Bifurcations and chaos of a delayed ecological model

In this paper, a complicated delayed ecological model is studied in detail. First, the parameter plane is divided into three different regions, in which there exist various fixed points of the model. Second, by using the center manifold theorem, it is shown that there exists the transcritical bifurcation rather than the saddle-node bifurcation with respect to the zero fixed point of the model. For the non-zero fixed points of the model, their stabilities and bifurcations are studied when some parameters are varied. Third, the numerical simulation results not only show the consistence with the theoretical analysis but also display some new interesting dynamical behaviors and various bifurcations, including periodic invariant n-cycles circles (n = 6, 7, 8, 16, 22), continuous closed invariant circles, transcritical bifurcations, Naimark–Sacker bifurcations and inverse Naimark–Sacker bifurcations. In particular, some false phenomena, such as a 3-cycle and many 4-cycles, obtained only from the bifurcation diagram are revised by our qualitative analysis. Finally, the computations of the maximum Lyapunov exponents confirm that there exist chaotic motions in this model.