Spatial Ecology & Singularly Perturbed Reaction-Diffusion Equations

Prof. Arjen Doelman, Mathematisch Instituut, Lorentz Center; Leiden University, The Netherlands

Abstract: Pattern formation in ecological systems is driven by counteracting feedback mechanisms on widely different spatial scales. Moreover, ecosystem models typically have the nature of reaction-diffusion systems: the dynamics of ecological patterns can be studied by the methods (geometric) singular perturbation theory. In this talk we give an overview of the surprisingly rich cross-fertilization between ecology, the physics of pattern formation and the mathematics of singular perturbations. We show how a mathematical approach uncovers mechanisms by which real-life ecosystems may evade (catastrophic) tipping under slowly varying climatological circumstances. This insight is based on two crucial ingredients: the careful study of Busse balloons in (parameter, wavenumber)-space associated to spatially periodic patterns and the validation of the model predictions by field observations. Vice versa, ecosystem models motivate the study of classes of singularly perturbed reaction-diffusion equations that exhibit much more complex behavior than the models so far studied by mathematicians: we present several novel research directions initiated by ecology.


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