412 research outputs found
Turing conditions for pattern forming systems on evolving manifolds
The study of pattern-forming instabilities in reaction-diffusion systems on
growing or otherwise time-dependent domains arises in a variety of settings,
including applications in developmental biology, spatial ecology, and
experimental chemistry. Analyzing such instabilities is complicated, as there
is a strong dependence of any spatially homogeneous base states on time, and
the resulting structure of the linearized perturbations used to determine the
onset of instability is inherently non-autonomous. We obtain general conditions
for the onset and structure of diffusion driven instabilities in
reaction-diffusion systems on domains which evolve in time, in terms of the
time-evolution of the Laplace-Beltrami spectrum for the domain and functions
which specify the domain evolution. Our results give sufficient conditions for
diffusive instabilities phrased in terms of differential inequalities which are
both versatile and straightforward to implement, despite the generality of the
studied problem. These conditions generalize a large number of results known in
the literature, such as the algebraic inequalities commonly used as a
sufficient criterion for the Turing instability on static domains, and
approximate asymptotic results valid for specific types of growth, or specific
domains. We demonstrate our general Turing conditions on a variety of domains
with different evolution laws, and in particular show how insight can be gained
even when the domain changes rapidly in time, or when the homogeneous state is
oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to
higher-order spatial systems are also included as a way of demonstrating the
generality of the approach
Amplitude death criteria for coupled complex Ginzburg-Landau systems
Amplitude death, which occurs in a system when one or more macroscopic
wavefunctions collapse to zero, has been observed in mutually coupled
solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a
few applications. While studies have considered amplitude death on oscillator
systems and in externally forced complex Ginzburg-Landau systems, a route to
amplitude death has not been studied in autonomous continuum systems. We derive
simple analytic conditions for the onset of amplitude death of one macroscopic
wavefunction in a system of two coupled complex Ginzburg-Landau equations with
general nonlinear self- and cross-interaction terms. Our results give a more
general theoretical underpinning for recent amplitude death results reported in
the literature, and suggest an approach for tuning parameters in such systems
so that they either permit or prohibit amplitude death of a wavefunction
(depending on the application). Numerical simulation of the coupled complex
Ginzburg-Landau equations, for examples including cubic, cubic-quintic, and
saturable nonlinearities, is used to illustrate the analytical results.Comment: 7 pages, 4 figure
Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media
We study dynamics emergent from a two-dimensional reaction--diffusion process
modelled via a finite lattice dynamical system, as well as an analogous PDE
system, involving spatially nonlocal interactions. These models govern the
evolution of cells in a bioactive porous medium, with evolution of the local
cell density depending on a coupled quasi--static fluid flow problem. We
demonstrate differences emergent from the choice of a discrete lattice or a
continuum for the spatial domain of such a process. We find long--time
oscillations and steady states in cell density in both lattice and continuum
models, but that the continuum model only exhibits solutions with vertical
symmetry, independent of initial data, whereas the finite lattice admits
asymmetric oscillations and steady states arising from symmetry-breaking
bifurcations. We conjecture that it is the structure of the finite lattice
which allows for more complicated asymmetric dynamics. Our analysis suggests
that the origin of both types of oscillations is a nonlocal reaction-diffusion
mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure
Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold
A contemporary procedure to grow artificial tissue is to seed cells onto a
porous biomaterial scaffold and culture it within a perfusion bioreactor to
facilitate the transport of nutrients to growing cells. Typical models of cell
growth for tissue engineering applications make use of spatially homogeneous or
spatially continuous equations to model cell growth, flow of culture medium,
nutrient transport, and their interactions. The network structure of the
physical porous scaffold is often incorporated through parameters in these
models, either phenomenologically or through techniques like mathematical
homogenization. We derive a model on a square grid lattice to demonstrate the
importance of explicitly modelling the network structure of the porous
scaffold, and compare results from this model with those from a modified
continuum model from the literature. We capture two-way coupling between cell
growth and fluid flow by allowing cells to block pores, and by allowing the
shear stress of the fluid to affect cell growth and death. We explore a range
of parameters for both models, and demonstrate quantitative and qualitative
differences between predictions from each of these approaches, including
spatial pattern formation and local oscillations in cell density present only
in the lattice model. These differences suggest that for some parameter
regimes, corresponding to specific cell types and scaffold geometries, the
lattice model gives qualitatively different model predictions than typical
continuum models. Our results inform model selection for bioactive porous
tissue scaffolds, aiding in the development of successful tissue engineering
experiments and eventually clinically successful technologies.Comment: 38 pages, 16 figures. This version includes a much-expanded
introduction, and a new section on nonlinear diffusion in addition to polish
throughou
Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
We formulate and study dynamics from a complex Ginzburg-Landau system with
saturable nonlinearity, including asymmetric cross-phase modulation (XPM)
parameters. Such equations can model phenomena described by complex
Ginzburg-Landau systems under the added assumption of saturable media. When the
saturation parameter is set to zero, we recover a general complex cubic
Ginzburg-Landau system with XPM. We first derive conditions for the existence
of bounded dynamics, approximating the absorbing set for solutions. We use this
to then determine conditions for amplitude death of a single wavefunction. We
also construct exact plane wave solutions, and determine conditions for their
modulational instability. In a degenerate limit where dispersion and
nonlinearity balance, we reduce our system to a saturable nonlinear
Schr\"odinger system with XPM parameters, and we demonstrate the existence and
behavior of spatially heterogeneous stationary solutions in this limit. Using
numerical simulations we verify the aforementioned analytical results, while
also demonstrating other interesting emergent features of the dynamics, such as
spatiotemporal chaos in the presence of modulational instability. In other
regimes, coherent patterns including uniform states or banded structures arise,
corresponding to certain stable stationary states. For sufficiently large yet
equal XPM parameters, we observe a segregation of wavefunctions into different
regions of the spatial domain, while when XPM parameters are large and take
different values, one wavefunction may decay to zero in finite time over the
spatial domain (in agreement with the amplitude death predicted analytically).
While saturation will often regularize the dynamics, such transient dynamics
can still be observed - and in some cases even prolonged - as the saturability
of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page
Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems
Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models
Heterogeneity induces spatiotemporal oscillations in reaction-diffusions systems
We report on a novel instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable, or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity, and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity
From One Pattern into Another: Analysis of Turing Patterns in Heterogeneous Domains via WKBJ
Pattern formation from homogeneity is well-studied, but less is known
concerning symmetry-breaking instabilities in heterogeneous media. It is
nontrivial to separate observed spatial patterning due to inherent spatial
heterogeneity from emergent patterning due to nonlinear instability. We employ
WKBJ asymptotics to investigate Turing instabilities for a spatially
heterogeneous reaction-diffusion system, and derive conditions for instability
which are local versions of the classical Turing conditions We find that the
structure of unstable modes differs substantially from the typical
trigonometric functions seen in the spatially homogeneous setting. Modes of
different growth rates are localized to different spatial regions. This
localization helps explain common amplitude modulations observed in simulations
of Turing systems in heterogeneous settings. We numerically demonstrate this
theory, giving an illustrative example of the emergent instabilities and the
striking complexity arising from spatially heterogeneous reaction-diffusion
systems. Our results give insight both into systems driven by exogenous
heterogeneity, as well as successive pattern forming processes, noting that
most scenarios in biology do not involve symmetry breaking from homogeneity,
but instead consist of sequential evolutions of heterogeneous states. The
instability mechanism reported here precisely captures such evolution, and
extends Turing's original thesis to a far wider and more realistic class of
systems.Comment: 23 pages, 7 Figure
Patterning of nonlocal transport models in biology: The impact of spatial dimension
Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction-diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with 'run-and-chase' interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells
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