Further investigations into the stability and bifurcation of a discrete predator-prey model

In this paper we revisit a discrete predator-prey model with a non-monotonic functional response, originally presented in Hu, Teng, and Zhang (2011) [6]. First, by citing several examples to illustrate the limitations and errors of the local stability of the equilibrium points E3 and E4 obtained in this article, we formulate an easily verified and complete discrimination criterion for the local stability of the two equilibria. Here, we present a very useful lemma, which is a corrected version of a known result, and a key tool in studying the local stability and bifurcation of an equilibrium point in a given system. We then study the stability and bifurcation for the equilibrium point E1 of this system, which has not been considered in any known literature. Unlike known results that present a large number of mathematical formulae that are not easily verified, we formulate easily verified sufficient conditions for flip bifurcation and fold bifurcation, which are explicitly expressed by the coefficient of the system. The center manifold theory and Project Method are the main tools in the analysis of bifurcations. The theoretical results obtained are further illustrated by numerical simulations.