An Algebraic Method on the Eigenvalues and Stability of Delayed Reaction-Diffusion Systems

Abstract

The eigenvalues and stability of the delayed reaction-diffusion systems are considered using the algebraic methods. Firstly, new algebraic criteria to determine the pure imaginary eigenvalues are derived by applying the matrix pencil and the linear operator methods. Secondly, a practical checkable criteria for the asymptotic stability are introduced.

1. Introduction

The reaction-diffusion system is a semilinear partial differential equation, which has been used for the study of morphogenesis, population dynamics, and autocatalytic oxidation reactions. The early reaction-diffusion models were constructed to describe the process of chemical reaction. For example, the Brusselator reaction-diffusion system and its improved systems had been researched by many scholars [1, 2]. Besides, during the past years, many results about the stability, steady state bifurcation, and the Hopf bifurcation on the reaction-diffusion systems had been derived [3–8]. From those results, we see that most of them had been mainly focused on two methods: the analytical methods and the numerical methods. The analytical methods, such as the center manifold theorem, the normal form theory, and the Laplace transformation, were mainly used to research the analytic solutions and dynamical property. The numerical methods, such as the Runge-Kutta methods and linear multistep methods, paid close attention to checking the changes and convergence of solutions. For example, Wu determined the direction and stability of periodic solutions occurring through the Hopf bifurcation by the center manifold theory and the normal form theory, which are the classic analytical methods in functional differential systems [1]. Wei et al. in papers [3–5] demonstrated the bifurcation and stability of different reaction-diffusion systems.

In this paper, we will introduce a new algebraic method. In fact, in the past recent years, many algebraic methods were applied to the functional differential systems, such as matrix theory, polynomial theory, the Lie group, and algebraic system. In particular for the study on the more complex systems, such as the high dimensional functional differential systems, the algebraic methods have their advancement. So in the following, we will discuss the delayed reaction-diffusion systems by the algebraic methods. We will study a general delayed reaction-diffusion system with spatial domain Ω in the following form: 𝜕𝑢₁𝜕𝑡=𝑑₁Δ𝑢₁+𝑓₁𝑢_𝑖;𝑢_𝑖(𝑥,𝑡−𝜏)𝜕𝑢₂𝜕𝑡=𝑑₂Δ𝑢₂+𝑓₂𝑢_𝑖;𝑢_𝑖(⋮𝑥,𝑡−𝜏)𝜕𝑢_𝑚𝜕𝑡=𝑑_𝑚Δ𝑢_𝑚+𝑓_𝑚𝑢_𝑖;𝑢_𝑖(𝑥,𝑡−𝜏)𝑥∈Ω,𝑡>0,𝑖=1,2,…,𝑚, (1) where the spatial domain Ω is restricted to one-dimensional interval (0,𝜋), that is, Ω=(0,𝜋). Also 𝑢_𝑖=𝑢_𝑖(𝑥,𝑡)>0  (𝑖=1,2,…,𝑚) stands for the concentrations of 𝑚 reactants or densities of 𝑚 species. Δ denotes the Laplace operator and Δ𝑢_𝑖=𝜕²𝑢_𝑖/𝜕𝑥². 𝑑_𝑖  (𝑖=1,2,…,𝑚) denotes the diffusion coefficients of 𝑢_𝑖, respectively. This paper will consider (1) subject to the origin condition and the homogeneous Neumann boundary condition on spatial domain Ω: 𝑢_𝑖(𝑥,0)=𝑢_𝑖(𝑥),𝑥∈Ω𝜕𝑢_𝑖(𝑥,𝑡)𝜕𝑥=0,𝑥=0,𝜋,𝑡>0𝑖=1,2,…,𝑚. (2) For the system (1) and the determined condition (2), we suppose that 𝑓_𝑖(0)=0, for any 𝑡≥0, and 𝑓_𝑖 is 𝐶^𝑘, 𝑘≥5. That is to say, the origin point (0,0,…,0) is an equilibrium of system (1) and (2). So the matrix form of system (1) can be written as 𝜕𝑈(𝑥,𝑡)=()(𝑈()(,𝜕𝑡Δ𝑈𝑥,𝑡+𝐹𝑥,𝑡,𝑈𝑥,𝑡−𝜏)) (3) where 𝑈()=𝑢𝑥,𝑡¹()𝑢𝑥,𝑡₂()⋮𝑢𝑥,𝑡_𝑚(),𝑑𝑥,𝑡Δ=¹Δ0⋯00𝑑₂Δ⋯0⋮⋮⋱⋮00⋯𝑑_𝑚Δ𝑓,𝐹=¹𝑓₂⋮𝑓_𝑚. (4) The linearization of system (3) at origin point (0,0,…,0) is 𝜕𝑈(𝑥,𝑡)=()()().𝜕𝑡Δ𝑈𝑥,𝑡+𝐴𝑈𝑥,𝑡+𝐵𝑈𝑥,𝑡−𝜏 (5) Furthermore, it is well known that the eigenvalue problem 𝜓^′′+𝜆𝜓=0,𝑥∈(0,𝜋),𝜓^′(0)=𝜓^′(𝜋)=0 (6) has eigenvalues 𝜆_𝑛=𝑛², (𝑛=0,1,2,…), with corresponding eigenfunctions 𝜓_𝑛=cos𝑛𝑥. Then eigenvalues of Δ are {−𝑑₁𝑛²,−𝑑₂𝑛²,…,−𝑑_𝑚𝑛²}, 𝑛=0,1,2,…. Let 𝑌=^∞_𝑛=0𝑦cos𝑛𝑥⋅𝑦,𝑦=^1𝑛𝑦_2𝑛⋮𝑦_𝑚𝑛. (7) Then 𝑌 is an eigenfunction for (5) with eigenvalue 𝑠. By straightforward analysis, we obtain 𝑝_𝑛(𝑠,𝜏)𝑦=𝑠𝐸−𝐷_𝑛−𝐴−𝐵𝑒^−𝑠𝜏𝑦=0,𝑛=0,1,2,…, (8) where 𝐷_𝑛=diag(−𝑑₁𝑛²,−𝑑₂𝑛²,…,−𝑑_𝑚𝑛²). So we can find all of the eigenvalues of system (5) from (8). Certainly, the stability, and Hopf bifurcation of system (1) can be described by studying the linear system (5).

The remaining parts of the paper are structured in the following way. In Section 2, we demonstrated the critical condition on the delay 𝜏 of the system (1) and got the algebraic criteria for determining the pure imaginary eigenvalues. In Section 3, we researched the stability and the Hopf bifurcation of the delayed reaction-diffusion equation (39) with Neumann boundary condition and derived the corresponding algebraic criteria. At last, we described a specific reaction-diffusion equation and simulated the results by MATLAB.

2. Algebraic Criteria for Determining the Pure Imaginary Eigenvalues

Firstly, we research an ordinary differential equation 𝑋̇(𝑡)=𝐴_𝑘𝑋(𝑡)(𝑡)𝑌̇(𝑡)(𝑡)𝐵+𝐵𝑌=−𝑋^T−𝑌(𝑡)𝐴^T_𝑘, (9) where 𝐴_𝑘,𝐵∈ℝ^𝑛×𝑛, 𝑋,𝑌∈ℂ^𝑛×𝑛. Let 𝑉 denote the vector space 𝑉=ℂ^𝑛×𝑛×ℂ^𝑛×𝑛, let 𝐼 denote the identity operator on 𝑉, and let operator 𝐌 on 𝑉 be given by 𝐌𝑋𝑌=𝐴_𝑘𝑋+𝐵𝑌−𝑋𝐵^T−𝑌𝐴^T_𝑘,∀𝑋,𝑌∈ℂ^𝑛×𝑛. (10) With 𝑍(𝑡)=^𝑋(𝑡)_𝑌(𝑡), the system (9) can be written as 𝑍̇(𝑡)=𝐌𝑍(𝑡). (11) Suppose that 𝑋𝑌𝑍(𝑡)==^{𝑋0𝑒𝑠𝑡}_{𝑌0𝑒𝑠𝑡} is a matrix solution of the system (9). We have 𝑠𝐸−𝐴_𝑘𝑋−𝐵𝑌=0𝑋𝐵^T+𝑌𝑠𝐸+𝐴^T_𝑘=0. (12) For any complex 𝑠, let 𝐓=𝐓(𝑠) be the operator 𝐓=𝑠𝐈−𝐌. Then 𝐓𝑋𝑌=𝑠𝐸−𝐴_𝑘𝑋−𝐵𝑌𝑋𝐵^T+𝑌𝑠𝐸+𝐴^T_𝑘,∀𝑋,𝑌∈ℂ^𝑛×𝑛. (13) For any complex 𝑠, let 𝐓⁺=𝐓⁺(𝑠)∶𝑉→𝑉, satisfying 𝐓⁺𝑋𝑌=𝑋𝑠𝐸+𝐴^T_𝑘+𝐵𝑌−𝑋𝐵^T+𝑠𝐸−𝐴_𝑘𝑌,∀𝑋,𝑌∈ℂ^𝑛×𝑛. (14) For any complex 𝑠, let Λ=Λ(𝑠), satisfying Λ𝑋=𝑠𝐸−𝐴_𝑘𝑋𝑠𝐸+𝐴^T_𝑘+𝐵𝑋𝐵^T,∀𝑋∈ℂ^𝑛×𝑛. (15) By simple computations, we can get 𝑠𝐸−𝐴_𝑘𝑋𝑠𝐸+𝐴^T_𝑘+𝐵𝑋𝐵^T=𝑋₀𝑠𝐸+𝐴^T_𝑘+𝐵𝑌₀𝑠𝐸−𝐴_𝑘𝑌𝑠𝐸+𝐴^T_𝑘+𝐵𝑌𝐵^T=−𝑋₀𝐵^T+𝑠𝐸−𝐴_𝑘𝑌₀. (16) By the operator language, that is, 𝐓⁺𝐓𝑋𝑌=Λ𝑋Λ𝑌. (17) In the following, we will convert matrix ordinary differential equation (10) to vector form. Let 𝜉 be the elementary transform, 𝜉∶ℂ^𝑛×𝑛→ℂ^𝑛2. That is, 𝑎𝜉𝐴=⋮^T1𝑎^T_𝑛𝑎,∀𝐴=⋮¹𝑎_𝑛,𝑎^T_𝑖∈ℂ^𝑛,𝑖=1,2,…,𝑛. (18) Let 𝑥=𝜉𝑋,𝑦=𝜉𝑌,𝑧=^𝑥_𝑦, and 𝐼₀=𝐸⊗𝐸00𝐸⊗𝐸,𝑀₀=𝐴_𝑘⊗𝐸𝐵⊗𝐸−𝐸⊗𝐵−𝐸⊗𝐴_𝑘. (19) Using the property of the Kronecker product, we have 𝜉(𝐴𝑋𝐵)=𝐴⊗𝐵^T𝜉𝑋, (20) where 𝐴,𝐵,𝑋∈ℂ^𝑛×𝑛. So (11) can be written as 𝐼₀𝑧̇(𝑡)=𝑀₀𝑧(𝑡). (21) Similarly, by denoting 𝑇₀=𝑇₀(𝑠), 𝑇⁺₀=𝑇⁺₀(𝑠), and Λ₀=Λ₀(𝑠) as follows: 𝑇₀=𝑇₀(𝑠)=𝑠𝐸−𝐴_𝑘⊗𝐸−𝐵⊗𝐸𝐸⊗𝐵𝐸⊗𝑠𝐸+𝐴_𝑘=𝑠𝐼₀−𝑀₀,𝑇⁺₀=𝑇⁺₀(𝑠)=𝐸⊗𝑠𝐸+𝐴_𝑘𝐵⊗𝐸−𝐸⊗𝐵𝑠𝐸−𝐴_𝑘,Λ⊗𝐸₀=Λ₀(𝑠)=𝑠𝐸−𝐴_𝑘⊗𝑠𝐸+𝐴_𝑘+𝐵⊗𝐵, (22) we have 𝑇⁺₀𝑇₀𝑥𝑦=Λ₀𝑥Λ₀𝑦,thatis,𝑇⁺₀𝑇₀=Λ₀00Λ₀. (23)

Lemma 1. For all complex 𝑠, det𝑇⁺₀(𝑠)=det𝑇₀(𝑠), and so det𝑇₀(𝑠)=±detΛ₀(𝑠).

Proof. Let 𝑇₀=^𝑎𝑏_𝑐𝑑, where 𝑎=(𝑠𝐸−𝐴_𝑘)⊗𝐸, 𝑏=−𝐵⊗𝐸, 𝑐=𝐸⊗𝐵, and 𝑑=𝐸⊗(𝑠𝐸+𝐴_𝑘). Noting the Kronecker product identities (𝐼⊗𝑀)(𝑁⊗𝐼)=(𝑁⊗𝐼)(𝐼⊗𝑀)=𝑁⊗𝑀, we have 𝑑𝑏=𝑏𝑑, 𝑏𝑐=𝑐𝑏. By regularity of (𝐸,𝐴_𝑘), we know that 𝑠𝐸+𝐴_𝑘 is nonsingular for enough complex 𝑠. So by the property of the polynomial and easy computations, we can get that det𝑇⁺₀(𝑠)=det𝑇₀(𝑠) for any complex 𝑠. Elsewhere, 𝑇⁺₀𝑇₀=Λ₀00Λ₀, (24) so det𝑇₀²=det𝑇⁺₀det𝑇₀=detΛ₀². (25) Thus det𝑇₀(𝑠)=±detΛ₀(𝑠). (26)

Theorem 2. Any pure imaginary eigenvalue of the system (5) or the system (1) is a zero point of detΛ₀(𝑠) and thus also one of the eigenvalues of the matrix pencil (𝐼₀,𝑀₀).

Proof . For (8), let 𝐷_𝑘+𝐴=𝐴_𝑘, 𝑘=0,1,2,…. Then we have 𝑝_𝑘𝑠,𝑒^−𝑠𝜏=𝑠𝐸−𝐴_𝑘−𝐵𝑒^−𝑠𝜏. (27) Let 𝑠=𝑖𝑤 be a pure imaginary eigenvalue of the system (5), let 𝑦 be associated eigenvector, and let ||𝑦||=1. We have 𝑝_𝑘(𝑠,𝑒^−𝑠𝜏)𝑦=0. By conjugating and transforming, we can get 𝑠𝐸−𝐴_𝑘𝑦𝑦^∗𝑠𝐸+𝐴^T_𝑘+𝐵𝑦𝑦^∗𝐵^T=0. (28) Via the elementary transform 𝜉, we get 𝑠𝐸−𝐴_𝑘⊗𝑠𝐸+𝐴_𝑘+𝐵⊗𝐵𝜉𝑦𝑦^∗=0. (29) That is Λ₀(𝑠)𝑢=0, 𝑢=𝜉(𝑦𝑦^∗). We know that detΛ₀(𝑠)=0, and so, det𝑠𝐼₀−𝑀₀=±detΛ₀(𝑠)=0. (30)

From Theorem 2, we know that all of the pure imaginary eigenvalues of the system (1) are zero points of the algebraic equation det𝑠𝐸−𝐴_𝑘⊗𝑠𝐸+𝐴_𝑘+𝐵⊗𝐵=0. (31) So the pure imaginary eigenvalues of the system (1) or (5) can be computed via the algebraic equation (31). In fact, (31) is usually called a polynomial eigenvalue problem. The classical and most widely used approach to research the polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. There are many forms for the linearization, but the companion form is most typically commission [9]. So we have the following results.

Theorem 3. The pure imaginary eigenvalues of the system (5) are the general eigenvalues of matrix pencil (𝑃,𝑄), where 𝑃=𝐼00𝐼,𝑄=−𝐺−𝐻𝐼0, (32)𝐼=𝐸⊗𝐸, 𝐺=(−𝐴_𝑘)⊕𝐴_𝑘, and 𝐻=𝐵⊗𝐵.

Proof . Equation (31) can be rewritten as 𝐸⊗𝐸𝑠²+𝐸⊗𝐴_𝑘−𝐴_𝑘⊗𝐸𝑠+𝐵⊗𝐵𝑣=0, (33) where 𝑣=𝜉(𝑦𝑦^∗). For −𝐴_𝑘⊗𝐸+𝐸⊗𝐴_𝑘=(−𝐴_𝑘)⊕𝐴_𝑘, then 𝐸⊗𝐸𝑠²+−𝐴_𝑘⊕𝐴_𝑘𝑠+𝐵⊗𝐵𝑣=0. (34) Let 𝐼=𝐸⊗𝐸, 𝐺=(−𝐴_𝑘)⊕𝐴_𝑘, and 𝐻=𝐵⊗𝐵; then we have 𝐼𝑠²+𝐺𝑠+𝐻𝑣=0. (35) Equation (35) is a quadratic eigenvalue problem, and its linearization is 𝑠−𝐼00𝐼−𝐺−𝐻𝐼0𝑢=0, (36) where 𝑢=(_𝑣^𝑠𝑣). So let 𝑃=𝐼00𝐼,𝑄=−𝐺−𝐻𝐼0. (37) Then (𝑠𝑃−𝑄)𝑢=0 has nonzero solutions if and only if det(𝑠𝑃−𝑄)=0. (38) So 𝑠 is the general eigenvalue of matrix pencil (𝑃,𝑄).

Suppose that the system (1) is stable at the initial time. Then the stability of the system will change as the positive real part root of characteristic equation emerged for the different parameter values of system (1). So the first pure imaginary eigenvalue is the critical condition and plays an important role in researching the stability and the Hopf bifurcation of the delayed reaction-diffusion equation. From the previous results, we can get all of the pure imaginary eigenvalues for a delayed reaction-diffusion equation. Here let the time delay 𝜏 be the parameter. Then we can get the stable or unstable condition of the delayed reaction-diffusion equation for different parameter values.

3. The Stability and the Hopf Bifurcation of a Delayed Reaction-Diffusion Equation

In the recent years, scholars discussed many different reaction-diffusion systems [10–12]. For different systems, they expounded different results to justify the stability. Here we will study a general high-dimensional reaction-diffusion equation with single delay. The considered reaction-diffusion system is 𝜕𝑢₁𝜕𝑡=𝑑₁Δ𝑢₁+𝑓₁𝑢_𝑖(𝑥,𝑡);𝑢_𝑖(𝑥,𝑡−𝜏)𝑥∈Ω,𝑡>0𝜕𝑢₂𝜕𝑡=𝑑₂Δ𝑢₂+𝑓₂𝑢_𝑖(𝑥,𝑡);𝑢_𝑖(⋮𝑥,𝑡−𝜏)𝑥∈Ω,𝑡>0𝜕𝑢_𝑚𝜕𝑡=𝑑_𝑚Δ𝑢_𝑚+𝑓_𝑚𝑢_𝑖(𝑥,𝑡);𝑢_𝑖(𝑥,𝑡−𝜏)𝑥∈Ω,𝑡>0𝜕𝑢_𝑖(𝑥,𝑡)𝑢𝜕𝑛=0𝑥∈𝜕Ω,𝑡>0_𝑖(𝑥,𝑡)=𝜑_𝑖(𝑥,𝑡)𝑥∈Ω,𝑡∈[−𝜏,0], (39) where 𝑖=1,2,…,𝑚 and Ω=[0,𝜋] is a bounded connected domain in 𝑅 with smooth boundary 𝜕Ω. The functions 𝑓_𝑖 are continuously differentiable, and 𝜕𝑢_𝑖/𝜕𝑛 is the out normal derivative of 𝑢_𝑖. We assume that 𝑢^∗_𝑖>0 such that 𝑓_𝑖𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑚=0. (40) That is to say, (𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑚) is the positive equilibrium of system (39). The linearization of system (39) is 𝜕𝑋(𝑥,𝑡)=()()(),𝑋()=𝑢𝜕𝑡Δ𝑋𝑥,𝑡+𝐴𝑋𝑥,𝑡+𝐵𝑋𝑥,𝑡−𝜏𝑥,𝑡¹()𝑢𝑥,𝑡₂()⋮𝑢𝑥,𝑡_𝑚(),𝑑𝑥,𝑡Δ=¹Δ0⋯00𝑑₂Δ⋯0⋮⋮⋱⋮00⋯𝑑_𝑚Δ,𝐴=𝜕𝑓₁𝜕𝑢₁𝜕𝑓₁𝜕𝑢₂⋯𝜕𝑓₁𝜕𝑢_𝑚𝜕𝑓₂𝜕𝑢₁𝜕𝑓₂𝜕𝑢₂⋯𝜕𝑓₂𝜕𝑢_𝑚⋮⋮⋱⋮𝜕𝑓_𝑚𝜕𝑢₁𝜕𝑓_𝑚𝜕𝑢₂⋯𝜕𝑓_𝑚𝜕𝑢_{𝑚(𝑢∗1,𝑢∗2,…,𝑢∗𝑚)},𝐵=𝜕𝑓₁𝜕𝑢_1𝜏𝜕𝑓₁𝜕𝑢_2𝜏⋯𝜕𝑓₁𝜕𝑢_𝑚𝜏𝜕𝑓₂𝜕𝑢_1𝜏𝜕𝑓₂𝜕𝑢_2𝜏⋯𝜕𝑓₂𝜕𝑢_𝑚𝜏⋮⋮⋱⋮𝜕𝑓_𝑚𝜕𝑢_1𝜏𝜕𝑓_𝑚𝜕𝑢_2𝜏⋯𝜕𝑓_𝑚𝜕𝑢_{𝑚𝜏(𝑢∗1,𝑢∗2,…,𝑢∗𝑚)}, (41) where 𝑢_𝑖=𝑢_𝑖(𝑥,𝑡) and 𝑢_𝑖𝜏=𝑢_𝑖(𝑥,𝑡−𝜏). For simplicity we suppose that the eigenvalues of the Laplace operator Δ by the Neumann boundary condition are simple. From Section 1, we know that the eigenvalues of Δ in (0,𝜋) with the Neumann boundary condition are 𝜇_𝑛=−𝑛², 𝑛=0,1,2,…. Let 𝑠 denote an eigenvalue of system (39). Then there exists a nonzero vector 𝑦∈Dom(Δ) such that 𝑝(𝑠,𝜏,𝑛)𝑦=𝑠𝐸−𝐷_𝑛−𝐴−𝐵𝑒^−𝑠𝜏𝑦=0. (42) The stability of the system (39) is determined by det𝑝(𝑠,𝜏,𝑛)=0. (43) We know that the system (39) is asymptotically stable when all of the eigenvalues of (42) are in the left part of the imaginary axis. Apparently, for nondiffusive case, that is, 𝑑₁=𝑑₂=⋯=𝑑_𝑚=0, the system (39) becomes a delay differential equation, and the stability has been studied by many scholars. Here we mainly discuss the diffusive system. Let 𝑆(𝜏,𝑛)=𝑠∈𝐶∣det𝑝(𝑠,𝜏,𝑛)=0. (44) Firstly, when 𝜏=0, the system (39) becomes a partial differential equation with no delay. So we can get the stability in the following.

Lemma 4. For 𝜏=0, let 𝑠 be any element of the set 𝑆(0,𝑛). That is, 𝑠∈𝜎(𝐷_𝑛+𝐴+𝐵).(i)The system (39) is eigenmode 𝑛-independently stable if Res<0, for all 𝑛∈𝑁;(ii)else, the system (39) is eigen-mode 𝑛-dependently stable if Res<0, for a fixed 𝑛∈𝑁.

Next we research the delay-independent and the delay-dependent stabilities of the system (39). Let 𝜎(𝑃,𝑄) denote the set which contains all the general eigenvalues of matrix pencil (𝑃,𝑄). From (38) in Section 2, we have the following results.

Theorem 5. For any 𝜏≥0 and a fixed eigen-mode 𝑛∈𝑁, if the coefficient matrices of the system (39) satisfy the following conditions:(i)Res<0,𝑠∈𝜎(𝐷_𝑛+𝐴+𝐵), (ii)Res≠0,𝑠∈𝜎(𝑃,𝑄), then the system (39) is asymptotically stable for all 𝜏≥0 with fixed eigen-mode 𝑛; for example, the stability is delay-independent and eigenmode 𝑛-dependent. On the other hand, if, for all eigen-mode 𝑛, the stability is delay independent, then stability of system (39) is delay independent and eigenmode 𝑛-independent.

Lemma 6. Given a fixed eigen-mode 𝑛∈𝑁, let Res<0, 𝑠∈𝜎(𝐷_𝑛+𝐴+𝐵). For all the pure imaginary roots 𝑠=𝑖𝑤 of (31) (or general pure imaginary eigenvalues of matrix pencil (𝑃,𝑄)), one can get the critical value of delay parameter 𝜏^∗_𝑛 for the fixed eigen-mode 𝑛.

Conclusion. Suppose that the system (39) is eigen-mode 𝑛-independently stable at 𝜏=0, and for multiple eigen mode 𝑛, the critical values of delay parameter are 𝜏^∗_𝑛, 𝑛=1,2,…,𝑘. Let 𝜏^∗=min{𝜏^∗_𝑛∣𝑛=1,2,…,𝑘}. One can get the minimal critical value of delay parameter 𝜏^∗. When 𝜏∈[0,𝜏^∗), the system (39) is asymptotically stable, and when 𝜏≥𝜏^∗, the stability of system (39) changes. That is to say, the positive equilibrium of system (39) (𝑢^∗,𝑣^∗) is delay dependently stable, and the system (39) first generates bifurcation at 𝜏=𝜏^∗.
From the above theorems, we can discuss the stability and the Hopf bifurcation of system (39). So the dynamical character of a generic class of nonlinear reaction-diffusion system with the Neumann boundary condition can be studied by the algebraic methods. It is well known that reaction and diffusion of physical chemistry and chemical or biochemical species can produce all kinds of spatial patterns. Next we will give a general example.

Example 7. Consider a delayed reaction-diffusion equation with the Neumann boundary condition: 𝜕𝑢𝜕𝑡=𝑑₁()𝑢Δ𝑢+𝑐−𝑎𝑢𝑥,𝑡+𝑘²(𝑥,𝑡−𝜏)𝑣(𝑥,𝑡−𝜏)𝑥∈Ω,𝑡>0,𝜕𝑣𝜕𝑡=𝑑₂Δ𝑣+𝛾𝑢²()()()𝑥,𝑡−𝜏−𝑏𝑣𝑥,𝑡𝑥∈Ω,𝑡>0,𝜕𝑢𝑥,𝑡()𝜕𝑥=0,𝜕𝑣𝑥,𝑡𝑢()𝜕𝑥=0,𝑥∈𝜕Ω,𝑡>0,𝑥,𝑡=𝑢₀(𝑥,𝑡),𝑣(𝑥,𝑡)=𝑣₀(𝑥,𝑡)𝑥∈Ω,𝑡∈[−𝜏,0], (45) where 𝑎, 𝑏, 𝑐, 𝑘, and 𝛾 especially denote the decay rate. This system is a biochemically mathematical model, such as the improved Gierer gene model. Apparently, (𝑢^∗,𝑣^∗)=((𝛾𝑐+𝑘𝑏)/𝑎𝛾,(𝛾𝑐+𝑘𝑏)²/𝑎²𝑏𝛾) is the unique constant positive equilibrium of system (45). Here we will only consider the numerical case of 𝑑₁=0.03, 𝑑₂=0.3, 𝑎=0.8, 𝑏=1, 𝑐=0.2, 𝑘=1, and 𝛾=1. Hence the positive equilibrium (𝑢^∗,𝑣^∗)=(1.5,2.25). Then (43) has no root in the right of the imaginary axis for 𝑛≥2, has two roots in the imaginary axis 𝑠₁₁≐±0.305𝑖 and 𝑠₁₂≐±0.702𝑖 for 𝑛=1, and has two roots in the imaginary axis 𝑠₀₁≐±0.648𝑖 and 𝑠₀₂≐±0.803𝑖 for 𝑛=0. The parameter 𝜏 has four sequences 𝜏₁≐1.625+2𝑘𝜋/0.305, 𝜏₂≐6.898+2𝑘𝜋/0.702, 𝜏₃≐0.424+2𝑘𝜋/0.648, and 𝜏₄≐5.782+2𝑘𝜋/0.803, 𝑘=0,1,…. So the smallest Hopf bifurcation value is 𝜏^∗=min{𝜏₁,𝜏₂,𝜏₃,𝜏₄}=0.424. Then the unique positive equilibrium of system (45) is asymptotically stable when 𝜏∈[0,𝜏^∗) and unstable when 𝜏>𝜏^∗. The simulation results are shown in Figures 1 and 2.

(a)

(b)

Figure 1 

For system (45), when 𝜏=0.2<𝜏∗, the equilibrium (𝑢∗,𝑣∗) is locally asymptotically stable.

(a)

(b)

Figure 2 

For system (45), when 𝜏=6>𝜏∗, the equilibrium (𝑢∗,𝑣∗) is unstable.

4. Conclusion

In this paper, we consider a general high-dimensional delayed reaction-diffusion system. By the algebraic methods, such as the matrix pencil and the linear operator, we discussed the eigenvalues and the stability of the delayed reaction-diffusion system (1) and (39). In fact, we only find the pure imaginary eigenvalues, which are a few parts of the infinite eigenvalues. So we can easily compute pure imaginary eigenvalues from the algebraic equation (31) by MATLAB. Certainly, applying the algebraic methods to analyze the dynamical properties of the reaction-diffusion system with delays is still a new and immature field, so we believe that the algebraic methods used to research the stability of the delayed reaction-diffusion systems would be of more interest in the future.

Acknowledgment

The research is funded by the Fundamental Research Funds for the Central Universities (DL12BB24).