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

Yue Xing; Weihua Jiang; Xun Cao; Yue Xing; Weihua Jiang; Xun Cao

doi:10.3934/mbe.2023818

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18413-18444. doi: 10.3934/mbe.2023818

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

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Academic Editor: Xiangsheng Wang

Received: 30 August 2023 Revised: 21 September 2023 Accepted: 21 September 2023 Published: 25 September 2023

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. First, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of the coexistence equilibrium via Turing/Turing-Turing bifurcations by describing the first Turing bifurcation curve; we also theoretically predict possible bistable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that the coexistence equilibrium can also be destabilized via Hopf, Hopf-Hopf and Turing-Hopf bifurcations; also possible stable/bistable spatially inhomogeneous staggered periodic patterns and bistable 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, strong 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., bistable spatially heterogeneous staggered/synchronous periodic patterns, thus diversifying populations' survival patterns.
- predator-prey model,
- pattern formations,
- Turing bifurcation,
- Hopf bifurcation,
- predator-taxis
Citation: Yue Xing, Weihua Jiang, Xun Cao. Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18413-18444. doi: 10.3934/mbe.2023818

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Abstract

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. First, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of the coexistence equilibrium via Turing/Turing-Turing bifurcations by describing the first Turing bifurcation curve; we also theoretically predict possible bistable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that the coexistence equilibrium can also be destabilized via Hopf, Hopf-Hopf and Turing-Hopf bifurcations; also possible stable/bistable spatially inhomogeneous staggered periodic patterns and bistable 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, strong 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., bistable spatially heterogeneous staggered/synchronous periodic patterns, thus diversifying populations' survival patterns.

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