Mathematics Colloquia and Seminars

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Description

patially periodic patterns are observed in a multitude of applications, including semi-arid vegetation, mussel beds, during multicellular development, and in reactions of chemical species. Importantly, the existence and stability of these patterns can have profound consequences for the system. For example, the arrangement into spatial patterns has been shown to be a productive strategy that can delay the extinction of vegetation and other populations experiencing harsh conditions. A core ingredient necessary for patterns to arise is the existence of processes that promote activation and inhibition over multiple spatial scales. The formation and persistence of patterns have traditionally been studied in nonlinear reaction diffusion equations, and more recently in models with nonlocal interactions as well. The addition of nonlocal terms introduces additional spatial scales into the model and therefore has interesting implications for pattern formation. In this talk, we will apply analytical and numerical ideas to determine characteristics of pattern onset and the behavior of patterns far from onset in a nonlocal reaction diffusion model that contains competition and facilitation terms. Of particular interest is how the choice of competition and facilitation kernels contributes to differences in pattern formation and persistence.