This paper discusses
a prey-predator system with cross-diffusion.
We can prove that the set of coexistence steady-states of this system
S or $\supset$-shaped branch with respect to a bifurcation parameter
in a large cross-diffusion case.
We give also some criteria on the stability of these positive
we find the Hopf bifurcation point
on the steady-state solution branch
in a certain case.