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An analysis of vegetation stripe formation in semi-arid landscapes. (English) Zbl 1068.92047
Summary: Vegetation stripes (“tiger bush”) are a characteristic feature of semi-arid environments. The stripes typically lie along the contours of gentle slopes, and some authors report a gradual uphill migration. A previous mathematical model of {\it C. A. Klausmeier} [Science 284, 1826--1828 (1999)] has shown that this phenomenon can be explained relatively simply by the downhill flow of rainwater coupled with the diffusive spread of the plant population. This paper presents a detailed analysis of pattern formation in the Klausmeier model. The author derives formulae for the wavelength and migration speed of the predicted patterns, and systematically investigates how these depend on model parameters. The results make new predictions and suggest possible approaches to testing the model.

92C15Developmental biology, pattern formation
92C80Plant biology
35Q80Applications of PDE in areas other than physics (MSC2000)
Full Text: DOI
[1] Couteron, P., Lejeune, O.: Periodic spotted patterns in semi-arid vegetation explained by a propagation-inhibition model. J. Ecol. 89, 616--628 (2001) · doi:10.1046/j.0022-0477.2001.00588.x
[2] Dunkerley, D.L.: Banded vegetation: development under uniform rainfall from a simple cellular automaton model. Plant Ecol 129, 103--111 (1997) · doi:10.1023/A:1009725732740
[3] Dunkerley, D.L., Brown, K.J.: Oblique vegetation banding in the Australian arid zone: implications for theories of pattern evolution and maintenance. J. Arid Environments 52, 163--181 (2002) · doi:10.1006/jare.2001.0940
[4] Hemming, C.F.: Vegetation arcs in Somaliland. J. Ecol. 53, 57--67 (1965) · doi:10.2307/2257565
[5] HilleRisLambers, R., Rietkerk, M., van de Bosch, F., Prins, H.H.T., de Kroon, H.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82, 50--61 (2001) · doi:10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2
[6] Jorné, J.: The effects of ionic migration on oscillations and pattern formation in chemical systems. J. Theor. Biol. 43, 375--380 (1974) · doi:10.1016/S0022-5193(74)80067-6
[7] Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826--1828 (1999) · doi:10.1126/science.284.5421.1826
[8] Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263--294 (1997) · Zbl 0903.92031 · doi:10.1007/BF02462004
[9] Lejeune, O., Tlidi, M.: A model for the explanation of vegetation stripes (tiger bush). J. Vegetation Science 10, 201--208 (1999) · doi:10.2307/3237141
[10] Mabbutt, J.A., Fanning, P.C.: Vegetation banding in arid Western Australia. J. Arid Environments 12, 41--59 (1987)
[11] MacFadyen, W.: Vegetation patterns in the semi-desert plains of British Somaliland. Geographical J. 115, 199--211 (1950) · doi:10.2307/1789384
[12] Mauchamp, A., Rambal, S., Lepart, J.: Simulating the dynamics of a vegetation mosaic: a spatialized functional model. Ecological Modelling 71, 107--130 (1994) · doi:10.1016/0304-3800(94)90078-7
[13] Meron, E., Gilad, E., von Hardenberg, J., Shachak, M., Zarmi, Y.: Vegetation patterns along a rainfall gradient. Chaos, Solitons and Fractals 19, 367--376 (2004) · Zbl 1083.92039 · doi:10.1016/S0960-0779(03)00049-3
[14] Montaña, C., Lopez-Portillo, J., Mauchamp, A.: The response of two woody species to the conditions created by a shifting ecotone in an arid ecosystem. J. Ecol. 78, 789--798 (1990) · doi:10.2307/2260899
[15] Montaña, C.: The colonization of bare areas in two-phase mosaics of an arid ecosystem. J. Ecol. 80, 315--327 (1992) · doi:10.2307/2261014
[16] Perumpanani, A.J., Sherratt, J.A., Maini, P.K.: Phase differences in reaction-diffusion-advection systems and applications to morphogenesis. IMA J. Appl. Math. 55, 19--33 (1995) · Zbl 0843.35041 · doi:10.1093/imamat/55.1.19
[17] Rietkerk, M., Boerlijst, M.C., van Langevelde, F., HilleRisLambers, R., van de Koppel, J., Prins, H.H.T., de Roos, A.: Self-organisation of vegetation in arid ecosystems. Am. Nat. 160, 524--530 (2002) · doi:10.1086/342078
[18] Rovinsky, A.B., Menzinger, M.: Self-organization induced by the differential flow of activator and inhibitor. Phys. Rev. Lett. 70, 778--781 (1993) · doi:10.1103/PhysRevLett.70.778
[19] Satnoianu, R.A., Maini, P.K., Menzinger, M.: Parameter space analysis, pattern sensitivity and model comparison for Turing and stationary flow-distributed waves. Physica D 160, 79--102 (2001) · Zbl 1001.35061 · doi:10.1016/S0167-2789(01)00345-1
[20] Thiéry, J.M., D’Herbès, J.-M., Valentin, C.: A model simulating the genesis of banded vegetation patterns in Niger. J. Ecol. 83, 497--507 (1995) · doi:10.2307/2261602
[21] van de Koppel, J., Rietkerk, M., van Langevelde, F., Kumar, L., Klausmeier, C.A., Fryxell, J.M., Hearne, J.W., van Andel, J., de Ridder, N., Skidmore, M.A., Stroosnijder, L., Prins, H.H.T.: Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems. Am. Nat. 159, 209--218 (2002) · doi:10.1086/324791
[22] von Hardenberg, J., Meron, E., Shachak, M., Zarmi, Y. Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87, art. no. 198101 (2001) · Zbl 1083.92039
[23] White, L.P.: Vegetation stripes on sheet wash surfaces. J. Ecol. 59, 615--622 (1971) · doi:10.2307/2258335
[24] Wickens, G.E., Collier, F.W.: Some vegetation patterns in the Republic of Sudan. Geoderma 6, 43--59 (1971) · doi:10.1016/0016-7061(71)90050-4
[25] Worrall, G.A.: The Butana grass patterns. J. Soil Sci. 10, 34--53 (1959) · doi:10.1111/j.1365-2389.1959.tb00664.x