Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis

Fengrong Zhang; Ruining Chen; Fengrong Zhang; Ruining Chen

doi:10.3934/era.2024215

Electronic Research Archive

2024, Volume 32, Issue 7: 4723-4740. doi: 10.3934/era.2024215

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Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis

Fengrong Zhang ^,,
Ruining Chen

College of Science, China University of Petroleum (East China), Qingdao 266580, China

Received: 21 May 2024 Revised: 11 July 2024 Accepted: 23 July 2024 Published: 29 July 2024

This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient $ \chi $ and time delay $ \tau $ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of $ \tau $ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of $ \tau $, spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes.
- prey-predator model,
- stability,
- prey-taxis,
- delay,
- nonhomogeneous Hopf bifurcation
Citation: Fengrong Zhang, Ruining Chen. Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis[J]. Electronic Research Archive, 2024, 32(7): 4723-4740. doi: 10.3934/era.2024215

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Abstract

This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient $ \chi $ and time delay $ \tau $ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of $ \tau $ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of $ \tau $, spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes.

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