Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

We perform a bifurcation analysis of a discrete predator–prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F1,F2F1,F2 and F3F3 and collect complete information on this in a single scheme. In the case of F2F2 we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM. Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.