Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays

This paper is concerned with three 3-species time-delayed Lotka-Volterra reaction-diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction-diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction-diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey-predator reaction-diffusion systems are included in the discussion.     

Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

Some coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature.     

Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates

An analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction–diffusion equations with global regulation possess the same major qualitative features as the corresponding local models. In particular, using the introduced notion of the stability in the mean integral sense we prove the competitive exclusion principle for the autocatalytic system and the permanence for the hypercycle system. Existence and stability of stationary solutions are studied. For some parameter values it is proved that stable spatially non-uniform solutions appear.     

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.     

Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model

Using the energy estimate and Gagliardo–Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for a strongly coupled reaction–diffusion system are proved. This system is the Shigesada–Kawasaki–Teramoto three-species cooperating model with self- and cross-population pressure. Meanwhile, some criteria on the global asymptotic stability of the positive equilibrium point for the model are also given by Lyapunov function. As a by-product, we proved that only constant steady states exist if the diffusion coefficients are large enough.     

Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.     

Effect of diffusion on a two-species eco-epidemiological model

This paper deals with the stabilizing effect of diffusion on a prey?–?predator system where the prey population is infected by a microparasite. The predator functional response is a concave-type function. Conditions for the local as well as global stability of the model without diffusion are derived in terms of system parameters. It is also shown that an unstable equilibrium of the model without diffusion can be made stable by increasing the diffusion coefficients appropriately.     

Well-posedness of a moving-boundary problem with two moving reaction strips

We deal with a one-dimensional coupled system of semi-linear reaction-diffusion equations in two a priori unknown moving phases driven by a non-local kinetic condition. The PDEs system models the penetration of gaseous carbon dioxide in unsaturated porous materials (like concrete). The main issue is that the strong competition between carbon dioxide diffusion and the fast reaction of carbon dioxide with calcium hydroxide–which are the main active reactants–leads to a sudden drop in the alkalinity of concrete near the steel reinforcement. This process–called concrete carbonation–facilitates chemical corrosion and drastically influences the lifetime of the material. We present details of a class of moving-boundary models with kinetic condition at the moving boundary and address the local existence, uniqueness and stability of positive weak solutions. We also point out our concept of global solvability. The application of such moving-boundary systems to the prediction of carbonation penetration into ordinary concrete samples is illustrated numerically.     

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.     

Avian–human influenza epidemic model with diffusion,nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

We consider a reaction-diffusion system which models a fast reversible reaction of type C ₁ + C ₂?C ₃ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L ²-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.     

Novel stability criteria for impulsive delayed reaction–diffusion Cohen–Grossberg neural networks via Hardy–Poincarè inequality

This work is devoted to the investigation of stability theory for impulsive delayed reaction–diffusion Cohen–Grossberg neural networks with Dirichlet boundary condition. By means of Hardy–Poincarè inequality and Gronwall–Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring global exponential stability of the equilibrium point. The presented stability criteria show that not only reaction–diffusion coefficients but also regional features as well as the first eigenvalue of the Dirichlet Laplacian will impact the stability. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.     

Degn-Harrison反应扩散系统的Turing不稳定性与Hopf分支

在齐次Neumann边界条件下研究一类Degn-Harrison反应扩散系统.首先讨论常微分系统正平衡点的稳定性和Hopf分支,其次研究扩散系统,给出扩散系数对正平衡点稳定性的影响,建立系统的Turing不稳定性,同时在扩散系数满足一定条件时给出Hopf分支的存在性.     

Dynamics of a spatially distributed logistic equation with small diffusion and small delay

For the spatially distributed Hutchinson equation with transport and small diffusion constant, we show that the loss of stability of the equilibrium can occur even for asymptotically small values of the delay coefficient. Here infinitely many roots of the characteristic quasipolynomial tend to the imaginary axis as the small parameter, the diffusion constant, tends to zero. Thus, the critical (in the problem on the equilibrium stability) case of infinite dimension is realized. We construct special quasinormal forms, namely, nonlinear parabolic systems and families of degenerate parabolic systems whose nonlocal dynamics describes the behavior of solutions of the original equation in a small neighborhood of the equilibrium. These quasinormal forms can have a rather complicated dynamics; moreover, the onset and disappearance of steady-state modes as the small parameter tends to zero is a typical phenomenon. Therefore, the local dynamics of the Hutchinson equation with and without transport are very distinct.     

Numerical modelling of an SIR epidemic model with diffusion

A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.     

Spreading Speeds of Time-Dependent Partially Degenerate Reaction-Diffusion Systems*

This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity. By using the principal Lyapunov exponent theory, the author first proves the existence, uniqueness and stability of spatially homogeneous entire positive solution for time dependent partially degenerate reaction-diffusion system. Then the author shows that such system has a finite spreading speed interval in any direction and there is a spreading speed...     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model

In the new social media era, it is becoming increasingly important to explore the propagation rules for rumors in social networks. This article is concerned with investigating a diffusive susceptible-infected rumor propagation model with a nonlinear propagation function in a spatially heterogeneous environment. We establish the uniform persistence and analyze the asymptotic behavior of the rumor-spreading steady state for the spatially heterogeneous model when one of the diffusion coefficients tends to zero. Moreover, to better reflect the effect of a time delay on the process of rumor propagation, we establish a spatially homogeneous model with a time delay and prove the existence and local stability of the corresponding equilibrium point. Furthermore, the optimal control in the spatially homogeneous environment case is derived. Finally, several numerical simulations are performed to verify the theoretical results in both spatially heterogeneous and spatially homogeneous systems.     

一类具有阶段结构的三次捕食者-食饵交错扩散系统的整体解

应用能量估计方法和Gagliardo-Nirenberg型不等式证明了一类强耦合反应扩散系统整体解的存在性和一致有界性,该系统是具有阶段结构的两种群Lotka-Volterra捕食者-食饵交错扩散模型的推广.通过构造Lyapunov函数给出了该系统正平衡点全局渐近稳定的充分条件.