We consider a predator–prey system of Leslie type with generalized Holling typeIII functional response p(x) =mx2ax2+bx+1.By allowing bto be negative (b ?2?a), p(x)is monotonic for b 0and nonmono-tonic for b 0when 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 codimension2) for some values of parameters and a degenerate Bogdanov–Takens singularity (focus or center case) of codimension3for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension2, we show that the model ex-hibits 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)astable limit cycle enclosing two non-hyperbolic positive equilibria; (ii)astable 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 codimension3, we prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension3. These results not only demonstrate that the dynamics of this model when b ?2?aare much more com-plex and far richer than the case when b 0but also provide new bifurcation phenomena for predator–prey systems.