November  2012, 17(8): 2815-2827. doi: 10.3934/dcdsb.2012.17.2815

Vegetation patterns and desertification waves in semi-arid environments: Mathematical models based on local facilitation in plants

1. 

Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom, United Kingdom

Received  March 2011 Revised  August 2011 Published  July 2012

In semi-arid regions, infiltration of rain water into the soil is significantly higher in vegetated areas than for bare ground. However, quantitative data on the dependence of infiltration capacity on plant biomass is very limited. In this paper, we use a simple reaction-diffusion-advection model to investigate the effects of varying the strength of this dependence. We begin by studying the formation of banded vegetation patterns on gentle slopes ("tiger bush"), which is a hallmark of semi-deserts. We calculate the range of rainfall parameter values over which such patterns occur, using numerical continuation methods. We then consider interfaces between vegetation and bare ground, showing that the vegetated region either expands or contracts depending on whether the rainfall parameter is above or below a critical value. We conclude by discussing the mathematical questions raised by our work.
Citation: Jonathan A. Sherratt, Alexios D. Synodinos. Vegetation patterns and desertification waves in semi-arid environments: Mathematical models based on local facilitation in plants. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2815-2827. doi: 10.3934/dcdsb.2012.17.2815
References:
[1]

M. R. Agular and O. E. Sala, Patch structure, dynamics and implications for the functioning ofarid ecosystems,, Trends Ecol. Evol., 14 (1999), 273. doi: 10.1016/S0169-5347(99)01612-2. Google Scholar

[2]

E. O. Alzahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models,, Math. Model. Nat. Phenom., 5 (2010), 13. Google Scholar

[3]

N. Barbier, P. Couteron, R. Lefever, V. Deblauwe and O. Lejeune, Spatial decoupling of facilitation, competition at the origin ofgapped vegetation patterns,, Ecology, 89 (2008), 1521. doi: 10.1890/07-0365.1. Google Scholar

[4]

S. S. Berg and D. L. Dunkerley, Patterned mulga near Alice Springs, central Australia, and thepotential threat of firewood collection on this vegetation community,, J. Arid Environ., 59 (2004), 313. doi: 10.1016/j.jaridenv.2003.12.007. Google Scholar

[5]

A. I. Borthagaraya, M. A. Fuentesa and P. A. Marque, Vegetation pattern formation in a fog-dependent ecosystem,, J. Theor. Biol., 265 (2010), 18. doi: 10.1016/j.jtbi.2010.04.020. Google Scholar

[6]

R. M. Callaway, Positive interactions among plants,, Botanical Rev., 61 (1995), 306. doi: 10.1007/BF02912621. Google Scholar

[7]

P. Couteron, A. Mahamane, P. Ouedraogo and J. Seghieri, Differences between banded thickets (tiger bush) at two sites in West Africa,, J. Veg. Sci., 11 (2000), 321. doi: 10.2307/3236624. Google Scholar

[8]

V. Deblauwe, P. Couteron, J. Bogaert and N. Barbier, Determinants and dynamics of banded vegetation pattern migration in arid climates,, Ecological monographs, 82 (2012), 3. doi: 10.5061/dryad.1qr41s56. Google Scholar

[9]

J. D. Dockery and R. Lui, Existence of traveling wave solutions for a bistable evolutionary ecology model,, SIAM J. Math. Anal., 23 (1992), 870. doi: 10.1137/0523046. Google Scholar

[10]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Cong. Numer., 30 (1981), 265. Google Scholar

[11]

E. J. Doedel, H. B. Keller and J. P. Kernévez, Numerical analysis and control of bifurcation problems. I. Bifurcation in finite dimensions,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 1 (1991), 493. Google Scholar

[12]

E. J. Doedel, W. Govaerts, Yu. A. Kuznetsov and A. Dhooge, Numerical continuation of branch points of equilibria and periodic orbits,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 841. Google Scholar

[13]

D. L. Dunkerley and K. J. Brown, Oblique vegetation banding in the Australian arid zone: Implications for theories of pattern evolution and maintenance,, J. Arid Environments, 52 (2002), 163. doi: 10.1006/jare.2001.0940. Google Scholar

[14]

J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, J. Dyn. Diff. Eq., 21 (2009), 663. doi: 10.1007/s10884-009-9152-7. Google Scholar

[15]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Rat. Mech. Anal., 65 (1977), 335. Google Scholar

[16]

A. Giannini, M. Biasutti and M. M. Verstraete, A climate model-based review of drought in the Sahel: Desertification, the re-greening and climate change,, Global Planet. Change, 64 (2008), 119. doi: 10.1016/j.gloplacha.2008.05.004. Google Scholar

[17]

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

[18]

V. Guttal and C. Jayaprakash, Self-organisation and productivity in semi-arid ecosystems: Implications of seasonality in rainfall,, J. Theor. Biol., 248 (2007), 290. Google Scholar

[19]

R. HilleRisLambers, M. Rietkerk, F. van de Bosch, H. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems,, Ecology, 82 (2001), 50. Google Scholar

[20]

R. C. Hills, The influence of land management and soil characteristics on infiltration and the occurrence of overland flow,, J. Hydrology, 13 (1971), 163. doi: 10.1016/0022-1694(71)90213-7. Google Scholar

[21]

Y. Jin and X.-Q. Zhao, Bistable waves for a class of cooperative reaction-diffusion systems,, J. Biol. Dyn., 2 (2008), 196. Google Scholar

[22]

S. Kéfi, M. Rietkerk, C. L. Alados, Y. Pueyo, A. ElAich, V. Papanastasis 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

[23]

S. Kéfi, M. Rietkerk and G. G. Katul, Vegetation pattern shift as a result of rising atmospheric CO2 in arid ecosystems,, Theor. Pop. Biol., 74 (2008), 332. doi: 10.1016/j.tpb.2008.09.004. Google Scholar

[24]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation,, Science, 284 (1999), 1826. doi: 10.1126/science.284.5421.1826. Google Scholar

[25]

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

[26]

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

[27]

R. Lefever, N. Barbier, P. Couteron and O. Lejeune, Deeply gapped vegetation patterns: On crown/root allometry, criticality and desertification,, J. Theor. Biol., 261 (2009), 194. doi: 10.1016/j.jtbi.2009.07.030. Google Scholar

[28]

W. MacFadyen, Vegetation patterns in the semi-desert plains of British Somaliland,, Geographical J., 115 (1950), 199. doi: 10.2307/1789384. Google Scholar

[29]

A. K. McDonald, R. J. Kinucan and L. E. Loomis, Ecohydrological interactions within banded vegetation in the northeastern Chihuahuan Desert, USA,, Ecohydrology, 2 (2009), 66. Google Scholar

[30]

C. Montaña, The colonization of bare areas in two-phase mosaics of an arid ecosystem,, J. Ecol., 80 (1992), 315. doi: 10.2307/2261014. Google Scholar

[31]

E. N. Mueller, J. Wainwright and A. J. Parsons, The stability of vegetation boundaries and the propagation ofdesertification in the American Southwest: A modelling approach,, Ecol.\ Model., 208 (2007), 91. doi: 10.1016/j.ecolmodel.2007.04.010. Google Scholar

[32]

Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems,, Oikos, 117 (2008), 1522. doi: 10.1111/j.0030-1299.2008.16735.x. Google Scholar

[33]

M. Rietkerk, P. Ketner, J. Burger, B. Hoorens and H. Olff, Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa,, Plant Ecology, 148 (2000), 207. doi: 10.1023/A:1009828432690. Google Scholar

[34]

M. Rietkerk, M. C. Boerlijst, F. van Langevelde, R. HilleRisLambers, J. van de Koppel, H. H. T. Prins and A. de Roos, Self-organisation of vegetation in arid ecosystems,, Am. Nat., 160 (2002), 524. doi: 10.1086/342078. Google Scholar

[35]

M. Rietkerk, S. C. Dekker, P. C. de Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems,, Science, 305 (2004), 1926. doi: 10.1126/science.1101867. Google Scholar

[36]

T. M. Scanlon, K. K. Caylor, 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

[37]

W. H. Schlesinger, J. F. Reynolds, G. L. Cunningham, L. F. Huenneke, W. M. Jarrell, R. A. Virginia and W. G. Whitford, Biological feedbacks in global desertification,, Science, 247 (1990), 1043. doi: 10.1126/science.247.4946.1043. Google Scholar

[38]

J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes,, J. Math. Biol., 51 (2005), 183. doi: 10.1007/s00285-005-0319-5. Google Scholar

[39]

J. A. Sherratt and G. J. Lord, Nonlinear dynamics, pattern bifurcations in a model for vegetation stripes in semi-arid environments,, Theor. Pop. Biol., 71 (2007), 1. Google Scholar

[40]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments I,, Nonlinearity, 23 (2010), 2657. doi: 10.1088/0951-7715/23/10/016. Google Scholar

[41]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments II. Patterns with the largest possible propagation speeds,, Proc. R. Soc. Lond. A}, 467 (2011), 3272. doi: 10.1098/rspa.2011.0194. Google Scholar

[42]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments III. The transition between homoclinic solutions,, submitted., (). Google Scholar

[43]

G.-Q. Sun, Z. Jin and Q. Tan, Measurement of self-organization in arid ecosystems,, J. Biol. Systems, 18 (2010), 495. doi: 10.1142/S0218339010003366. Google Scholar

[44]

D. J. Tongway and J. A. Ludwig, Theories on the origins, maintainance, dynamics, and functioning ofbanded landscapes,, in, (2001), 20. Google Scholar

[45]

N. Ursino and S. Contarini, Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity,, Adv. Water Resour., 29 (2006), 1556. doi: 10.1016/j.advwatres.2005.11.006. Google Scholar

[46]

N. Ursino, Modeling banded vegetation patterns in semiarid regions: Inter-dependence between biomass growth rate and relevant hydrological processes,, Water Resour. Res., 43 (2007). Google Scholar

[47]

N. Ursino, Above and below ground biomass patterns in arid lands,, Ecol. Model., 220 (2009), 1411. doi: 10.1016/j.ecolmodel.2009.02.023. Google Scholar

[48]

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

[49]

C. Valentin and J. M. d'Herbès, Niger tiger bush as a natural water harvesting system,, Catena, 37 (1999), 231. doi: 10.1016/S0341-8162(98)00061-7. Google Scholar

[50]

J. van de Koppel, M. Rietkerk, F. van Langevelde, L. Kumar, C. A. Klausmeier, J. M. Fryxell, J. W. Hearne, J. van Andel, N. de Ridder, M. A. Skidmore, L. Stroosnijder and H. H. T. Prins, Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems,, Am. Nat., 159 (2002), 209. doi: 10.1086/324791. Google Scholar

[51]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'', Translations of Mathematical Monographs, 140 (1994). Google Scholar

[52]

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. Google Scholar

[53]

M. Yu, Q. Gao, H. E. Epstein and X. S. Zhang, An ecohydrological analysis for optimal use of redistributed water among vegetation patches,, Ecol. Appl., 18 (2008), 1679. doi: 10.1890/07-0640.1. Google Scholar

[54]

N. Zeng and J. Yoon, Expansion of the world's deserts due to vegetation-albedo feedback under global warming,, Geophys. Res. Lett., 36 (2009). Google Scholar

show all references

References:
[1]

M. R. Agular and O. E. Sala, Patch structure, dynamics and implications for the functioning ofarid ecosystems,, Trends Ecol. Evol., 14 (1999), 273. doi: 10.1016/S0169-5347(99)01612-2. Google Scholar

[2]

E. O. Alzahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models,, Math. Model. Nat. Phenom., 5 (2010), 13. Google Scholar

[3]

N. Barbier, P. Couteron, R. Lefever, V. Deblauwe and O. Lejeune, Spatial decoupling of facilitation, competition at the origin ofgapped vegetation patterns,, Ecology, 89 (2008), 1521. doi: 10.1890/07-0365.1. Google Scholar

[4]

S. S. Berg and D. L. Dunkerley, Patterned mulga near Alice Springs, central Australia, and thepotential threat of firewood collection on this vegetation community,, J. Arid Environ., 59 (2004), 313. doi: 10.1016/j.jaridenv.2003.12.007. Google Scholar

[5]

A. I. Borthagaraya, M. A. Fuentesa and P. A. Marque, Vegetation pattern formation in a fog-dependent ecosystem,, J. Theor. Biol., 265 (2010), 18. doi: 10.1016/j.jtbi.2010.04.020. Google Scholar

[6]

R. M. Callaway, Positive interactions among plants,, Botanical Rev., 61 (1995), 306. doi: 10.1007/BF02912621. Google Scholar

[7]

P. Couteron, A. Mahamane, P. Ouedraogo and J. Seghieri, Differences between banded thickets (tiger bush) at two sites in West Africa,, J. Veg. Sci., 11 (2000), 321. doi: 10.2307/3236624. Google Scholar

[8]

V. Deblauwe, P. Couteron, J. Bogaert and N. Barbier, Determinants and dynamics of banded vegetation pattern migration in arid climates,, Ecological monographs, 82 (2012), 3. doi: 10.5061/dryad.1qr41s56. Google Scholar

[9]

J. D. Dockery and R. Lui, Existence of traveling wave solutions for a bistable evolutionary ecology model,, SIAM J. Math. Anal., 23 (1992), 870. doi: 10.1137/0523046. Google Scholar

[10]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Cong. Numer., 30 (1981), 265. Google Scholar

[11]

E. J. Doedel, H. B. Keller and J. P. Kernévez, Numerical analysis and control of bifurcation problems. I. Bifurcation in finite dimensions,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 1 (1991), 493. Google Scholar

[12]

E. J. Doedel, W. Govaerts, Yu. A. Kuznetsov and A. Dhooge, Numerical continuation of branch points of equilibria and periodic orbits,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 841. Google Scholar

[13]

D. L. Dunkerley and K. J. Brown, Oblique vegetation banding in the Australian arid zone: Implications for theories of pattern evolution and maintenance,, J. Arid Environments, 52 (2002), 163. doi: 10.1006/jare.2001.0940. Google Scholar

[14]

J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, J. Dyn. Diff. Eq., 21 (2009), 663. doi: 10.1007/s10884-009-9152-7. Google Scholar

[15]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Rat. Mech. Anal., 65 (1977), 335. Google Scholar

[16]

A. Giannini, M. Biasutti and M. M. Verstraete, A climate model-based review of drought in the Sahel: Desertification, the re-greening and climate change,, Global Planet. Change, 64 (2008), 119. doi: 10.1016/j.gloplacha.2008.05.004. Google Scholar

[17]

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

[18]

V. Guttal and C. Jayaprakash, Self-organisation and productivity in semi-arid ecosystems: Implications of seasonality in rainfall,, J. Theor. Biol., 248 (2007), 290. Google Scholar

[19]

R. HilleRisLambers, M. Rietkerk, F. van de Bosch, H. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems,, Ecology, 82 (2001), 50. Google Scholar

[20]

R. C. Hills, The influence of land management and soil characteristics on infiltration and the occurrence of overland flow,, J. Hydrology, 13 (1971), 163. doi: 10.1016/0022-1694(71)90213-7. Google Scholar

[21]

Y. Jin and X.-Q. Zhao, Bistable waves for a class of cooperative reaction-diffusion systems,, J. Biol. Dyn., 2 (2008), 196. Google Scholar

[22]

S. Kéfi, M. Rietkerk, C. L. Alados, Y. Pueyo, A. ElAich, V. Papanastasis 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

[23]

S. Kéfi, M. Rietkerk and G. G. Katul, Vegetation pattern shift as a result of rising atmospheric CO2 in arid ecosystems,, Theor. Pop. Biol., 74 (2008), 332. doi: 10.1016/j.tpb.2008.09.004. Google Scholar

[24]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation,, Science, 284 (1999), 1826. doi: 10.1126/science.284.5421.1826. Google Scholar

[25]

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

[26]

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

[27]

R. Lefever, N. Barbier, P. Couteron and O. Lejeune, Deeply gapped vegetation patterns: On crown/root allometry, criticality and desertification,, J. Theor. Biol., 261 (2009), 194. doi: 10.1016/j.jtbi.2009.07.030. Google Scholar

[28]

W. MacFadyen, Vegetation patterns in the semi-desert plains of British Somaliland,, Geographical J., 115 (1950), 199. doi: 10.2307/1789384. Google Scholar

[29]

A. K. McDonald, R. J. Kinucan and L. E. Loomis, Ecohydrological interactions within banded vegetation in the northeastern Chihuahuan Desert, USA,, Ecohydrology, 2 (2009), 66. Google Scholar

[30]

C. Montaña, The colonization of bare areas in two-phase mosaics of an arid ecosystem,, J. Ecol., 80 (1992), 315. doi: 10.2307/2261014. Google Scholar

[31]

E. N. Mueller, J. Wainwright and A. J. Parsons, The stability of vegetation boundaries and the propagation ofdesertification in the American Southwest: A modelling approach,, Ecol.\ Model., 208 (2007), 91. doi: 10.1016/j.ecolmodel.2007.04.010. Google Scholar

[32]

Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems,, Oikos, 117 (2008), 1522. doi: 10.1111/j.0030-1299.2008.16735.x. Google Scholar

[33]

M. Rietkerk, P. Ketner, J. Burger, B. Hoorens and H. Olff, Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa,, Plant Ecology, 148 (2000), 207. doi: 10.1023/A:1009828432690. Google Scholar

[34]

M. Rietkerk, M. C. Boerlijst, F. van Langevelde, R. HilleRisLambers, J. van de Koppel, H. H. T. Prins and A. de Roos, Self-organisation of vegetation in arid ecosystems,, Am. Nat., 160 (2002), 524. doi: 10.1086/342078. Google Scholar

[35]

M. Rietkerk, S. C. Dekker, P. C. de Ruiter and J. van de Koppel, Self-organized patchiness and catastrophic shifts in ecosystems,, Science, 305 (2004), 1926. doi: 10.1126/science.1101867. Google Scholar

[36]

T. M. Scanlon, K. K. Caylor, 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

[37]

W. H. Schlesinger, J. F. Reynolds, G. L. Cunningham, L. F. Huenneke, W. M. Jarrell, R. A. Virginia and W. G. Whitford, Biological feedbacks in global desertification,, Science, 247 (1990), 1043. doi: 10.1126/science.247.4946.1043. Google Scholar

[38]

J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes,, J. Math. Biol., 51 (2005), 183. doi: 10.1007/s00285-005-0319-5. Google Scholar

[39]

J. A. Sherratt and G. J. Lord, Nonlinear dynamics, pattern bifurcations in a model for vegetation stripes in semi-arid environments,, Theor. Pop. Biol., 71 (2007), 1. Google Scholar

[40]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments I,, Nonlinearity, 23 (2010), 2657. doi: 10.1088/0951-7715/23/10/016. Google Scholar

[41]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments II. Patterns with the largest possible propagation speeds,, Proc. R. Soc. Lond. A}, 467 (2011), 3272. doi: 10.1098/rspa.2011.0194. Google Scholar

[42]

J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments III. The transition between homoclinic solutions,, submitted., (). Google Scholar

[43]

G.-Q. Sun, Z. Jin and Q. Tan, Measurement of self-organization in arid ecosystems,, J. Biol. Systems, 18 (2010), 495. doi: 10.1142/S0218339010003366. Google Scholar

[44]

D. J. Tongway and J. A. Ludwig, Theories on the origins, maintainance, dynamics, and functioning ofbanded landscapes,, in, (2001), 20. Google Scholar

[45]

N. Ursino and S. Contarini, Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity,, Adv. Water Resour., 29 (2006), 1556. doi: 10.1016/j.advwatres.2005.11.006. Google Scholar

[46]

N. Ursino, Modeling banded vegetation patterns in semiarid regions: Inter-dependence between biomass growth rate and relevant hydrological processes,, Water Resour. Res., 43 (2007). Google Scholar

[47]

N. Ursino, Above and below ground biomass patterns in arid lands,, Ecol. Model., 220 (2009), 1411. doi: 10.1016/j.ecolmodel.2009.02.023. Google Scholar

[48]

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

[49]

C. Valentin and J. M. d'Herbès, Niger tiger bush as a natural water harvesting system,, Catena, 37 (1999), 231. doi: 10.1016/S0341-8162(98)00061-7. Google Scholar

[50]

J. van de Koppel, M. Rietkerk, F. van Langevelde, L. Kumar, C. A. Klausmeier, J. M. Fryxell, J. W. Hearne, J. van Andel, N. de Ridder, M. A. Skidmore, L. Stroosnijder and H. H. T. Prins, Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems,, Am. Nat., 159 (2002), 209. doi: 10.1086/324791. Google Scholar

[51]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'', Translations of Mathematical Monographs, 140 (1994). Google Scholar

[52]

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. Google Scholar

[53]

M. Yu, Q. Gao, H. E. Epstein and X. S. Zhang, An ecohydrological analysis for optimal use of redistributed water among vegetation patches,, Ecol. Appl., 18 (2008), 1679. doi: 10.1890/07-0640.1. Google Scholar

[54]

N. Zeng and J. Yoon, Expansion of the world's deserts due to vegetation-albedo feedback under global warming,, Geophys. Res. Lett., 36 (2009). Google Scholar

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