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Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response. (English) Zbl 1239.39011
Summary: 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 ${F}_{1},{F}_{2}$ and ${F}_{3}$ and collect complete information on this in a single scheme. In the case of ${F}_{2}$ 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.
##### MSC:
 39A28 Bifurcation theory (difference equations) 92D25 Population dynamics (general)
##### Keywords:
iterated map; normal form; hyperbolic response; stable cycle
MATCONT