A nonstandard Finite Difference Approach Preserving Dynamic Consistency For Predator–Prey Systems

Abstract

We develop a nonstandard finite–difference (NSFD) approach for predator–prey systems that preserves dynamic consistency with their continuous counterparts. Focusing on models with a Holling type-III functional response, the discrete scheme is constructed so that fundamental qualitative features are retained for any time step: solutions remain positive and uniformly bounded, and population persistence is ensured. We characterize all equilibria of the continuous model and of the induced discrete map, and we prove that the stability type is inherited by the discretization. In particular, an interior equilibrium of the differential system may generate a Hopf bifurcation, while the corresponding fixed point of the NSFD map exhibits a Neimark–Sacker bifurcation under analogous parameter conditions. Numerical experiments corroborate the analysis and illustrate the robustness of the proposed method compared with standard schemes.