68 research outputs found
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 change arcing-direction from
convex-downslope when growing on top of a ridge to convex-upslope when growing
in a valley. This behavior 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 behavior.Comment: 26 pages, 13 figures, 2 table
Wavelength selection beyond Turing
Spatial patterns arising spontaneously due to internal processes are
ubiquitous in nature, varying from regular patterns of dryland vegetation to
complex structures of bacterial colonies. Many of these patterns can be
explained in the context of a Turing instability, where patterns emerge due to
two locally interacting components that diffuse with different speeds in the
medium. Turing patterns are multistable, such that many different patterns with
different wavelengths are possible for the same set of parameters, but in a
given region typically only one such wavelength is dominant. In the Turing
instability region, random initial conditions will mostly lead to a wavelength
that is similar to that of the leading eigenvector that arises from the linear
stability analysis, but when venturing beyond, little is known about the
pattern that will emerge. Using dryland vegetation as a case study, we use
different models of drylands ecosystems to study the wavelength pattern that is
selected in various scenarios beyond the Turing instability region, focusing
the phenomena of localized states and repeated local disturbances
Predicting the emergence of localised dihedral patterns in models for dryland vegetation
Localised patterns are often observed in models for dryland vegetation, both
as peaks of vegetation in a desert state and as gaps within a vegetated state,
known as `fairy circles'. Recent results from radial spatial dynamics show that
approximations of localised patterns with dihedral symmetry emerge from a
Turing instability in general reaction--diffusion systems, which we apply to
several vegetation models. We present a systematic guide for finding such
patterns in a given reaction--diffusion model, during which we obtain four key
quantities that allow us to predict the qualitative properties of our solutions
with minimal analysis. Our results are complemented by numerical simulations
for various localised states in four well-established vegetation models,
highlighting the universality of such solutions.Comment: 30 pages, 12 figure
Analysis and Simulations of a Nonlocal Gray-Scott Model
The Gray-Scott model is a set of reaction-diffusion equations that describes
chemical systems far from equilibrium. Interest in this model stems from its
ability to generate spatio-temporal structures, including pulses, spots,
stripes, and self-replicating patterns. We consider an extension of this model
in which the spread of the different chemicals is assumed to be nonlocal, and
can thus be represented by an integral operator. In particular, we focus on the
case of strictly positive, symmetric, convolution kernels that have a
finite second moment. Modeling the equations on a finite interval, we prove the
existence of small-time weak solutions in the case of nonlocal Dirichlet and
Neumann boundary constraints. We then use this result to develop a finite
element numerical scheme that helps us explore the effects of nonlocal
diffusion on the formation of pulse solutions.Comment: 28 pages, 2 figure
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