Nonstandard finite difference schemes for a general predator–prey system

Abstract

In this paper, we transform a continuous-time predator–prey system with general functional response and recruitment for both species into a discrete-time model by nonstandard finite difference (NSFD) scheme. We prove theoretically and confirm by numerical simulations that the constructed NSFD schemes preserve the essential qualitative properties including positivity and stability of the continuous model for any finite step size. We also show that the standard finite difference schemes such as the Euler scheme, the second order Runge-Kutta scheme and the classical fourth order Runge-Kutta scheme cannot preserve the properties of the continuous model for large step sizes. They can generate the numerical solutions which are completely different from the solutions of the continuous model. Especially, the global stability of a non-hyperbolic equilibrium point of the constructed NSFD schemes is proved by the Lyapunov stability theorem. The performed numerical simulations confirm the validity of the obtained theoretical results as well as the advantages of NSFD schemes over standard ones.