Numerical simulations of reaction–diffusion equations modeling prey–predator interaction with delay

Abstract

To model biological systems one often uses ordinary and partial differential equations. These equations can be quite good at approximating observed behavior, but they suffer from the downfall of containing many parameters, often signifying quantities which cannot be determined experimentally. For the better understanding of complicated phenomena, the delay differential equation approach to model such phenomena is becoming more and more essential to capture the rich variety of dynamics observed in natural systems. In this study, we investigated numerically the influence of delay on the dynamics of nonlinear reaction–diffusion equations modeling prey–predator interaction using finite difference scheme subject to appropriate initial and boundary conditions. We first consider the prey–predator with Holling type II functional response where the growth of prey is assumed to be logistic in the sense of predator in one-dimensional space. The effect of delay was investigated with the help of simulations and is compared with the model equation without delay. The proposed method is then extended to two-dimensional space.