Misc,
Limit Theorems for Competitive Density Dependent Population Processes
(2010)cite arxiv:1005.0010 Comment: This is a very early draft of a manuscript in preparation for future submission. It is submitted in support of a second manuscript, discussing the biological and statistical implications of these theorems, currently in submission..
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.
