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When does colonisation of a semi-arid hillslope generate vegetation patterns?

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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.

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Notes

  1. 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).

  2. 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 \(10^{-3}\).

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Acknowledgments

I am grateful to Eleanor Tanner for valuable discussions.

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  1. Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Jonathan A. Sherratt

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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

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