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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 , 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.
The focus of this dissertation is a class of random processes known as interacting measure-valued stochastic processes. These processes are related to another class of stochastic processes known as superprocesses. Both superprocesses and interacting measure-valued stochastic processes arise naturally from branching particle systems as scaling limits. A branching particle system is a collection of particles that propagate randomly through space, and that upon death give birth to a random number of particles (children). Therefore when the populations of the particle system and branching rate are large one can often use a superprocess to approximate it and carry out calculations that would be very difficult otherwise.There are many branching particle systems which do not satisfy the strong independence assumptions underlying superprocesses and thus are more difficult to study mathematically. This dissertation attempts to address two measure-valued processes with different types of dependencies (interactions) that the associated particles exhibit. In both cases, the method used to carry out this work is called Perkins' historical stochastic calculus, and has never before been used to investigate interacting measure-valued processes of these types. That is, we construct the measure-valued stochastic process associated with an interacting branching particle system directly without taking a scaling limit.The first type of interaction we consider is when all particles share a common chaotic drift from being immersed in the same medium, as well as having other types of individual interactions. The second interaction involves particles that attract to or repel from the center of mass of the entire population. For measure-valued processes with this latter interaction, we study the long-term behavior of the process and show that it displays some types of equilibria.
No abstract available.