We study the impact of the Allee effect and prey refuge on the stability of a discrete time predatorprey model. In this study, we focus on stability behavior of the system with the Allee effect in predator, prey and both populations by using center manifold theorem and type of bifurcations. Based on our analytical and numerical results, we observe that the Allee effects stabilizes the systems dynamics in a moderate value of the prey reguge.

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The discrete-time Holling type II prey-predator models with the refuge and Allee effects are formulated and studied. The existence of fixed points and their stabilities are investigated for both hyperbolic and non-hyperbolic cases. Numerical simulations are conducted to demonstrate the theoretical results.

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We examine a discrete-time host–parasitoid model incorporating simultaneously an Allee and a refuge effect on the host. We investigate existence of a positive fixed point, local asymptotic stability, global stability of the fixed points and bifurcations. Numerical examples are given for verification of the theoretical results and we compare the model with existing data from the literature.