were found by using numerical techniques to search for stable Hopf bifurcations. One consequence ofthe results is that apparent cases ofdirectional selection may be due to stable cycling.The causes ofcycling in populations, which include predator-prey oscillations, the role of time delays, and amplification of environmental disturbances have long been a topic of research in population biology (1, 2). Selection within a single population would be an alternate, and possibly quite general, explanation ofcycling. These cycles could provide an explanation ofcurious behavior observed in one-locus genetic experiments. In this paper, one example of stable cycling in a discrete-time, constant-fitness, two-locus, two-allele model will be discussed, and general features of all the examples I have found thus far will also be included. In ref. 3 cycling that results from genotype-environment interactions is discussed. The only other reported instance of stable cycling due to genetic causes alone is the independent work ofAkin (4), who recently proved the existence of stable cycles arising from a Hopfbifurcation in the continuous time, two-locus, two-allele model. This paper complements and extends the work of Akin by dealing with a discrete-time model and examining the behavior of the cycling populations, rather than proving the existence of cycles. Akin employed an asymptotic, analytic approach, whereas I have used a numerical approach. BACKGROUND AND METHODSThe model used in this study is the usual deterministic two-allele discrete-generation model (5) with alleles A and a at the A locus and B and b at the B locus. The frequencies (and "names") of the four chromosomal types AB, Ab, aB, ab are x1, X2, X3, X4,

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“…Moreover, these cycles disappear at a second Hopf bifurcation, as the fitness parameter is further changed. We know of no other single-locus population-genetic models exhibiting such behavior, although single Hopf bifurcations do arise in models of (i) constant viability selection and recombination for two loci each with two alleles (Hastings 1981;Akin 1982Akin , 1983 and (ii) constant selection and multilocus mutation with selfing (Yang and Kondrashov 2003).…”