This study examines the complexities of a discrete-time predator–prey model by integrating the impact of prey refuge, with the goal of providing a more realistic understanding of predator–prey interactions. We explore the existence and stability of fixed points within the model, offering a thorough examination of these critical aspects. Furthermore, we use center manifold and bifurcation theory to thoroughly analyze the presence and direction of period-doubling and Neimark–Sacker bifurcations. We also provide numerical simulations to validate our theoretical findings and demonstrate the intricacy of the model. The findings suggest that the inclusion of prey refuge has a notable stabilizing impact on the predator–prey model, hence enhancing the overall stability and resilience of the ecosystem.

In this paper, the dynamics of a discrete-time chemostat model were investigated. The discretization was obtained using the piecewise constant argument method. An analysis was performed to determine the existence and stability of fixed points. In addition, we have shown that the model experiences transcritical and period-doubling bifurcations. Two chaos control techniques, feedback control and hybrid control, were employed to control bifurcation and chaos in the model. Moreover, we provided numerical simulations to substantiate our theoretical results. This study illustrates that the piecewise constant argument method is more dynamically consistent than the forward Euler method.

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: $ \begin{equation} S_{n+1} = \dfrac{T_{n} S_{n-2}}{S_{n-2} + T_{n-1}},\quad T_{n+1} = \dfrac{S_{n} T_{n-2}}{\pm T_{n-2} \pm S_{n-1}},\quad \;\;\;\; n = 0,1,2,...., \quad\quad\quad(0.1)\end{equation} $ where the initial conditions $ s_{-2}, s_{-1}, s_0, t_{-2}, t_{-1}, t_0 $ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.