Diffusive predator-prey model with Beddington-DeAngelis functional response: theoretical analysis and numerical results

Abstract

In this work, we consider a one-dimensional diffusive predator–prey model with a Beddington–DeAngelis functional response, formulated as a system of partial differential equations with appropriate initial and boundary conditions. An equilibrium analysis is carried out to determine the conditions for a unique coexisting equilibrium. The local stability of the equilibrium point ( u * , v * ) is then examined, and it is shown to be locally asymptotically stable under certain criteria. Conditions for the onset of Turing instability and Hopf bifurcation are also obtained. The model is solved numerically using the Forward Time Central Space (FTCS) scheme and the corresponding results are presented for five different cases. We present some profiles for the prey and predator densities when we have: Turing instability, Hopf bifurcation and neither of them. As there is no known exact solution, numerical rate of convergence in time is computed and compared with the theoretical rate of convergence to verify the accuracy and reliability of the numerical schemes.