January  2012, 11(1): 261-273. doi: 10.3934/cpaa.2012.11.261

Ostwald ripening in dryland vegetation

1. 

Department of Physics, Ben-Gurion University, Beer-Sheva, 84105, Israel

2. 

Institute of Atmospheric Sciences and Climate - CNR, C.so Fiume 4, 10133, Torino, Italy

3. 

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

Received  February 2010 Revised  September 2010 Published  September 2011

Dryland landscapes self-organize to form various patterns of vegetation patchiness. Two major classes of patterns can be distinguished: regular patterns with characteristic length scales and scale-free patterns. The latter form under conditions of global competition over the water resource. In this paper we show that the asymptotic dynamics of scale-free vegetation patterns involve patch coarsening similar to Ostwald ripening in two-phase mixtures. We demonstrate it numerically, using a spatially explicit model for water-limited vegetation, and further study it by drawing an analogy to an activator-inhibitor system that shares many properties with the vegetation system. The ecological implications of patch coarsening may not be highly significant due to the long time scales involved. The reported results, however, raise an interesting pattern formation question associated with the incompatibility of mechanisms that stabilize vegetation spots and the condition of global competition.
Citation: Assaf Y. Kletter, Jost von Hardenberg, Ehud Meron. Ostwald ripening in dryland vegetation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 261-273. doi: 10.3934/cpaa.2012.11.261
References:
[1]

R. A. Brown, Longitudinal instabilities and secondary flows in the planetary boundary layer: A review,, Rev. of Geophysics and Space Physics, 18 (1980), 683.  doi: 10.1029/RG018i003p00683.  Google Scholar

[2]

C. Valentin, J. M. d'Herbès and J. Poesen, Soil and water components of banded vegetation patterns,, Catena, 37 (1999), 1.  doi: 10.1016/S0341-8162(99)00053-3.  Google Scholar

[3]

T. M. Scanlon, K. C. Kelly, S. A. Levin and I. Rodriguez-Iturbe, Positive feedbacks promote power-law clustering of Kalahari vegetation,, Nature, 449 (2007), 209.  doi: 10.1038/nature06060.  Google Scholar

[4]

S. Kéfi, M. Rietkerk, C. L. Alados, Y. Pueyo, V. P. Papanastasis, A. ElAich and P. C. de Ruiter, Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems,, Nature, 449 (2007), 213.  doi: 10.1038/nature06111.  Google Scholar

[5]

A. Manor and N. Shnerb, Facilitation, competition, and vegetation patchiness: From scale free distributions to patterns,, J. Theoretical Biology, 253 (2008), 838.  doi: 10.1016/j.jtbi.2008.04.012.  Google Scholar

[6]

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

[7]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habitat creation,, Phys. Rev. Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.098105.  Google Scholar

[8]

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

[9]

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

[10]

R. Lefever and O. Lejeune, On the Origin of tiger bush,, B. Math. Biol., 59 (1997), 263.  doi: 10.1007/BF02462004.  Google Scholar

[11]

O. Lejeune, M. Tlidi and R. Lefever, Vegetation spots and stripes: dissipative structures in arid landscapes,, International Journal of Quantum Chemistry, 98 (2004), 261.  doi: 10.1002/qua.10878.  Google Scholar

[12]

M. Rietkerk, M. C. Boerlijst, F. Van Langevelde, R. HilleRisLambers, J. Van de Koppel, L. Kumar, H. H. T. Prins and A. M. De Roos, Self-organization of vegetation in arid ecosystems,, American Naturalist, 160 (2002), 524.  doi: 10.1086/342078.  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

[14]

E. Gilad, M. Shachak and E. Meron, Dynamics and spatial organization of plant communities in water limited systems,, Theoretical Population Biology, 72 (2007), 214.  doi: 10.1016/j.tpb.2007.05.002.  Google Scholar

[15]

E. Sheffer, H. Yizhaq, E. Gilad, M. Shachak and E. Meron, Why do plants in resource deprived environments form rings?, Ecological Complexity, 4 (2007), 192.  doi: 10.1016/j.ecocom.2007.06.008.  Google Scholar

[16]

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

[17]

A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency,, J. Theoretical Biology, 256 (2009), 574.  doi: 10.1016/j.jtbi.2008.10.020.  Google Scholar

[18]

E. Meron, Modeling dryland landscapes,, Math. Model. Nat. Phenom., 6 (2011), 163.  doi: 10.1051/mmnp/20116109.  Google Scholar

[19]

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

[20]

E. Knobloch, Spatially localized structures in dissipative systems: open problems,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/4/T02.  Google Scholar

[21]

D. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation,, Nonlinearity, 22 (2009), 485.  doi: 10.1088/0951-7715/22/2/013.  Google Scholar

[22]

P. W. Voorhees, Ostwald ripening of two-phase mixtures,, Annu. Rev. Mat. Sci., 22 (1992), 197.  doi: 10.1146/annurev.ms.22.080192.001213.  Google Scholar

[23]

B. Meerson and P. V. Sasorov, Domain stability, competition, growth, and selection in globally constrained bistable systems,, Phys. Rev. E, 53 (1996), 3491.  doi: 10.1103/PhysRevE.53.3491.  Google Scholar

[24]

L. Schimansky-Geier, Ch. Zülicke1 and E. Schöll, Domain formation due to Ostwald ripening in bistable systems far from equilibrium,, Z. Phys. B, 84 (1991), 433.  doi: 10.1007/BF01314019.  Google Scholar

[25]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation,, IMA J. Appl. Math., 48 (1992), 249.  doi: 10.1093/imamat/48.3.249.  Google Scholar

[26]

M. Conti, B. Meerson, A. Peleg and P. V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.046117.  Google Scholar

[27]

P. Coullet, C. Elphick and D. Repaux, Nature of spatial chaos,, Phys. Rev. Lett., 58 (1987), 431.  doi: 10.1103/PhysRevLett.58.431.  Google Scholar

[28]

C. Elphick, E. Meron and E. A. Spiegel, Spatiotemporal complexity in traveling wavetrains,, Phys. Rev. Lett., 61 (1988), 496.  doi: 10.1103/PhysRevLett.61.496.  Google Scholar

[29]

A. Hagberg and E. Meron, Order parameter equations for front transitions: Nonuniformly curved fronts,, Physica D, 123 (1998), 460.  doi: 10.1016/S0167-2789(98)00143-2.  Google Scholar

[30]

A. Y. Kletter, "Dynamics of Vegetation Patterns in Water-limited Systems,", Ph.D. thesis, (2010).   Google Scholar

show all references

References:
[1]

R. A. Brown, Longitudinal instabilities and secondary flows in the planetary boundary layer: A review,, Rev. of Geophysics and Space Physics, 18 (1980), 683.  doi: 10.1029/RG018i003p00683.  Google Scholar

[2]

C. Valentin, J. M. d'Herbès and J. Poesen, Soil and water components of banded vegetation patterns,, Catena, 37 (1999), 1.  doi: 10.1016/S0341-8162(99)00053-3.  Google Scholar

[3]

T. M. Scanlon, K. C. Kelly, S. A. Levin and I. Rodriguez-Iturbe, Positive feedbacks promote power-law clustering of Kalahari vegetation,, Nature, 449 (2007), 209.  doi: 10.1038/nature06060.  Google Scholar

[4]

S. Kéfi, M. Rietkerk, C. L. Alados, Y. Pueyo, V. P. Papanastasis, A. ElAich and P. C. de Ruiter, Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems,, Nature, 449 (2007), 213.  doi: 10.1038/nature06111.  Google Scholar

[5]

A. Manor and N. Shnerb, Facilitation, competition, and vegetation patchiness: From scale free distributions to patterns,, J. Theoretical Biology, 253 (2008), 838.  doi: 10.1016/j.jtbi.2008.04.012.  Google Scholar

[6]

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

[7]

E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: from pattern formation to habitat creation,, Phys. Rev. Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.098105.  Google Scholar

[8]

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

[9]

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

[10]

R. Lefever and O. Lejeune, On the Origin of tiger bush,, B. Math. Biol., 59 (1997), 263.  doi: 10.1007/BF02462004.  Google Scholar

[11]

O. Lejeune, M. Tlidi and R. Lefever, Vegetation spots and stripes: dissipative structures in arid landscapes,, International Journal of Quantum Chemistry, 98 (2004), 261.  doi: 10.1002/qua.10878.  Google Scholar

[12]

M. Rietkerk, M. C. Boerlijst, F. Van Langevelde, R. HilleRisLambers, J. Van de Koppel, L. Kumar, H. H. T. Prins and A. M. De Roos, Self-organization of vegetation in arid ecosystems,, American Naturalist, 160 (2002), 524.  doi: 10.1086/342078.  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

[14]

E. Gilad, M. Shachak and E. Meron, Dynamics and spatial organization of plant communities in water limited systems,, Theoretical Population Biology, 72 (2007), 214.  doi: 10.1016/j.tpb.2007.05.002.  Google Scholar

[15]

E. Sheffer, H. Yizhaq, E. Gilad, M. Shachak and E. Meron, Why do plants in resource deprived environments form rings?, Ecological Complexity, 4 (2007), 192.  doi: 10.1016/j.ecocom.2007.06.008.  Google Scholar

[16]

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

[17]

A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency,, J. Theoretical Biology, 256 (2009), 574.  doi: 10.1016/j.jtbi.2008.10.020.  Google Scholar

[18]

E. Meron, Modeling dryland landscapes,, Math. Model. Nat. Phenom., 6 (2011), 163.  doi: 10.1051/mmnp/20116109.  Google Scholar

[19]

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

[20]

E. Knobloch, Spatially localized structures in dissipative systems: open problems,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/4/T02.  Google Scholar

[21]

D. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation,, Nonlinearity, 22 (2009), 485.  doi: 10.1088/0951-7715/22/2/013.  Google Scholar

[22]

P. W. Voorhees, Ostwald ripening of two-phase mixtures,, Annu. Rev. Mat. Sci., 22 (1992), 197.  doi: 10.1146/annurev.ms.22.080192.001213.  Google Scholar

[23]

B. Meerson and P. V. Sasorov, Domain stability, competition, growth, and selection in globally constrained bistable systems,, Phys. Rev. E, 53 (1996), 3491.  doi: 10.1103/PhysRevE.53.3491.  Google Scholar

[24]

L. Schimansky-Geier, Ch. Zülicke1 and E. Schöll, Domain formation due to Ostwald ripening in bistable systems far from equilibrium,, Z. Phys. B, 84 (1991), 433.  doi: 10.1007/BF01314019.  Google Scholar

[25]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation,, IMA J. Appl. Math., 48 (1992), 249.  doi: 10.1093/imamat/48.3.249.  Google Scholar

[26]

M. Conti, B. Meerson, A. Peleg and P. V. Sasorov, Phase ordering with a global conservation law: Ostwald ripening and coalescence,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.046117.  Google Scholar

[27]

P. Coullet, C. Elphick and D. Repaux, Nature of spatial chaos,, Phys. Rev. Lett., 58 (1987), 431.  doi: 10.1103/PhysRevLett.58.431.  Google Scholar

[28]

C. Elphick, E. Meron and E. A. Spiegel, Spatiotemporal complexity in traveling wavetrains,, Phys. Rev. Lett., 61 (1988), 496.  doi: 10.1103/PhysRevLett.61.496.  Google Scholar

[29]

A. Hagberg and E. Meron, Order parameter equations for front transitions: Nonuniformly curved fronts,, Physica D, 123 (1998), 460.  doi: 10.1016/S0167-2789(98)00143-2.  Google Scholar

[30]

A. Y. Kletter, "Dynamics of Vegetation Patterns in Water-limited Systems,", Ph.D. thesis, (2010).   Google Scholar

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