In this paper, we consider a discrete-time SIR epidemic model. Codimension-two bifurcations associated with 1:2, 1:3 and 1:4 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Numerical simulations are carried out to verify and illustrate these theoretical results. More precisely, two kinds of high-resolution stability phase diagrams are exhibited to describe how the system's complexity unfolds with control parameters varying.

We study the dynamical properties of a discrete-time host-parasitoid model with Holling type III functional response. It is shown that flip bifurcation and Neimark-Sacker bifurcation occur in certain parameter regimes. A sufficient condition based on the model parameters for which both populations can coexist is derived. The boundedness, existence and local stability of the unique equilibrium are proved. In addition, the numerical simulations have been done, in addition to supporting the analytical findings, more behaviors are extracted from the model in a two-dimensional parameter space. Finally, we emphasize the importance of clearly presenting biological assumptions that are inherent to the structure of a discrete model.