A deterministic model of cyclical selection

Abstract

SUMMARYA deterministic model of cyclical selection in randomly mating populations is studied. Sufficient conditions for a protected polymorphism, which are for the special case of alternating selection also necessary conditions, are obtained using a simple graphical approach. The most important condition requires ‘marginal overdominance’ (Wallace, 1968); the other conditions seem hard to satisfy in a natural situation. Furthermore it is shown that the cyclical selection model can be regarded as a special case of a frequency-dependent selection model (Cockerhamet al.1972). Using this property, a mean fitness function for the cyclical selection model is derived. Generally, the mean fitness will not be maximized under cyclical selection. The relevance of the model to the problem of the role of cyclical selection in the maintenance of genetic polymorphism in natural populations is discussed. It is concluded that this relevance is probably rather limited with regard to the creation of protected polymorphism, but that the influence of cyclical selection on transient polymorphisms might be more significant. An approximate formula for the time needed for a given change in gene frequency under cyclical selection is derived. It appears that cyclical selection can extend considerably the time during which a transient polymorphism persists, especially if the selective differences in the different environments are of the same order of magnitude and of opposite sign.