The bifurcation analysis of a generalized
predator-prey model depending on all parameters is carried out in
this paper. The model, which was first proposed by Hanski et al.
, has a degenerate saddle of codimension 2 for some
parameter values, and a Bogdanov-Takens singularity (focus case)
of codimension 3 for some other parameter values. By using normal
form theory, we also show that saddle bifurcation of codimension 2
and Bogdanov-Takens bifurcation of codimension 3 (focus case)
occur as the parameter values change in a small neighborhood of
the appropriate parameter values, respectively. Moreover, we
provide some numerical simulations using XPPAUT to show that the
model has two limit cycles for some parameter values, has one
limit cycle which contains three positive equilibria inside for
some other parameter values, and has three positive equilibria but
no limit cycles for other parameter values.