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Interactions of cross-diffusion and nonlocal delay induce spatial vegetation patterning in semi-arid environments

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Abstract

In semi-arid regions, the positive feedback effect of vegetation and soil moisture plays an indispensable role in the process of water absorption by vegetation roots, and vegetation can also absorb water through the nonlocal interaction of roots. So in this paper, we consider how the interactions of cross-diffusion and nonlocal delay affect the growth of vegetation. By mathematical analysis, the conditions under which Turing patterns occur are obtained for this vegetation-water model. At the same time, we obtain the amplitude equation near the Turing bifurcation boundary by using the multi-scale analysis method. Conditions for the emergence of Turing patterns such as stripes, spots, and mixtures of stripes and spots are identified through the stability analysis of amplitude equation. Some numerical simulations are given to illustrate the analytical results, especially the evolution processes of vegetation patterns depicted under different parameters.

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Funding

The work is supported by the National Natural Science Foundation of China (61872227, 12126420, 52276028), Postdoctoral Science Foundation of China (Grant No. 2022M711854), and Shuimu Tsinghua Scholar Program (Grant No. 2021SM062).

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Authors and Affiliations

  1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Gaihui Guo, Jingjing Wang & Shihan Zhao

  2. School of Science, Beijing University of Civil Engineering and Architecture, Beijing, 102616, People’s Republic of China

    Conghui Zhang

Authors
  1. Gaihui Guo
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  2. Jingjing Wang
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  3. Shihan Zhao
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  4. Conghui Zhang
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Contributions

GG and JW contributed to the conception of the study and wrote the main manuscript text. SZ and CZ prepared figures. All authors reviewed the manuscript.

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Correspondence to Gaihui Guo.

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The authors declare that the work is our original research, which has not been fully or partially published before. There is no Conflict of interest in submitting this manuscript, and we have no relevant financial or non-financial interests to disclose.

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Guo, G., Wang, J., Zhao, S. et al. Interactions of cross-diffusion and nonlocal delay induce spatial vegetation patterning in semi-arid environments. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09460-8

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  • DOI: https://doi.org/10.1007/s11071-024-09460-8

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