Application of method of small parameters to multi-niche population genetic models

Abstract

In theoretical population genetics it has been customary to analyze models in which most or all of the parameters are assigned specific values. For example, one of the earliest models discussed by Haldane was concerned with the rate of elimination of a recessive lethal, and in this model the viabilities assigned to genotypes AA, Au, au are 1, 1, 0, respectively. Implicit in the study of such models is the assumption that the conclusions derived, in their general qualitative aspects, and to some extent in their quantitative features, will be applicable to more “realistic” models in which the parameter values are only approximately the same as those assigned in the model. In the above example an assignment of b, , b, , 15, as the respective viabilities, with b, and b, nearly 1 and b, nearly zero, should not alter the qualitative features of the model and should result in only small changes of the important quantitative aspects. A gene perfectly dominant in all its pleiotropic effects coupled with a recessive allele perfectly lethal when homozygous, even in the complex natural environment, comprise a structure of little interest, except to the extent that it may serve as an idealized approximation for cases in which the dominance and the lethality are not quite perfect. This paper is one of several companion papers in which a theory of perturbation of parameters in deterministic genetic systems is developed. The most naive application of this perturbation theory would be to show that the qualitative and quantitative conclusions derived from models, such as the aforementioned model of Haldane, do indeed persist under small perturbations of the parameters in the model. However, the theory has more valuable and less obvious applications in the analysis of complex genetic population systems composed of a number of interacting subpopulations, or generally subsystems. A number of examples are presented below which illustrate this kind of application.