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 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.

[1]

Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399

[2]

Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275

[3]

Jiebao Sun, Boying Wu, Jing Li, Dazhi Zhang. A class of doubly degenerate parabolic equations with periodic sources. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1199-1210. doi: 10.3934/dcdsb.2010.14.1199

[4]

Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047

[5]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[6]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[7]

Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations and Control Theory, 2020, 9 (2) : 375-398. doi: 10.3934/eect.2020010

[8]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[9]

J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136

[10]

Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415

[11]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. The cost of controlling weakly degenerate parabolic equations by boundary controls. Mathematical Control and Related Fields, 2017, 7 (2) : 171-211. doi: 10.3934/mcrf.2017006

[12]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (4) : 607-635. doi: 10.3934/cpaa.2004.3.607

[13]

Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203

[14]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455

[15]

El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control and Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013

[16]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[17]

Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487

[18]

Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873

[19]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[20]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (5)

[Back to Top]