Bifurcations In A Predator-prey System Of Leslie Type With Holling Type Ⅳ Functional Response

We consider a predator-prey system of Leslie type with Holling type IV func-tional response p(x)= mx/ax2+bx+1· By allowing b to be negative(b> - 2(?)a), p(x) is nonmonotonic when x≥ 0. The model has two non-hyperbolic positive equilibri-a(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 simul-taneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer nu-merical simulations which demonstrate that the model can have:(ⅰ)a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ⅱ)one stable limit cycle en-closing three hyperbolic positive equilibria; or(ⅲ)system has two limit cycles,one unstable limit cycle enclosing a positive equilibrium, one stable limit cycle enclos-ing all the positive equilibria point and the unstable limit cycle. When b= O we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3.These results not only complete the bifurcation analysis of Li and Xiao [16], but also provide new bifurcation phgnomena for predator-prey systems.