Stability Criteria for Reaction-Diffusion Systems with Skew-Gradient Structure

This paper deals with reaction-diffusion systems with skew-gradient structure. In connection with calculus of variations, we show that there is a close relation between the stability of a steady state and its relative Morse index. The stability criteria presented here were partially motivated by some recent works of Yanagida.     

Bifurcation points for a reaction-diffusion system with two inequalities

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type which is subject to diffusion-driven instability. We show that obstacles (e.g. a unilateral membrane) for both quantities modeled in terms of inequalities introduce a new bifurcation of spatially non-homogeneous steady states in the domain of stability of the trivial solution of the corresponding classical problem without obstacles.     

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.     

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.     

Destabilization for quasivariational inequalities of reaction-diffusion type

We consider a reaction-diffusion system of the activator-inhibitor type with unilateral boundary conditions leading to a quasivariational inequality. We show that there exists a positive eigenvalue of the problem and we obtain an instability of the trivial solution also in some area of parameters where the trivial solution of the same system with Dirichlet and Neumann boundary conditions is stable. Theorems are proved using the method of a jump in the Leray-Schauder degree.     

A variational approach to bifurcation points of a reaction-diffusion system with obstacles and neumann boundary conditions

Given a reaction-diffusion system which exhibits Turing’s diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the classical case without unilateral obstacles. The study is based on a variational approach to a non-variational problem which even after transformation to a variational one has an unusual structure for which usual variational methods do not apply.     

Turing patterns of a strongly coupled predator-prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.     

New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type which is subject to diffusion-driven instability. We show that an obstacle (e.g. a unilateral membrane) modeled either in terms of inequalities or of inclusions, introduces whole beams of new global bifurcation points of spatially non-homogeneous stationary solutions which lie in parameter domains which are excluded as bifurcation points for the problem without the obstacle.     

Non-constant positive steady state of one resource and two consumers model with diffusion

In this paper, a predator-prey reaction-diffusion system with one resource and two consumers is considered. Assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington-DeAngelis functional response, and they compete for the common resource. First, it is proved that the unique positive constant steady state is stable for the ODE system and the reaction-diffusion system. Second, a prior estimates of positive steady state is given. Finally, the non-existence of non-constant positive steady state, the existence and bifurcation of non-constant positive steady state are studied.     

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low (n?10) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n?10.Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction-diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.     

Point condensations of a model for fungal development

We study the stationary solutions for a reaction-diffusion system of activator-inhibitor type which arises as a model for fungal development. Under the condition that the activator diffuses slowly and the inhibitor diffuses very quickly we rigorously construct solutions which show single peak pattern near the boundary or in the interior in the activator component and have nearly constant values in the other. We also establish the linear stability and instability of such solutions.     

Spherically symmetric internal layers for activator-inhibitor systems II. Stability and symmetry breaking bifurcations

A reaction-diffusion system of activator-inhibitor type is studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. We analyze the stability property of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions which are not spherically symmetric.     

Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

两点边值问题解的存在性

本文讨论两点边值问题解的存在性,在没有孤立性条件下,获得了不动点定理,作为应用实例,给出了反应扩散方程稳态解的存在性证明。     

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.     

Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.     

Convergence in competition models with small diffusion coefficients

It is well known that for reaction-diffusion 2-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the reaction system implies global stability for the reaction-diffusion system. This is not in general true for spatially inhomogeneous models. We show here that for an important range of such models, for small enough diffusion coefficients, global convergence to an equilibrium holds for the reaction-diffusion system, if for each point in space the reaction system has a globally attracting hyperbolic equilibrium. This work is planned as an initial step towards understanding the connection between the asymptotics of reaction-diffusion systems with small diffusion coefficients and that of the corresponding reaction systems.     

QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.     

一类捕食-食饵-互惠模型非常数正平衡解的存在性及分歧

讨论了一类捕食-食饵-互惠反应扩散系统的非常数正平衡解.首先分析了常数正平衡解的稳定性,其次;利用最大值原理和Harnack不等式给出了正解的失验估计.在此基础上,利用积分性质进一步讨论了非常数正解的不存在性,相应地证明了当扩散系数d_2 d_3大于特定正常数且扩散系数d_1有界时此模型没有非常数正解.同时利用度理论证明了当模型的线性化算子的正特征值的代数重数是奇数且扩散系数d_3不小于给定正常数时此模型至少存在一个非常数正解,最后研究了非常数正解的分歧.     

The steady states of a non-cooperative model of nuclear reactors

This paper characterizes the existence of coexistence states in a reaction-diffusion model arising in the theory of nuclear reactors. From a mathematical point of view, the importance of this model relies upon the fact that the associated variational systems are of non-cooperative type and, consequently, the comparison techniques available for cooperative systems fail to work out. Although in higher spatial dimensions the dynamics of the model might be rather involved, by the absence of limitations for the number of steady states, we can prove the uniqueness of the steady state in the one-dimensional prototype model. Our results complement and eventually sharpen the findings of Arioli [G. Arioli, Long term dynamics of a reaction-diffusion system, J. Differential Equations 235 (2007) 298-307].