Bifurcation Analysis In A Host-generalist Parasitoid Model With Holling ? Functional Response

In this paper,we study a host-generalist paxasitoids model with Holling-? func-tional response,where the generalist paxasitoid are introduced to control the invasion of the hosts.It is shown that there are a nilpotent cusp or focus or elliptic-type equi-librium of codimension three and a weak focus of multiplicity at most two for various parameter values,and the model undergoes a cusp type Bogdanov-Takens bifurcation of codimension three,focus and elliptic types degenerate Bogdanov-Takens bifurca-tion of codimension three,a Hopf bifurcation,and a degenerate Hopf bifurcation of codimension two as the parameters vary.Moreover,there exists a critical value for the carrying capacity of generalist parasitoids such that:(i)when the carrying capacity of the generalist paxasitoids is smaller than the critical value,the invasion hosts can always persist in spite of the predation by the generalist parasitoids,i.e.,the generalist parasitoids cannot control the invasion of the invading hosts;(ii)when the carrying capacity for the generalist parasitoids is larger than the critical value,the invasion hosts can tend to extinction,or persist in the form of multiple coexistent steady states or multiple coexistent periodic oscillations depending on the initial host populations,i.e.,whether the invasion can be stopped and reversed by teh generalist parasitoids depends on the initial host populations;(iii)In both cases,the general-ist parasitoids always persit.Numerical simulations are presented to illustrate the theoretical results.