152 research outputs found
On localized vegetation patterns, fairy circles and localized patches in arid landscapes
We investigate the formation of localized structures with a varying width in
one and two-dimensional systems. The mechanism of stabilization is attributed
to strong nonlocal coupling mediated by a Lorentzian type of Kernel. We show
that, in addition to stable dips found recently [see, e.g., C. Fernandez-Oto,
M. G. Clerc, D. Escaff, and M. Tlidi, Phys. Rev. Lett. {\bf{110}}, 174101
(2013)], exist stable localized peaks which appear as a result of strong
nonlocal coupling, i.e. mediated by a coupling that decays with the distance
slower than an exponential. We applied this mechanism to arid ecosystems by
considering a prototype model of a Nagumo type. In one-dimension, we study the
front that connects the stable uniformly vegetated state with the bare one
under the effect of strong nonlocal coupling. We show that strong nonlocal
coupling stabilizes both---dip and peak---localized structures. We show
analytically and numerically that the width of localized dip, which we
interpret as fairy circle, increases strongly with the aridity parameter. This
prediction is in agreement with filed observations. In addition, we predict
that the width of localized patch decreases with the degree of aridity.
Numerical results are in close agreement with analytical predictions
Introduction: Localized Structures in Dissipative Media: From Optics to Plant Ecology
Localised structures in dissipative appears in various fields of natural
science such as biology, chemistry, plant ecology, optics and laser physics.
The proposed theme issue is to gather specialists from various fields of
non-linear science toward a cross-fertilisation among active areas of research.
This is a cross-disciplinary area of research dominated by the nonlinear optics
due to potential applications for all-optical control of light, optical
storage, and information processing. This theme issue contains contributions
from 18 active groups involved in localized structures field and have all made
significant contributions in recent years.Comment: 14 pages, 0 figure, submitted to Phi. Trasaction Royal Societ
Plant clonal morphologies and spatial patterns as self-organized responses to resource-limited environments
We propose here to interpret and model peculiar plant morphologies (cushions,
tussocks) observed in the Andean altiplano as localized structures. Such
structures resulting in a patchy, aperiodic aspect of the vegetation cover are
hypothesized to self-organize thanks to the interplay between facilitation and
competition processes occurring at the scale of basic plant components
biologically referred to as 'ramets'. (Ramets are often of clonal origin.) To
verify this interpretation, we applied a simple, fairly generic model (one
integro-differential equation) emphasizing via Gaussian kernels non-local
facilitative and competitive feedbacks of the vegetation biomass density on its
own dynamics. We show that under realistic assumptions and parameter values
relating to ramet scale, the model can reproduce some macroscopic features of
the observed systems of patches and predict values for the inter-patch distance
that match the distances encountered in the reference area (Sajama National
Park in Bolivia). Prediction of the model can be confronted in the future to
data on vegetation patterns along environmental gradients as to anticipate the
possible effect of global change on those vegetation systems experiencing
constraining environmental conditions.Comment: 14 pages, 6figure
Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion
In the limit of large diffusivity ratio, spot-like solutions in the
two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion are
studied. It is shown analytically that such spots undergo an instability as the
diffusivity ratio is decreased. An instability threshold is derived. For spots
of small radius, it is shown that this instability leads to a spot splitting
into precisely two spots. For larger spots, it leads to deformation, fingering
patterns and space-filling curves. Numerical simulations are shown to be in
close agreement with the analytical predictions.Comment: To appear, PR
Cavity solitons in vertical-cavity surface-emitting lasers
We investigate a control of the motion of localized structures of light by
means of delay feedback in the transverse section of a broad area nonlinear
optical system. The delayed feedback is found to induce a spontaneous motion of
a solitary localized structure that is stationary and stable in the absence of
feedback. We focus our analysis on an experimentally relevant system namely the
Vertical-Cavity Surface-Emitting Laser (VCSEL). In the absence of the delay
feedback we present experimental evidence of stationary localized structures in
a 80 m aperture VCSEL. The spontaneous formation of localized structures
takes place above the lasing threshold and under optical injection. Then, we
consider the effect of the time-delayed optical feedback and investigate
analytically the role of the phase of the feedback and the carrier lifetime on
the self-mobility properties of the localized structures. We show that these
two parameters affect strongly the space time dynamics of two-dimensional
localized structures. We derive an analytical formula for the threshold
associated with drift instability of localized structures and a normal form
equation describing the slow time evolution of the speed of the moving
structure.Comment: 7 pages, 5 figure
Homoclinic snaking in bounded domains
Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of “snaking without bistability”, recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint)
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