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 χ and ξ, 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 χ exceeds some critical value. In model A, it is shown that prey-taxis may destabilize the coexistence steady state provided χ and ξ 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.