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 and 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), 683-697. doi: 10.1029/RG018i003p00683.

[2]

C. Valentin, J. M. d'Herbès and J. Poesen, Soil and water components of banded vegetation patterns, Catena, 37 (1999), 1-24. doi: 10.1016/S0341-8162(99)00053-3.

[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), 209-212. doi: 10.1038/nature06060.

[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), 213-216. doi: 10.1038/nature06111.

[5]

A. Manor and N. Shnerb, Facilitation, competition, and vegetation patchiness: From scale free distributions to patterns, J. Theoretical Biology, 253 (2008), 838-842. doi: 10.1016/j.jtbi.2008.04.012.

[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), 1771-1776. doi: 10.1098/rspb.2009.2208.

[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), 0981051. doi: 10.1103/PhysRevLett.93.098105.

[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), 680-691. doi: 10.1016/j.jtbi.2006.08.006.

[9]

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

[10]

R. Lefever and O. Lejeune, On the Origin of tiger bush, B. Math. Biol., 59 (1997), 263-294. doi: 10.1007/BF02462004.

[11]

O. Lejeune, M. Tlidi and R. Lefever, Vegetation spots and stripes: dissipative structures in arid landscapes, International Journal of Quantum Chemistry, 98 (2004), 261-271. doi: 10.1002/qua.10878.

[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), 524-530. doi: 10.1086/342078.

[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.

[14]

E. Gilad, M. Shachak and E. Meron, Dynamics and spatial organization of plant communities in water limited systems, Theoretical Population Biology, 72 (2007), 214-230. doi: 10.1016/j.tpb.2007.05.002.

[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), 192-200. doi: 10.1016/j.ecocom.2007.06.008.

[16]

E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: forms and functions, Chaos, 17 (2007), 037109. doi: 10.1063/1.2767246.

[17]

A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency, J. Theoretical Biology, 256 (2009), 574-583. doi: 10.1016/j.jtbi.2008.10.020.

[18]

E. Meron, Modeling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187. doi: 10.1051/mmnp/20116109.

[19]

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

[20]

E. Knobloch, Spatially localized structures in dissipative systems: open problems, Nonlinearity, 21 (2008), T45-T60. doi: 10.1088/0951-7715/21/4/T02.

[21]

D. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation, Nonlinearity, 22 (2009), 485-524. doi: 10.1088/0951-7715/22/2/013.

[22]

P. W. Voorhees, Ostwald ripening of two-phase mixtures, Annu. Rev. Mat. Sci., 22 (1992), 197-215. doi: 10.1146/annurev.ms.22.080192.001213.

[23]

B. Meerson and P. V. Sasorov, Domain stability, competition, growth, and selection in globally constrained bistable systems, Phys. Rev. E, 53 (1996), 3491-3494. doi: 10.1103/PhysRevE.53.3491.

[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) 433-441. doi: 10.1007/BF01314019.

[25]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249.

[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), 046117. doi: 10.1103/PhysRevE.65.046117.

[27]

P. Coullet, C. Elphick and D. Repaux, Nature of spatial chaos, Phys. Rev. Lett., 58 (1987) 431-434. doi: 10.1103/PhysRevLett.58.431.

[28]

C. Elphick, E. Meron and E. A. Spiegel, Spatiotemporal complexity in traveling wavetrains, Phys. Rev. Lett., 61 (1988), 496-499. doi: 10.1103/PhysRevLett.61.496.

[29]

A. Hagberg and E. Meron, Order parameter equations for front transitions: Nonuniformly curved fronts, Physica D, 123 (1998), 460-473. doi: 10.1016/S0167-2789(98)00143-2.

[30]

A. Y. Kletter, "Dynamics of Vegetation Patterns in Water-limited Systems," Ph.D. thesis, Ben-Gurion University, 2010.

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), 683-697. doi: 10.1029/RG018i003p00683.

[2]

C. Valentin, J. M. d'Herbès and J. Poesen, Soil and water components of banded vegetation patterns, Catena, 37 (1999), 1-24. doi: 10.1016/S0341-8162(99)00053-3.

[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), 209-212. doi: 10.1038/nature06060.

[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), 213-216. doi: 10.1038/nature06111.

[5]

A. Manor and N. Shnerb, Facilitation, competition, and vegetation patchiness: From scale free distributions to patterns, J. Theoretical Biology, 253 (2008), 838-842. doi: 10.1016/j.jtbi.2008.04.012.

[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), 1771-1776. doi: 10.1098/rspb.2009.2208.

[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), 0981051. doi: 10.1103/PhysRevLett.93.098105.

[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), 680-691. doi: 10.1016/j.jtbi.2006.08.006.

[9]

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

[10]

R. Lefever and O. Lejeune, On the Origin of tiger bush, B. Math. Biol., 59 (1997), 263-294. doi: 10.1007/BF02462004.

[11]

O. Lejeune, M. Tlidi and R. Lefever, Vegetation spots and stripes: dissipative structures in arid landscapes, International Journal of Quantum Chemistry, 98 (2004), 261-271. doi: 10.1002/qua.10878.

[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), 524-530. doi: 10.1086/342078.

[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.

[14]

E. Gilad, M. Shachak and E. Meron, Dynamics and spatial organization of plant communities in water limited systems, Theoretical Population Biology, 72 (2007), 214-230. doi: 10.1016/j.tpb.2007.05.002.

[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), 192-200. doi: 10.1016/j.ecocom.2007.06.008.

[16]

E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: forms and functions, Chaos, 17 (2007), 037109. doi: 10.1063/1.2767246.

[17]

A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency, J. Theoretical Biology, 256 (2009), 574-583. doi: 10.1016/j.jtbi.2008.10.020.

[18]

E. Meron, Modeling dryland landscapes, Math. Model. Nat. Phenom., 6 (2011), 163-187. doi: 10.1051/mmnp/20116109.

[19]

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

[20]

E. Knobloch, Spatially localized structures in dissipative systems: open problems, Nonlinearity, 21 (2008), T45-T60. doi: 10.1088/0951-7715/21/4/T02.

[21]

D. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation, Nonlinearity, 22 (2009), 485-524. doi: 10.1088/0951-7715/22/2/013.

[22]

P. W. Voorhees, Ostwald ripening of two-phase mixtures, Annu. Rev. Mat. Sci., 22 (1992), 197-215. doi: 10.1146/annurev.ms.22.080192.001213.

[23]

B. Meerson and P. V. Sasorov, Domain stability, competition, growth, and selection in globally constrained bistable systems, Phys. Rev. E, 53 (1996), 3491-3494. doi: 10.1103/PhysRevE.53.3491.

[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) 433-441. doi: 10.1007/BF01314019.

[25]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249.

[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), 046117. doi: 10.1103/PhysRevE.65.046117.

[27]

P. Coullet, C. Elphick and D. Repaux, Nature of spatial chaos, Phys. Rev. Lett., 58 (1987) 431-434. doi: 10.1103/PhysRevLett.58.431.

[28]

C. Elphick, E. Meron and E. A. Spiegel, Spatiotemporal complexity in traveling wavetrains, Phys. Rev. Lett., 61 (1988), 496-499. doi: 10.1103/PhysRevLett.61.496.

[29]

A. Hagberg and E. Meron, Order parameter equations for front transitions: Nonuniformly curved fronts, Physica D, 123 (1998), 460-473. doi: 10.1016/S0167-2789(98)00143-2.

[30]

A. Y. Kletter, "Dynamics of Vegetation Patterns in Water-limited Systems," Ph.D. thesis, Ben-Gurion University, 2010.

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