The Diffusive Spread of Alleles in Heterogeneous Populations

Abstract

Abstract. The spread of genes and individuals through space in populations is relevant in many biological contexts. I study, via systems of reaction-diffusion equations, the spatial spread of advantageous alleles through structured populations. The results show that the temporally asymptotic rate of spread of an advantageous allele, a kind of invasion speed, can be approximated for a class of linear partial differential equations via a relatively simple formula, c ϭ 2, that is reminiscent of a classic formula attributed to R. A. Fisher. The parameters r and D represent an ͙rD asymptotic growth rate and an average diffusion rate, respectively, and can be interpreted in terms of eigenvalues and eigenvectors that depend on the population's demographic structure. The results can be applied, under certain conditions, to a wide class of nonlinear partial differential equations that are relevant to a variety of ecological and evolutionary scenarios in population biology. I illustrate the approach for computing invasion speed with three examples that allow for heterogeneous dispersal rates among different classes of individuals within model populations. 2 ‫ץ‬t ‫ץ‬x can be interpreted as describing the frequency of a focal allele (e.g., genes of type 1), P ϭ P(x,t), as a function of onedimensional space, x, and time, t, in a biallelic diploid population of constant size (constant across space and time), where D Ͼ 0 is the diffusion coefficient (assumed to be the same for all individuals in the population), and m is the difference in the additive effects of the two alleles on Malthusian fitness. This description of Fisher's equation holds under the assumptions that the population is in Hardy-Weinberg equilibrium (at every point in space) and that there are no dominance effects on Malthusian fitness. Equation (1) is exact under the above conditions, but also applies approximately in more general settings (Nagylaki and Crow 1974;Aronson and Weinberger 1975;Smouse 1976;Fife 1979;Dockery andLui 1992, 1994;Lui and Selgrade 1993). The focal allele is assumed to be advantageous (i.e., m Ͼ 0) so that, when initially rare, it comes to spread through the population, resulting in an allelic invasion. In this setting, solutions to Fisher's equation, as time progresses, take the form of waves traveling across space, and are called traveling wave solutions having associated wave speeds (in the present context wave speed and invasion speed are treated as equivalent concepts). Although Fisher (1937) provided essentially no derivation of equation (1) his central argument that a rare, advantageous allele comes to spread through its population as a wave with invasion speedwhich was proven in a more general setting by Kolmogoroff et al. (1937), is foundational in theoretical biology, demonstrating that the spatial spread of a rare allele through a population depends on both its relative effect on reproductive capacity, m, and its mobility, D. While many studies have extended the basic form of Fisher's equation to more complex ecological and ...