The role of boundary constraints in simulating a nonlocal Gray-Scott model

Abstract

We present a second-order algorithm for approximating solutions to nonlocal diffusive processes in reaction-diffusion equations. The numerical scheme relies on a quadrature method for the spatial discretization and a second-order Adams-Bashford method for the time marching. This algorithm is then used to simulate a nonlocal Gray-Scott model, known for generating interesting structures including periodic patterns, traveling waves, pulse and multi-pulse solutions. Our main goal is to study the impact of boundary constraints on the formation of stationary pulse solutions. We consider nonlocal Dirichlet and Neumann boundary constraints, as well as what we refer to as `free' boundary conditions. In addition, we investigate the effects of using different convolution kernels, fat- or thin-tailed, on the formation of these localized solutions. Our numerical results show that when the spread of the kernel is large, i.e. when the model is nonlocal, both the type of kernel and the type of boundary constraint used have a strong impact on the solution's profile.