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Stability and cross-diffusion-driven instability for a water-vegetation model with the infiltration feedback effect

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Abstract

This paper is devoted to a mathematical model with diffusion and cross-diffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and cross-diffusion-driven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small water-uptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the water-uptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12301634, 12061081, 61872227, 12126420). The authors wish to express grateful thanks to the anonymous referees for their careful reading and some valuable comments and suggestions which greatly improved this work.

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Authors and Affiliations

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

    Gaihui Guo, Shihan Zhao & Danfeng Pang

  2. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, 221018, China

    Youhui Su

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  1. Gaihui Guo
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  2. Shihan Zhao
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  4. Youhui Su
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Correspondence to Gaihui Guo or Youhui Su.

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The work is supported by the National Natural Science Foundation of China (12301634, 12061081, 61872227, 12126420).

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Guo, G., Zhao, S., Pang, D. et al. Stability and cross-diffusion-driven instability for a water-vegetation model with the infiltration feedback effect. Z. Angew. Math. Phys. 75, 33 (2024). https://doi.org/10.1007/s00033-023-02167-7

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  • DOI: https://doi.org/10.1007/s00033-023-02167-7

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