Stability, chaos and codimension-one bifurcation of a discrete prey-predator model with prey refuge and Holling type-II functional response

In this paper, we analyze the complex dynamics of a discrete prey-predator model with prey refuge and Holling type-II functional response. Our focus is on investigating the stability, chaotic behavior, and codimension-one bifurcation properties of the model. We begin by establishing the nonnegativity conditions required for the model's biological relevance, followed by the derivation of equilibrium solutions. Specifically, we show that the model always admits trivial and semitrivial equilibria, and an interior equilibrium exists under specific parametric conditions. Next, we apply the linear stability theory to examine the local dynamics at these biologically meaningful equilibria. This analysis reveals the conditions under which the equilibria are stable or unstable, providing a clear understanding of the system's behavior at these points. Furthermore, we conduct a comprehensive bifurcation analysis, identifying the codimension-one bifurcation sets and exploring the occurrence of bifurcations as system's parameter, namely the bifurcation parameter, varies. Our results highlight the key parameters that drive transitions in system behavior, leading to phenomena such as stability loss, periodic solutions, and chaotic dynamics. We also address the chaotic behavior observed in the model by implementing control strategies, including the OGY (Ott-Grebogi-Yorke) method and a hybrid control approach. These methods effectively mitigate chaos, restoring stability to the system under certain conditions. Importantly, each theoretical result is supplemented with a biological interpretation, elucidating the implications of our findings in the context of ecological systems. Finally, we illustrate our theoretical results with numerical simulations, demonstrating the model's complex dynamics, including bifurcations, maximum Lypunov function, phase portraits, and chaotic regimes. These simulations not only validate our analytical results but also provide deeper insights into the rich dynamics exhibited by discrete prey-predator interactions with Holling type-II functional response and prey refuge.