The nk model of fitness interactions is examined. This model has been used by previous authors to investigate the effects of fitness epistasis on substitution dynamics in molecular evolution, and to make broader claims about the importance of epistasis. To examine these claims, an infinite-allele approximation is introduced. In this limit, it is shown that the nk model is, at an appropriate level of description, formally identical to the non-epistatic House-of-Cards model—a well-studied model in theoretical population genetics. It is further shown that in many parameter regimes, the analytical results obtained from this infinite-allele approximation are very close to results from the full nk model (with a finite number of alleles per locus). The findings presented shed light on a number of previous results.
Evaluates "patterns of introgression across the hybrid zone for 13 diagnostic X-linked loci with known chromosomal positions using a maximum likelihood model". Finds different patterns. Correlates cline width and cline center at different loci.
M. Bramson, R. Durrett, and G. Swindle. Ann. Probab., 17 (2):
444--481(1989)This paper examines a version of the contact process with a large range. Particles die at rate 1, and a particle is created at an empty site $x$ at rate $łambda$ times the fraction of occupied sites in $y:||x-y||M$. This contact process is dominated by a branching random walk with death rate 1 and birth rate $łambda$, and it is shown that in many ways these two processes are very similar when $M$ is large. In particular, as $M\toınfty$, the critical value for the contact process converges to 1, which is the critical value for branching random walks. The authors obtain precise rates for this convergence, in every dimension, enabling them to describe the ``crossover'' from contact process to branching process behavior in terms of the survival probability of a process started from a single particle. The proofs of the main results use many estimates for branching random walks, further detailing the nature of this crossover behavior..