A Semilinear Parabolic System for Migration and Selection in Population Genetics

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The selection coefficients depend on position; the drift and diffusion coefficients may do so. For two alleles (the scalar case), the global analysis of D. Henry (1981, “Geometric Theory of Semilinear Parabolic Equations,” Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin) is extended from homogeneous, isotropic migration (corresponding to the Laplacian) to arbitrary migration (corresponding to an arbitrary elliptic operator). For multiple alleles, sufficient conditions are given for the global loss of an allele that is nowhere the fittest. In the case of no dominance with at least one change in the direction of selection, sufficient conditions are established for global convergence to a stable equilibrium with all the intermediate alleles absent and one or two extreme alleles present. Sufficient conditions on the migration pattern for casting the elliptic operator into variational form are proved; in this case, the above results become more explicit.