Local adaptation under demographic and genetic fluctuations

Evolution frequently plays out over ecological timescales. Local adaptation under the joint action of evolutionary and ecological processes frequently leads to novel outcomes, as is evidenced by the theoretical work on adaptation at species' borders. However, to date this body of work does not have a theory for the effect of stochastic processes on local adaptation. The primary goal of this thesis is to show that demographic and genetic fluctuations can significantly impact upon local adaptation. In addition, the effect of polygenic evolution is also analysed. Specifically, three types of models are considered. First a deterministic mainland-island, subject to hard directional selection, maladaptive gene flow and density regulation is solved for two different trait architectures: an explicit multilocus trait and a quantitative trait. The maladaptive and adaptive steady states can be bistable. This depends on the underlying architecture of the trait, as well as locus number and ploidy. Sourcesink structure can emerge, accompanied by a novel, upper critical threshold above which maladaptation occurs. The most favourable condition for local adaptation occurs for few loci and low migration. Second, a stochastic version of the mainland-island model is analysed as a diffusion process. This is the central premise of the thesis and is explored by examining properties of the stationary distributions of both trait architectures, and the first-passage properties of the single locus case. It is found that across a range of migration rates that depend on locus number and migrant polymorphism, local adaptation may be reversed or escape from maladaptation becomes possible at varying transition rates. The diffusion model is compared to a similar discrete model. The continuous model is in good qualitative agreement with the discrete model. Third, the stochastic model is generalised to the infinite island model, which evolves deterministically. Under deterministic dynamics a range of equilibria are possible, depending on whether habitat size varies or is fixed. Multilocus dynamics restrict the conditions for polymorphism. Stochastic dynamics can have potentially detrimental consequences for the persistence of the island population when drift is strong. The relevance of the stochastic model to border populations is discussed. Although the diffusion process imposes severe constraints on the permissible parameter ranges, it is still able to provide a good qualitative understanding of the impact demographic and genetic fluctuations have on local adaptation