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A new reaction-diffusion-advection model with long-range inhibition for vegetation-desertification pattern-formation as a unified approach

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  • Franco, Rebeca
  • Morales, Marco A.
  • Rodríguez-Mora, J.Isrrael
  • Agustín-Serrano, Ricardo

Abstract

This work proposes a new mathematical model for reproducing desertification and vegetation patterns existing in nature. The model consists of two nonlinear partial differential equations. One of them describes the spatio-temporal dynamic of vegetation in an analogous way to Lefever's model, while Hardenberg's model describes precipitation dynamics. The model's equations are solved using a numerical-functional difference method for the spatio-temporal terms. The numerical results reproduce various bi-dimensional (2D) patterns observed in water-limited regions, including stripes, spots, hollows, and labyrinths. 2D patterns with these morphologies are characterized by their Fourier spectra and quantified their fractal dimensions. The numerical solutions of the model also predict transitions from bare soil at low precipitation to homogeneous vegetation at high rainfalls. These results reveal an underlying mechanism for the local desertification process and the vegetation self-organization distribution, broad context of matter order-disorder transitions. The proposed model even reproduces the desertification patterns for local instabilities of the hydrodynamic type beyond the instability induced by diffusion, which suggests that it is a reduced mathematical model with a unifying general framework for patterns found in nature.

Suggested Citation

  • Franco, Rebeca & Morales, Marco A. & Rodríguez-Mora, J.Isrrael & Agustín-Serrano, Ricardo, 2024. "A new reaction-diffusion-advection model with long-range inhibition for vegetation-desertification pattern-formation as a unified approach," Ecological Modelling, Elsevier, vol. 492(C).
    • Handle: RePEc:eee:ecomod:v:492:y:2024:i:c:s0304380024001108
    DOI: 10.1016/j.ecolmodel.2024.110722
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    References listed on IDEAS

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    1. Marco A. Morales & Irving Fernández-Cervantes & Ricardo Agustín-Serrano & Andrés Anzo & Mercedes P. Sampedro, 2016. "Patterns formation in ferrofluids and solid dissolutions using stochastic models with dissipative dynamics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 89(8), pages 1-17, August.
    2. Corina E. Tarnita & Juan A. Bonachela & Efrat Sheffer & Jennifer A. Guyton & Tyler C. Coverdale & Ryan A. Long & Robert M. Pringle, 2017. "A theoretical foundation for multi-scale regular vegetation patterns," Nature, Nature, vol. 541(7637), pages 398-401, January.
    3. Meron, Ehud, 2012. "Pattern-formation approach to modelling spatially extended ecosystems," Ecological Modelling, Elsevier, vol. 234(C), pages 70-82.
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