Mathematical study of two-variable systems with adaptive numerical methods

In this paper, we consider reaction-diffusion systems arising from two-component predator-prey models with Smith growth functional response. The mathematical approach used here is twofold, since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with a fourth-order difference operator, advance the resulting nonlinear system of ordinary differential equations with a family of two competing exponential time differencing (ETD) schemes, and analyze them for stability. A numerical comparison of these two methods for solving various predator-prey population models with functional responses is also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.