Turing instability for a attraction-repolsion chemotaxis system with logistic growth

1. Introduction

In this paper , we deal with attraction-repolsion chemotaxis system

in a \(d\)-dimensional box \(\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\) is a bounded domain with smooth boundary \(\alpha,\beta,\mu,\chi,\xi,\beta,\gamma,\eta>0\). In the model (1) \(u_{1}\), \(u_{2}\) and \(u_{3}\) represent the cell density, the concentration of the chemoattractant (attractive signal) and the concentration of the chemorepellent (repulsive signal) respectively, \(\mathbf g(u)\) is logistic source. The classical Keller-Segel system can be obtained by setting \(d_{i} = 1 ,(i=1,2,3), \xi = 0, u_{3}\equiv0, \mathbf g(u)\equiv 0 \) in (1) which models the mechanism of chemotaxis and has been extensively studied since 1970, we refer to [1, 2, 3, 4] and the references therein. Apart form the aforementioned system a source of logistic type is included in (1) to describe the spontaneous growth of cells. The effect of preventing ultimate growth has been widely studied.

Chemotaxis is a chemosensitive movement of species which may detect and respond to chemical substances in the environment. The first model about chemotaxis was proposed by Keller and Segel [5]

which describes the aggregation process of the slime mold formation in Dictyostelium Discoidium, where \(v\) denotes the chemical concentration and u is the concentration of species. For this system, there have been abundant results. Osaki and Yagi [6] found that when \(n = 1\), all the solutions are global and bounded. When \(n \geq 2\), blow-up may happen (see Horstmann and Wang [7]; Herrero et al., [8]; Winkler et al., [9]). A detailed introduction into the mathematic of the Keller-Segel model for chemotaxis is presented in Horstmann [1, 10, 11].

In the study of chemotaxis-diffusion-growth models, the pattern dynamics is another mathematically challenging and physically important research project (see Tello and Winkler [12], Aida and Yagi [13], Kurata et al., [14], Painter and Hillen [15], Okuda and Osaki [16], Kuto et al., [17] and Banerjee et al., [18]. Guo and Hwang [19] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Their result can be interpreted as a rigorous mathematical characterization for pattern formation in the Keller-Segel model. By using the similar method, Fu and Liu [20] proved that the linear unstable positive constant equilibrium in the Keller-Segel model with a logistic source is also unstable in the full nonlinear sense. The emergence of patterns is a phenomenon frequently observed in the physical world [21].

Many authors have investigated the formation of patterns by using self-diffusive reaction-diffusion models [21, 22, 23, 24, 25]. Recently, some researchers made attempts to discover the effect of cross-diffusion on the pattern formation, and found that with appropriate cross-diffusion coefcients, linear reaction terms are sufficient to produce pattern formation [26, 27, 28], but there is only few attention having been paid to this direction. Therefore, based on the model (1): First, we analyse criteria of linear stability and instability of the positive constant equilibrium \({\mathbf w_{c}}\) (see Theorem 1). Second, by applying the higher-order energy estimates, the embedding theorem and the Guo-Strauss’ bootstrap technique (see Guo and Strauss [29]), it is proved that for given any general perturbation of magnitude \(\delta\), its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of \(\ln{\frac{1}{\delta}}\) (see Theorem 2). We assert further that the positive constant equilibrium point \({\mathbf w_{c}}\) is nonlinearly unstable in the above conditions (Corollary 1). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a quantitative characterization for the nonlinear evolution of early-stage spatiotemporal pattern formation in the model (1).

The organization of this paper is as follows: in Section 2, we first prove Turing instability does not take place in the absence of chemotactic effect. Second, we give linear stability and instability criterions for the model (1), and discuss some properties of solutions for the corresponding linearized system. In Section 3, we consider the growing modes of (1), and prove the Bootstrap lemma. In Section 4, quantitative characterization for pattern formation and proof of nonlinear instability are given.

2. Linear stability and instability criterions

In this section, we study in detail linear Stability,linear instability of positive constant equilibrium point \(\mathbf w_{c}=(1,\frac{\alpha}{\beta},\frac{\gamma}{\eta})\) to the model (1) in a \(d\)-dimensional box \(\Omega =\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\), and \(\mathbf{g(u)}=\mu \mathbf{u_{1}(1-u_{1})}\).

2.1. Stability of positive constant equilibrium point for (1) without chemotaxis

We consider the stability of \({\mathbf w_{c}}\) for the corresponding system (1) without chemotaxis For sake convenience, take \(\mathbf{ w(x,t)= (U_{1}(x,t),U_{2}(x,t),U_{3}(x,t))^T}\) and \[\begin{array}{ll} G({w})=\left(% \begin{array}{ccc} g_{1}({w}) & \\ g_{2}({w}) &\\ g_{3}({w})) \end{array}% \right) = \left(% \begin{array}{ccc} \mu{u_{1}}(1-u_{1}) & \\ \alpha{u_{1}-\beta{u_{2}}} &\\ \gamma{u_{1}-\eta{u_{3}}}) \end{array}% \right), \end{array} \] then \[\frac{\partial{G}}{\partial{w}}|_{ w_{c}} \equiv{G_{w}({w_{c}})}= \left(% \begin{array}{ccc} -\mu & 0 & 0\\ \alpha & -\beta & 0 \\ \nu & 0 & -\eta \end{array} \right). \]

Lemma 1. The positive equilibrium point \(\mathbf w_{c}\) of (3) is locally asymptotically stable.

Proof. Let \(0=k_{1}< k_{2}< k_{3}< \cdot\cdot\cdot\) be the eigenvalues of the operator \(-\Delta\) on \(\mathbb{T}^{d}\) with the homogeneous Neumann boundary condition, and \(E(k_{i})\) be the eigenspace corresponding to \(k_{i}\) in \(H^{1}(\mathbb{T}^{d})\). Let \({\mathbf X}=[H^{1}(\mathbb{T}^{d})]^{3}\) and \({\mathbf X}_{ij}=\left\{c\cdot\phi_{ij} | c\in \mathbb{R}^{3}\right\}\), where \(\left\{\phi_{ij}, j=1,\cdot\cdot\cdot,\dim E(k_{i}) \right\} \) is an orthonormal basis of \(E(k_{i})\). Then \({\mathbf X}=\oplus_{i=1}^\infty {\mathbf X}_i\), \({\mathbf X}_i=\oplus_{j=1}^{\dim E(\mu_i)}{\mathbf X}_{ij}\). Let \(D= diag(d_{1}, d_{2}, d_{3})\). The linearization of (3) at \(\mathbf w_{c}\) is $$ {\mathbf{w}}_{t}=\left(D\Delta+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\right){\mathbf w}. $$ For each \(i\geq1\), \({\mathbf X}_i\) is invariant under the operator \(D\Delta+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\), and \(\lambda\) is an eigenvalue of this operator on \({\mathbf{X}}_i\) if and only if it is an eigenvalue of the matrix \( -k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\). The characteristic polynomial of \(-k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})\) is given by $$\begin{array}{ll} det (\lambda{I}-( -k_{i}D+\mathbf{G}_{\mathbf{w}}({\mathbf{w_{c}}})))= \left(% \begin{array}{ccc} \lambda+ k_{i}d_{1} +\mu& 0 & 0\\ -\alpha & \lambda+ k_{i}d_{2}+\beta & 0 \\ -\gamma & 0 & \lambda+ k_{i}d_{3}+\eta \end{array} \right) = 0 \end{array} $$ implies \(\Psi(\lambda)= (\lambda+ k_{i}d_{1}+\mu)(\lambda+ k_{i}d_{2}+\beta)(\lambda+ k_{i}d_{3}+\eta)=0\), then \(\lambda_{1}=-(k_{i}d_{1}+\mu) \), \(\lambda_{2}=-(k_{i}d_{2}+\beta)\) and \(\lambda_{3} = -(k_{i}d_{3}+\eta)\). So all the eigenvalues are negative, hence \( \mathbf w_{c}\) is locally asymptotically stable, this complete the proof.

2.2. Criteria of linear stability and instability

Let \(\hat{u}_{1}({\mathbf x},t)=u_{1}({\mathbf x},t)-u_{1c}, \hat{u}_{2}({\mathbf x},t)=u_{2}({\mathbf x},t)-u_{2c}, \hat{u_{3}}({\mathbf x},t)=u_{3}({\mathbf x},t)-u_{3c}\) be nonlinear evolution of a perturbation around \((u_{1c},u_{2c},u_{3c})=(1,\frac{\alpha}{\beta},\frac{\nu}{\eta})\), and omitting the symbol \(“ \wedge”\), then we rewrite (3) with The corresponding linearized system can be written as Let \({\mathbf w}({\mathbf x},t)\equiv(u_{1}({\mathbf x},t),u_{2}({\mathbf x},t),u_{3}({\mathbf x},t))^{\mathrm{T}}\), \({\mathbf q}=(q_{1},\ldots,q_{d})\in \mathbb{N}^{d}\) and \(e_{{\mathbf q}}({\mathbf x})=\prod^{d}\limits_{i=1}\cos(q_{i}x_{i})\). Then \(\{e_{{\mathbf q}}({\mathbf x})\}_{{\mathbf q}\in\mathbb{N}^{d}}\) forms a basis of the space of functions in \(\mathbb{T}^{d}\) that satisfy the homogeneous Neumann boundary condition. We try to find a normal mode to the linearized system (5) of the following form where \({\mathbf r_{{\mathbf q}}}\) is a vector depending on \({\mathbf q}\). Substituting (6) into (5), we have \[ \lambda_{{\mathbf q}}{\mathbf r_{{\mathbf q}}}=\left(% \begin{array}{ccc} -d_{1}q^2-\mu & \chi q^2 & -\xi q^2\\ \alpha & -d_{2}q^2 -\beta & 0\\ \gamma &0 & -d_{3}q^2-\eta \end{array}% \right){\mathbf r_{{\mathbf q}}}:={\mathbf L_{q}}{\mathbf r_{{\mathbf q}}}, \] where \(q^2=|{\mathbf q}|^{2}=\sum^{d}\limits_{i=1}q^2_{i}\). Then the corresponding characteristic equation of \({\mathbf L_{q}}\) is where and In order to consider instability of \({\mathbf w_{c}}\), we make the following basic assumptions:

• (H\(_{1}\)) There exists \({\textbf q}\in \mathbb{N}^{d}\) such that the matrix \({\textbf L_{q}}\) has at least one eigenvalue with positive real part;
• (H\(_{2}\)) \(d_{1}, d_{2}, d_{3}>0\) and \(d_{i}\neq d_{j}\), \(i\neq j\), \(i,j=1,2,3\).

It is know that a first necessary condition for Turing instability to happen is that \(d_{i}\neq d_{j} (i\neq j)\), implying that \(u_{1}, u_{2}\) and \(u_{3}\) must move with different diffusion constants.
For every \(\lambda_{1}({\mathbf q}), \lambda_{2}({\mathbf q}), \lambda_{3}({\mathbf q})\) be the solutions of \(\det({ \lambda_{\textbf q}}\mathrm{I}-{\mathbf L_{q}})=0\). It will be state by Lemma 3 that there exist finitely many values \({\mathbf q}\in \mathbb{N}^{d}\) such that \[\max\left\{\mathrm{Re}\lambda_{1}({\mathbf q}), \mathrm{Re}\lambda_{2}({\mathbf q}), \mathrm{Re}\lambda_{3}({\mathbf q})\right\}>0.\] Hence there exists one \(q^{2}\) having the largest eigenvalue

• (H\(_{3}\)) At \(\mathbf{\overline{q}}=(\overline{q}_{1},\cdot\cdot\cdot,\overline{q}_{d})\in \mathbb{N}^{d}\) which attains \(\lambda_{\max}=\mathrm{Re}\lambda_{i}({\mathbf {\overline{q}}})\), we assume that the Jordan canonical form of the matrix \({\mathbf L_{\overline{q}}}={\mathbf G}_{\mathbf w}({\mathbf w_{c}})+{\mathbf Q}(\overline{q}^2)\) is \(J=\mathrm{diag}(\lambda_{1}({\mathbf{\overline{q}}}),\lambda_{2}({\mathbf{\overline{q}}}),\lambda_{3}({\mathbf{\overline{q}}}))\), where \(\overline{q}^2=\sum^{d}\limits_{i=1}\overline{q}^2_{i}\) and \[ {\mathbf Q}(\overline{q}^2):=\left(% \begin{array}{ccc} -d_{1}\overline{q}^2 &\chi\overline{q}^2 & -\xi\overline{q}^2\\ 0 & -d_{2}\overline{q}^2 & 0\\ 0 & 0 & -d_{3}\overline{q}^2 \end{array}% \right). \]

Let us carry on discussion on the characteristic equation (7). Denote \[A:=\bar{B}^{2}_{2}-3\bar{B}_{1}, B:=\bar{B}_{2}\bar{B}_{1}-9\bar{B}_{0}, C:=\bar{B}^{2}_{1}-3\bar{B}_{2}\bar{B}_{0}\] and \begin{eqnarray*}\Delta&=& B^{2}-4AC=3\left\{4\bar{B}^{3}_{1}+4\bar{B}^{3}_{2}\bar{B}_{0}+27\bar{B}^{2}_{0}-\bar{B}^{2}_{2}\bar{B}^{2}_{1}-18\bar{B}_{2}\bar{B}_{1}\bar{B}_{0}\right\}\\ &:=& Q_{6}q^{12}+Q_{5}q^{10}+Q_{4}q^{8}+Q_{3}q^{6}+Q_{2}q^{4}+Q_{1}q^{2}+Q_{0}, \end{eqnarray*} where \begin{eqnarray*} Q_{6}&=&3\left\{4C^{3}_{21}C_{01}+27C_{01}-C^{2}_{21}C^{2}_{11}-18C_{21}C_{11}C_{01}\right\},\\ Q_{5}&=&6\left\{27C_{01}C_{02}+2C^{3}_{21}C_{02}+6C^{2}_{21}C_{22}C_{01}-C^{2}_{21}C_{11}C_{12}-C_{21}C^{2}_{11}C_{22}\right.\\ &&\left.-9C_{21}C_{11}C_{02}-9C_{21}C_{12}C_{01}-9C_{22}C_{11}C_{01}\right\}, \end{eqnarray*} \begin{eqnarray*} Q_{4}&=&3\left\{27C_{02}+54C_{01}C_{03}+4C^{2}_{21}C_{03}+12C^{2}_{21}C_{22}C_{02}+12C_{21}C^{2}_{22}C_{01}+4C^{2}_{11}\right.\\ &&\left.-C^{2}_{21}C^{2}_{12}-2C_{11}C_{13}C^{2}_{21}-4C_{21}C_{22}C_{11}C_{03}-C^{2}_{22}C^{2}_{11}-18C_{21}C_{11}C_{03}\right.\\ &&\left.-18C_{21}C_{12}C_{02}-18C_{21}C_{13}C_{01}-18C_{22}C_{11}C_{02}-18C_{22}C_{12}C_{01}\right\},\\ Q_{3}&=&6\left\{27C_{01}C_{04}+27C_{02}C_{03}+2C^{3}_{21}C_{04}+12C^{3}_{22}C_{01}+6C^{2}_{21}C_{22}C_{03}\right.\\ &&\left.+6C_{21}C^{2}_{22}C_{02}+4C_{11}C_{12}-C^{2}_{21}C_{12}C_{13}-C_{21}C_{22}C^{2}_{12}-2C_{21}C_{22}C_{11}C_{13}\right.\\ &&\left.-C^{2}_{22}C_{11}C_{12}-9C_{21}C_{11}C_{04}-9C_{21}C_{12}C_{03}-9C_{21}C_{13}C_{02}\right.\\ &&\left.-9C_{22}C_{11}C_{03}-9C_{22}C_{12}C_{02}-9C_{22}C_{13}C_{01}\right\},\\ Q_{2}&=&3\left\{27C_{03}+54C_{02}C_{04}+4C^{3}_{22}C_{02}+12C^{2}_{21}C_{22}C_{04}+4C^{2}_{12}+12C_{21}C^{2}_{22}C_{03}\right.\\ &&\left.+8C_{11}C_{13}-C^{2}_{21}C^{2}_{13}-4C_{21}C_{22}C_{12}C_{13}-C^{2}_{22}C^{2}_{12}-2C^{2}_{22}C_{11}C_{13}-18C_{21}C_{12}C_{04}\right.\\ &&\left.-18C_{21}C_{13}C_{03}-18C_{22}C_{11}C_{04}-18C_{22}C_{13}C_{02}-18C_{22}C_{12}C_{03}\right\},\\ Q_{1}&=&6\left\{27C_{03}C_{04}+2C^{3}_{22}C_{03}+6C_{21}C^{2}_{22}C_{04}+4C_{12}C_{13}-C_{21}C_{22}C^{2}_{13}-C_{12}C_{13}C^{2}_{22}\right.\\ &&\left.-9C_{21}C_{13}C_{04}-9C_{22}C_{12}C_{04}-9C_{22}C_{13}C_{03}\right\},\\ Q_{0}&=&3\left\{ 4C^{3}_{22}C_{04}+4C^{2}_{13}+27C^{2}_{04}-C^{2}_{22}C^{2}_{13}-18C_{22}C_{13}C_{04}\right\}. \end{eqnarray*} The derivative of \(\psi(\lambda_{{\mathbf q}})\) is \(\psi'(\lambda_{{\mathbf q}})=3\lambda^{2}_{{\mathbf q}}+2\bar{B}_{2}\lambda_{{\mathbf q}}+\bar{B}_{1}.\) Obviously, equation \(\psi'(\lambda_{{\mathbf q}})=0\) has two roots as follows Next, let us give one result concerning the cubic equation in Hu et al., [30] (which was first introduced in Fan [31]), which is used to discuss the linear stability and instability of positive constant equilibrium solution for the model (1).

Lemma 2. Let equation \(x^{3} + bx^{2} +cx + d = 0\), where \(b, c, d \in \mathbb{R}\). Let further \(A=b^{2}-3c, B=bc-9d, C=c^{2}-3bd\) and \(\Delta= B^{2 }- 4AC\). Then the equation has three real roots if and only if \(\Delta \leq 0\); the equation has one real root and a pair of conjugate complex roots if and only if \(\Delta>0\). Furthermore, the conjugate complex roots are \(w=\frac{-2b+Y^{1/3}_{1}+Y^{1/3}_{2}}{6}\pm i\frac{\sqrt{3}\left({Y^{1/3}_{1}-Y^{1/3}_{2}}\right)}{6}\), where \(Y_{1,2}=bA+\frac{3\left(-B\pm\sqrt{B^{2}-4AC}\right)}{2}.\)

According to Lemma 2, on the one hand, if \(\Delta\leq0\), then (7) has three real roots \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\), and denote \(\lambda_{1}({\mathbf q})\leq\lambda_{2}({\mathbf q})\leq\lambda_{3}({\mathbf q})\). From this, we further infer that \(\lambda^{\ast}_{1,2}({\mathbf q})\) also are real. Moreover, recall that \(\bar{B}_{2}=-(\lambda_{1}({\mathbf q})+\lambda_{2}({\mathbf q})+\lambda_{3}({\mathbf q}))>0\), it means that (7) has at least one eigenvalue with negative real part. On the other hand, if \(\Delta>0\), then Equation (7) has one real root \(\lambda_{1}({\mathbf q})\) and a pair of conjugate complex roots \[\lambda_{2,3}({\mathbf q})=\frac{-2\bar{B}_{2}+Y^{1/3}_{1}+Y^{1/3}_{2}}{6}\pm \mathrm{i}\frac{\sqrt{3}\left({Y^{1/3}_{1}-Y^{1/3}_{2}}\right)}{6}\] with \[Y_{1,2}=\bar{B}_{2}A+\frac{3\left(-B\pm\sqrt{B^{2}-4AC}\right)}{2}.\] Notice by the Routh-Hurwitz criterion that \({\mathbf q}=0\), in the case of \(C_{22}C_{13}>C_{04}\), then (8) has three negative roots. So we consider the case \({\mathbf q}\neq0\) in the sequel. In this section, our first main purpose is to give criteria for linear stability and instability of \({\mathbf w_{c}}\).

Theorem 1. (Linear stability and instability). Let \({\mathbf w_{c}}\) be positive constant equilibrium solution of (1). Assume that \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) are three roots of \(\psi(\lambda)=\lambda^{3}+\bar{B}_{2}\lambda^{2}+\bar{B}_{1}\lambda+\bar{B}_{0}=0\), and that \(\lambda^{\ast}_{1}\) and \(\lambda^{\ast}_{2}\) are two roots of \(\psi'(\lambda)=3\lambda^{2}+2\bar{B}_{2}\lambda+\bar{B}_{1}=0\), then we have the following conclusions:

• (1) If one of the following conditions holds, then \({\mathbf w_{c}}\) is linearly stable.
  • H\(_{S1}\) \(\Delta\leq0\), \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{1}< \lambda^{\ast}_{2}< 0\).
  • H\(_{S2}\) \(\Delta>0\), \(\bar{B}_{0}>0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}< 0\), \(\mathrm{Re}\lambda_{3}< 0\).
• (2) If one of the following conditions holds, then \({\mathbf w_{c}}\) is linearly unstable.
  • H\(_{U1}\) \(\Delta\leq0\), and one of the following conditions holds:
  • H\(_{U11}\) \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{2}>\lambda^{\ast}_{1}>0\).
  • H\(_{U12}\) \(\bar{B}_{0}>0\) and \(\lambda^{\ast}_{2}>0>\lambda^{\ast}_{1}\).
  • H\(_{U13}\) \(\bar{B}_{0}0>\lambda^{\ast}_{1}\).
  • H\(_{U14}\)\(\bar{B}_{0}< 0\) and \(\lambda^{\ast}_{1}< \lambda^{\ast}_{2}< 0\).
  • H\(_{U2}\) \(\Delta >0\), and one of the following conditions holds:
  • H\(_{U21}\) \(\bar{B}_{0}>0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}>0\), \(\mathrm{Re}\lambda_{3}>0\).
  • H\(_{U22}\) \(\bar{B}_{0}< 0\) and the conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\) satisfy \(\mathrm{Re}\lambda_{2}< 0\), \(\mathrm{Re}\lambda_{3}< 0\).

Here \(\Delta= B^{2}-4AC\), \(A:=\bar{B}^{2}_{2}-3\bar{B}_{1}\), \(B:=\bar{B}_{2}\bar{B}_{1}-9\bar{B}_{0}\), \(C:=\bar{B}^{2}_{1}-3\bar{B}_{2}\bar{B}_{0}\), in particular, \(\bar{B}_{0}=\psi(0)=-\lambda_{1}\lambda_{2}\lambda_{3}\).

Proof. Let \(\Delta\leq0\). By Lemma 2, the equation \(\psi(\lambda)=\lambda^{3}+\bar{B}_{2}\lambda^{2}+\bar{B}_{1}\lambda+\bar{B}_{0}=0\) has three real roots \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) and assume \(\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\). Moreover, the equation \(\psi'(\lambda)=3\lambda^{2}+2\bar{B}_{2}\lambda+\bar{B}_{1}=0\) has also two real roots \(\lambda^{\ast}_{1}\) and \(\lambda^{\ast}_{2}\) with \(\lambda^{\ast}_{1}\leq\lambda^{\ast}_{2}\), and \[ \begin{array}{ll} \psi'(\lambda)>0, \forall \lambda\in(-\infty, \lambda^{\ast}_{1})\cup(\lambda^{\ast}_{2}, +\infty),\\ \psi'(\lambda)0\), then \(\lambda_{2}>0\), \(\lambda_{3}>0\). Since \(\psi(\lambda)\) is increasing for all \(\lambda\in(-\infty, \lambda^{\ast}_{1}]\) and \(\psi(0)=\bar{B}_{0}>0\), one has \(\lambda_{1}0\), this contradicts \(\bar{B}_{2}>0\). Hence, \({\mathbf w_{c}}\) is linearly unstable. Under the condition ( H\(_{U12}\)), if \(\lambda^{\ast}_{1}0\), then \(\lambda_{1}0\), we have \(\lambda_{2}>0\), \(\lambda_{3}>0\). This means that \({\mathbf w_{c}}\) is linearly unstable.
Similarly, it is proved that when condition ( H\(_{U13}\)) or ( H\(_{U14}\)) holds, eigenvalues \(\lambda_{1}< 0\), \(\lambda_{2}0\), that is, \({\mathbf w_{c}}\) is linearly unstable.
In the case ( H\(_{S1}\)), By monotonicity of \(\psi(\lambda)\) for all \(\lambda\in(\lambda^{\ast}_{2},+\infty)\), it holds \(\lambda_{1}< 0\), \(\lambda_{2}< 0\) and \(\lambda_{3}< 0\). Hence, \({\mathbf w_{c}}\) is linearly stable.
We now let \(\Delta>0\). In view of Lemma 2, \(\psi(\lambda)=0\) has one real root \(\lambda_{1}\) and a pair of conjugate complex roots \(\lambda_{2}\), \(\lambda_{3}\). If condition ( H\(_{U21}\)) holds, then it follows from \(\bar{B}_{0}>0\) that real root \(\lambda_{1}0\), \(\mathrm{Re}\lambda_{3}>0\). Similarly, we can also prove that if condition ( H\(_{U22}\)) holds, then \({\textbf w_{c}}\) is linearly unstable. If condition (H\(_{S2}\)) holds, it is easily to obtain that \({\textbf w_{c}}\) is linearly stable. This completes the proof.

2.3. Some properties of solutions of the linearized system (5)

Lemma 3. If \({\mathbf q}\in \mathbb{N}^{d}\) and \(q^2\) sufficiently large, then all eigenvalues of \({\textbf L_{q}}\) have negative real parts.

Proof. Notice that \(C_{21}\), \(C_{22}\), \(C_{11}\), \(C_{13}\), \(C_{01}\), \(C_{04}\) and \(\bar{B}_{2}\) are all positive, where the parameters are mentioned in (8) and (9). In addition, \(\bar{B}_{2}\), \(\bar{B}_{1}\), \(\bar{B}_{0}\) and \(\bar{B}_{2}\bar{B}_{1}-\bar{B}_{0}\) are positive if \({\mathbf q}\in \mathbb{N}^{d}\) sufficiently large. It is follows from the Routh-Hurwitz criterion that all eigenvalues of \({\mathbf L_{q}}\) have negative real parts for \({\mathbf q}\in \mathbb{N}^{d}\) sufficiently large.

For given \({\mathbf q}\in \mathbb{N}^{d}\), let \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) be the eigenvalues of \({\mathbf L_{q}}\) and the corresponding eigenvectors by \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\), \({\mathbf r}_{3}({\mathbf q})\). According to eigenvectors, we divide \({\mathbf q}\) into the following four cases to analyze:
Case 1: \({\mathbf q}\in \mathbb{N}^{d}_{R1}\): \({\mathbf L_{q}}\) has three real eigenvalues \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\) and \(\lambda_{3}({\mathbf q})\), and three corresponding linearly independent eigenvectors \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\) and \({\mathbf r}_{3}({\mathbf q})\). In the case we arrange \(\lambda_{1}({\mathbf q})\leq\lambda_{2}({\mathbf q})\leq\lambda_{3}({\mathbf q})\).
Case 2: \({\mathbf q}\in \mathbb{N}^{d}_{R2}\): \({\mathbf L_{q}}\) has a single root \(\lambda_{1}({\mathbf q})=\lambda_{s}({\mathbf q})\) and a double root \(\lambda_{2}({\mathbf q})=\lambda_{3}({\mathbf q})=\lambda_{d}({\mathbf q})\) (or \({\mathbf L_{q}}\) has three repeated real root \(\lambda_{s}({\mathbf q})=\lambda_{d}({\mathbf q})\)), meanwhile, there are only two linearly independent real eigenvectors \({\mathbf r}_{s}({\mathbf q})\) and \({\mathbf r}_{d}({\mathbf q})\). In this case we need find another independent vector \({\mathbf r}’_{d}({\mathbf q})\) satisfying \[({\mathbf L_{q}}-\lambda_{d}({\mathbf q}) \mathrm{I}){\mathbf r}’_{d}({\mathbf q})={\mathbf r}_{d}({\mathbf q}). \] Case 3: \({\mathbf q}\in \mathbb{N}^{d}_{R3}\): (7) has a triple eigenvalue \(\lambda({\mathbf q})\) which only corresponding one linearly independent eigenvector \({\mathbf r}({\mathbf q})\). In this case, we need to supplement another two independent vectors \({\mathbf r}'({\mathbf q})\) and \({\mathbf r}”({\mathbf q})\), which satisfy \[({\mathbf L_{q}}-\lambda({\mathbf q}) \mathrm{I}){\mathbf r}'({\mathbf q})={\mathbf r}({\mathbf q}), ({\mathbf L_{q}}-\lambda({\mathbf q}) \mathrm{I}){\mathbf r}”({\mathbf q})={\mathbf r}'({\mathbf q}).\] Case 4: \({\mathbf q}\in\mathbb{N}^{d}_{C}=\mathbb{N}^{d}-(\mathbb{N}^{d}_{R1}\bigcup\mathbb{N}^{d}_{R2}\bigcup\mathbb{N}^{d}_{R3})\): The characteristic equation (7) has one real root and a pair of conjugate complex roots. The eigenvalues and the corresponding eigenvectors are denoted by \(\lambda_{r}({\mathbf q})\), \(\mathrm{Re}\lambda_{c}({\mathbf q})+i\mathrm{Im}\lambda_{c}({\mathbf q})\), \(\mathrm{Re}\lambda_{c}({\mathbf q})-i\mathrm{Im}\lambda_{c}({\mathbf q})\) and \({\mathbf r}({\mathbf q})\), \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})+i\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\), \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})-i\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\), respectively. Notice that \(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\) and \(\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\) are linearly independent vectors. Given any initial perturbation \({\mathbf w}({\mathbf x}, 0)\), it can be expressed as where \(w_{i}({\mathbf q}), w_{d}({\mathbf q}), w’_{d}({\mathbf q}), w_{s}({\mathbf q}), w({\mathbf q}), w'({\mathbf q}), w”({\mathbf q}), w^{\mathrm{Re}}({\mathbf q}), w^{\mathrm{Im}}({\mathbf q}), w_{r}({\mathbf q})\in\mathbb{R}\), \(i=1,2,3\) and Thus, the unique solution \({\mathbf w}({\mathbf x}, t)\) to the linearized system (5) can be written in the following form. Recall that \[\lambda_{\max}= \max\limits_{{\mathbf q}\in \mathbb{N}^{d}}\max\limits_{1\leq i\leq3} \mathrm{Re}\lambda_{i}({\mathbf q})>0,\] where \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are the solutions of (7). Denote By the assumption ( H\(_{3}\)), the largest eigenvalue \(\lambda_{\max}\) can be obtained, provided that \({\mathbf q}\) belongs to \(\mathbb{N}^{d}_{R1}\) or \(\mathbb{N}^{d}_{C}\).
In the sequel, we define \[I=\{i | 1 \leq i \leq 3\}, I_{1}=\{i | \lambda_{i}({\mathbf q})=\lambda_{\max}, 1 \leq i \leq 3\},\] and \[ \begin{array}{ll} \Lambda_{R1}=\mathbb{N}^{d}_{R1}\cap{\mathbb{N}^{d}}_{\max}, \Lambda_{C}=\mathbb{N}^{d}_{C}\cap{\mathbb{N}^{d}}_{\max},\\ \Lambda_{C1}=\{{\mathbf q}\in\Lambda_{C} | \mathrm{Re}\lambda_{c}({\mathbf q})=\lambda_{\max}\},\\ \Lambda_{C2}=\{{\mathbf q}\in\Lambda_{C} | \lambda_{r}({\mathbf q})=\lambda_{\max}\},\\ \Lambda_{C3}=\{{\mathbf q}\in\Lambda_{C} | \mathrm{Re}\lambda_{c}({\mathbf q})=\lambda_{\max}, \lambda_{r}({\mathbf q})=\lambda_{\max}\}. \end{array} \] Let \(e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})\) be the dominant part of the solution \(e^{\mathfrak{L}t}{\mathbf w}_{0}({\mathbf x})\) of the linearied system (5) and \begin{eqnarray*} e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})&=&\sum\limits_{{\mathbf q}\in\Lambda_{R1}}\sum\limits_{i\in I_{1}}w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q})e^{\lambda_{\max}t}e_{{\mathbf q}}({\mathbf x})\notag\\ && +\sum\limits_{{\mathbf q}\in\Lambda_{C1}}\left[w^{\mathrm{Re}}({\mathbf q})\left(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\cos[(\mathrm{Im}\lambda_{c}({\mathbf q}))t] -\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\sin[(\mathrm{Im}\lambda_{c}({\mathbf q}))t]\right)\right.\notag\\ &&\left.+w^{\mathrm{Im}}({\mathbf q})\left(\mathrm{Re}{\mathbf r}_{c}({\mathbf q})\sin[(\mathrm{Im}\lambda_{c}({\mathbf q}))t] +\mathrm{Im}{\mathbf r}_{c}({\mathbf q})\cos[(\mathrm{Im}\lambda_{c}({\mathbf q}))t]\right)\right]e^{\lambda_{\max}t}\notag\\ && +\sum\limits_{{\mathbf q}\in\Lambda_{C2}}w_{r}({\mathbf q}){\mathbf r}_{r}({\mathbf q})e^{\lambda_{\max}t}e_{{\mathbf q}}({\mathbf x})\notag \end{eqnarray*} Since \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are the roots of (7), let \(\beta_{i}({\mathbf q})=\frac{1}{q^{2}}\lambda_{i}({\mathbf q})\), then \(\beta_{1}({\mathbf q})\), \(\beta_{2}({\mathbf q})\), \(\beta_{3}({\mathbf q})\) are the three roots of \({\mathbf F}_{q}(\beta_{{\mathbf q}})=\det\left(\beta_{{\mathbf q}}\mathrm{I}-\frac{1}{q^{2}}{\mathbf L_{q}}\right)=0\) and \begin{eqnarray*} {\mathbf F}_{q}(\beta_{{\mathbf q}})&=&\det\left(% \begin{array}{ccc} \beta_{{\mathbf q}}+d_{1}+\frac{\mu}{q^2}& -\chi &\xi\\ -\alpha & \beta_{{\mathbf q}}+d_{2}+\frac{\beta}{q^2} & 0\\ -\gamma & 0&\beta_{{\mathbf q}}+d_{3}+\frac{\eta}{q^2} \end{array}% \right)\\ &=&\beta^{3}_{{\mathbf q}}+\bar{b}_{2}({\mathbf q})\beta^{2}_{{\mathbf q}}+\bar{b}_{1}({\mathbf q})\beta_{{\mathbf q}}+\bar{b}_{0}({\mathbf q}) \end{eqnarray*} with Moreover, One can define a function \({\mathbf F}^{\ast}(\beta_{{\mathbf q}})\) of the form \[{\mathbf F}^{\ast}(\beta_{{\mathbf q}}):=\beta^{3}_{{\mathbf q}}+\bar{b}_{2}\beta^{2}_{{\mathbf q}}+\bar{b}_{1}\beta_{{\mathbf q}}+\bar{b}_{0} =(\beta_{{\mathbf q}}+d_{1})(\beta_{{\mathbf q}}+d_{2})(\beta_{{\mathbf q}}+d_{3}).\] It is clear from the assumption ({\textbf H\(_{2}\)}) that the equation \({\textbf F}^{\ast}(\beta_{{\mathbf q}})=0\) has different negative roots \(-d_{1}\), \(-d_{2}\), \(-d_{3}\). For \(q^{2}\) sufficiently large, it follows from Lemma 3 that \(\mathrm{Re}\beta_{i}({\mathbf q})< 0\), \(\forall 1\leq i\leq 3\). Thus and For \(q^{2}\) large enough, by (18) and (19), we have Again combining (18) and (20) yields for \(q^{2}\) sufficiently large Applying (21) and (22), for every sequence \(\{{\mathbf q}_{m}\}\in \mathbb{N}^{d}\), there exists a subsequence of \(\{{\mathbf q}_{n}\}\) such that for \(1 \leq i\leq 3\) there exist limits \[\lim\limits_{n\rightarrow\infty}\mathrm{Re}\beta_{i}({\mathbf q}_{n}), \lim\limits_{n\rightarrow\infty}\mathrm{Im}\beta_{i}({\mathbf q}_{n}).\] Hence Notice by (18) and (23) that This means that \(\{\beta_{1},\beta_{2},\beta_{3}\}\) is a permutation of \(\{-d_{1},-d_{2},-d_{3}\}\). So for every sequence \(\{{\mathbf q}_{n}\}\in\mathbb{N}^{d}\), there exists a subsequence \(\{{\mathbf q}_{n_{j}}\}\) such that \[\lim\limits_{j\rightarrow\infty}\beta_{i}({\mathbf q}_{n_{j}})=\beta_{i}.\] Hence we can assume that \[ \lim\limits_{q^2\rightarrow\infty}\beta_{i}({\mathbf q})=-d_{i}, \forall 1 \leq i \leq3, \] or equivalently Using the similar arguments of Lemma 4 in Hoang [1], the following lemma can be derived.

Lemma 4. If \({\mathbf q}\in \mathbb{N}^{d}\) and \(q^2\) sufficiently large, then \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are real numbers and \(\lambda_{i}({\mathbf q})\neq\lambda_{j}({\mathbf q})\), \(i\neq j, i,j=1,2,3\).

Proof. It follows from the assumptions ({\textbf H\(_{2}\)}) and (25) that \(\mathrm{Re}\lambda_{i}({\mathbf q})\neq\mathrm{Re}\lambda_{j}({\mathbf q}), i\neq j.\) If there exists a sequence \(\{{\mathbf q}_{n}\}\in\mathbb{N}^{d}\) such that the sequence \(\lambda_{i_{n}}({\mathbf q}_{n})\notin\mathbb{R}\), then we can choose a subsequence \(\{n_{m}\}\) of \(\{n\}\) and an integer \(j, 1 \leq j\leq 3\) such that \(i_{n_{m}}\equiv j\). Hence \[\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\lambda_{j}({\mathbf q}_{n_{m}})=-d_{j},\] and \[\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\overline{\lambda_{j}({\mathbf q}_{n_{m}})}=-d_{j},\] where \(\overline{\lambda_{j}({\mathbf q}_{n_{m}})}\) is the complex conjugation of \(\lambda_{j}({\mathbf q}_{n_{m}})\). Notice that \(\overline{\lambda_{1}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{2}({\mathbf q}_{n_{m}}), \lambda_{3}({\mathbf q}_{n_{m}})\right\}\), \(\overline{\lambda_{2}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{1}({\mathbf q}_{n_{m}}), \lambda_{3}({\mathbf q}_{n_{m}})\right\}\) and \(\overline{\lambda_{3}({\mathbf q}_{n_{m}})}\in\left\{\lambda_{1}({\mathbf q}_{n_{m}}), \lambda_{2}({\mathbf q}_{n_{m}})\right\}\), then there exists a subsequence of \(\{n_{m}\}\), still denoted by \(\{n_{m}\}\) and \(1 \leq l\leq 3, l\neq j\) such that \(\overline{\lambda_{j}({\mathbf q}_{n_{m}})}=\lambda_{l}({\mathbf q}_{n_{m}})\), one can obtain \[-d_{j}=\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\overline{\lambda_{j}({\mathbf q}_{n_{m}})} =\lim\limits_{q^2_{n_{m}}\rightarrow\infty}\frac{1}{q^{2}_{n_{m}}}\lambda_{l}({\mathbf q}_{n_{m}})=-d_{l}, \forall m\in \mathbb{N}.\] So \(d_{j}=d_{l}\) and \(j\neq l\), in contradiction to the assumption ({\textbf H\(_{2}\)}). Therefore, for \(q^{2}\) sufficiently large \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) are real numbers, and we deduce by \(\mathrm{Re}\lambda_{i}({\mathbf q})\neq\mathrm{Re}\lambda_{j}({\mathbf q})\) that \(\lambda_{i}({\mathbf q})\neq\lambda_{j}({\mathbf q})\) whenever \(i\neq j\), which completes the proof.

3. Growing modes and Bootstrap lemma

3.1. Growing modes in the model (1)

For convenience we will always denote universal positive constants depending on \(d_{i}\), \(\chi\), \(\xi\), \(\mu\), \(\alpha\), \(\beta\), \( \gamma \), \(\eta\) \((i=1,2,3)\) by \(C_{k}(k=1,2,\cdot\cdot\cdot)\). Norm in \(L^{2}(\mathbb{T}^{d})\) is denoted by \(\|\cdot\|\).

Lemma 5. Suppose that ({\textbf H\(_{1}\)}) and ({\textbf H\(_{3}\)}) hold, and \({\mathbf w}({\mathbf x},t)\equiv{e^{\mathfrak{L}t}{\mathbf w}_{0}({\mathbf x})\) is a solution to the linearized system (5) with initial condition \({\mathbf w}_{0}({\mathbf x})\). Then there exists a constant \(\hat{C}_{1}>0\) depending on \(d_{i}\), \(\chi\), \(\xi\), \(\mu\), \(\alpha\), \(\beta\), \( \gamma\), \(\eta\) \((i=1,2,3)\) such that}

Proof. We will proceed in the following two cases.
Case 1: For \({\mathbf t\geq 0}\) ,\({\mathbf q}\in \mathbb{N}^{d}\), \(q^{2}\) sufficiently large. By Lemma 4, for \(q^{2}\) sufficiently large, the matrix \({\mathbf L_{q}}\) has three distinct eigenvalues \(\lambda_{1}({\mathbf q})\), \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\) and the corresponding linearly independent eigenvectors \({\mathbf r}_{1}({\mathbf q})\), \({\mathbf r}_{2}({\mathbf q})\), \({\mathbf r}_{3}({\mathbf q})\). We first look for eigenvector \({\mathbf r}_{1}({\mathbf q})\) such that \[{\mathbf r}_{1}({\mathbf q})=(1, r_{12}({\mathbf q}), r_{13}({\mathbf q}))^{\mathrm{T}},\] where \(r_{12}({\mathbf q})\), \(r_{13}({\mathbf q})\) are the solutions of the linear system \[\begin{array}{ll} (-d_{2}q^2-\beta -\lambda_{1}({\mathbf q}))r_{12}({\mathbf q})+ 0=-\alpha,\\ 0+(-d_{3}q^2-\eta -\lambda_{1}({\mathbf q}))r_{13}({\mathbf q})=-\gamma. \end{array} \] \[\begin{array}{ll} r_{12}({\mathbf q})=\frac{\alpha}{(d_{2}q^2+\beta +\lambda_{1}({\mathbf q}))}, \end{array} \] \[\begin{array}{ll} r_{13}({\mathbf q})=\frac{\gamma}{(d_{3}q^2+\eta +\lambda_{1}({\mathbf q}))}, \end{array} \] \[\begin{array}{ll} \displaystyle\lim\limits_{q^2\rightarrow\infty}r_{12}=0,\\ \displaystyle\lim\limits_{q^2\rightarrow\infty}r_{13}=0,\end{array}\] hence

Let \({\mathbf r}_{2}({\mathbf q})=(r_{21}({\mathbf q}), 1, r_{23}({\mathbf q}))^{\mathrm{T}}\), \({\mathbf r}_{3}({\mathbf q})=(r_{31}({\mathbf q}), r_{32}({\mathbf q}), 1)^{\mathrm{T}}\) be eigenvectors corresponding to the eigenvalues \(\lambda_{2}({\mathbf q})\), \(\lambda_{3}({\mathbf q})\), respectively. Then \[\lim\limits_{q^2\rightarrow\infty}r_{21}({\mathbf q})q^{2}=\frac{\chi}{(d_{2}-d_{1})}, \lim\limits_{q^2\rightarrow\infty}r_{23}({\mathbf q})q^{2}=0,\] and \[\lim\limits_{q^2\rightarrow\infty}r_{31}({\mathbf q})q^{2}=\frac{-\xi}{(d_{3}-d_{1})}, \lim\limits_{q^2\rightarrow\infty}r_{32}({\mathbf q})q^{2}=0.\] Therefore By (27) and (28), we deduce that there exists a constant \(C_{1}>0\) such that For \(q^{2}\) sufficiently large, it is follows from (13) that \({\mathbf w}_{{\mathbf q}}=\sum\limits^{3}_{i=1}w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q}).\) Based on Cramer’s Rule and Hadamard inequality, we have In terms of (27) and (28), one can obtain Applying (30) and (31) yields where \( C_{2}:=\max\left\{1, \sqrt{(\frac{\chi}{d_{2}-d_{1}})^{2}+1}, \sqrt {(\frac{\xi}{d_{2}-d_{3}})^{2}+1}\right\}>0\). Then, using (29), (32) and \( \lambda_{i}({\mathbf q})\leq \lambda_{\max}\), this shows that for \(q^{2}\) sufficiently large there exists a constant \(C_{3}>0\) independent of \({\mathbf q}\) such that \[\left|w_{i}({\mathbf q}){\mathbf r}_{i}({\mathbf q})e^{\lambda_{i}({\mathbf q})t}\right|\leq C_{1}C_{2}e^{\lambda_{\max}t}|{\mathbf w}_{{\mathbf q}}|,\] which leads to Case 2: For \(t\leq1\). It is sufficiently to derive standard estimate in \({\mathbf L^{2}}\). From Neumann boundary condition , we can multiplying the first equation in (6) by \(u_{1}\), the second equation by \(k u_{2}\) and the third by \(u_{3}\), adding them together, and integrating the result in \(\mathbb{T}^{d}\), we have \[\begin{array}{ll} \displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{T}^{d}}\{|u_{1}|^2+k|u_{2}|^2+|u_{3}|^2\}{\mathbf d x}+\int_{\mathbb{T}^{d}}\{d_{1}|\nabla{u+{1}}|^2+kd_{2}|\nabla{u_{2}}|^2+d_{3}|\nabla{u_{3}}|^2- \chi (\nabla{u_{1}}\nabla{u_{2}}) \displaystyle+\xi (\nabla{u_{1}}\nabla{u_{3}})\}{\mathbf d x}\\ \displaystyle=-\mu\int_{\mathbb{T}^{d}}u_{1}^2{\mathbf d x}-k\beta\int_{\mathbb{T}^{d}}u_{2}^2{\mathbf d x}- \eta\int_{\mathbb{T}^{d}}u_{3}^2{\mathbf d x}+\alpha k\int_{\mathbb{T}^{d}}u_{1}u_{2}{\mathbf d x}+\gamma \int_{\mathbb{T}^{d}}u_{1}u_{3}{\mathbf d x}. \end{array} \] where \(k=\frac{\chi^{2}d_{3}}{d_{1}d_{2}d_{3}+d_{2}\xi^{2}}\). Then the integrand of the second integral can be estimated as follows Using Young inequality, we deduce that Then \[\begin{array}{ll} \displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{T}^{d}}\{|u_{1}|^2+k|u_{2}|^2+|u_{3}|^2\}{\mathbf d x}\leq \displaystyle\max(-\mu+\frac{k\alpha^{2}}{2\beta}+\frac{\gamma^2}{2\eta},-\frac{\beta}{2}, \frac{\eta}{2})\int_{\mathbb{T}^{d}}(|u_{1}|^2+ k|u_{2}|^2+|u_{3}|^2){\mathbf d x}. \end{array}\] By Grownwall inequality, we can obtain \(\|{\mathbf w}({\mathbf \cdot},t)\|\leq{\hat{C}_{1}e^{\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|},\) where \(\hat{C}_{1}= \max(-\mu+\frac{k\alpha^{2}}{2\beta}+\frac{\gamma^2}{2\eta},-\frac{\beta}{2},\frac{\eta}{2})\). This completes the proof.

3.2. Bootstrap lemma and \(H^{2}\)-estimate in the model (1)

Denote \[\partial_{ x_{i} x_{j}}u=\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}, \partial_{ x_{i}}u=\frac{\partial u}{\partial x_{i}}, D^\alpha u=\frac{\partial^{|\alpha|}u}{\partial x^{\alpha_1}_{1}\cdots\partial x^{\alpha_d}_{d}},\] where \(\alpha=(\alpha_1,\cdots,\alpha_d)\), \(|\alpha|=\sum^{d}\limits_{i=1}\alpha_i\), \(i,j=1,\cdots,d\). Let us introduce By standard theory of parabolic equation, we can establish the existence of local solutions for the model (4).

Lemma 6. (Local existence). For \(s\geq1 (d=1)\) and \(s\geq2 (d=2,3)\), there exist a \(T_{0}>0\) such that the problem (4) with \(u_{1}(\cdot,0), u_{2}(\cdot,0), u_{3}(\cdot,0)\in{H^{s}(\mathbb{T}^{d})}\) has a unique solution \({\mathbf w}(\cdot,t)\) on \((0,T_{0})\) which satisfies \[ \|{\mathbf w}(t)\|_{H^{s}(\mathbb{T}^{d})}\leq{C}{\|{\mathbf w}(0)\|_{H^{s}(\mathbb{T}^{d})}}, \] where \(C\) is a positive constant depending on \(d_{i}, \xi, \chi, \alpha, \beta, \gamma, \eta (i=1,2,3)\).

Lemma 7. Let \({\mathbf w}({\mathbf x},t)=(u_{1}({\mathbf x},t),u_{2}({\mathbf x},t),v({\mathbf x},t))^{\mathrm{T}}\) be a solution of the nonlinear perturbation systemare the generic constants > (3). Then \begin{eqnarray*} &&\frac{1}{2}\frac{d}{dt}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{|D^{\alpha}u_{1}|^2+k|D^{\alpha}u_{2}|^2 +|D^{\alpha}u_{3}|^2\right\}d{\mathbf x}\\ &&+\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{\frac{d_{1}}{4}|\nabla (D^{\alpha}u_{1})|^2+\frac{kd_{2}}{2}|\nabla (D^{\alpha}u_{2})|^2 +\frac{3d_{3}}{2}|\nabla( D^{\alpha}u_{3})|^2\right\}d{\mathbf x}\\ &&+\frac{\beta{k}}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{2}|^2d{\mathbf x}+\frac{\eta}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{3}|^2d{\mathbf x}\\ &&\leq\hat{C}_2\|{\mathbf w}\|_{H^2(\mathbb{T}^{d})}\|\nabla^3{\mathbf w}\|^2+\hat{C}_{3}\|u_{1}\|^2, \end{eqnarray*} where \(\hat{C}_{2}\) and \(C_{0}\) are the generic constants and \(\hat{C}_{3}=(\frac{\alpha^{2}\eta k +\nu ^{2}}{8 \beta\eta a^{2}})c_{0}\).

Proof. Let \({\mathbf w}({\mathbf x},t)\) be a solution of (4). It is not hard to verify that if \({\tilde{\mathbf w}}({\mathbf x},t)=(\tilde{u}_{1}({\mathbf x},t),\tilde{u}_{2}({\mathbf x},t),\tilde{u}_{3}({\mathbf x},t))^{\mathrm{T}}\) is the even extension of \({\mathbf w}({\mathbf x},t)\) on \(2\mathbb{T}^{d}=(-\pi,\pi)^{d} (d=1,2,3)\). The \({\tilde{\mathbf w}}({\mathbf x},t)\) is also the solution of (4) with the homogeneous Neumann boundary conditions and periodical boundary conditions on \(2\mathbb{T}^{d}\). Therefore,

Using Young inequality, we get The nonlinear term \(J_1\) is bounded by Recalling that the Sobolev imbedding \(H^{2}(\mathbb{T}^{d})\hookrightarrow L^{\infty}(\mathbb{T}^{d})\) for \(d\leq3\), we have Notice that Moreover, if \(g\in H^{1}(2\mathbb{T}^{d})\) with \(\int_{2\mathbb{T}^{d}} g=0\), then It follows from (43) and (44) that \[\|\partial_{x_{i}}g\|\leq C_{9}\|\nabla(\partial_{x_{i}}g)\|, \|\partial_{x_{i}x_{j}}g\|\leq C_{9}\|\nabla(\partial_{x_{i}x_{j}}g)\|\] and Together with (40) and (45), we further get Then as a consequence of(40) and (45), one can obtain where \(C_{12}:=C_{4}+(1+2d)C_{9}\).
Applying interpolation, we can deduce that for all \(\varepsilon>0\), By the choice of \(\varepsilon>0\) in (48) such that \(\left(\frac{\alpha^{2}k\eta+\beta\nu^{2}}{2\beta\eta}\right)C_{0}\varepsilon=d_{1}/4\), then Then as a consequence of(40) ,(41),(42)and (45), one can obtain Substituting (47), (49)-(50) into (37), we have \begin{eqnarray*} &&\frac{1}{2}\frac{d}{dt}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{|D^{\alpha}u_{1}|^2+k|D^{\alpha}u_{2}|^2 +|D^{\alpha}u_{3}|^2\right\}d{\mathbf x}\\ &&+\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}\left\{\frac{d_{1}}{4}|\nabla (D^{\alpha}u_{1})|^2+\frac{kd_{2}}{2}|\nabla (D^{\alpha}u_{2})|^2 +\frac{3d_{3}}{2}|\nabla( D^{\alpha}u_{3})|^2\right\}d{\mathbf x}\\ &&+\frac{\beta{k}}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{2}|^2d{\mathbf x}+\frac{\eta}{2}\sum_{|\alpha|=2}\int_{\mathbb{T}^{d}}|D^{\alpha}u_{3}|^2d{\mathbf x}\\ &&\leq\hat{C}_2\|{\mathbf w}\|_{H^2(\mathbb{T}^{d})}\|\nabla^3{\mathbf w}\|^2+\hat{C}_{3}\|u_{1}\|^2, \end{eqnarray*} where \(\hat{C}_{2}\) and \(C_{0}\) are the generic constants and \(\hat{C}_{3}=(\frac{\alpha^{2}\eta k +\gamma^{2}}{8 \beta\eta a^{2}})c_{0}.\)This completes the proof of Lemma 7.

Lemma 8. Let \({\textbf w}({\textbf x},t)\) be a solution to the system (4) such that for \(0\leq t\leq T\),

and Then for \(0\leq{t}\leq{T}\), where \(\hat{C}_4=\max\{(1+C^2_{9})k, 4\hat{C}^2_1[1+ \hat{C}_3(1+C^2_{9})/(2\lambda_{\max})]\}\geq1\), if \(k\geq1\). \(\hat{C}_4=\max\{(1+C^2_{9})/k, 4\hat{C}^2_1[1+\hat{C}_3(1+C^2_{9})/(2\lambda_{\max}k)]\}\geq 1\), if \(k<1\).

Proof. It follows from (45) that

So By Lemma 7 and (51), we infer Integrating (57) and using (52), we conclude We first consider the case \(k\geq1\). By (57), we have \[\sum\limits_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},t)\|^2\leq k\sum\limits_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},0)\|^2 +\frac{4\hat{C}^2_1\hat{C}_3}{\lambda_{\max}}e^{2\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|^2.\] We see from this estimate and (55) that where \(\hat{C}_4:=\max\left\{(1+C^2_{9})k, 4\hat{C}^2_1\left[1+\frac{\hat{C}_3(1+C^2_{9})}{\lambda_{\max}}\right]\right\}\). On the other hand, for \(K< 1\), we deduce by (57) that \[ \sum_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},t)\|^2\leq\frac{1}{K}\left(\sum_{|\alpha|=2}\|D^{\alpha}{\mathbf w}({\mathbf \cdot},0)\|^2 +\frac{4\hat{C}^2_1\hat{C}_3}{\lambda_{\max}}e^{2\lambda_{\max}t}\|{\mathbf w}({\mathbf \cdot},0)\|^2\right). \] This estimate, combined with (52) and (55) gives where \(\hat{C}_4:=\max\left\{\frac{1+C^2_{9}}{k}, 4\hat{C}^2_1\left[1+\frac{\hat{C}_3(1+C^2_{9})}{\lambda_{\max}k}\right]\right\}\). This completes the proof of Lemma 8.

4. Main result

Assume \(\theta\) be a small fixed constant. For \(\delta>0\) arbitrary small, we define the escape time \(T^{\delta}\) by where \(\lambda_{\max}\) is the dominant eigenvalue which is the maximal growth rate (see (10)). Obviously, Our main result in this paper is as follows:

Theorem 2. Suppose that (H\(_{1}\)),(H\(_{2}\))and(H\(_{3}\)) are satisfied. Let \({\textbf w}_{0}({\textbf x})\in{H^2(\mathbb{T}^{d})}\) with \(\|{\mathbf w}_0({\mathbf x})\|=1\). Then there exist constants \(\delta_0>0, \hat{C}>0\), and \(\theta>0\) depending on \(d_{i}, \chi, \xi, \mu, \alpha, \beta, \eta, \gamma, (i=1,2,3)\) such that \(\forall 0< \delta\leq\delta_0\), if the initial perturbation of the steady state \({\mathbf w_{c}}\) is \({\mathbf w}^{\delta}({\mathbf \cdot},0)=\delta{\mathbf w}_0\), then its nonlinear evolution \({\mathbf w}^{\delta}({\mathbf \cdot},t)\) satisfies

for \(0\leq{t}\leq{T^{\delta}}\), and \(\rho>0\) is the gap between the largest growth rate \(\lambda_{\max}\) and the rest of \(\mathrm{Re}\lambda_{i}({\mathbf q})\) in (7), \(e^{\mathfrak{M}t}{\mathbf w}_{0}({\mathbf x})\) defined in (16) is the dominant part of the solution of the linearized system (5).

Proof. Let \({\mathbf w}^{\delta}({\mathbf x},t)\) be the solutions to (4) with initial data \({\mathbf w}^{\delta}({\mathbf \cdot},0)=\delta{\mathbf w}_0\). Define

From the definition of \(T^{\ast}\) and Lemma 5, for \(\forall 0\leq t\leq{T^{\ast}}\), we can obtain Furthermore, by Lemma 8 and the bootstrap argument, we possess Applying Duhamel’s principle, we know that the solution of (4) It follows from Lemma 5, (40), (44) and Lemma 8 that for \(0\leq{t}\leq\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\), where \(\hat{C}_{5}=\max C^2_{9}\{\chi+\chi\frac{C_{4}}{C^2_{9}}+\xi+\xi\frac{C_{4}}{C^2_{9}}+\mu C_{1}\}\). By (66) and (68), we see that for \(t\leq\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\), We now prove that if \(\delta_{0}\) and \(\theta\) are chosen such that and as well as then \(T^{\delta}=\min{\{T^{\delta},T^{\ast},T^{\ast\ast}\}}\) for \(\delta\leq\delta_{0}\).
If \(T^{\ast\ast}\) is the smallest, we can let \(t=T^{\ast\ast}\leq{T^{\delta}}\) in (67). By (70) and (71) we have \[\begin{array}{ll} \displaystyle\left\|{\mathbf w}^{\delta}(T^{\ast\ast})\right\|_{H^2(\mathbb{T}^{d})} \leq\sqrt{\hat{C}_4}\left\|{\mathbf w}^{\delta}_0\right\|_{H^2(\mathbb{T}^{d})}+\sqrt{C_4} \theta \displaystyle< \frac{1}{\hat{C}_{2}}\min\left\{\frac{d_{1}}{4}, \frac{d_{2}}{4}, \frac{d_{3}K}{2}\right\}, \end{array}\] for \(\delta\) sufficiently small and \(\hat{C}_4\geq1\), in contradiction to the definition of \(T^{\ast\ast}\). On the other hand, if \(T^{\ast}\) is the minimum, we can let \(t=T^{\ast}\) in (67), so that \[\begin{array}{ll} \displaystyle\left\|{\mathbf w}^{\delta}({\mathbf \cdot},T^{\ast})-\delta{e}^{\mathfrak{L}T^{\ast}}{\mathbf w}_0\right\| \leq \hat{C}_1\hat{C}_4\hat{C}_5\left\{\frac{\delta\|{\mathbf w}_0\|^{2}_{H^2(\mathbb{T}^{d})}}{\lambda_{\max}} +\frac{\theta}{\lambda_{\max}}\right\}\delta e^{\lambda_{\max}T^{\ast}} \displaystyle< \frac{\hat{C}_1}{2}\delta e^, \end{array} \] for sufficiently small \(\delta_{0}\) in (73) and \(\hat{C}_5/{\hat{C}}_{2}\leq1\). This again contradicts the definition of \(T^{\ast}\). Therefore, the desired assertion follows. Finally, we prove the inequality (62). Notice by (14) that We next estimate each term \(J_{i} (i=6,7,8,\cdots,12)\) on the right-hand sides of (73). It is not difficult to know that there are finitely many values \({\mathbf q}\in \mathbb{N}^{d}\) satisfying \(\mathrm{Re}\lambda_{i}({\mathbf q})=\lambda_{\max}\) and \(|{\mathbf q}|\) is bounded for each \({\mathbf q}\in{\mathbb{N}^{d}}_{\max}\). For each \({\mathbf q}\in \mathbb{N}^{d}\), \(q^{2} 0\) such that By the similar method to prove (32), using (74) and (13) there exists a constant \(C_{**}>0\) such that and By (29), (32),(74), (75) and \(\|{\mathbf w}_{0}\|=1\), there exists a constant \(\hat{C}_{6}>0\) such that \[\displaystyle J^{2}_7\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t}\left(\frac{\pi}{2}\right)^{d}\sum\limits_{{\mathbf q}\in\Lambda_{R1}}|{\mathbf w}_{{\mathbf q}}|^{2}\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t}\|{\mathbf w}_{0}\|^{2}\leq\delta^{2} \hat{C}^{2}_6 e^{2(\lambda_{\max}-\rho)t},\] that is, Moreover, Similarly, there exists a constant \(\hat{C}_{7}>0\) such that Substituting (69), (77)-(81) into (73) yields \begin{eqnarray*} \displaystyle\left\|{\mathbf w}^{\delta}({\mathbf \cdot},t)-\delta{e}^{\mathfrak{M}t}{\mathbf w}_0\right\| &\leq&\hat{C}_1\hat{C}_4\hat{C}_5\left\{\frac{\delta\|{\mathbf w}_{0}\|^2_{H^2}}{\lambda_{\max}}+\frac{\delta e^{\lambda_{\max}t}}{\lambda_{\max}}\right\}\delta e^{\lambda_{\max}t} +\hat{C}_6\delta e^{(\lambda_{\max}-\rho)t}+\delta e^{(\lambda_{\max}-\rho)t}+5\hat{C}_7 \delta e^{(\lambda_{\max}-\rho)t}\\ &\leq&\left\{(1+\hat{C}_{6}+5\hat{C}_7)e^{-\rho t}+\frac{\hat{C}_1\hat{C}_4\hat{C}_5}{\lambda_{\max}}\left(\delta\|{\mathbf w}_0\|^2_{H^2(\mathbb{T}^{d})} +\delta e^{\lambda_{\max}t}\right)\right\}\delta e^{\lambda_{\max}t}\\ &\leq&{\hat{C}}\left\{e^{-\rho t}+\delta\|{\mathbf w}_0\|^2_{H^2(\mathbb{T}^{d})}+\delta e^{\lambda_{\max}t}\right\}\delta e^{\lambda_{\max}t}, \forall 0\leq{t}\leq{T^{\delta}}, \end{eqnarray*} where \(\hat{C}:=\max\{1+\hat{C}_{6}+5\hat{C}_7, \frac{\hat{C}_1\hat{C}_4\hat{C}_5}{\lambda_{\max}}\}\) and thereby completes the proof.

Corollary 1.(Nonlinear instability). Let the conditions ( \({\mathbf H_{1}}\)),(\({\mathbf H_{2}}\)) and (\({\mathbf H_{3}}\) ) are holds. Then the positive constant equilibrium point \({\mathbf w_{c}}\) of the problem (1) is nonlinearly unstable in the sense of the \(L^{2}\)-norm.

Proof. Notice that \({\mathbf L}_{{\mathbf q_{0}}}\) has an eigenvalue \(\mathrm{Re}\lambda_{{\mathbf q_{0}}} = \lambda_{\max}\), if there exists \({\mathbf q}_0=(q_{01},\ldots,q_{0d})\in{\mathbb{N}^{d}}_{\max}\), and denote the corresponding eigenvector by \({\mathbf r}_{{\mathbf q_{0}}}\). Assume \[{\mathbf w}_0({\mathbf x})=\kappa\frac{{\mathbf r}({\mathbf q}_0)}{|{\mathbf r}({\mathbf q}_0)|}e_{{\mathbf q}_0}({\mathbf x})\] with \(\kappa=1/\|e_{{\mathbf q}_0}\|=\sqrt{(2/\pi)^{d}}\) so that \(\|{\mathbf w}_{0}(x)\|=1\). In addition, if \(t=T^{\delta}\) then for \(\delta\) sufficiently small, we require

It follows from Theorem 2 and(80) that Notice that the dominant part of the solution of the linearized system (5) satisfies By (81) and (82), we deduce that \[\|{\mathbf w}^{\delta}({\mathbf \cdot},T^{\delta})\|>\frac{1}{2}\theta>0.\]

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.