Diffusion in Fisher's population model

Abstract

We consider Fisher's selection model for a single locus with n alleles. The population is distributed in a spatial domain, migration is represented by a diffusion term. For the resulting parabolic system with zero flux boundary conditions we show that if the selection model has a stable polymorphism, then every solution of the parabolic system converges to a spatially homogeneous stationary solution. If initially all genes are present in the population, then this solution is the polymorphism. The results carry over to some types of convolution equations. Diffusion in Fisher's population model. We consider the well-known Fisher-Wright-Haldane model of population genetics for n alleles ah a2, . . . , an, where Pj is the frequency of they-th allele, nak. We assume continuous time, i.e., overlapping generations. The differential equations read n n (0 Pj = TlfjkPkPj TifrsPrPsPji J = U 2, . . . , li. k= r,5=l For the derivation of the model and for results on the ordinary differential equation we refer to [1], [2], [4], or [3]. The last reference contains an account of the earlier literature and more detailed results. We prefer to use a condensed notation. Let F = (fJk) be the matrix of viabilities and p = (pj) the column vector of frequencies. Furthermore define the diagonal matrix P = (PjoJk). Always 0* (transpose) is the row vector corresponding to the column vector a, in particular e* = ( 1 , . . . , 1). Then a*b is a scalar product, ba* is a dyad, and a*Fa is a quadratic form. With this notation equations (1) read (2) p = PFp — p*Fp • p. We consider the population distributed in a bounded spatial domain Q of R with smooth (say C) boundary and describe migration and reproductive interaction of neighboring individuals by a diffusion term (3) pt = PFp p*Fp p + Ap. Received by the editors on August 7,1979. Copyright © 1980 Rocky Mountain Mathematics Consortium 39