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A non-standard numerical scheme for a generalized Gause-type predator–prey model. (English) Zbl 1043.92040

Summary: A non-standard finite-difference scheme is constructed to simulate a predator–prey model of Gause-type with a functional response. Using fixed-point analysis, it is shown that the scheme preserves the physical properties of the model and gives results that are qualitatively equivalent to the real dynamics of the model. It is also shown that the scheme undergoes a supercritical Hopf bifurcation for a specific value of the bifurcation parameter (${k}_{0}$). This leads to the existence of a stable limit cycle created by the scheme when the bifurcation parameter passes through ${k}_{0}$, as predicted by the continuous model.

The scheme is used to simulate a model with functional responses of Holling-types II and III. The simulation results generated by the non-standard finite-difference scheme are compared with those obtained from the standard methods such as forward-Euler and Runge–Kutta methods. These comparisons show that the standard methods give erroneous results that disagree with the theoretical predictions of the model. However, it is proved that the proposed non-standard finite-difference scheme is consistent with the asymptotic dynamics of the model. Numerical simulations are presented to support these facts.

##### MSC:
 92D40 Ecology 65L12 Finite difference methods for ODE (numerical methods) 34C60 Qualitative investigation and simulation of models (ODE) 92D25 Population dynamics (general) 34C23 Bifurcation (ODE) 37N25 Dynamical systems in biology 65L05 Initial value problems for ODE (numerical methods)