Nonlocal and Anisotropic Partial Differential Equations in Mathematical Biology

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In this research I consider nonlocal and anisotropic differential equations that arise as mathematical models for biological processes. These models are generalizations of well known reaction-advection-diffusion models and a rich qualitative theory is waiting to be explored. I expect to find new forms of pattern formation that include singular solutions, global patterns and accelerated invasion fronts. I will use these versatile models for several specific biological applications. For example, I use the models to analyse how wolves use linear features in the forest environment to change their hunting strategies, and how we can understand the impact of this change on wolf-ungulate dynamics. Another example involves sea turtles, who use an internal compass to navigate long distances across the ocean to find their breeding beaches. Mathematical models help to understand their navigational abilities by quantifying directional cues (magnetic, chemotactic), and they are used to develop protection strategies. The nonlocal models also apply to forest fire spread, and they are used to estimate the probability that a fire breaches an obstacle to enter human settlements. Finally, another application of these class of models lies in cancer research. While cancer research is not directly part of this NSERC grant, results and methods developed here will impact cancer modelling and provide a framework to develop better treatment strategies. Nonlocal and anisotropic continuum models for spatial movement obtain their beauty from a rich mathematical theory and wide reaching applications.