Evolution with Stochastic Fitness and Stochastic Migration
Migration between local populations plays an important role in evolution – influencing local adaptation, speciation, extinction, and the maintenance of genetic variation. Like other evolutionary mechanisms, migration is a stochastic process, involving both random and deterministic elements. Many models of evolution have incorporated migration, but these have all been based on simplifying assumptions, such as low migration rate, weak selection, or large population size. We thus have no truly general and exact mathematical description of evolution that incorporates migration.
We derive an exact equation for directional evolution, essentially a stochastic Price equation with migration, that encompasses all processes, both deterministic and stochastic, contributing to directional change in an open population. Using this result, we show that increasing the variance in migration rates reduces the impact of migration relative to selection. This means that models that treat migration as a single parameter tend to be biassed – overestimating the relative impact of immigration. We further show that selection and migration interact in complex ways, one result being that a strategy for which fitness is negatively correlated with migration rates (high fitness when migration is low) will tend to increase in frequency, even if it has lower mean fitness than do other strategies. Finally, we derive an equation for the effective migration rate, which allows some of the complex stochastic processes that we identify to be incorporated into models with a single migration parameter.
As has previously been shown with selection, the role of migration in evolution is determined by the entire distributions of immigration and emigration rates, not just by the mean values. The interactions of stochastic migration with stochastic selection produce evolutionary processes that are invisible to deterministic evolutionary theory.
Figure 1. The consequences of a negative correlation between fitness and immigration rate:
<top figure in picture
Figure 2. The influence of the variance in immigration rates, on the expected change in mean phenotype:
<bottom-left in picture
Essentially, when the variance in the immigration rate is small it results in a negative phenotypic change on the mean phenotype of a population, as the migrant phenotypes predominate.
Figure 3. The relation between variance in immigration, selection, and in an island-continent model:
>bottom-right in picture.
>The curves show the change in mean phenotypic value assuming a mean phenotypic value as decided by a population’s selection.
>The variance in migration is shown in descending value on the curves.
Though the mean rate of migration is the same on all the curves, it is clear that the the island can still get close to fixation of the phenotype favoured there, so long as the variance in immigration is high.
So for this figure, assuming that migration is an indefinitely persisting condition, migrants will prevent the local island favoured phenotype from remaining intact, as they are essentially working against selection.