Construction and analysis of exponential time differencing methods for the robust simulation of ecological models

Abstract

In this thesis, we consider some interesting mathematical models arising in ecology. Our focus is on the exploration of robust numerical solvers which are applicable to models arising in mathematical ecology. To begin with, we consider a simple but nonlinear second-order time-dependent partial differential equation, namely, the Allen-Cahn equation. We discuss the construction of a class of exponential time differencing methods to solve this particular problem. This is then followed by a discussion on the extension of this approach to solve a more difficult fourth-order time-dependent partial differential equation, namely, Kuramoto-Sivashinsky equation. This equation is nonlinear. Further applications include the extension of this approach to solve a complex predator-prey system which is a system of fourth-order time-dependent non-linear partial differential equations. For each of these differential equation models, we presented numerical simulation results.