Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package wavetrain (www.ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall-migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.