Multifarious bifurcations in a discrete Gause type predator-prey model with Allee effect and Holling type-III functional response

By using the semi-discretization method, this paper discretizes a Gause type predator-prey model with the combined outcome of the Allee effect and Holling type-III functional response, and studies the complex dynamical properties of the discrete system. Besides considering the existence and local stability of its fixed points, one is mainly devoted to deriving the sufficient conditions for the occurrences of multifarious bifurcations, such as the transcritical bifurcation, period-doubling bifurcation and Neimark-Sacker bifurcation, etc, by employing the Center Manifold Theorem and bifurcation theory. The results obtained not only clearly show that the direction and stability of the bifurcated closed orbits, but also complement the corresponding ones in a known literature. Furthermore, numerical simulations are given to confirm the existence of period-doubling bifurcation and Neimark-Sacker bifurcation. Especially, the corresponding biological meanings are also formulated.