### Evolutionary Dynamics

I'm interested in the evolution of populations that are invading or spreading into new territory.  Such populations are called range expansions.
 A range expansion of E. coli on a Petri dish. (Image courtesy of Bryan Weinstein)
When populations spread on a surface or in three-dimensional space (such as a solid tumor in healthy tissue), spatial fluctuations may play an important role in the evolutionary dynamics.  We can see in the image above that an initially well-mixed population of red, blue, and green-fluorescent E. coli segregate into single-colored sectors as the organisms spread across the surface of a Petri dish (forming a colony).This is a result of the small effective population size at the frontier of the colony, where a small fraction of the total population divides into the new territory.  Small number fluctuations, then, rapidly fix the population to a single color at the frontier.

We studied such range expansions in the presence of selection and mutation.  In the simplest instance, we consider a blue strain with a selective advantage $s$ over a yellow strain, which may convert to the yellow strain with some rate $\mu$.  In this case, depending on the mutation rate $\mu$ and the selection coefficient $s$, the blue strain may either survive in the population at long times, or go extinct.  Spatial fluctuations strongly suppress the blue strain, leading to more extinction, as shown in the phase diagram below.
 In a two-dimensional range expansion (a microbial colony growing across a Petri dish, for example), extinction is enhanced due to spatial fluctuations.  We see in this phase diagram that a blue strain with a selective advantage $s$ will go extinct faster in a range expansion for a fixed mutation rate $\mu$ than in the well-mixed case (dashed line).  Such spatial extinction transitions may be described by directed percolation, which has concrete predictions for quantities such as the opening angle $\theta$ of blue genetic sectors [1,2].
References
[1] M. O. Lavrentovich, M. E. Wahl, D. R. Nelson, and A. W. Murray Spatially constrained growth enhances conversional meltdown Biophysical Journal 110(12)  271 (2016)
[2] M. O. Lavrentovich, K. S. Korolev, and D. R. Nelson Radial Domany-Kinzel models with mutation and selection Physical Review E 87 012103 (2013)