Limit Theorems for Competitive Density Dependent Population Processes

Abstract: Near the beginning of the century, Wright and Fisher devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for mathematical, and not biological reasons - makes assumptions that are violated in most real-world populations. In this paper, I consider an alternative framework that merges P. A. P. Moran's continuous-time Markov chain model of allele frequency with the density dependent models of ecological competition proposed by Gause, Lotka and Volterra, that, unlike Moran's model allow for a stochastically varying -- but bounded -- population size. I require that allele numbers vary according to a density-dependent population process, for which the limiting law of large numbers is a dissipative, irreducible, competitive dynamical system. Under the assumption that this limiting system admits a codimension one submanifold of attractive fixed points -- a condition that naturally generalises the weak selection regime of classical population dynamics -- it is shown that for an appropriate rescaling of time, the finite dimensional distributions of the original process converge to those of a diffusion process on the submanifold. Weak convergence results are also obtained for a related process.