### 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.

Original language | English |
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Pages (from-to) | 199–226 |

Number of pages | 28 |

Journal | Journal of Mathematical Biology |

Volume | 73 |

Issue number | 1 |

Early online date | 7 Nov 2015 |

DOIs | |

Publication status | Published - Jul 2016 |

### Keywords

- Colonization
- Desert
- Pattern formation
- Periodic travelling wave
- Reaction–diffusion–advection
- Semi-arid

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics
- Modelling and Simulation

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## Profiles

## Jonathan Adam Sherratt

- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor

Person: Academic (Research & Teaching)