In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter
m
m
in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when
m
m
is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large
m
m
reveal interesting contrast with that for the well-studied case
m
=
0
m=0
, i.e., the classical Lotka-Volterra predator-prey model.