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

Existence theorem for a model of dryland vegetation

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.
Citation: Yukie Goto, Danielle Hilhorst, Ehud Meron, Roger Temam. Existence theorem for a model of dryland vegetation. Discrete & 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.   Google Scholar

[2]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.tpb.2007.05.002.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.003.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2006.08.006.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1098/rspb.2009.2208.  Google Scholar

[13]

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

show all references

References:
[1]

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

[2]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.tpb.2007.05.002.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.003.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2006.08.006.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1098/rspb.2009.2208.  Google Scholar

[13]

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

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