Erin Ellefsen, Department of Applied Mathematics, University of Colorado Boulder
Nonlocal Models in Ecology
Nonlocal models can be very useful to describe some phenomena in Ecology. However, they also pose both analytical and computational challenges. We investigate territory development of meerkats by studying a system of nonlocal continuum equations. We perform a long-wave approximation of this system to investigate a local approximation, and take advantage of the structure of the local and nonlocal systems in order to find steady state solutions. We compare these steady states to determine if the local approximation is an appropriate approximation for future work. We also study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species. A key finding is that in some cases a population, which is initially above the Allee threshold in some area, will actually survive.