Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response

We consider a predator-prey system of Leslie type with generalized Holling type III functional response p(x)=mx2ax2+bx+1. By allowing b to be negative (b>-2a), p(x) is monotonic for b>. 0 and nonmonotonic for b<. 0 when x≥. 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov-Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov-Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>-2a are much more complex and far richer than the case when b>. 0 but also provide new bifurcation phenomena for predator-prey systems.