PDE Methods for Nonlocal Models

We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integro-differential equations (PIDEs) defined on a spatially extended domain. Our primary focus is on scalar equations in two spatial dimensions. These models arise in a variety of neuronal modeling problems and also occur in material science. We first derive a PDE which is equivalent to the PIDE. We then find circularly symmetric solutions of the resultant PDE; the linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable, our analysis predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE. We illustrate our results with specific numerical examples and discuss other systems for which this technique can be used.