Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction
type: Item
instance of: scholarly article
title: Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction
zbMATH Open document ID: 1472.35035
author: Xiaoli Wang
author: Junping Shi
author: Guohong Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-03-03T00:00:00Z
review text: A reaction-diffusion model describing water and plant interaction in a flat environment is studied. The system is governed by the Klausmeier-Gray-Scott equations \begin{align*} \frac{\partial W}{\partial t} &= d_1 \Delta W + a - W B^2 - W, \\ \frac{\partial B}{\partial t} &= d_2 \Delta B + W B^2 - mB \end{align*} with Neumann boundary conditions. After carefully investigating the existence and stability of uniform steady states the authors study the bifurcations from these steady states to large amplitude spatial patterned solutions. In the case of small rain fall \(a\) the authors construct a simpler shadow system by considering the singular limit \(d_1\rightarrow \infty\). The authors also carry out numerical investigations on linear and rectangular domains, which demonstrate the occurence of complicated vegetation patterns. The article shows, that slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating in a small area or ``spots'', even when the rainfall is too low to support uniform vegetation.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 92C80
Mathematics Subject Classification ID: 35B36
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K51
zbMATH DE Number: 7317190
zbMATH Keywords: reaction-diffusion model
zbMATH Keywords: water-plant interaction
zbMATH Keywords: spatial pattern formation
zbMATH Keywords: steady state bifurcation
zbMATH Keywords: shadow system
reviewed by: Alois Steindl
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124860
OpenAlex ID: W3112730456
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DOI: 10.1016/J.JMAA.2020.124860