133 research outputs found
Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
We consider reaction-diffusion equations that are stochastically forced by a
small multiplicative noise term. We show that spectrally stable travelling wave
solutions to the deterministic system retain their orbital stability if the
amplitude of the noise is sufficiently small.
By applying a stochastic phase-shift together with a time-transform, we
obtain a semilinear sPDE that describes the fluctuations from the primary wave.
We subsequently develop a semigroup approach to handle the nonlinear stability
question in a fashion that is closely related to modern deterministic methods
Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
We establish the existence and nonlinear stability of travelling pulse
solutions for the discrete FitzHugh-Nagumo equation with infinite-range
interactions close to the continuum limit. For the verification of the spectral
properties, we need to study a functional differential equation of mixed type
(MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and
phase spaces, by building on a technique developed by Bates, Chen and Chmaj for
the discrete Nagumo equation. This allows us to transfer several crucial
Fredholm properties from the PDE setting to our discrete setting
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
Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations (survey)
Analysis and Stochastic
Single-cell Hi-C reveals cell-to-cell variability in chromosome structure.
Large-scale chromosome structure and spatial nuclear arrangement have been linked to control of gene expression and DNA replication and repair. Genomic techniques based on chromosome conformation capture (3C) assess contacts for millions of loci simultaneously, but do so by averaging chromosome conformations from millions of nuclei. Here we introduce single-cell Hi-C, combined with genome-wide statistical analysis and structural modelling of single-copy X chromosomes, to show that individual chromosomes maintain domain organization at the megabase scale, but show variable cell-to-cell chromosome structures at larger scales. Despite this structural stochasticity, localization of active gene domains to boundaries of chromosome territories is a hallmark of chromosomal conformation. Single-cell Hi-C data bridge current gaps between genomics and microscopy studies of chromosomes, demonstrating how modular organization underlies dynamic chromosome structure, and how this structure is probabilistically linked with genome activity patterns
Multifactorial Origins of Heart and Gut Defects in nipbl-Deficient Zebrafish, a Model of Cornelia de Lange Syndrome
nipbl-deficient zebrafish provide evidence that heart and gut defects in Cornelia de Lange Syndrome are caused by combined effects of multiple gene expression changes that occur during early embryonic development
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