Stochastic Spatial Models

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We study random processes distributed over space which evolve in time. Examples include populations undergoing random reproduction and migration, diseases spreading through a population, species competing for resources, or oil flowing through a porous medium (percolation). Although the local rules of evolution can appear complicated, often it is easier to analyze the process at large space and time scales where these processes can exhibit universal behaviour. Much of my research is to find, justify and analyze a mathematically precise picture of this large scale evolution. Certain epidemic models turn out to be mathematically equivalent to certain kinds of percolation. Both models have a critical parameter (infection probability or percolation probability) above which the epidemic can survive in the long term, or oil will percolate. One objective is to find how this critical parameter can behave in a certain regime, corresponding say to a disease which can perform long range infection. For competing species our results can sometimes tell us when can they co-exist in equilibrium or when one type can drive the other out. The general mathematical methods designed to handle this question also tell us how cooperative behaviour can evolve out of simple Darwinian selection--a central problem in modern evolutionary theory. From a mathematical perspective it turns out that these questions can be harder in a 2-dimensional environment (which is most relevant) than in higher dimensions and we will try to address this question in this challenging and crucial setting. For the evolution of cooperation we will consider multi-type models where the coexistence of many types (clearly present in nature) should be possible. We also study the limiting models that arise from viewing these stochastic spatial models in the large scale. A central question is whether or not these processes capture all the underlying information or if there are hidden variables which require a richer structure. For the continuum epidemic models, we believe the disease will survive in stochastic waves and we will study the qualitative and quantitative nature of these waves. A final objective will be to tighten the connection between many of the original spatial models and their scaling continuum limits. For some models we know that if we take snapshots at fixed times then the probabilities of the rescaled spatial models will converge to those of a certain well-known universal limit called super-Brownian motion. We want to show that the entire time evolution of the rescaled models converge to that of this ubiquitous limit.