Stability, bifurcation analysis and chaos control in a discrete predator-prey system incorporating prey immigration

In this paper, we explore the complex dynamical behavior of a discrete predator-prey system incorporating the prey immigration effect, which is transformed from a continuous model to a discrete system by utilizing nonstandard finite difference scheme. We analyze the stability conditions to better understand the behavior of the system when we include or exclude the immigration effect in the discrete system. Furthermore, we demonstrate that the discrete system undergoes supercritical Neimark-Sacker bifurcation when the bifurcation parameter passes through a critical value. We also study the state feedback chaos control strategy for the discrete system and we obtain the triangular region restricted by the lines that contain stable eigenvalues. Moreover, we illustrate phase portraits, maximum Lyapunov exponents, and bifurcation diagrams for the discrete system. We present the numerical simulations to validate the theoretical findings. Finally, with the advantage of the nonstandard finite difference discretization method, we eliminate the flip bifurcation that occurs when Euler discretization is used.