Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Abstract

Existing literature established stability in a delayed logistic Lotka–Volterra predator–prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size $h=\frac{1}{m},m\in Z^{+}$. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay $\tau -$sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.