Establishment in a new habitat by polygenic adaptation.

Maladapted individuals can only colonise a new habitat if they can evolve a positive growth rate fast enough to avoid extinction, a process known as evolutionary rescue. We treat log fitness at low density in the new habitat as a single polygenic trait and use the infinitesimal model to follow the evolution of the growth rate; this assumes that the trait values of offspring of a sexual union are normally distributed around the mean of the parents' trait values, with variance that depends only on the parents' relatedness. The probability that a single migrant can establish depends on just two parameters: the mean and genetic variance of the trait in the source population. The chance of success becomes small if migrants come from a population with mean growth rate in the new habitat more than a few standard deviations below zero; this chance depends roughly equally on the probability that the initial founder is unusually fit, and on the subsequent increase in growth rate of its offspring as a result of selection. The loss of genetic variation during the founding event is substantial, but highly variable. With continued migration at rate M, establishment is inevitable; when migration is rare, the expected time to establishment decreases inversely with M. However, above a threshold migration rate, the population may be trapped in a 'sink' state, in which adaptation is held back by gene flow; above this threshold, the expected time to establishment increases exponentially with M. This threshold behaviour is captured by a deterministic approximation, which assumes a Gaussian distribution of the trait in the founder population with mean and variance evolving deterministically. By assuming a constant genetic variance, we also develop a diffusion approximation for the joint distribution of population size and trait mean, which extends to include stabilising selection and density regulation. Divergence of the population from its ancestors causes partial reproductive isolation, which we measure through the reproductive value of migrants into the newly established population.