American Institute of Mathematical Sciences

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July  2011, 16(1): 197-224. doi: 10.3934/dcdsb.2011.16.197

Existence theorem for a model of dryland vegetation

Yukie Goto 1, , Danielle Hilhorst 2, , Ehud Meron 3, and  Roger Temam 4,

1. 

18-20 Avenue De La République, 92400, Courbevoie, France
2. 

Laboratoire d'Analyse Numérique, Université Paris Sud, Orsay, France
3. 

Institute for Dryland Environmental Research, BIDR, Ben-Gurion University, Sede Boqer campus 84990, Israel
4. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, 47205, United States

Received  June 2010 Revised  January 2011 Published  April 2011

In this article, we consider the dryland vegetation model proposed by Gilad et al [6, 7]. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation [3, 7], and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
Keywords: porous media equations, Desertification, Galerkin approximation, parabolic equations, degenerate parabolic equations, Dryland vegetation, compactness theorems..
Mathematics Subject Classification: Primary: 35Q92; Secondary: 35K6.
Citation: Yukie Goto, Danielle Hilhorst, Ehud Meron, Roger Temam. Existence theorem for a model of dryland vegetation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 197-224. doi: 10.3934/dcdsb.2011.16.197
References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1968), 5042-5044.

[2]

F. Borgogno, P. D'Odorico, F. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophysics, 47 (2009), RG1005. doi: 10.1029/2007RG000256.

[3]

E. Feireisl, D. Hilhorst, M. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295.

[4]

E. Gilad, M. Shachak and E. Meron, Dynamicsa and spatial organization of plant communities in water limites systems, Ther. Popul. Biol., 72 (2007), 214-230. doi: 10.1016/j.tpb.2007.05.002.

[5]

E. Gilad and J. von Hardenberg, A fast algorithm for convolution integrals with space and time variant kernels, J. Comput. Phys., 216 (2006), 326-336. doi: 10.1016/j.jcp.2005.12.003.

[6]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habitat creation, Phy. Rev. Lett., 98 (2004), 098105-1-098105-4.

[7]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, A mathematical model of plants as ecosystem engineers, J. Ther. Biol., 244 (2007), 680-691. doi: 10.1016/j.jtbi.2006.08.006.

[8]

E. Meron, H. Yizhanq and E. Gilad, Localized structures in dryland vegetaion: forms and functions, Chaos, 17 (2007), 139-144. doi: 10.1063/1.2767246.

[9]

M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walkerk, Catastrophic shifts in ecosystem, Nature, 413 (2001), 591-596. doi: 10.1038/35098000.

[10]

R. Temam, "Navier Stokes Equations and Nonlinear Functional Analysis," American Mathematical Society, 2001.

[11]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetatio patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101-1-198101-4. doi: 10.1103/PhysRevLett.87.198101.

[12]

J. von Hardenberg, A. Y. Kletter, H. Yizhaq, J. Nathan and E. Meron, Periodic vs. scale-free patterns in dryland vegetation, Proc. R. Soc. B., 277 (2010), 1771-1776. doi: 10.1098/rspb.2009.2208.

[13]

H. Yizhaq, E. Gilad and E. Meron, Banded vegetation: Biological productivity and resilience, Physica A, 356 (2005), 139-144. doi: 10.1016/j.physa.2005.05.026.

show all references

References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1968), 5042-5044.

[2]

F. Borgogno, P. D'Odorico, F. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophysics, 47 (2009), RG1005. doi: 10.1029/2007RG000256.

[3]

E. Feireisl, D. Hilhorst, M. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295.

[4]

E. Gilad, M. Shachak and E. Meron, Dynamicsa and spatial organization of plant communities in water limites systems, Ther. Popul. Biol., 72 (2007), 214-230. doi: 10.1016/j.tpb.2007.05.002.

[5]

E. Gilad and J. von Hardenberg, A fast algorithm for convolution integrals with space and time variant kernels, J. Comput. Phys., 216 (2006), 326-336. doi: 10.1016/j.jcp.2005.12.003.

[6]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habitat creation, Phy. Rev. Lett., 98 (2004), 098105-1-098105-4.

[7]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, A mathematical model of plants as ecosystem engineers, J. Ther. Biol., 244 (2007), 680-691. doi: 10.1016/j.jtbi.2006.08.006.

[8]

E. Meron, H. Yizhanq and E. Gilad, Localized structures in dryland vegetaion: forms and functions, Chaos, 17 (2007), 139-144. doi: 10.1063/1.2767246.

[9]

M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walkerk, Catastrophic shifts in ecosystem, Nature, 413 (2001), 591-596. doi: 10.1038/35098000.

[10]

R. Temam, "Navier Stokes Equations and Nonlinear Functional Analysis," American Mathematical Society, 2001.

[11]

J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetatio patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101-1-198101-4. doi: 10.1103/PhysRevLett.87.198101.

[12]

J. von Hardenberg, A. Y. Kletter, H. Yizhaq, J. Nathan and E. Meron, Periodic vs. scale-free patterns in dryland vegetation, Proc. R. Soc. B., 277 (2010), 1771-1776. doi: 10.1098/rspb.2009.2208.

[13]

H. Yizhaq, E. Gilad and E. Meron, Banded vegetation: Biological productivity and resilience, Physica A, 356 (2005), 139-144. doi: 10.1016/j.physa.2005.05.026.

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