The instability of periodic solutions for a population model with cross-diffusion

Weiyu Li; Hongyan Wang; Weiyu Li; Hongyan Wang

doi:10.3934/math.20231529

AIMS Mathematics

2023, Volume 8, Issue 12: 29910-29924. doi: 10.3934/math.20231529

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Research article

The instability of periodic solutions for a population model with cross-diffusion

Weiyu Li ,
Hongyan Wang ^,

College of Science, Heilongjiang Bayi Agricultural University, Daqing 163319, Heilongjiang, China

Received: 07 September 2023 Revised: 22 October 2023 Accepted: 29 October 2023 Published: 03 November 2023
MSC : 35B32, 35K57, 35Q92

This paper is concerned with a population model with prey refuge and a Holling type Ⅲ functional response in the presence of self-diffusion and cross-diffusion, and its Turing pattern formation problem of Hopf bifurcating periodic solutions was studied. First, we discussed the stability of periodic solutions for the ordinary differential equation model, and derived the first derivative formula of periodic functions for the perturbed model. Second, applying the Floquet theory, we gave the conditions of Turing patterns occurring at Hopf bifurcating periodic solutions. Additionally, we determined the range of cross-diffusion coefficients for the diffusive population model to form Turing patterns at the stable periodic solutions. Finally, our research was summarized and the relevant conclusions were simulated numerically.
- refuge,
- population model,
- cross-diffusion,
- periodic solutions,
- Turing pattern
Citation: Weiyu Li, Hongyan Wang. The instability of periodic solutions for a population model with cross-diffusion[J]. AIMS Mathematics, 2023, 8(12): 29910-29924. doi: 10.3934/math.20231529

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Abstract

This paper is concerned with a population model with prey refuge and a Holling type Ⅲ functional response in the presence of self-diffusion and cross-diffusion, and its Turing pattern formation problem of Hopf bifurcating periodic solutions was studied. First, we discussed the stability of periodic solutions for the ordinary differential equation model, and derived the first derivative formula of periodic functions for the perturbed model. Second, applying the Floquet theory, we gave the conditions of Turing patterns occurring at Hopf bifurcating periodic solutions. Additionally, we determined the range of cross-diffusion coefficients for the diffusive population model to form Turing patterns at the stable periodic solutions. Finally, our research was summarized and the relevant conclusions were simulated numerically.

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