How does spatial structure of a population influence the dynamics of their genetic evolution? This question rang clear through the threeweek miniseries delivered this past month by Alison Etheridge. Through analysis of a variety of agentbased models, as well as their limiting populationbased stochastic differential equation models, Etheridge painted a vivid picture of how population density, expansion, and migration all play vital roles in genetic drift and diversity.
Etheridge started her first lecture by reviewing some of the fundamental nonspatial models of evolution, namely the WrightFisher model and its diffusion limit. She quickly turned, though, to consider the effect of spatial structure through the following type of model: In each generation individuals give birth to a random, mean 1, number of offspring whose locations are randomly distributed in some region around the parent’s location. This dynamic leads to interesting effects such as spatial population clumping. Further refining this type of model lead Etheridge (with her collaborators N. Barton and A. Veber) to introduce a class of spatial LambdaFlemingViot processes and associated coalescents in which selection occurs in an evolving population according to spatially influenced rules. The natural question in the context of this model is to understand on what spatial scales can we expect to observe a signature of natural selection. Etheridge provided an answer to this question in certain scaling limits.
Whereas the models in the first lecture dealt with stable population sizes, Etheridge’s second lecture opened the horizon to the potential impact of range expansion and population growth. Such an expanding population can be modeled in some cases through spatial branching random walks in which the mean number of offspring exceeds one in regions where the population density is small. The population density roughly follows the wellstudied FisherKPP equation which has traveling wave solutions. On top of this expansion, Etheridge imposed a model for selection between different allelic types. Prevalence of mutations over time in the population are affected by two main types of events: (1) population waves and (2) genetic waves. Population waves lead to founder effects whereby deleterious mutations become common in a region based on the fact that they were present in the first individuals to arrive there. Over time, individuals at the front of the expansion become less and less fit, and those from the bulk outcompete them, forming an inner genetic wave. The balance of these two effects was explored in this lecture through numerical simulations. Figure 1 starkly illustrates how even without additional mutation and selection range expansion can lead to segregation of different types of genetic populations in Pseudomonas aeruginosa.
