# 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 & Continuous Dynamical Systems - B, 2012, 17 (8) : 2815-2827. doi: 10.3934/dcdsb.2012.17.2815
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