  The role of predator evasion mediated by chemical signaling is studied in a
diffusive prey-predator model when prey-taxis is taken into account (model A)
or not (model B) with taxis strength coefficients $\chi$ and $\xi$
respectively. In the kinetic part of the models it is assumed that the rate of
prey consumption includes functional responses of Holling, Bedington-DeAngelis
or Crowley-Martin. Existence of global-in-time classical solutions to model A
is proved in space dimension n=1 while to model B for any $n\geq 1$. The
Crowley-Martin response combined with bounded rate of signal production
precludes blow-up of solution in model A for $n\leq 3$. Local and global
stability of a constant coexistence steady state which is stable for ODE and
purely diffusive model are studied along with mechanism of Hopf bifurcation for
Model B when $\chi$ exceeds some critical value. In model A it is shown that
prey taxis may destabilize the coexistence steady state provided $\chi$ and
$\xi$ are big enough. Numerical simulation depicts emergence of complex
space-time patterns for both models and indicate existence of solutions to
model A which blow-up in finite time for $n=2$
