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The role of predator evasion mediated by chemical signaling is studied in adiffusive 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 ofprey consumption includes functional responses of Holling, Bedington-DeAngelisor Crowley-Martin. Existence of global-in-time classical solutions to model Ais proved in space dimension n=1 while to model B for any $n\geq 1$. TheCrowley-Martin response combined with bounded rate of signal productionprecludes blow-up of solution in model A for $n\leq 3$. Local and globalstability of a constant coexistence steady state which is stable for ODE andpurely diffusive model are studied along with mechanism of Hopf bifurcation forModel B when $\chi$ exceeds some critical value. In model A it is shown thatprey taxis may destabilize the coexistence steady state provided $\chi$ and$\xi$ are big enough. Numerical simulation depicts emergence of complexspace-time patterns for both models and indicate existence of solutions tomodel A which blow-up in finite time for $n=2$

Repulsive chemotaxis and predator evasion in predator prey models with diffusion and prey taxis