Klausmeier Model Research Articles

Stabilizing a homoclinic stripe

For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts

Landscape-scale vegetation stripes (tiger bush) observed on the gentle slopes of semi-arid regions are useful indicators of future ecosystem degradation and catastrophic shifts towards desert. Mathematical models like the Klausmeier model—a set of coupled partial differential equations describing vegetation and water densities in space and time—are central to understanding their formation and development. One assumption made for mathematical simplicity is the local dispersal of seeds via a diffusion term. In fact, a large amount of work focuses on fitting dispersal ‘kernels’, probability density functions for seed dispersal distance, to empirical data of different species and modes of dispersal. In this paper, we address this discrepancy by analysing an extended Klausmeier model that includes long-distance seed dispersal via a non-local convolution term in place of diffusion, and assessing its effect on the resilience of striped patterns. Many authors report a slow uphill migration of stripes; but others report no detectable migration speed. We show that long-distance seed dispersal permits the formation of patterns with a very slow (possibly undetectable) migration speed, and even stationary patterns which could explain the inconsistencies in the empirical data. In general, we show that the resilience of patterns to reduced rainfall may vary significantly depending on the rate of seed dispersal and the width of the dispersal kernel, and compare a selection of ecologically relevant kernels to examine the variation in pattern resilience.

A topographic mechanism for arcing of dryland vegetation bands.

Banded patterns consisting of alternating bare soil and dense vegetation have been observed in water-limited ecosystems across the globe, often appearing along gently sloped terrain with the stripes aligned transverse to the elevation gradient. In many cases, these vegetation bands are arced, with field observations suggesting a link between the orientation of arcing relative to the grade and the curvature of the underlying terrain. We modify the water transport in the Klausmeier model of water-biomass interactions, originally posed on a uniform hillslope, to qualitatively capture the influence of terrain curvature on the vegetation patterns. Numerical simulations of this modified model indicate that the vegetation bands arc convex-downslope when growing on top of a ridge, and convex-upslope when growing in a valley. This behaviour is consistent with observations from remote sensing data that we present here. Model simulations show further that whether bands grow on ridges, valleys or both depends on the precipitation level. A survey of three banded vegetation sites, each with a different aridity level, indicates qualitatively similar behaviour.

Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal.

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.

Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation

Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential equations and is often referred to as the Klausmeier model. This paper provides analytical and numerical investigations regarding the influence of different parameters, including the so-far not contemplated evaporation, on the long-term model results. Another focus is set on the influence of different initial conditions and on environmental noise, which has been added to the model. It is shown that patterning is beneficial for semi-arid ecosystems, that is, vegetation is present for a broader parameter range. Both parameter variability and environmental noise have only minor impacts on the model results. Increasing mortality has a high, nonlinear impact underlining the importance of further studies in order to gain a sufficient understanding allowing for suitable management strategies of this natural phenomenon.

When does colonisation of a semi-arid hillslope generate vegetation patterns?

Patterned vegetation occurs in many semi-arid regions of the world. Most previous studies have assumed that patterns form from a starting point of uniform vegetation, for example as a response to a decrease in mean annual rainfall. However an alternative possibility is that patterns are generated when bare ground is colonised. This paper investigates the conditions under which colonisation leads to patterning on sloping ground. The slope gradient plays an important role because of the downhill flow of rainwater. One long-established consequence of this is that patterns are organised into stripes running parallel to the contours; such patterns are known as banded vegetation or tiger bush. This paper shows that the slope also has an important effect on colonisation, since the uphill and downhill edges of an isolated vegetation patch have different dynamics. For the much-used Klausmeier model for semi-arid vegetation, the author shows that without a term representing water diffusion, colonisation always generates uniform vegetation rather than a pattern. However the combination of a sufficiently large water diffusion term and a sufficiently low slope gradient does lead to colonisation-induced patterning. The author goes on to consider colonisation in the Rietkerk model, which is also in widespread use: the same conclusions apply for this model provided that a small threshold is imposed on vegetation biomass, below which plant growth is set to zero. Since the two models are quite different mathematically, this suggests that the predictions are a consequence of the basic underlying assumption of water redistribution as the pattern generation mechanism.

Irregular Striped Vegetation Patterns in Arid and Semiarid Regions Resulted from Oscillatory Instability

Striped vegetation pattern formation in arid and semiarid regions has attracted many researchers in ecological field. In this research, we modify the classical Klausmeier model and study nonlinear mechanism of oscillatory instability which can also possibly lead to pattern formation in ecological systems. Via simulation, irregular striped vegetation patterns are numerically obtained for the modified model. From the mechanism of oscillatory instability, this research provides a theoretical explanation for the formation of irregular striped vegetation patterns in nature.

Quasi-Linear Parabolic Reaction-Diffusion Systems: A User's Guide to Well-Posedness, Spectra, and Stability of Travelling Waves

This paper is concerned with quasi-linear parabolic reaction-diffusion-advection systems on extended domains. Frameworks for well-posedness in Hilbert spaces and spaces of continuous functions are presented, based on known results using maximal regularity. It is shown that spectra of travelling waves on the line are meaningfully given by familiar tools for semilinear equations, such as dispersion relations, and basic connections of spectra to stability and instability are considered. In particular, a principle of linearized orbital instability for manifolds of equilibria is proven. Our goal is to provide easy access for practitioners to these rigorous methods. As a guiding example the Gray--Scott--Klausmeier model for vegetation-water interaction is considered in detail.

Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semiarid Environments V: The Transition from Patterns to Desert

Vegetation in semideserts often self-organizes into spatial patterns. On gentle slopes, these typically consist of stripes of vegetation running parallel to the contours, separated by stripes of bare ground. The Klausmeier model is one of the oldest and most established of a number of mathematical models for this “banded vegetation.” The model is a system of reaction-diffusion-advection equations. Under the standard nondimensionalization, one of its dimensionless parameters ($\nu$) reflects the relative rates of water flow downhill and plant dispersal and is therefore very large. This paper is the fifth and last in a series in which the author provides a detailed analytical understanding of the existence and form of pattern solutions (periodic travelling waves) of the Klausmeier model, to leading order as $\nu\rightarrow\infty$. The problem is a very rich one because the underlying mathematics depends fundamentally on the way in which the migration speed $c$ scales with $\nu$. This paper concerns the case $1=O(c)$ and $c=o(\nu^{1/2})$ as $\nu\rightarrow\infty$. The author derives leading order expressions for the curves bounding the parameter region giving patterns, and for the pattern forms in this region. An important consequence of this is leading order formulae for the maximum and minimum rainfall levels for which patterns exist. The author demonstrates via numerical simulations that a decrease in rainfall through the minimum level for patterns causes a transition to full-blown desert that cannot be reversed by increasing the rainfall again.

Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semiarid Environments IV: Slowly Moving Patterns and Their Stability

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.

Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments III: The transition between homoclinic solutions

Self-organised patterns of vegetation are a characteristic feature of semi-deserts. On hillsides, these typically comprise vegetation bands running parallel to the contours, separated by regions of bare ground (“tiger bush”). The present study concerns the Klausmeier mathematical model for this phenomenon [C.A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science 284 (1999) 1826–1828], which is one of the earliest and most influential of the various theoretical models for banded vegetation. The model is a system of reaction–diffusion–advection equations, and after rescaling it contains three dimensionless parameters, one of which (the slope parameter) is much larger than the other two. The present study is the third in a series of papers in which the author exploits the large value of the slope parameter to obtain leading order approximations to the parameter regions in which patterns exist, and to the form of these patterns. The boundary of the parameter region giving patterns consists in part of two loci of homoclinic solutions, that are homoclinic to different steady states. The present paper concerns behaviour for parameters close to the intersection point of these loci. The author shows that this part of parameter space divides naturally into three regions, with a different solution structure in each. In one region, the solution corresponds to a limit cycle of a reduced system of ordinary differential equations; the other two regions involve multiple matched layers. As part of the analysis, the author derives formulae for the homoclinic solution loci, and for the location of their intersection. All of the results are valid to leading order for large values of the slope parameter. The author presents a detailed numerical verification of his analytical results. The paper concludes with discussions of the ecological implications of the results, and the main outstanding mathematical questions.

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.

Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds

Pattern formation at the ecosystem level is a rapidly growing area of spatial ecology. The best studied example is vegetation stripes running along contours in semi-arid regions. Theoretical models are a widely used tool for studying these banded vegetation patterns, and one important model is the system of advection–diffusion equations proposed by Klausmeier. The present study is part of a series of papers whose objective is a comprehensive understanding of patterned solutions of the Klausmeier model. The author focuses on the region of parameter space in which the propagation speed of the patterns is close to its maximum possible value. Exploiting the large value of one of the model parameters, a leading order approximation is obtained for the maximum propagation speed, and the author undertakes a detailed investigation of the parameter region in which there are patterns with speeds close to this maximum.

Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments I

In many semi-arid environments, vegetation cover is sparse, and is self-organized into large-scale spatial patterns. In particular, banded vegetation is typical on hillsides. Mathematical modelling is widely used to study these banded patterns, and many models are effectively extensions of a coupled reaction–diffusion–advection system proposed by Klausmeier (1999 Science 284 1826–8). However, there is currently very little mathematical theory on pattern solutions of these equations. This paper is the first in a series whose aim is a comprehensive understanding of these solutions, which can act as a springboard both for future simulation-based studies of the Klausmeier model, and for analysis of model extensions. The author focusses on a particular part of parameter space, and derives expressions for the boundaries of the parameter region in which patterns occur. The calculations are valid to leading order at large values of the ‘slope parameter’, which reflects a comparison of the rate of water flow downhill with the rate of vegetation dispersal. The form of the corresponding patterns is also studied, and the author shows that the leading order equations change close to one boundary of the parameter region in which there are patterns, leading to a homoclinic solution. Conclusions are drawn on the way in which changes in mean annual rainfall affect pattern properties, including overall biomass productivity.

An Analysis of Vegetation Stripe Formation in Semi-Arid Landscapes

Vegetation stripes ("tiger bush") are a characteristic feature of semi-arid environments. The stripes typically lie along the contours of gentle slopes, and some authors report a gradual uphill migration. A previous mathematical model (Klausmeier, Science, 284:1826, 1999) has shown that this phenomenon can be explained relatively simply by the downhill flow of rainwater coupled with the diffusive spread of the plant population. This paper presents a detailed analysis of pattern formation in the Klausmeier model. The author derives formulae for the wavelength and migration speed of the predicted patterns, and systematically investigates how these depend on model parameters. The results make new predictions and suggest possible approaches to testing the model.