一类耦合分数阶反应–扩散系统的边界控制

针对带有空间变化的反应项的耦合分数阶反应–扩散系统边界镇定问题,利用反步法设计了用于Robin边界条件的状态反馈控制.通过可逆的积分变换将原耦合系统转化为一个稳定的目标系统.利用变量代换和逐次逼近法分析了核函数矩阵的存在唯一性.借助分数阶Lyapunov直接法证明了闭环系统的Mittag-Leffler稳定性.数值仿真验证了所提出方法的有效性.

Nonlinear predator-prey singularly perturbed Robin Problems for reaction diffusion systems

The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.

A Reduced Mechanism for Flame Inhibition by Phosphorus-containing Compounds Based on Level of Importance Analysis

A reduced mechanism for propane/air combustion and its flame inhibition by phosphorus-containing compounds (PCCs) is constructed with the level of importance (LOI) method. The analysis is performed on solutions of freely propagating premixed flames with detailed chemical kinetics involving 121 species and 682 reactions proposed by Jayaweera et al. For the non-homogeneous reaction-diffusion system, the chemical lifetime of each species is weighted by its diffusion timescale, and the characteristic flame timescale is used to normalize the chemical lifetime. The definition of sensitivity in LOI is extended so that multi-parameters can be used as sensitivity targets. Propane, oxygen, dimethyl methylphosphonate (DMMP), and flame speed are selected to be perturbed for sensitivity analysis, the species with low LOI index are removed, and reactions involving the redundant species are excluded from the mechanism. A skeletal mechanism is obtained, which consists of 57 species and 268 elementary reactions. Calculations for laminar flame speeds, key flame radicals and catalytic cycles using the skeletal mechanism are in good agreement with those by using the detailed mechanism over a wide range of equivalence ratio undoped and doped with DMMP.

Global attractors associated with general partly dissipative reaction-diffusion systems in unbounded domains

The asymptotic behaviour of solutions for general partly dissipative reaction-diffusion systems in Rn is studied. The asymptotic compactness of the solutions and then the existence of the global attractor are proved in L2(Rn )× L2(Rn ) .

Wolbachia infection dynamics by reaction-diffusion equations

Dengue fever is caused by the dengue virus and transmitted by Aedes mosquitoes.A promising avenue for eradicating the disease is to infect the wild aedes population with the bacterium Wolbachia driven by cytoplasmic incompatibility(CI).When releasing Wolbachia infected mosquitoes for population replacement,it is essential to not ignore the spatial inhomogeneity of wild mosquito distribution.In this paper,we develop a model of reaction-diffusion system to investigate the infection dynamics in natural areas,under the assumptions supported by recent experiments such as perfect maternal transmission and complete CI.We prove non-existence of inhomogeneous steady-states when one of the diffusion coefficients is sufficiently large,and classify local stability for constant steady states.It is seen that diffusion does not change the criteria for the local stabilities.Our major concern is to determine the minimum infection frequency above which Wolbachia can spread into the whole population of mosquitoes.We find that diffusion drives the minimum frequency slightly higher in general.However,the minimum remains zero when Wolbachia infection brings overwhelming fitness benefit.In the special case when the infection does not alter the longevity of mosquitoes but reduces the birth rate by half,diffusion has no impact on the minimum frequency.

THE GLOBAL STABILITY OF CONSTANT EQUILIBRIUM OF REACTION-DIFFUSION SYSTEMS——DIFFERENTIAL INEQUALITIES AND LIAPUNOV FUNCTIONAL

In this paper methods of differential inequalities and Liapunov functionals for proving the global stability of constant equilibria of reaction-diffusion systems are given, and examples for showing how to use these methods are also given.

Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems

In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of three- dimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotk,u-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.

Qualitative Analysis on a Reaction-Diffusion Prey-Predator Model and the Corresponding Steady-States

The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem. The local and global stability of the positive constant steady-state are discussed, and then some results for non- existence of positive non-constant steady-states are derived.

Stability of planar waves in reaction-diffusion system

This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t →∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.

Existence of spatiotemporal patterns in the reaction-diffusion predator-prey model incorporating prey refuge

The pattern formation in reaction-diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal exis- tence and importance. The present investigation deals with a spatial dynamics of the Beddington-DeAngelis predator-prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique pos- itive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion- driven instability of the spatiotemporal model are investigated. Based on the appro- priate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes repli- cation. The results obtained appear to enrich the findings of the model system under consideration.

ON THE EXISTENCE OF GLOBAL ATTRACTOR FOR A CLASS OF INFINITE DIMENSIONAL DISSIPATIVE NONLINEAR DYNAMICAL SYSTEMS

By means of a nonstandard estimation about the energy functional, the authors prove the existence of a global attractor for an abstract nonlinear evolution equation. As an application, the existence of a global attractor for some nonlinear reaction-diffusion equations with some distribution derivatives in inhomogeneous terms is obtained.

Entire solutions for some reaction-diffusion systems

This paper is concerned with the existence of entire solutions of some reaction-diffusion systems. We first consider Belousov-Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.

On a non-autonomous reaction-convection diffusion model to study the bacteria distribution in a river

In this paper, we propose a non-autonomous convection-reaction diffusion system （CDI） with a nonlinear reaction source function. This model refers to the quantification and the distribution of antibiotic resistant bacteria （ARB） in a river. The main contributions of this paper are： （i） the determination of the limit set of the system by applying the semigroups theory, it is shown that it is reduced to the solutions of the associated elliptic system （CDI）e, （ii） sufficient conditions for the existence of a positive solution of （CDI）e based on the Leray-Schauder＇s degree theory. Numerical simulations which support our theoretical analysis are also given.

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.

Composite Waves for a Cell Population System Modeling Tumor Growth and Invasion

In the recent biomechanical theory of cancer growth,solid tumors are considered as liquid-like materials comprising elastic components.In this fluid mechanical view,the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate,with the latter depending on the local cell density(contact inhibition) or/and on the mechanical stress in the tumor.For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling,the authors prove that there are always traveling waves above a minimal speed,and analyse their shapes.They appear to be complex with composite shapes and discontinuities.Several small parameters allow for analytical solutions,and in particular,the incompressible cells limit is very singular and related to the Hele-Shaw equation.These singular traveling waves are recovered numerically.

Simultaneous Identification of Two Parameters on the Reaction Diffusion System from Discrete Measurement Data

This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the help of the smooth interpolation technique.The main objective of the article is to analyse the asymptotic behavior of the solution of the inverse problem for the linearly coupled reaction diffusion system with respect to the homogeneous Dirichlet boundary condition.

Null Controllability of Some Reaction-Diffusion Systems with Only One Control Force in Moving Domains

In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani＇s fixed point theorem combined with observability estimates for the associated lineaxized system.

A periodic reaction-diffusion system modelling man-environment-man epidemics

The purpose of this work is to study the spatial dynamics of a periodic reaction-diffusion epidemic model arising from the spread of oral-faecal transmitted diseases. We first show that the disease spreading speed is coincident with the minimal wave speed for monotone periodic travelling waves. Then we obtain a threshold result on the global attractivity of either zero or the positive periodic solution in a bounded spatial domain.

PATTERN FORMATION IN CHEMOTAXIC REACTION-DIFFUSION SYSTEMS

In this study, we investigate two-dimensional patterns generated by chemotaxis reaction-diffusion systems. We numerically examine the Keller-Segel models with the volume-filling aggregation term and the receptor aggregation term in two dimensions. Spotted, striped and reversed spotted patterns are obtained as stable motionless equi- librium patterns. The relative stability of these patterns is studied numerically on the basis of the derived free energy. The intuitive understanding of these generated patterns and the relation with three-dimensional patterns are also discussed.

The asymptotic stability model with for an SIQS epidemic diffusion

In this paper, an SIQS epidemic model with constant recruitment and standard inci- dence is investigated. Quarantine is taken into consideration on the basis of SIS model. The asymptotic stability of the equilibrium to a reaction^diffusion system with homo- geneous Neumann boundary conditions is considered. Sufficient conditions for the local and global asymptotic stability are given by linearization and the method of upper and lower solutions and its associated monotone iterations. The result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.