Pattern solutions of the Klausmeier Model for banded vegetation in semiarid environments IV: slowly moving patterns and their stability

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Abstract

Banded vegetation is a characteristic feature of semiarid environments, comprising stripes of vegetation running parallel to the contours on hillsides, separated by stripes of bare ground. Mathematical modelling plays an important role in the study of this phenomenon, and the Klausmeier model is one of the oldest and most established, consisting of coupled reaction-diffusion-advection equations for plant biomass and water density. In this model the dimensionless parameter corresponding to slope gradient is much larger than the other parameters, and this paper is part of a series investigating the asymptotic form of pattern solutions for large values of the slope parameter. The pattern solutions move uphill with a constant speed, $c$ say, and the focus of this paper is $c=O(1)$. I begin by deriving the leading order form of the patterns, and the region of parameter space in which they occur, for $c=o(1)$. Using this, I show that all patterns with $c=o(1)$ are unstable as model solutions for sufficiently large values of the slope parameter. I then consider patterns with $c=O_s(1)$, showing that this region of parameter space contains both stable and unstable low amplitude patterns. I conclude by discussing the ecological implications of my results.


Read More: http://epubs.siam.org/doi/abs/10.1137/120862648
Original languageEnglish
Pages (from-to)330-350
Number of pages21
JournalSIAM Journal on Applied Mathematics
Volume73
Issue number1
DOIs
Publication statusPublished - 1 Jan 2013

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vegetation
advection-diffusion equation
parameter
biomass
modeling
water

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title = "Pattern solutions of the Klausmeier Model for banded vegetation in semiarid environments IV: slowly moving patterns and their stability",
abstract = "Banded vegetation is a characteristic feature of semiarid environments, comprising stripes of vegetation running parallel to the contours on hillsides, separated by stripes of bare ground. Mathematical modelling plays an important role in the study of this phenomenon, and the Klausmeier model is one of the oldest and most established, consisting of coupled reaction-diffusion-advection equations for plant biomass and water density. In this model the dimensionless parameter corresponding to slope gradient is much larger than the other parameters, and this paper is part of a series investigating the asymptotic form of pattern solutions for large values of the slope parameter. The pattern solutions move uphill with a constant speed, $c$ say, and the focus of this paper is $c=O(1)$. I begin by deriving the leading order form of the patterns, and the region of parameter space in which they occur, for $c=o(1)$. Using this, I show that all patterns with $c=o(1)$ are unstable as model solutions for sufficiently large values of the slope parameter. I then consider patterns with $c=O_s(1)$, showing that this region of parameter space contains both stable and unstable low amplitude patterns. I conclude by discussing the ecological implications of my results.Read More: http://epubs.siam.org/doi/abs/10.1137/120862648",
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N2 - Banded vegetation is a characteristic feature of semiarid environments, comprising stripes of vegetation running parallel to the contours on hillsides, separated by stripes of bare ground. Mathematical modelling plays an important role in the study of this phenomenon, and the Klausmeier model is one of the oldest and most established, consisting of coupled reaction-diffusion-advection equations for plant biomass and water density. In this model the dimensionless parameter corresponding to slope gradient is much larger than the other parameters, and this paper is part of a series investigating the asymptotic form of pattern solutions for large values of the slope parameter. The pattern solutions move uphill with a constant speed, $c$ say, and the focus of this paper is $c=O(1)$. I begin by deriving the leading order form of the patterns, and the region of parameter space in which they occur, for $c=o(1)$. Using this, I show that all patterns with $c=o(1)$ are unstable as model solutions for sufficiently large values of the slope parameter. I then consider patterns with $c=O_s(1)$, showing that this region of parameter space contains both stable and unstable low amplitude patterns. I conclude by discussing the ecological implications of my results.Read More: http://epubs.siam.org/doi/abs/10.1137/120862648

AB - Banded vegetation is a characteristic feature of semiarid environments, comprising stripes of vegetation running parallel to the contours on hillsides, separated by stripes of bare ground. Mathematical modelling plays an important role in the study of this phenomenon, and the Klausmeier model is one of the oldest and most established, consisting of coupled reaction-diffusion-advection equations for plant biomass and water density. In this model the dimensionless parameter corresponding to slope gradient is much larger than the other parameters, and this paper is part of a series investigating the asymptotic form of pattern solutions for large values of the slope parameter. The pattern solutions move uphill with a constant speed, $c$ say, and the focus of this paper is $c=O(1)$. I begin by deriving the leading order form of the patterns, and the region of parameter space in which they occur, for $c=o(1)$. Using this, I show that all patterns with $c=o(1)$ are unstable as model solutions for sufficiently large values of the slope parameter. I then consider patterns with $c=O_s(1)$, showing that this region of parameter space contains both stable and unstable low amplitude patterns. I conclude by discussing the ecological implications of my results.Read More: http://epubs.siam.org/doi/abs/10.1137/120862648

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