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UBC Theses and Dissertations

Stochastic models for spatial populations Chen, Yu-Ting


This thesis is dedicated to the study of various spatial stochastic processes from theoretical biology. For finite interacting particle systems from evolutionary biology, we study two of the simple rules for the evolution of cooperation on finite graph in Ohtsuki, Hauert, Lieberman, and Nowak [Nature 441 (2006) 502-505] which were first discovered by clever, but non-rigorous, methods. We resort to the notion of voter model perturbations and give a rigorous proof, very different from the original arguments, that both of the rules of Ohtsuki et al. are valid and are sharp. Moreover, the generality of our method leads to a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs in terms of the voter model fixation probabilities. This should be of independent interest for other voter model perturbations. For spatial branching processes from population biology, we prove pathwise non-uniqueness in the stochastic partial differential equations (SPDE’s) of some one-dimensional super-Brownian motions with immigration and zero initial value. In contrast to a closely related case studied in a recent work by Mueller, Mytnik, and Perkins [30], the solutions of the present SPDE’s are assumed to be nonnegative and are unique in law. In proving possible separation of solutions, we use a novel method, called continuous decomposition, to validate natural immigrant-wise semimartingale calculations for the approximating solutions, which may be of independent interest in the study of superprocesses with immigration.

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