  This paper focus on the diffusive eco-epidemiological prey-predator model
with infectious diseases in prey, and with the homogeneous Neumann and
Dirichlet boundary conditions, respectively. When boundary conditions are
homogeneous Neumann boundary conditions, we give a complete conclusion about
the stabilities of nonnegative constant equilibrium solutions. The results show
that such a problem has neither periodic solutions nor Turing patterns. When
boundary conditions are homogeneous Dirichlet boundary conditions, we first
establish the necessary and sufficient conditions for the existence of positive
equilibrium solutions, and prove that the positive equilibrium solution is
unique when it exists. Then we study the global asymptotic stabilities of
trivial and semi-trivial nonnegative equilibrium solutions.
