Simple efficient simulation of the complex dynamics of some nonlinear hyperbolic predator–prey models with spatial diffusion

Abstract

In this manuscript, we will consider a parabolic partial differential equation that models the spatial-temporal dynamics in predator–prey systems with Michaelis–Menten-type functional response. We study numerically an extension of those systems by employing finite differences. The extension includes the presence of inertial times with extended reaction components in a diffusive system consisting of two populations. The spatial domain is a closed and bounded rectangle, and initial-boundary conditions are imposed on the model. An easy-to-implement fully discrete finite-difference methodology is designed here. We establish analytically the existence and uniqueness of discrete solutions as a consequence of the explicit character of the scheme, along with the second-order consistency. To prove the stability and the second-order convergence of the technique, the energy method will be applied conveniently in a discrete manner. From the point of view of the applications, we provide various numerical simulations that show the appearance of complex patterns in the parabolic case. The results will resemble qualitatively those of some papers already published in the specialized literature. Our simulations also show that different patterns are present in the hyperbolic scenario. We also perform a numerical analysis of the convergence of the method.