Turing patterns in a networked vegetation model

Xiaomei Bao; Canrong Tian; Xiaomei Bao; Canrong Tian

doi:10.3934/mbe.2024334

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 11: 7601-7620. doi: 10.3934/mbe.2024334

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Turing patterns in a networked vegetation model

Xiaomei Bao ^1,,
Canrong Tian ^{2
,
,}

1.
School of Foreign Languages, Yancheng Institute of Technology, Yancheng 224003, China
2.
School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224003, China

Academic Editor: Frithjof Lutscher

Received: 30 August 2024 Revised: 31 October 2024 Accepted: 05 November 2024 Published: 19 November 2024

A vegetation model composed of water and plants was proposed by introducing a weighted graph Laplacian operator into the reaction-diffusion dynamics. We showed the global existence and uniqueness of the solution via monotone iterative sequence. The parameter space of Turing patterns for plant behavior is obtained based on the analysis of the eigenvalues of the Laplacian of weighted graph, while the amplitude equation determining the stability of Turing patterns is obtained by weakly nonlinear analysis. We also show that the optimal rainfall is only determined by the density of the water. By some numerical simulations, we examine the individual effect of diffusion term on the formation of regular Turing patterns. We show that the large diffusion induces stable Turing patterns.
- networked vegetation model,
- Turing pattern,
- graph Laplace,
- amplitude equation,
- optimal control
Citation: Xiaomei Bao, Canrong Tian. Turing patterns in a networked vegetation model[J]. Mathematical Biosciences and Engineering, 2024, 21(11): 7601-7620. doi: 10.3934/mbe.2024334

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Abstract

A vegetation model composed of water and plants was proposed by introducing a weighted graph Laplacian operator into the reaction-diffusion dynamics. We showed the global existence and uniqueness of the solution via monotone iterative sequence. The parameter space of Turing patterns for plant behavior is obtained based on the analysis of the eigenvalues of the Laplacian of weighted graph, while the amplitude equation determining the stability of Turing patterns is obtained by weakly nonlinear analysis. We also show that the optimal rainfall is only determined by the density of the water. By some numerical simulations, we examine the individual effect of diffusion term on the formation of regular Turing patterns. We show that the large diffusion induces stable Turing patterns.

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