Finite One-dimensional Domain Research Articles

Size matters: tissue size as a marker for a transition between reaction-diffusion regimes in spatio-temporal distribution of morphogens.

The reaction–diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction–diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction–diffusion length, and identified two reaction–diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction–diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction–diffusion model to study tissue morphogenesis.

Where and how do localized perturbations affect stream and coastal ocean populations with nonlinear growth dynamics?

Chaotic systems are sensitive to small changes in parameters and initial conditions, but are spatially distributed chaotic populations sensitive to the location of perturbations within their domain? Here, we examine the transient responses to perturbation in a density-dependent population model with asymmetrical dispersal, where offspring are moved some distance downstream of the parent population and diffusively spread, an appropriate approximation for species in streams, rivers, and ocean currents. We find that over a portion of the studied parameter space, our system displays extreme convective instability; that is, an arbitrarily small point-source perturbation to the upstream region grows and propagates downstream (in our finite one-dimensional domain, the current moves offspring preferentially in the downstream direction). We can then divide our domain into distinct upstream and downstream regions based on the response to perturbation. A perturbation to the upstream region will create transient downstream instability, and a perturbation to the downstream region has no qualitative effect on the population. We investigate the factors affecting the existence/size of the sensitive upstream region, discuss the ecologically relevant time scales of the transient dynamics following a perturbation, and consider the ecological implications of managing and predicting a population with transient amplification of perturbations.

Extended Stefan Problem for Solidification of Binary Alloys in a Finite Planar Domain

We consider the extended Stefan problem for the solidification of a binary alloy in a finite one-dimensional domain accounting for constitutional supercooling. In this problem, solidification front...

Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

This contribution proposes a numerical procedure capable of performing nonsmooth modal analysis (mode shapes and corresponding frequencies) of the autonomous wave equation defined on a finite one-dimensional domain with one end subject to a Dirichlet condition and the other end subject to a frictionless time-independent unilateral contact condition. Nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the linear and nonlinear boundary conditions. The analysis is performed using the wave finite element method, which is a shock-capturing finite volume method. As opposed to the traditional finite element method with time-stepping schemes, potentially discontinuous deformation, stress and velocity wave fronts induced by the unilateral contact condition are here accurately described, which is critical for seeking periodic orbits. Additionally, the proposed strategy introduces neither numerical dispersion nor artificial dissipation of energy, as required for modal analysis. As a consequence of the mixed time–space discretization, no impact law is needed for the well-posedness of the problem in line with the continuous framework. The frequency–energy dependency nonlinear spectrum of vibration, shown in the form of backbone curves, provides valuable insight on the dynamics. In contrast to the linear system (without the unilateral contact condition) whose modes of vibration are standing harmonic waves, the nonsmooth modes of vibrations are traveling waves stemming from the unilateral contact condition. It is also shown that the vibratory resonances of the periodically driven system with light structural damping are well predicted by nonsmooth modal analysis. Furthermore, the initially unstressed and prestressed configurations exhibit stiffening and softening behaviors, respectively, as expected.

MIXING TIME OF A + B− > 0 IN ONE DIMENSION

A mixing time density of [Formula: see text] on a finite one-dimensional domain is defined for general initial and boundary conditions in which [Formula: see text] and [Formula: see text] diffuse at the same rate. The density is a measure of the number of [Formula: see text] and [Formula: see text] particles that mix through the center of the reaction zone. It also corresponds to the reaction density for the special case in which [Formula: see text] and [Formula: see text] annihilate upon contact. An exact expression is found for the generating function of the mixing time. The analysis is extended to multiple reaction fronts and finitely ramified fractals. The method involves using the kernel of the Laplace transform integral operator to map and analyze a moving homogeneous Dirichlet interior point condition.

Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

We present three examples of delayed bifurcations for spike solutions of reaction–diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analyzed in the context of ODE’s in Mandel and Erneux (1987). It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been “delayed” relative to the threshold value computed directly from a linear stability analysis. In contrast to the study of Mandel and Erneux, we analyze the delay effect in systems of partial differential equations (PDE’s). In particular, for spike solutions of singularly perturbed generalized Gierer–Meinhardt and Gray–Scott models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the Gierer–Meinhardt model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation of the Gierer–Meinhardt model on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities triggered by the extrinsic tuning of a feed rate parameter. In all three cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through the analysis of certain explicitly solvable nonlocal eigenvalue problems (NLEP’s). The theory is confirmed by numerical solutions of the full PDE systems.

Kinematics of age mixing in advection-dispersion models

This paper investigates age mixing processes arising in advection-dispersion models, where large-scale travel and residence time distributions can be explicitly calculated based on the underlying velocity field. In particular, we analyze spatially integrated age mixing dynamics by comparing the age distributions of the storage and of the outflow(s). The relevance of the work lies in the impact of age mixing dynamics on the shape of travel time distributions (TTDs), which ultimately control the long-term memory of catchment transport processes. We set up a theoretical framework that bridges previous Lagrangian and Eulerian water age theories in heterogeneous media. The framework allows for the analysis of the dynamical connection between water age distributions in large-scale volume and flux samples. Theoretical advances are then illustrated through the application to a finite one-dimensional domain with constant advection and dispersion coefficient. Therein, we analyze the type of mixing emerging for different Peclet numbers and diverse spatiotemporal patterns of solute input. We find that in spite of the enhanced nonstationarity of TTDs, the type of mixing is markedly invariant. Moreover, for relatively low Peclet numbers the different ages available in the control volume are systematically removed from the domain at a rate nearly proportional to their relative abundance (random sampling). Emerging large-scale patterns of age mixing yield theoretical and practical implications for watershed hydrology, where TTDs can be used to infer general patterns of catchment response across scales.

Encounter distribution of two random walkers on a finite one-dimensional interval

We analyse the first-passage properties of two random walkers confined to a finite one-dimensional domain. For the case of absorbing boundaries at the endpoints of the interval, we derive the probability that the two particles meet before either one of them becomes absorbed at one of the boundaries. For the case of reflecting boundaries, we obtain the mean first encounter time of the two particles. Our approach leads to closed-form expressions that are more easily tractable than a previously derived solution in terms of the Weierstrass’ elliptic function.

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.

Density profiles in open superdiffusive systems

We numerically solve a discretized model of Lévy random walks on a finite one-dimensional domain with a reflection coefficient r and in the presence of sources. At the domain boundaries, the steady-state density profile is nonanalytic. The meniscus exponent μ, introduced to characterize this singular behavior, uniquely identifies the whole profile. Numerical data suggest that μ = α/2 + r(α/2 - 1), where α is the Lévy exponent of the step-length distribution. As an application, we show that this model reproduces the temperature profiles obtained for a chain of oscillators displaying anomalous heat conduction. Remarkably, the case of free-boundary conditions in the chain corresponds to a Lévy walk with negative reflection coefficient.

Analytical Solutions for Contaminant Diffusion in Double-Layered Porous Media

Analytical solutions for conservative solute diffusion in one-dimensional double-layered porous media are presented in this paper. These solutions are applicable to various combinations of fixed solute concentration and zero-flux boundary conditions (BC) applied at each end of a finite one-dimensional domain and can consider arbitrary initial solute concentration distributions throughout the media. Several analytical solutions based on several initial and BCs are presented based on typical contaminant transport problems found in geoenvironmental engineering including (1) leachate diffusion in a compacted clay liner (CCL) and an underlying stratum; (2) contaminant removal from soil layers; and (3) contaminant diffusion in a capping layer and underlying contaminated sediments. The analytical solutions are verified against numerical solutions from a finite-element method based model. Problems related to leachate transport in a CCL and an underlying stratum of a landfill and contaminant transport through a capping layer over contaminated sediments are then investigated, and the suitable definition of the average degree of diffusion is considered.

Quantum motion over a finite one-dimensional domain: With and without dissipation

Quantum motion of a single particle over a finite one-dimensional spatial domain is considered for the generalized four parameter infinity of boundary conditions (GBC) of Carreauet al [1]. The boundary conditions permit complex eigenfunctions with nonzero current for discrete states. Explicit expressions are obtained for the eigenvalues and eigenfunctions. It is shown that these states go over to plane waves in the limit of the spatial domain becoming very large. Dissipation is introduced through Schrodinger-Langevin (SL) equation. The space and time parts of the SL equation are separated and the time part is solved exactly. The space part is converted to nonlinear ordinary differential equation. This is solved perturbatively consistent with the GBC. Various special cases are considered for illustrative purposes.

Oscillations of fronts and front pairs in two- and three-component reaction-diffusion systems

We investigate multi-co,ponent reaction-diffusion systems on a finite one-dimensional domain. The first component is assumed both to react on a short time-scale, and to have a small diffusion length leading to patterns with sharp fronts. To analyze stability, we apply the SLEP method by Nishiura and Fujii. This method is superior to standard perturbation methods. We illustrate this using a two-component system. In the three-component system, in a situation where both the influence of the first component on the other components and the interaction of the latter is only weak, we show that there is a unique Hopf destabilization of stationary fronts when changing time constants. As an example we treat a system with global coupling used e.g. for semi-conductor devices. For filaments (front pairs) we find two types of Hopf destabilizations: breathing and swinging. The global coupling controls which type occurs. This corresponds to recent experimental results and is confirmed via numerical calculations.