A probabilistic approach to fractional reaction-diffusion equations

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This research project will use techniques from probability and analysis to answer questions arising in biology. In particular, the question of how populations expand their range provides us with sophisticated mathematical models to study. There are several critical deficiencies in the standard model (the so-called Fisher-KPP equation) that would provide interesting research topics. For instance, the Fisher-KPP equation does not account for certain phenomena observed in populations. An example of one such phenomenon is the "Allee effect", under which the optimal conditions for an individual may not occur when the population density is small (for example, competition for resources might decrease in areas of low population density, but other advantages, such as protection from the herd, will also be compromised). The Allee effect can impact the range expansion of a population in a number of ways. One topic that we should like to investigate is that of `expansion load'. This is the name given to the accumulation of deleterious mutations at the front of an expanding population. In the absence of an Allee effect, expansion load is attributed to the advantage that an individual enjoys from being at the front (where population density is small and growth rates are maximal) more than offsetting the disadvantage conferred by a deleterious mutation. In the presence of noise (corresponding to the randomness due to reproduction in a finite population), the fittest types can be completely lost from the expanding population front. However, under an Allee effect, population growth favours areas of higher density, and deleterious mutations at the population front no longer enjoy this advantage. In the presence of an Allee effect, the maximum growth rate of the population sits behind the expanding front. One might therefore expect fitter individuals in the population's bulk to recolonise the front. We shall investigate the accumulation of deleterious mutations in different models of expanding populations (in a spatial analogue of Muller's ratchet). Mathematically, we will account for the presence of the Allee effect in a population by incorporating a `cooperation term' into our models. Of particular interest is to understand the interplay between selection, range expansion and what is known as genetic drift (the randomness due to reproduction in a finite population). Simulations of this scenario present an intriguing picture that will help us better understand these interactions.