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Traveling Waves in Models of Population Dynamics with Nonlocal Interactions

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This thesis focuses on ecological models of population dynamics and the traveling, migratory waves that can result when a stable state either displaces an unstable state, or displaces another stable state. We consider the effect of nonlocal interactions, where members of the species interact over a distance. This gives rise to integro-partial differential equations of reaction-diffusion type. Our work has encompassed two projects. The first considers a single species competing nonlocally with itself, while the second considers a food chain system of three species. In both, we consider traveling waves and determine how nonlocality can affect the speed of propagation, the stability of the equilibria, and the shape of the fronts. In our first project, we developed a piecewise linear, reaction-diffusion model describing the growth and movement of a single species, u, so that when we considered a particular form of nonlocality, we were able to reduce the integro-PDE to a system of algebraic equations. This allowed a full description of the traveling wave solutions. We also considered the effects of asymmetric nonlocality, where the distance over which the nonlocal interactions occurred was different to the left and to the right of a given location. We were able to show how the extent of the nonlocality and the strength of the asymmetry affected the speed of propagation of the traveling fronts, and how they could cause a loss of monotonicity in the solutions. Finally, we considered cases where the waves could propagate in either direction. In our second project, we considered a three species food chain model, where species u was preyed upon by species v, which in turn was preyed upon by species w. Our primary focus was on biological control, where the bottom species u is an important crop, while v is a pest that has infested the crop. The superpredator w is introduced into this pest-infested environment in an attempt to restore the system to a pest-free state. For this model, we considered two types of nonlocality: one where the crop species u competes nonlocally with itself, and the other where the superpredator w is assumed to be highly mobile and therefore preys upon the pest v in a nonlocal fashion. In this context, we examined how biological control could prove to be highly susceptible to noise, and could fail outright if the pest species was highly diffusive. We showed, however, that control could be restored if the superpredator was sufficiently diffusive, and that the control could be made robust if the superpredator behaved nonlocally. Since our focus was on biological control, where the superpredator is generally introduced artificially, our results point to properties of the superpredator which can lead to successful control.

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  • 11/06/2019
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