Evolution in the Face of Spatial Heterogeneity

Abstract

Populations experience selection pressures which are heterogeneous over space, yet when modelling evolution there are often assumptions employed in the literature which act to reduce the impact of such spatial heterogeneities. This is especially prevalent in cases where resistance evolution is modelled, such as in the agricultural context, where control measures such as pesticides induce a selection pressure. In this thesis I address three examples of models of evolution and demonstrate that incorporating spatial heterogeneity can generate novel dynamics.

In the first project, we study a stochastic, lattice-based growth model of a population consisting of two sub-populations: a `wild-type' sub-population and a slower-spreading `mutant' sub-population. The population undergoes a range expansion into uninhabited territory, in which there are circular patches which may only be invaded by the mutant sub-population. For each combination of mutant spreading speed and patch proportion over space, we measure the probability of whether an isolated mutant sub-population will dominate the population frontier, and demonstrate that this can be predicted analytically by employing statistical and geometrical arguments. These results can act as a starting point for determining an optimal pesticide distribution which minimises the rate at which resistance emerges in the face of an expanding population frontier.

In the second project, we employ a 1D reaction-diffusion model of organism dispersal in the presence of a spatio-temporally heterogeneous control measure. We consider a population consisting of a susceptible sub-population that dies in the presence of the control measure and a resistant sub-population that is not affected by the control measure. We demonstrate that a fixed amount of control measure can be distributed in such a way as to minimise the initial rate at which the resistant sub-population rises in frequency, subject to various spatial constraints. This will be useful in the agricultural context, where such spatial tuning will prolong the effective lifetime of a pesticide.

In the third project, we study a stochastic, graph-based model in which two sub-populations compete. The susceptible sub-population spreads more easily, but only the resistant sub-population may occupy `treated' regions in which the control measure is applied. Therefore, the treated regions can be interpreted as `zealots' which act as a constant source of the resistant sub-population. We investigate how the topology of the graph, the relative spreading strength of the resistant sub-population and the distribution of the control measure interact to inform the long-time proportion of the resistant sub-population over the graph. Analytical results can be determined in the regime where most connected components of the graph are small or when the mean degree of nodes is large. These results will be a useful starting point to help tune the distribution of pesticides applied on the geographical scale to reduce the rate at which resistance emerges.