Abstract
We analyse the model for vegetation growth in a semiarid landscape proposed by von Hardenberg et al [Phys. Rev. Lett. 87:198101, 2001], which consists of two parabolic partial differential equations that describe the evolution in space and time of the water content of the soil and the level of vegetation. This model is a generalisation of one proposed by Klausmeier but it contains additional terms that capture additional physical effects. By considering the limit in which the diffusion of water in the soil is much faster than the spread of vegetation, we reduce the system to an asymptotically simpler parabolicelliptic system of equations that describes small amplitude instabilities of the uniform vegetated state. We carry out a thorough weakly nonlinear analysis to investigate bifurcations and pattern formation in the reduced model. We find that the pattern forming instabilities are subcritical except in a small region of parameter space. In the original model at large amplitude there are localised solutions, organised by homoclinic snaking curves. The resulting bifurcation structure is well known from other models for pattern forming systems. Taken together our results describe how the von Hardenberg model displays a sequence of (often hysteretic) transitions from a nonvegetated state, to localised patches of vegetation that exist with uniform lowlevel vegetation, to periodic patterns, to higherlevel uniform vegetation as the precipitation parameter increases.
Original language  English 

Pages (fromto)  6390 
Journal  Journal of Mathematical Biology 
Volume  73 
Issue number  1 
DOIs  
Publication status  Published  Jul 2016 
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Jonathan Dawes
 Department of Mathematical Sciences  Professor
 Centre for Networks and Collective Behaviour
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Water Innovation and Research Centre (WIRC)
 Institute for Policy Research (IPR)
 Centre for Mathematical Biology
 Centre for Nonlinear Mechanics
Person: Research & Teaching