ABSTRACT:

Population and vegetation models often use nonlocal forms of dispersal to describe the spread of individuals and plants. When these long-range effects are modeled by spatially extended convolution kernels, the mathematical analysis of solutions can be simplified by posing the relevant equations on unbounded domains. However, in order to numerically validate these results, these same equations then need to be restricted to bounded sets. Thus, it becomes important to understand what effects, if any, do the different boundary constraints have on the solution. To address this question we present a quadrature method valid for convolution kernels with finite second moments. This scheme is designed to approximate at the same time the convolution operator together with the prescribed nonlocal boundary constraints, which can be Dirichlet, Neumann, or what we refer to as free boundary constraints. We then apply this scheme to study pulse solutions of an abstract 1-d nonlocal Gray-Scott model as a case study. We consider convolution kernels with exponential and with algebraic decay. We find that, surprisingly, boundary effects can be more prominent when using exponentially decaying kernels.

BIO:

Dr. Jaramillo is an Associate Professor in the Department of Mathematics at the University of Houston. She completed her PhD at the University of Minnesota in 2015, and before her current position she was an NSF postdoctoral Fellow at the University of Arizona. Her research is in the area of partial and ingetro-differential equations. She uses tools from functional analysis to develop analytical and numerical methods to study coherent structures arising in a variety of applications. Her current work focuses on understanding how nonlocal interactions affect pattern formation in biological and chemical systems.