Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

Hongfei Xu; Jinfeng Wang; Xuelian Xu

doi:10.3934/dcdsb.2021237

Article Contents

2022, Volume 27, Issue 8: 4473-4489. Doi: 10.3934/dcdsb.2021237

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Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

1.
School of Mathematics Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China
2.
College of Teacher Education, Harbin Normal University, Harbin, Heilongjiang, 150025, China

^* Corresponding author: Jinfeng Wang
^* Corresponding author: Jinfeng Wang

Received: March 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: August 2022

Partially supported by National NSFC grant 11971135, NSFH grant LH2019A017, LH2020A019 and 2018-KYYWF-0999

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Abstract

This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an $ n $-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35K59.

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Figure 1. Change of $ \chi_i $ with non-negative integer $ i $

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Figure 2. $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ is asymptotically stable for $ \chi = 0 $

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Figure 3. $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ remains stable for $ \chi = 200<\hat{\chi} $

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Figure 4. Spatial heterogenous and time-periodic patterns for $ \chi = 308>\hat{\chi} $

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