Bifurcation Analysis of a Predator-Prey Model with Alternative Prey and Prey Refuges

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov-Takens bifurcation of a predator-prey model with alternative prey and prey refuges, which was proposed by Chen et al. [2023]. The results show that the predator-prey model can undergo a supercritical Hopf bifurcation or a Bogdanov-Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator-prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov-Takens bifurcation is at most two.