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.