Roughening interfaces in spatial population dynamics

The spatial structure and geometry of biological systems can have a strong effect on that system’s evolutionary dynamics. In particular, spatially structured populations may invade one another, giving rise to invasion fronts that may exhibit qualitatively different evolutionary dynamics in different dimensions or geometric configurations. For examples of invasion fronts arising in nature, one might think of a thin layer of bacteria cells growing on a Petri dish, an animal species expanding into new territory, or a cancerous tumor growing into and competing with the surrounding healthy tissue. Perhaps the most well-studied class of invasion fronts in population genetics is the Fisher wave, which was developed to explain how an advantageous gene sweeps throughout a population.

In this thesis, I will focus on the study of invasion fronts which develop an enhanced roughness due to internal dynamics of the invading population; namely, I make use of simple lattice and analytic models to explore how the interface between an unstable, mutating population and a healthy bystander population develops an enhanced roughness as the mutating population approaches a population collapse via mutational meltdown. I discuss the differences in this roughening behavior for populations in different dimensions and geometries and show that universal aspects of the roughness may be captured by studying how the characteristic size of the interface changes with time.