7,346 research outputs found
Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems
This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120880628We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables
Global attractors and extinction dynamics of cyclically competing species
Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases
Anisotropic surface reaction limited phase transformation dynamics in LiFePO4
A general continuum theory is developed for ion intercalation dynamics in a
single crystal of a rechargeable battery cathode. It is based on an existing
phase-field formulation of the bulk free energy and incorporates two crucial
effects: (i) anisotropic ionic mobility in the crystal and (ii) surface
reactions governing the flux of ions across the electrode/electrolyte
interface, depending on the local free energy difference. Although the phase
boundary can form a classical diffusive "shrinking core" when the dynamics is
bulk-transport-limited, the theory also predicts a new regime of
surface-reaction-limited (SRL) dynamics, where the phase boundary extends from
surface to surface along planes of fast ionic diffusion, consistent with recent
experiments on LiFePO4. In the SRL regime, the theory produces a fundamentally
new equation for phase transformation dynamics, which admits traveling-wave
solutions. Rather than forming a shrinking core of untransformed material, the
phase boundary advances by filling (or emptying) successive channels of fast
diffusion in the crystal. By considering the random nucleation of SRL
phase-transformation waves, the theory predicts a very different picture of
charge/discharge dynamics from the classical diffusion-limited model, which
could affect the interpretation of experimental data for LiFePO4.Comment: 15 pages, 10 figure
A reaction-diffusion model of cholinergic retinal waves
Prior to receiving visual stimuli, spontaneous, correlated activity called
retinal waves drives activity-dependent developmental programs. Early-stage
waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of
action potentials. They are believed to be initiated by the spontaneous firing
of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then
propagates this activity laterally. Their extended inter-wave intervals and
shifting wave boundaries are the result of the slow after-hyperpolarization of
the SACs creating an evolving mosaic of recruitable and refractory cells, which
can and cannot participate in waves, respectively. Recent evidence suggests
that cholinergic waves may be modulated by the extracellular concentration of
ACh. Here, we construct a simplified, biophysically consistent,
reaction-diffusion model of cholinergic retinal waves capable of recapitulating
wave dynamics observed in mice retina recordings. The dense, recurrent
connectivity of SACs is modeled through local, excitatory coupling occurring
via the volume release and diffusion of ACh. In contrast with previous,
simulation-based models, we are able to use non-linear wave theory to connect
wave features to underlying physiological parameters, making the model useful
in determining appropriate pharmacological manipulations to experimentally
produce waves of a prescribed spatiotemporal character. The model is used to
determine how ACh mediated connectivity may modulate wave activity, and how the
noise rate and sAHP refractory period contributes to critical wave size
variability.Comment: 38 pages, 10 figure
Projective and Coarse Projective Integration for Problems with Continuous Symmetries
Temporal integration of equations possessing continuous symmetries (e.g.
systems with translational invariance associated with traveling solutions and
scale invariance associated with self-similar solutions) in a ``co-evolving''
frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with
the evolving solution) leads to improved accuracy because of the smaller time
derivative in the new spatial frame. The slower time behavior permits the use
of {\it projective} and {\it coarse projective} integration with longer
projective steps in the computation of the time evolution of partial
differential equations and multiscale systems, respectively. These methods are
also demonstrated to be effective for systems which only approximately or
asymptotically possess continuous symmetries. The ideas of projective
integration in a co-evolving frame are illustrated on the one-dimensional,
translationally invariant Nagumo partial differential equation (PDE). A
corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is
used to illustrate the coarse-grained method. A simple, one-dimensional
diffusion problem is used to illustrate the scale invariant case. The
efficiency of projective integration in the co-evolving frame for both the
macroscopic diffusion PDE and for a random-walker particle based model is again
demonstrated
Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane
We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne
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