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Abstract: In this paper, the dynamical behaviors of a class of discrete-time SIRS epidemic models are discussed. The conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. The numerical simulations not only illustrate the validity of our results, but also exhibit more complex dynamical behaviors, such as flip bifurcation, Hopf bifurcation and chaos phenomenon. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models. [Copyright &y& Elsevier]

Abstract: The paper studies the dynamical behaviors of a discrete predator–prey system with nonmonotonic functional response. The local stability of equilibria of the model is obtained. The model undergoes flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model, such as the period-doubling bifurcation in periods 2, 4 and 8, and quasi-periodic orbits and chaotic sets. The most interesting aspect is choosing the same parameters and the initial value of the model; then we vary the parameter , and obtain series bifurcations, such as flip bifurcation and Hopf bifurcation. [Copyright &y& Elsevier]