Abstract:

Two-locus models are a major subject of study in the field of mathematical population genetics. In comparison to one-locus model, their analysis is much more complicated due to the occurrence of genetic interactions (epistasis) and genetic linkage which introduce additional non-linearities. In this talk, I will first give an overview over already published results of deter-ministic analyses, invoking among others index theory, of the ODE’s corresponding to two different types of two-locus models. One assumes two alleles at both loci under a simplifying assumption and the other is concerned with an arbitrary number of alleles. The main focus of this talk concerns a model of the effects of linkage and genetic interactions on the fate of a new weakly beneficial mutation that arises in a large population. Here, stochasticity is essential and therefore, we apply a Markov chain model or, more precisely, a two-type branching process. First, I characterize when this new mutation can successfully invade the population and become established. Then, I show how the probability that the mutation is successful depends on the parameters. For instance, under some conditions increasing the positive effect of genetic interactions is more effective to boost the invasion pobabilities, than increasing linkage. Under other conditions it is the other way around. Most of this analysis is based on the theory of slightly supercritical branching processes.

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