Turing bifurcation in a system with cross diffusion

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction–diffusion equations subject to Neumann boundary data. Hence, we assume that there are two substances in a two-dimensional bounded spatial domain where they are diffusing according to Fick's law: the velocity of the flow of diffusing substance is directed opposite to the (spatial) gradient of the density and is proportional to its modulus, but the spatial flow of each substance is influenced not only by its own but also by the other one's density (cross diffusion). The domains in which the substances are diffusing are of three type: a regular hexagon, a rectangle and an isosceles rectangular triangle. It will be assumed that there is no migration across the boundary of these domains. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises.     

Pattern selection in the 2D FitzHugh–Nagumo model

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

     

On the Convergence of Certain Numerical Characteristics of Variational Dirichlet Problems in Variable Domains

We prove two theorems that enable one to reduce the problem of convergence of general characteristics of variational Dirichlet problems in variable domains to the problem of convergence of simpler characteristics of these problems. We describe the case where the convergence of simpler characteristics takes place.     

Bifurcations in a human migration model of Scheurle-Seydel type-II: Rotating waves

This paper treats the conditions for the existence of rotating wave solutions of a system modelling the behavior of students in graduate programs at neighbouring universities near each other which is a modified form of the model proposed by Scheurle and Seydel. We assume that both types of individuals are continuously distributed throughout a bounded two-dimension spatial domain of two types (circle and annulus), across whose boundaries there is no migration, and which simultaneously undergo simple (Fickian) diffusion. We will show that at a critical value of a system-parameter bifurcation takes place: a rotating wave solution arises.     

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.     

Turing instability for a ratio-dependent predator-prey model with diffusion

Ratio-dependent predator-prey models have been increasingly favored by field ecologists where predator-prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction-diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included.     

Diffusion driven instability in an inhomogeneous circular domain

Classical reaction-diffusion systems have been extensively studied and are now well understood. Most of the work to date has been concerned with homogeneous models within one-dimensional or rectangular domains. However, it is recognised that in most applications nonhomogeneity, as well as other geometries, are typically more important. In this paper, we present a two chemical reaction-diffusion process which is operative within a circular region and the model is made nonhomogeneous by supposing that one of the diffusion coefficients varies with the radial variable. Linear analysis leads to the derivation of a dispersion relation for the point of onset of instability and our approach enables both axisymmetric and nonaxisymmetric modes to be described. We apply our workings to the standard Schnackenberg activator-inhibitor model in the case when the variable diffusion coefficient takes on a step-function like profile. Some fully nonlinear simulations show that the linear analysis captures the essential details of the behaviour of the model.     

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction–diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction–diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.     

Diffusion and growth in an evolving network

We study a simple model of a population of agents whose interaction network co-evolves with knowledge diffusion and accumulation. Diffusion takes place along the current network and, reciprocally, network formation depends on the knowledge profile. Diffusion makes neighboring agents tend to display similar knowledge levels. On the other hand, similarity in knowledge favors network formation. The cumulative nonlinear effects induced by this interplay produce sharp transitions, equilibrium co-existence, and hysteresis, which sheds some light on why multiplicity of outcomes and segmentation in performance may persist resiliently over time in knowledge-based processes.     

市场导向、多主体协同与创新扩散:基于复杂网络的动态仿真

市场导向与多主体协同关系密切,是提升创新扩散效率的主要驱动要素。本文首先从协同关系和创新收益两个层面,构建空间结构和有效预期的演化机制,生成复杂网络模型,对多主体协同的创新扩散过程进行动态仿真。其次通过细化市场导向理论在用户需求、竞争驱动和职能协同等不同维度的作用路径,深入分析了市场导向对多主体协同的影响机制。研究表明:(1)市场导向对多主体协同的影响与网络结构动态特征具有高度相关性;(2)少量的用户需求与竞争驱动导向对多主体协同的效益提升最为显著,职能协同导向的影响则呈现周期性“倒U型”波动特征;(3)用户需求导向对多主体协同的创新收益驱动效应最明显,竞争驱动导向的推动效果次之。     

Estimation of the Diffusion Coefficient from Crossings

In a previous work we proposed a kernel method for estimating the value of a state‐dependent diffusion coefficient σ(x) from discrete time observations. We propose an estimator of the value σ(x), based on the squared length of the transitions where a crossing past the level x takes place and we investigate its limit properties. Using the Poisson equation and other limit theorems for discrete martingales, we prove consistency and asymptotic mixed normality of this estimator. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dynamical properties of the reaction–diffusion type model of fast synaptic transport

The modulation of a signal that is transmitted in the nerve system takes place in chemical synapses. This article focuses on the phenomena undergone in the presynaptic part of the synapse. A diffusion–reaction type model based on the partial differential equation is proposed. Through an averaging procedure this model is reduced to a model based on ordinary differential equations with control, which is then analyzed according to its dynamical properties—controllability, observability and stability. The system is strongly connected to the one introduced by Aristizabal and Glavinovic (2004) [13]. The biological implications of the obtained mathematical results are also discussed.     

On the Uniform Convergence of the Empirical Density of an Ergodic Diffusion

We investigate the uniform convergence of the density of the empirical measure of an ergodic diffusion. It is known that under certain conditions on the drift and diffusion coefficients of the diffusion, the empirical density f _t converges in probability to the invariant density f, uniformly on the entire real line. We show that under the same conditions, uniform convergence of f _t to f on compact intervals takes place almost surely. Moreover, we prove that under much milder conditions (the usual linear growth condition on the drift and diffusion coefficients and a finite second moment of the invariant measure suffice), we have the uniform convergence of f _t to f on compacta in probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Spatiotemporal dynamics of reaction–diffusion models of interacting populations

The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator–prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.     

The stability of equilibria for a reaction–diffusion–ODE system on convex domains

A reaction–diffusion equation coupled to an ODE on convex domains is proposed to model a single species with a non-mobile state and an active state. Under general cooperative/competitive interactions, a trivial stability on convex domains is shown and the reaction–diffusion–ODE system does not support interesting patterns.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

Dead-core solutions for slightly non-isothermal diffusion-reaction problems with power-law kinetics

The paper deals with dead-core solutions to a non-isothermal reaction- diffusion problem with power-law kinetics for a single reaction that takes place in a catalyst pellet along with mass and heat transfer from the bulk phase to the outer pellet surface. The model boundary value problem for two coupled non-linear diffusion-reaction equations is solved using the semi-analytical method. The exact solutions are established under the assumption of a small temperature gradient in the pellet. The nonlinear algebraic expressions are derived for the critical Thiele modulus, dead-zone length, reactant concentration, and temperature profiles in catalyst pellets of planar geometry. The effects of the reaction order, Arrhenius number, energy generation function, Thiele modulus, and Biot numbers are investigated on the concentration and temperature profiles, dead-zone length, and critical Thiele modulus.     

Diffusion in a locally stationary random environment

This paper deals with homogenization of diffusion processes in a locally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims at giving a macroscopic approximation that takes into account the microscopic heterogeneities.     

System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary

Summary We study a system of particles and the nonlinear McKean-Vlasov diffusion that is its limit for weak interactions in Statistical Mechanics, reflecting in a domain with sticky boundary. The interaction takes place in particular in the sojourn condition. We show existence and uniqueness for the nonlinear martingale problem, by a contraction argument on time-change. Then we construct the system of particles by a limiting procedure, and show propagation of chaos towards the nonlinear diffusion.

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Résumé Nous étudions un système de particules et la diffusion non-linéaire de type McKean-Vlasov qui en est la limite en Mécanique Statistique pour des interactions faibles, en réflexion dans un domaine à bord collant. L'interaction réside en particulier dans la condition de séjour. Nous montrons l'existence et l'unicité pour le problème de martingales non-linéaire, par une méthode de contraction sur le changement de temps. Nous construisons le système de particules en tant que limite en loi, et démontrons la propagation du chaos vers la diffusion non-linéaire.

     

The vibration of a rotating heavy non-uniform thread and its stability

The linear vibration of a heavy non-uniform thread is investigated for different boundary conditions at the ends and taking an arbitrary additional tension into account. The thread is assumed to be ideal and inextensible and the motion takes place in a plane which may rotate about the vertical axis at constant angular velocity. A general scheme for solving the initial- and boundary value problem is proposed. Attention is focused mainly on the effective computation of the natural frequencies and mode of vibration. Given specific parametric types of mass distribution for the thread, sufficiently complete families of solutions describing the principal modes of vibration are constructed. Based on these families, stability and instability domains are constructed effectively, in terms of the system parameters, for a plane vibration of a rotating heavy thread subject to concentrated tension. New mechanical effects, of possible interest in practice, are observed and discussed.