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This paper studies a reaction–advection–diffusion prey–predator system in one spatial dimension. Adapting the Lotka–Volterra-type functional response, we prove the global existence and boundedness of solutions of the system in a bounded open interval. In view of asymptotic behavior of solutions, we show that if the predation is weak, the semi-trivial steady state at which prey only survive is globally asymptotically stable. In case of strong predation, the positive steady state is globally asymptotically stable when the predator-taxis is weak.

Global dynamics of a Lotka-Volterra type prey–predator model with diffusion and predator-taxis