一类具有遮阳效应的植被模型的稳态分支研究

Acta mathematica scientia,Series A ›› 2025, Vol. 45 ›› Issue (3): 875-887.

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Steady-State Bifurcation for a Vegetation Model with Shading Effect

Liang Juan^1,^*(),Guo Zunguang¹,Zhang Hongtao²

¹Department of Science, Taiyuan Institute of Technology, Taiyuan 030008
²School of Mathematics, North University of China, Taiyuan 030051

Received:2024-07-31 Revised:2025-01-13 Online:2025-06-26 Published:2025-06-20
Supported by:
NSFC(42075029);Fundamental Research Program of Shanxi Province(202203021212327);Fundamental Research Program of Shanxi Province(202203021211213);Program for the (Reserved) Discipline Leaders of Taiyuan Institute of Technology(2024KJ012)

Abstract

Abstract:

A vegetation-water reaction-diffusion model with shading effect under no-flux boundary conditions is studied. The existence of steady-state bifurcation of the model is firstly proved, and the conditions for the generation of steady-state bifurcation are obtained. Then the structure of the non-constant steady-state solution in the case of single eigenvalues is obtained by using the Crandall-Rabinowitz bifurcation theorem. By adopting the implicit function theorem and the techniques of space decomposition, the structure of the non-constant steady-state solution in the case of double eigenvalues is obtained. Finally, numerical simulations are shown to verify the theoretical analysis results.

Key words: vegetation model, steady-state bifurcation, space decomposition, non-constant solutions

CLC Number:

O175

Liang Juan, Guo Zunguang, Zhang Hongtao. Steady-State Bifurcation for a Vegetation Model with Shading Effect[J].Acta mathematica scientia,Series A, 2025, 45(3): 875-887.

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References

[1]	Tarnita C E, Bonachela J A, Sheffer E, et al. A theoretical foundation for multi-scale regular vegetation patterns. Nature, 2017, 541(7637): 398-401
[2]	Carter P, Doelman A. Traveling stripes in the Klausmeier model of vegetation pattern formation. SIAM J Appl Math, 2018, 78(6): 3213-3237
[3]	Liang J, Liu C, Sun G Q, et al. Nonlocal interactions between vegetation induce spatial patterning. Appl Math Comput, 2022, 428: 127061
[4]	Klausmeier C A. Regular and irregular patterns in semiarid vegetation. Science, 1999, 284(5421): 1826-1828
[5]	HilleRisLambers R, Rietkerk M, Van Den Bosch F, et al. Vegetation pattern formation in semi-arid grazing systems. Ecology, 2001, 82(1): 50-61
[6]	von Hardenberg J, Meron E, Shachak M, et al. Diversity of vegetation patterns and desertification. Phys Rev Lett, 2001, 87(19): 198101
[7]	Wang X L, Shi J P, Zhang G H. Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction. J Math Anal Appl, 2021, 497(1): 124860
[8]	Yi F Q, Wei J J, Shi J P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J Differ Equations, 2009, 246(5): 1944-1977
[9]	Yi F Q, Liu J X, Wei J J. Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model. Nonlinear Anal-Real, 2010, 11(5): 3770-3781
[10]	Wang M X. Non-constant positive steady states of the Sel'kov model. J Differ Equations, 2003, 190(2): 600-620
[11]	Crandall M G, Rabinowitz P H. Bifurcation from simple eigenvalues. J Funct Anal, 1971, 8(2): 321-340
[12]	Guo G H, Li B F, Wei M H, et al. Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model. J Math Anal Appl, 2012, 391(1): 265-277
[13]	Li S B, Wu J H, Dong Y Y. Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme. J Differ Equation, 2015, 259(5): 1990-2029
[14]	Li S B, Wu J H, Nie H. Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model. Comput Math Appl, 2015, 70(12): 3043-3056
[15]	Wang Y, Wu J, Jia Y. Steady-state bifurcation for a biological depletion model. Int J Bifurcat Chaos, 2016, 26(4): 1650066

Steady-State Bifurcation for a Vegetation Model with Shading Effect

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