Effects of Degenerate Bifurcations and their Applications to a Holling-Type II Predator–Prey System

Under the violated crossing condition, a degenerate Hopf bifurcation in ODE is generalized to an abstract evolution equation, and a degenerate steady-state bifurcation is established in the evolution equation as well, especially in the presence of symmetry. In the degenerate simple bifurcations, the existence, bifurcation direction and stability of bifurcating periodic solutions and steady-state solutions are carried out in detail. In order to illustrate the effectiveness of the newly developed procedure, a Holling-type II predator–prey system is taken as an example to investigate the degenerate simple bifurcations, the results of which complement and correct the existing works. Actually, the main novelty on degenerate bifurcations derived by the Lyapunov–Schmidt reduction and the singularity theory, complementing the traditional bifurcation theorems by Crandall and Rabinowitz or Hassard and Kazarinoff, leads to the usual pitchfork bifurcation is changed to parabolic or transcritical bifurcations, and the exchange of stability is different, yielding the quasi-global property for a small perturbation of system parameter.