Bifurcation analysis of a predator-prey model with Beddington-DeAngelis functional response and predator competition

In this paper, we consider a predator-prey model with Beddington-DeAngelis functional response and predator competition, which is a five-parameter family of planar vector field. It is shown that the model can undergo a sequence of bifurcations including focus type degenerate Bogdanov-Takens bifurcation of codimension 3 and Hopf bifurcation of codimension at least 2 as the parameters vary. Our theoretical results indicate that predator competition can cause richer dynamics such as two limit cycles enclosing one or three hyperbolic positive equilibria and three kinds of homoclinic orbits (homoclinic to hyperbolic saddle, saddle-node, or neutral saddle). Moreover, there exists a threshold value m0$$ {m}_0 $$ for predator capturing rate m$$ m $$, below or equal to which the predators always tend to extinction, above which the predators and preys will coexist in the form of multiple steady states or periodic oscillations for all positive initial populations. Numerical simulations are presented to illustrate the theoretical results.