Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model

Jichun Li; Gaihui Guo; Hailong Yuan; Jichun Li; Gaihui Guo; Hailong Yuan

doi:10.3934/mbe.2024200

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 3: 4521-4553. doi: 10.3934/mbe.2024200

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Research article

Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model

School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China

Received: 10 December 2023 Revised: 09 January 2024 Accepted: 17 January 2024 Published: 28 February 2024

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.
- vegetation-sand model,
- nonlocal delay,
- multiple scale analysis,
- amplitude equation
Citation: Jichun Li, Gaihui Guo, Hailong Yuan. Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 4521-4553. doi: 10.3934/mbe.2024200

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Abstract

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.

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