# American Institute of Mathematical Sciences

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 and 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), 273-277. doi: 10.1016/S0169-5347(99)01612-2. [2] E. O. Alzahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35. [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), 1521-1531. doi: 10.1890/07-0365.1. [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), 313-350. doi: 10.1016/j.jaridenv.2003.12.007. [5] A. I. Borthagaraya, M. A. Fuentesa and P. A. Marque, Vegetation pattern formation in a fog-dependent ecosystem, J. Theor. Biol., 265 (2010), 18-26. doi: 10.1016/j.jtbi.2010.04.020. [6] R. M. Callaway, Positive interactions among plants, Botanical Rev., 61 (1995), 306-349. doi: 10.1007/BF02912621. [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), 321-328. doi: 10.2307/3236624. [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), 3-21. doi: 10.5061/dryad.1qr41s56. [9] J. D. Dockery and R. Lui, Existence of traveling wave solutions for a bistable evolutionary ecology model, SIAM J. Math. Anal., 23 (1992), 870-888. doi: 10.1137/0523046. [10] E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Numer., 30 (1981), 265-284. [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), 493-520. [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), 841-860. [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), 163-181. doi: 10.1006/jare.2001.0940. [14] J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dyn. Diff. Eq., 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7. [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), 335-361. [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), 119-128. doi: 10.1016/j.gloplacha.2008.05.004. [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), 680-691. doi: 10.1016/j.jtbi.2006.08.006. [18] V. Guttal and C. Jayaprakash, Self-organisation and productivity in semi-arid ecosystems: Implications of seasonality in rainfall, J. Theor. Biol., 248 (2007), 290-500. [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), 50-61. [20] R. C. Hills, The influence of land management and soil characteristics on infiltration and the occurrence of overland flow, J. Hydrology, 13 (1971), 163-181. doi: 10.1016/0022-1694(71)90213-7. [21] Y. Jin and X.-Q. Zhao, Bistable waves for a class of cooperative reaction-diffusion systems, J. Biol. Dyn., 2 (2008), 196-207. [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), 213-217. doi: 10.1038/nature06111. [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), 332-344. doi: 10.1016/j.tpb.2008.09.004. [24] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. doi: 10.1126/science.284.5421.1826. [25] A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency, J. Theor. Biol., 256 (2009), 574-583. doi: 10.1016/j.jtbi.2008.10.020. [26] R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294. doi: 10.1007/BF02462004. [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), 194-209. doi: 10.1016/j.jtbi.2009.07.030. [28] W. MacFadyen, Vegetation patterns in the semi-desert plains of British Somaliland, Geographical J., 115 (1950), 199-211. doi: 10.2307/1789384. [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), 66-71. [30] C. Montaña, The colonization of bare areas in two-phase mosaics of an arid ecosystem, J. Ecol., 80 (1992), 315-327. doi: 10.2307/2261014. [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), 91-101. doi: 10.1016/j.ecolmodel.2007.04.010. [32] Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems, Oikos, 117 (2008), 1522-1532. doi: 10.1111/j.0030-1299.2008.16735.x. [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), 207-224. doi: 10.1023/A:1009828432690. [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), 524-530. doi: 10.1086/342078. [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), 1926-1929. doi: 10.1126/science.1101867. [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), 209-212. doi: 10.1038/nature06060. [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), 1043-1048. doi: 10.1126/science.247.4946.1043. [38] J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes, J. Math. Biol., 51 (2005), 183-197. doi: 10.1007/s00285-005-0319-5. [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), 1-11. [40] J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments I, Nonlinearity, 23 (2010), 2657-2675. doi: 10.1088/0951-7715/23/10/016. [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), 3272-3294. doi: 10.1098/rspa.2011.0194. [42] J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments III. The transition between homoclinic solutions, submitted. [43] G.-Q. Sun, Z. Jin and Q. Tan, Measurement of self-organization in arid ecosystems, J. Biol. Systems, 18 (2010), 495-508. doi: 10.1142/S0218339010003366. [44] D. J. Tongway and J. A. Ludwig, Theories on the origins, maintainance, dynamics, and functioning ofbanded landscapes, in "Banded Vegetation Patterning in Arid and Semi-Arid Environments''(eds. D. J. Tongway, C. Valentin and J. Seghieri), Springer, New York, (2001), 20-31. [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), 1556-1564. doi: 10.1016/j.advwatres.2005.11.006. [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), W04412. [47] N. Ursino, Above and below ground biomass patterns in arid lands, Ecol. Model., 220 (2009), 1411-1418. doi: 10.1016/j.ecolmodel.2009.02.023. [48] 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. [49] C. Valentin and J. M. d'Herbès, Niger tiger bush as a natural water harvesting system, Catena, 37 (1999), 231-256. doi: 10.1016/S0341-8162(98)00061-7. [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), 209-218. doi: 10.1086/324791. [51] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994. [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), 1771-1776; American Mathematical Society, Providence RI, 1994. [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), 1679-1688. doi: 10.1890/07-0640.1. [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), art. no. L17401.

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), 273-277. doi: 10.1016/S0169-5347(99)01612-2. [2] E. O. Alzahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35. [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), 1521-1531. doi: 10.1890/07-0365.1. [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), 313-350. doi: 10.1016/j.jaridenv.2003.12.007. [5] A. I. Borthagaraya, M. A. Fuentesa and P. A. Marque, Vegetation pattern formation in a fog-dependent ecosystem, J. Theor. Biol., 265 (2010), 18-26. doi: 10.1016/j.jtbi.2010.04.020. [6] R. M. Callaway, Positive interactions among plants, Botanical Rev., 61 (1995), 306-349. doi: 10.1007/BF02912621. [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), 321-328. doi: 10.2307/3236624. [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), 3-21. doi: 10.5061/dryad.1qr41s56. [9] J. D. Dockery and R. Lui, Existence of traveling wave solutions for a bistable evolutionary ecology model, SIAM J. Math. Anal., 23 (1992), 870-888. doi: 10.1137/0523046. [10] E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Numer., 30 (1981), 265-284. [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), 493-520. [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), 841-860. [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), 163-181. doi: 10.1006/jare.2001.0940. [14] J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dyn. Diff. Eq., 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7. [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), 335-361. [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), 119-128. doi: 10.1016/j.gloplacha.2008.05.004. [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), 680-691. doi: 10.1016/j.jtbi.2006.08.006. [18] V. Guttal and C. Jayaprakash, Self-organisation and productivity in semi-arid ecosystems: Implications of seasonality in rainfall, J. Theor. Biol., 248 (2007), 290-500. [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), 50-61. [20] R. C. Hills, The influence of land management and soil characteristics on infiltration and the occurrence of overland flow, J. Hydrology, 13 (1971), 163-181. doi: 10.1016/0022-1694(71)90213-7. [21] Y. Jin and X.-Q. Zhao, Bistable waves for a class of cooperative reaction-diffusion systems, J. Biol. Dyn., 2 (2008), 196-207. [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), 213-217. doi: 10.1038/nature06111. [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), 332-344. doi: 10.1016/j.tpb.2008.09.004. [24] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828. doi: 10.1126/science.284.5421.1826. [25] A. Y. Kletter, J. von Hardenberg, E. Meron and A. Provenzale, Patterned vegetation and rainfall intermittency, J. Theor. Biol., 256 (2009), 574-583. doi: 10.1016/j.jtbi.2008.10.020. [26] R. Lefever and O. Lejeune, On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294. doi: 10.1007/BF02462004. [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), 194-209. doi: 10.1016/j.jtbi.2009.07.030. [28] W. MacFadyen, Vegetation patterns in the semi-desert plains of British Somaliland, Geographical J., 115 (1950), 199-211. doi: 10.2307/1789384. [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), 66-71. [30] C. Montaña, The colonization of bare areas in two-phase mosaics of an arid ecosystem, J. Ecol., 80 (1992), 315-327. doi: 10.2307/2261014. [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), 91-101. doi: 10.1016/j.ecolmodel.2007.04.010. [32] Y. Pueyo, S. Kéfi, C. L. Alados and M. Rietkerk, Dispersal strategies and spatial organization of vegetation in arid ecosystems, Oikos, 117 (2008), 1522-1532. doi: 10.1111/j.0030-1299.2008.16735.x. [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), 207-224. doi: 10.1023/A:1009828432690. [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), 524-530. doi: 10.1086/342078. [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), 1926-1929. doi: 10.1126/science.1101867. [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), 209-212. doi: 10.1038/nature06060. [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), 1043-1048. doi: 10.1126/science.247.4946.1043. [38] J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes, J. Math. Biol., 51 (2005), 183-197. doi: 10.1007/s00285-005-0319-5. [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), 1-11. [40] J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments I, Nonlinearity, 23 (2010), 2657-2675. doi: 10.1088/0951-7715/23/10/016. [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), 3272-3294. doi: 10.1098/rspa.2011.0194. [42] J. A. Sherratt, Pattern solutions of the Klausmeier model for banded vegetationin semi-arid environments III. The transition between homoclinic solutions, submitted. [43] G.-Q. Sun, Z. Jin and Q. Tan, Measurement of self-organization in arid ecosystems, J. Biol. Systems, 18 (2010), 495-508. doi: 10.1142/S0218339010003366. [44] D. J. Tongway and J. A. Ludwig, Theories on the origins, maintainance, dynamics, and functioning ofbanded landscapes, in "Banded Vegetation Patterning in Arid and Semi-Arid Environments''(eds. D. J. Tongway, C. Valentin and J. Seghieri), Springer, New York, (2001), 20-31. [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), 1556-1564. doi: 10.1016/j.advwatres.2005.11.006. [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), W04412. [47] N. Ursino, Above and below ground biomass patterns in arid lands, Ecol. Model., 220 (2009), 1411-1418. doi: 10.1016/j.ecolmodel.2009.02.023. [48] 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. [49] C. Valentin and J. M. d'Herbès, Niger tiger bush as a natural water harvesting system, Catena, 37 (1999), 231-256. doi: 10.1016/S0341-8162(98)00061-7. [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), 209-218. doi: 10.1086/324791. [51] A. I. Volpert, V. A. Volpert and V. A. Volpert, "Travelling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994. [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), 1771-1776; American Mathematical Society, Providence RI, 1994. [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), 1679-1688. doi: 10.1890/07-0640.1. [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), art. no. L17401.
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