Dynamics and Control of a Discrete Predator–Prey Model with Prey Refuge: Holling Type I Functional Response

In this study, we examine the dynamics of a discrete-time predator–prey system with prey refuge. We discuss the stability prerequisite for effective fixed points. The existence criteria for period-doubling (PD) bifurcation and Neimark–Sacker (N–S) bifurcation are derived from the center manifold theorem and bifurcation theory. Examples of numerical simulations that demonstrate the validity of theoretical analysis, as well as complex dynamical behaviors and biological processes, include bifurcation diagrams, maximal Lyapunov exponents, fractal dimensions (FDs), and phase portraits, respectively. From a biological perspective, this suggests that the system can be stabilized into a locally stable coexistence by the tiny integral step size. However, the system might become unstable because of the large integral step size, resulting in richer and more complex dynamics. It has been discovered that the parameter values have a substantial impact on the dynamic behavior of the discrete prey–predator model. Finally, to control the chaotic trajectories that arise in the system, we employ a feedback control technique.