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

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, Beddington-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 >= 1. The Crowley-Martin response combined with bounded rate of signal production precludes blow-up of solution in model A for n <= 3. Local and global stability of a constant coexistence steady state which is stable for the corresponding ordinary differential equation (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 indicates existence of solutions to model A which blow-up in finite time for n = 2.