This thesis concerns three models of deterministic and stochastic population invasions, starting from individual-level interactions and deducing population level behaviour. Firstly, we model a bacteria population near obstacles using the 2D FisherKolmogorov-Petrovskii-Piscounov (FKPP) equation with mixed boundary conditions along a corridor and in the half-plane. For a deterministic population, we calculate the smallest corridor width required for survival, the angle the population level sets make with the boundaries, and the population speed. As the hostility of the mixed boundaries increases, the condition for collapse behind the front is achieved before the condition to achieve speed zero ahead of the front. Secondly, we model an invasive ﬁsh population using the 1D FKPP equation and explore the eﬀect that sexual conﬂict between individuals has on the diffusion rate, and hence the invasion speed, of the population. After introducing a stochastic model for the microscopic movement, we demonstrate how sexual conﬂict can increase the eﬀective diﬀusion rate of a pair of individuals by determining the mean speed, separation, and time required for a direction change. In large populations, sexual conﬂict can increase the diﬀusion rate ahead of the front, where the speed of the invasion is determined. Finally, we model the spread of an opinion using the voter model with nonlocal interaction and diﬀusion. Individuals can either persuade others who are close by very strongly or persuade others who are far away very weakly. In low density populations, we determine the probability of either individual persuading the other when two diﬀerent individuals meet in a pair. In a high density population, a small noise expansion determines whether the proportion of either type in the population increases or decreases on average. In both regimes, we ﬁnd that wide and weakly persuading individuals have an advantage.
|Date of Award||21 Feb 2018|
|Supervisor||Tim Rogers (Supervisor) & Jonathan Dawes (Supervisor)|