Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay

Abstract

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. Firstly, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of coexistence equilibrium through Turing/Turing-Turing bifurcations via describing the first Turing bifurcation curve, and theoretically predict possible bi-stable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that coexistence equilibrium can also be destabilized through Hopf, Hopf-Hopf, Turing-Hopf bifurcations, and possible stable/bi-stable spatially inhomogeneous staggered periodic patterns, bi-stable spatially inhomogeneous synchronous periodic patterns, are theoretically predicted. Finally, numerical experiments also support theoretical predictions and partially extend them. In a word, theoretical analyses indicate that, on the one hand, large predator-taxis can eliminate spatial patterns caused by self-diffusion; on the other hand, the joint effects of predator-taxis and conversion time delay can induce complex survival patterns, e.g., bi-stable spatially heterogeneous staggered/synchronous periodic patterns, thus diversify populations’ survival patterns.