Abstract
Patterned vegetation occurs in many semi-arid regions of the world. Most previous studies have assumed that patterns form from a starting point of uniform vegetation, for example as a response to a decrease in mean annual rainfall. However an alternative possibility is that patterns are generated when bare ground is colonised. This paper investigates the conditions under which colonisation leads to patterning on sloping ground. The slope gradient plays an important role because of the downhill flow of rainwater. One long-established consequence of this is that patterns are organised into stripes running parallel to the contours; such patterns are known as banded vegetation or tiger bush. This paper shows that the slope also has an important effect on colonisation, since the uphill and downhill edges of an isolated vegetation patch have different dynamics. For the much-used Klausmeier model for semi-arid vegetation, the author shows that without a term representing water diffusion, colonisation always generates uniform vegetation rather than a pattern. However the combination of a sufficiently large water diffusion term and a sufficiently low slope gradient does lead to colonisation-induced patterning. The author goes on to consider colonisation in the Rietkerk model, which is also in widespread use: the same conclusions apply for this model provided that a small threshold is imposed on vegetation biomass, below which plant growth is set to zero. Since the two models are quite different mathematically, this suggests that the predictions are a consequence of the basic underlying assumption of water redistribution as the pattern generation mechanism.
Similar content being viewed by others
Notes
Assessment of vegetation band migration using satellite imagery was made possible by the declassification in 1995 of images from the US satellite missions Corona (1959–1972), Argon (1961–1964) and Lanyard (1963).
The numerical details of my implementation are as follows. I solve (1) using a semi-implicit finite difference scheme with upwinding, using a grid spacing \(\delta x=0.5\) and a time step \(\delta t=\min \{0.8\delta x/\nu ,\,0.1\delta x^2/\max \{D,1\}\}\); here the factor of 0.8 ensures that the CFL number is less than 1. I solve on a space domain of length 500 with Dirichlet conditions \((u,w)=(0,A)\) at \(x=-250\) and \((u,w)=(u_+,w_+)\) at \(x=250\). I solve over a time interval of length 1000. For the first iteration of the bisection method I use initial conditions \((u,w)=(0,A)\) on \(-250<x<0\) and \((u,w)=(u_+,w_+)\) on \(0<x<250\). For subsequent iterations I use the final solution form from the previous iteration, translated to be centred at \(x=0\): this accelerates convergence to the travelling wave profile. I estimate the velocity of this wave via the distance travelled over the final 100 time units, or over an earlier 100 time units if the front reaches an end of the domain before the end of the solution period. I terminate my numerical bisection method when two successive values of A differ by less than \(10^{-3}\).
References
Anderson KE, Hilker FM, Nisbet RM (2012) Directional biases and resource-dependence in dispersal generate spatial patterning in a consumer–producer model. Ecol Lett 15:209–217
Archer NAL, Quinton JN, Hess TM (2012) Patch vegetation and water redistribution above and below ground in south-east Spain. Ecohydrology 5:108–120
Atkinson K, Weimin H, Stewart DE (2009) Numerical solution of ordinary differential equations. Wiley, Hoboken
Barbier N, Couteron P, Lejoly J, Deblauwe V, Lejeune O (2006) Self-organized vegetation patterning as a fingerprint of climate and human impact on semi-arid ecosystems. Ecology 94:537–547
Baudena M, Rietkerk M (2013) Complexity and coexistence in a simple spatial model for arid savanna ecosystems. Theor Ecol 6:131–141
Bel G, Hagberg A, Meron E (2012) Gradual regime shifts in spatially extended ecosystems. Theor Ecol 5:591–604
Benguria RD, Depassier MC (2014) Shift in the speed of reaction–diffusion equation with a cut-off: pushed and bistable fronts. Phys D 280:38–43
Berg SS, Dunkerley DL (2004) Patterned mulga near Alice Springs, central Australia, and the potential threat of firewood collection on this vegetation community. J Arid Environ 59:313–350
Boaler SB, Hodge CAH (1964) Observations on vegetation arcs in the northern region, Somali Republic. J Ecol 52:511–544
Bochet E, García-Fayos P, Poesen J (2009) Topographic thresholds for plant colonization on semi-arid eroded slopes. Earth Surf Proc Land 34:1758–1771
Bonachela JA, Pringle RM, Sheffer E, Coverdale TC, Guyton JA, Caylor KK, Levin SA, Tarnita CE (2015) Termite mounds can increase the robustness of dryland ecosystems to climatic change. Science 347:651–655
Borthagaray AI, Fuentes MA, Marquet PA (2010) Vegetation pattern formation in a fog-dependent ecosystem. J Theor Biol 265:18–26
Buis E, Veldkamp A, Boeken B, Van Breemen N (2009) Controls on plant functional surface cover types along a precipitation gradient in the Negev Desert of Israel. J Arid Environ 73:82–90
Cantón Y, Del Barrio G, Solé-Benet A, Lázaro R (2004) Topographic controls on the spatial distribution of ground cover in the Tabernas badlands of SE Spain. Catena 55:341–365
Cartení F, Marasco A, Bonanomi G, Mazzoleni S, Rietkerk M, Giannino F (2012) Negative plant soil feedback and ring formation in clonal plants. J Theor Biol 313:153–161
Caylor KK, Okin GS, Turnbull L, Wainwright J, Wiegand T, Franz TE, Parsons AJ (2014) Integrating short- and long-range processes into models: the emergence of pattern. In: Mueller EV, Wainwright J, Parsons AJ, Turnbull L (eds) Patterns of land degradation in drylands—understanding self-organised ecogeomorphic systems. Springer, Dordrecht, pp 141–170
Corrado R, Cherubini AM, Pennetta C (2014) Early warning signals of desertification transitions in semiarid ecosystems. Phys Rev E 90:062705
Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford
Cruickshank I, Gurney WS, Veitch AR (1999) The characteristics of epidemics and invasions with thresholds. Theor Popul Biol 56:279–292
Dagbovie AS, Sherratt JA (2014) Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments. J R Soc Interface 11:20140465
Davis HG, Taylor CM, Lambrinos JG, Strong DR (2004) Pollen limitation causes an Allee effect in a wind-pollinated invasive grass (Spartina alterniflora). PNAS USA 101:13804–13807
Deblauwe V, Barbier N, Couteron P, Lejeune O, Bogaert J (2008) The global biogeography of semi-arid periodic vegetation patterns. Glob Ecol Biogoegr 17:715–723
Deblauwe V, Couteron P, Lejeune O, Bogaert J, Barbier N (2011) Environmental modulation of self-organized periodic vegetation patterns in Sudan. Ecography 34:990–1001
Deblauwe V, Couteron P, Bogaert J, Barbier N (2012) Determinants and dynamics of banded vegetation pattern migration in arid climates. Ecol Monogr 82:3–21
Dembélé F, Picard N, Karembé M, Birnbaum P (2006) Tree vegetation patterns along a gradient of human disturbance in the Sahelian area of Mali. J Arid Environ 64:284–297
Dralle D, Boisrame G, Thompson SE (2014) Spatially variable groundwater recharge and the hillslope hydrologic response: analytical solutions to the linearized hillslope Boussinesq equation. Water Resour Res 50:8515–8530
Duffy KJ, Patrick KL, Johnson SD (2013) Does the likelihood of an Allee effect on plant fecundity depend on the type of pollinator? J Ecol 101:953–962
Dunkerley DL, Brown KJ (2002) Oblique vegetation banding in the Australian arid zone: implications for theories of pattern evolution and maintenance. J Arid Environ 52:163–181
Galle S, Ehrmann M, Peugeot C (1999) Water balance in a banded vegetation pattern: a case study of tiger bush in western Niger. Catena 37:197–216
Gilad E, von Hardenberg J, Provenzale A, Shachak M, Meron E (2004) Ecosystem engineers: from pattern formation to habitat creation. Phys Rev Lett 93:098105
Gilad E, Von Hardenberg J, Provenzale A, Shachak M, Meron E (2007) A mathematical model of plants as ecosystem engineers. J Theor Biol 244:680–691
Gowda K, Riecke H, Silber M (2014) Transitions between patterned states in vegetation models for semiarid ecosystems. Phys Rev E 89:022701
Greenwood JEGW (1957) The development of vegetation patterns in Somaliland Protectorate. Geogr J 123:465–473
Gurney WSC, Veitch AR, Cruickshank I, McGeachin G (1998) Circles and spirals: population persistence in a spatially explicit predator–prey model. Ecology 79:2516–2530
Guttal V, Jayaprakash C (2007) Self-organization and productivity in semi-arid ecosystems: implications of seasonality in rainfall. J Theor Biol 248:290–500
Hagan PS (1981) The instability of non-monotonic wave solutions of parabolic equations. Stud Appl Math 64:57–88
Hejcmanová P, Hejcman M, Camara AA, Antonínová M (2010) Exclusion of livestock grazing and wood collection in dryland savannah: an effect on long-term vegetation succession. Afr J Ecol 48:408–417
Hemming CF (1965) Vegetation arcs in Somaliland. J Ecol 53:57–67
Henry D (1981) Geometric theory of semilinear parabolic equations. Springer, Berlin
HilleRisLambers R, Rietkerk M, van de Bosch F, Prins HHT, de Kroon H (2001) Vegetation pattern formation in semi-arid grazing systems. Ecology 82:50–61
Hooper DU, Johnson L (1999) Nitrogen limitation in dryland ecosystems: responses to geographical and temporal variation in precipitation. Biogeochemistry 46:247–293
Istanbulluoglu E, Bras RL (2006) On the dynamics of soil moisture, vegetation, and erosion: implications of climate variability and change. Water Resour Res 42:W06418
Kealy BJ, Wollkind DJ (2012) A nonlinear stability analysis of vegetative Turing pattern formation for an interaction–diffusion plant-surface water model system in an arid flat environment. Bull Math Biol 74:803–833
Kéfi S, Rietkerk M, van Baalen M, Loreau M (2007) Local facilitation, bistability and transitions in arid ecosystems. Theor Popul Biol 71:367–379
Kéfi S, Rietkerk M, Katul GG (2008) Vegetation pattern shift as a result of rising atmospheric \({\rm CO}_2\) in arid ecosystems. Theor Popul Biol 74:332–344
Klausmeier CA (1999) Regular and irregular patterns in semiarid vegetation. Science 284:1826–1828
Kletter AY, von Hardenberg J, Meron E, Provenzale A (2009) Patterned vegetation and rainfall intermittency. J Theor Biol 256:574–583
Kusserow H, Haenisch H (1999) Monitoring the dynamics of “tiger bush” (brousse tigrée) in the West African Sahel (Niger) by a combination of Landsat MSS and TM, SPOT, aerial and kite photographs. Photogramm Fernerkund Geoinf 2:77–94
Lefever R, Barbier H, Couteron P, Lejeune O (2009) Deeply gapped vegetation patterns: on crown/root allometry, criticality and desertification. J Theor Biol 261:194–209
Lefever R, Lejeune O (1997) On the origin of tiger bush. Bull Math Biol 59:263–294
Liu Q-X, Jin Z, Li BL (2008) Numerical investigation of spatial pattern in a vegetation model with feedback function. J Theor Biol 254:350–360
Marasco A, Iuorio A, Cartení F, Bonanomi G, Tartakovsky DM, Mazzoleni S, Giannino F (2014) Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback. Bull Math Biol 76:2866–2883
Martínez-García R, Calabrese JM, Garcia EH, López C (2014) Minimal mechanisms for vegetation patterns in semiarid regions. Philos Trans R Soc A 372:20140068
Meron E, Gilad E, von Hardenberg J, Shachak M, Zarmi Y (2004) Vegetation patterns along a rainfall gradient. Chaos Solitons Fractals 19:367–376
Meron E (2012) Pattern-formation approach to modelling spatially extended ecosystems. Ecol Model 234:70–82
Moreno-de las Heras M, Saco PM, Willgoose GR, Tongway DJ (2012) Variations in hydrological connectivity of Australian semiarid landscapes indicate abrupt changes in rainfall-use efficiency of vegetation. J Geophys Res 117:G03009
Müller J (2013) Floristic and structural pattern and current distribution of tiger bush vegetation in Burkina Faso (West Africa), assessed by means of belt transects and spatial analysis. Appl Ecol Environ Res 11:153–171
Murray JD (2003) Mathematical biology II: spatial models and biomedical applications. Springer, New York
Nippert JB, Knapp AK (2007a) Soil water partitioning contributes to species coexistence in tallgrass prairie. Oikos 116:1017–1029
Nippert JB, Knapp AK (2007b) Linking water uptake with rooting patterns in grassland species. Oecologia 153:261–272
Pelletier JD, DeLong SB, Orem CA, Becerra P, Compton K, Gressett K, Lyons-Baral J, McGuire LA, Molaro JL, Spinler JC (2012) How do vegetation bands form in dry lands? Insights from numerical modeling and field studies in southern Nevada. USA. J Geophys Res 117:F04026
Penny GG, Daniels KE, Thompson SE (2013) Local properties of patterned vegetation: quantifying endogenous and exogenous effects. Philos Trans R Soc A 371:20120359
Popović N (2011) A geometric analysis of front propagation in a family of degenerate reaction–diffusion equations with cutoff. ZAMM 62:405–437
Pueyo Y, Kéfi S, Alados CL, Rietkerk M (2008) Dispersal strategies and spatial organization of vegetation in arid ecosystems. Oikos 117:1522–1532
Pueyo Y, Moret-Fernández D, Saiz H, Bueno CG, Alados CL (2013) Relationships between plant spatial patterns, water infiltration capacity, and plant community composition in semi-arid mediterranean ecosystems along stress gradients. Ecosystems 16:452–466
Rietkerk M, Ketner P, Burger J, Hoorens B, Olff H (2000) Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa. Plant Ecol 148:207–224
Rietkerk M, Boerlijst MC, van Langevelde F, HilleRisLambers R, van de Koppel J, Prins HHT, de Roos A (2002) Self-organisation of vegetation in arid ecosystems. Am Nat 160:524–530
Rietkerk M, Dekker SC, de Ruiter PC, van de Koppel J (2004) Self-organized patchiness and catastrophic shifts in ecosystems. Science 305:1926–1929
Saco PM, Willgoose GR, Hancock GR (2007) Eco-geomorphology of banded vegetation patterns in arid and semi-arid regions. Hydrol Earth Syst Sci 11:1717–1730
Schwinning S (2010) The ecohydrology of roots in rocks. Ecohydrology 3:238–245
Sheffer E, Hardenberg J, Yizhaq H, Shachak M, Meron E (2013) Emerged or imposed: a theory on the role of physical templates and self-organisation for vegetation patchiness. Ecol Lett 16:127–139
Sherratt JA (2005) An analysis of vegetation stripe formation in semi-arid landscapes. J Math Biol 51:183–197
Sherratt JA (2010) Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments I. Nonlinearity 23:2657–2675
Sherratt JA, Smith MJ, Rademacher JDM (2010) Patterns of sources and sinks in the complex Ginzburg–Landau equation with zero linear dispersion. SIAM J Appl Dyn Syst 9:883–918
Sherratt JA (2011) Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II. Patterns with the largest possible propagation speeds. Proc R Soc Lond A 467:3272–3294
Sherratt JA, Synodinos AD (2012) Vegetation patterns and desertification waves in semi-arid environments: mathematical models based on local facilitation in plants. Discret Contin Dyn Syst Ser B 17:2815–2827
Sherratt JA (2013a) History-dependent patterns of whole ecosystems. Ecol Complex 14:8–20
Sherratt JA (2013b) Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments III: the transition between homoclinic solutions. Phys D 242:30–41
Sherratt JA (2013c) Pattern solutions of the Klausmeier model for banded vegetation semi-arid environments IV: slowly moving patterns and their stability. SIAM J Appl Math 73:330–350
Sherratt JA (2013d) Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments V: the transition from patterns to desert. SIAM J Appl Math 73:1347–1367
Sherratt JA (2015) Using wavelength and slope to infer the historical origin of semi-arid vegetation bands. PNAS USA 112:4202–4207
Sherratt JA, Lord GJ (2007) Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. Theor Popul Biol 71:1–11
Siero E, Doelman A, Eppinga MB, Rademacher J, Rietkerk M, Siteur K (2015) Stripe pattern selection by advective reaction–diffusion systems: resilience of banded vegetation on slopes. Chaos 25:036411
Siteur K, Eppinga MB, Karssenberg D, Baudena M, Bierkens MFP, Rietkerk M (2014a) How will increases in rainfall intensity affect semiarid ecosystems? Water Resour Res 50:5980–6001
Siteur K, Siero E, Eppinga MB, Rademacher J, Doelman A, Rietkerk M (2014b) Beyond Turing: the response of patterned ecosystems to environmental change. Ecol Complex 20:81–96
Stewart J, Parsons AJ, Wainwright J, Okin GS, Bestelmeyer B, Fredrickson EL, Schlesinger WH (2014) Modelling emergent patterns of dynamic desert ecosystems. Ecol Monogr 84:373–410
Thompson SE, Harman CJ, Heine P, Katul GG (2010) Vegetation–infiltration relationships across climatic and soil type gradients. J Geophys Res Biogeosci 115:G02023
Thompson S, Katul G, Konings A, Ridolfi L (2011) Unsteady overland flow on flat surfaces induced by spatial permeability contrasts. Adv Water Res 34:1049–1058
Thompson S, Katul G (2009) Secondary seed dispersal and its role in landscape organization. Geophys Res Lett 36:L02402
Tongway DJ, Ludwig JA (2001) Theories on the origins, maintainance, dynamics, and functioning of banded landscapes. In: Tongway DJ, Valentin C, Seghieri J (eds) Banded vegetation patterning in arid and semi-arid environments. Springer, New York, pp 20–31
Ursino N (2005) The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments. Adv Water Resour 28:956–963
Ursino N, Contarini S (2006) Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity. Adv Water Resour 29:1556–1564
Valentin C, d’Herbès JM, Poesen J (1999) Soil and water components of banded vegetation patterns. Catena 37:1–24
van der Stelt S, Doelman A, Hek G, Rademacher JDM (2013) Rise and fall of periodic patterns for a generalized Klausmeier–Gray–Scott model. J Nonlinear Sci 23:39–95
Vezzoli R, De Michele C, Pavlopoulos H, Scholes RJ (2008) Dryland ecosystems: the coupled stochastic dynamics of soil water and vegetation and the role of rainfall seasonality. Phys Rev E 77:051908
von Hardenberg J, Meron E, Shachak M, Zarmi Y (2001) Diversity of vegetation patterns and desertification. Phys Rev Lett 87:198101
White LP (1969) Vegetation arcs in Jordan. J Ecol 57:461–464
Worrall GA (1959) The Butana grass patterns. J Soil Sci 10:34–53
Wu XB, Thurow TL, Whisenant SG (2000) Fragmentation and changes in hydrologic function of tiger bush landscapes, south-west Niger. J Ecol 88:790–800
Yizhaq H, Sela S, Svoray T, Assouline S, Bel G (2014) Effects of heterogeneous soil-water diffusivity on vegetation pattern formation. Water Resour Res 50:5743–5758
Zelnik YR, Kinast S, Yizhaq H, Bel G, Meron E (2013) Regime shifts in models of dryland vegetation. Philos Trans R Soc A 371:20120358
Acknowledgments
I am grateful to Eleanor Tanner for valuable discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sherratt, J.A. When does colonisation of a semi-arid hillslope generate vegetation patterns?. J. Math. Biol. 73, 199–226 (2016). https://doi.org/10.1007/s00285-015-0942-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-015-0942-8