Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 497, Issue 1, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2020.124860
Keywords
Reaction-diffusion model; Water-plant interaction; Spatial pattern formation; Steady state bifurcation; Shadow system
Categories
Funding
- National Science Foundation of China [11701472, 11871403, 11871060]
- China Scholarship Council
- US-NSF [DMS-1853598]
- Fundamental Research Funds for the Central Universities [XDJK2020B050]
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The reaction-diffusion model proposed by Klausmeier, describing water and plant interaction, shows the existence of non-constant steady state solutions and large amplitude spatial patterned solutions. It is proven that non-homogeneous patterned vegetation states exist when the rainfall is at a lower level.
A reaction-diffusion model describing water and plant interaction proposed by Klausmeier is studied. The existence of non-constant steady state solutions is shown through bifurcation methods, and the existence of large amplitude spatial patterned solutions is proved using associated shadow system. It is rigorously shown that non-homogeneous patterned vegetation states exist when the rainfall is at a lower level in which homogeneous vegetation state cannot survive. Even when the rainfall is very low, slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating on a small area. This provides an example of diffusion-induced persistence that non-constant steady states may exist in a reaction-diffusion system when there are no positive constant steady states. (C) 2020 Elsevier Inc. All rights reserved.
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