Stability of Solutions to Certain Nonlinear Difference Equations of Population Genetics

Abstract

In the study of the relative frequencies of genotypes in a population one generally encounters nonlinear mathematics. A source of the nonlinearity is the interaction between genotypes (the mating process) which generates the changes of the genotypic frequencies. Mathematically the changes are expressed nonlinearly in terms of products of frequencies; a two-frequency product corresponds to an interaction between two genotypes. If a discrete-time mating process is considered, the above ideas can be summarized by a system of nonlinear difference equations. Of genetic interest are equilibrium states for these equationis, the stability of the equilibrium states, and the nature of the approach to equilibrium. In principle, the equilibrium states are determined from a system of nonlinear algebraic equations; however, the equations are often difficult to solve analytically. Nevertheless, exact results have been obtained for special cases, and one may employ computer iteration of the difference equations for specified initial frequencies to obtain numerical approximations for the equilibrium states. Now given an equilibrium state, one may ask questions about its stability. In the context of differential equations a summary of basic ideas was given by La Salle [1962]. It would be useful to know the totality of initial states which would, according to the difference equations, approach a particular equilibrium state. In practice, such a question is not readily 'answered by presently known mathematical methods, although some relevant general theorems have been proved by Hurt [1967]. On the other hand, it is often practical to determine for the initial states in the immediate neighborhood of equilibrium, whether all decay to equilibrium (stable case), all escape from equilibrium (unstable case), or some may escape from equilibrium (metastable case).