Dynamics in a discrete time model of logistic type

In this paper, we investigate the qualitative properties and bifurcations of a discrete-time logistic type model for the competitive interaction of two species. Applying polynomial symbolic algebraic theory to deal with complex high-order semi-algebraic systems, and using the bifurcation theory, we give not only the topological structure of the orbits near each fixed point but also the parameter conditions such that the model produces transcritical bifurcation, supercritical (or subcritical) flip bifurcation and supercritical (or subcritical) Neimark-Sacker bifurcation, respectively. Besides, the corresponding mapping is proven to be chaotic in the sense of Marotto. At last, we simulate the stable orbits of period 2 produced from the supercritical flip bifurcation, the stable invariant circle resulting from the Neimark-Sacker bifurcation and the chaos in the sense of Marotto to verify our results.