Cross-diffusion-driven Turing instability and weakly nonlinear analysis of Turing patterns in a uni-directional consumer-resource system

The aforementioned models only describe ideal populations and neglect spatial heterogeneity. In reality, according to Fick’s law, the species is spatially heterogeneous and hence individuals will tend to migrate towards regions of lower population density to increase the possibility of survival, and hence the species are distributed over space and interact with each other within their spatial domain [9]. Generally speaking, diffusion deals with transport from a region of higher concentration to one of lower concentration by random motion. The diffusion of individuals may be connected with other things, for example, the individuals look for food, the preys keep away from predators in order to not be caught, the individuals escape from high infection risks and so on (for example, see [10–16] and references therein). Considering the effects of spatial diffusion, Huang et al. [17] studied bifurcation and temporal periodic patterns in a delayed plant-pollinator model with a diffusion effect.

More generally, there exists a motional state of interaction as well, one that recognizes the possible bias, say, of the motion of one species toward or away from another species [18, 19]. This phenomenon is called cross-diffusion. The value of the cross-diffusion coefficient can be positive or negative. The term including positive cross-diffusion coefficients denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficients denote that one species tends to diffuse in the direction of a higher concentration. Since the pioneer work of [19], the concern on cross-diffusion in ecological models has attracted the attention of biologists, chemical experimenters and applied mathematicians and cross-diffusion has been one of the dominant topics in both ecology and mathematical ecology because of its universal existence and importance ([1, 9]). Shigesada et al. [20] investigated the effect of cross-diffusion in a competitive population system first. Then, many works have proposed to investigate the Turing instability driven by cross-diffusion and prove the existence of inhomogeneous steady states that induce the emergence of spatial patterns ([21–37] and [38–42]). Hence it is of great interest to explore the role of cross-diffusion in driving Turing instability and spatial pattern formation.

In our current work we first investigate the effect of cross-diffusion on the spatial inhomogeneous distribution of the two species in a uni-directional C-R system. Our results show that the resource supply rate has an important effect on the Turing bifurcation direction. We also show how the weakly nonlinear analysis of the C-R system (1.3) is able to point out some interesting phenomena like stable supercritical and subcritical Turing patterns and a hysteretic-type phenomenology due to the presence of a multiplicity of real stable equilibria for the amplitude equation in the subcritical case. The rest of the paper is organized as follows. Section 2 discusses the existence and unique positive spatially homogeneous steady state of system (1.3). The linear stability analysis of the proposed system and its corresponding non-spatial system are presented in Section 3. Weakly nonlinear pattern analysis and the related numerical simulations are given in Section 4. A brief conclusion and discussion are presented in Section 5. In the Appendix, we give the coefficient expressions of amplitude equations emerged in Section 4.

In this section, we shall investigate the impact of cross-diffusion on the stability of the unique positive spatially homogeneous steady state \(E_{\ast}\) of the system (1.3). Specifically, we are interested in investigating the Turing instability of the system (1.3), which is induced by cross-diffusion, that is, the positive steady state \(E_{\ast}\) of the model without cross-diffusion is stable, but it is unstable in the presence of the cross-diffusion term \(buv\).

3.3 Turing instability induced by cross-diffusion

In this subsection, we shall prove that the only potential destabilizing mechanism in our system (1.3) is the presence of the cross-diffusion. Linearizing system (1.3) along the positive steady state \(E_{\ast}=(u^{\ast}, v^{\ast})\) yields

$$ {\dot{\mathbf {w}}}=K{\mathbf {w}}+D^{b}\nabla^{2}{\mathbf {w}}, \quad\quad{\mathbf {w}}=\left ( \begin{matrix} u-u_{\ast}\\ v-v_{\ast} \end{matrix} \right ) , $$

(3.8)

where \(D^{b}=\bigl( {\scriptsize\begin{matrix}{} a_{1} & 0\cr bv_{\ast}& a_{2}+bu_{\ast} \end{matrix}} \bigr) \) and K is given by (3.1).

Substituting (3.5) into (3.8) leads to the following dispersion relation, which gives the eigenvalue λ as a function of the wave number \(k= \vert {\mathbf {k}} \vert \):

$$ \lambda^{2}+g\bigl(k^{2}\bigr)\lambda+h\bigl(k^{2} \bigr)=0, $$

(3.9)

where

$$ g\bigl(k^{2}\bigr)=k^{2}\operatorname {tr}\bigl(D^{b}\bigr)-\operatorname {tr}(K),\quad \quad h\bigl(k^{2}\bigr)=\operatorname {det}\bigl(D^{b} \bigr)k^{4}+qk^{2}+\operatorname {det}(K), $$

(3.10)

and

$$ q=a_{1}d_{2}v_{\ast}+a_{2}d_{1}u_{\ast}+bu_{\ast} \biggl( d_{1}u_{\ast}-\beta_{1}v_{\ast}+ \frac{\alpha_{12}c_{2}v_{\ast }}{(c_{2}+v_{\ast})^{2}} \biggr) . $$

(3.11)

Theorem 3.3

Cross-diffusion induced Turing instability

Assume that \((H_{1})\) and (3.3) hold. Then the unique positive spatially homogeneous steady state \(E_{\ast }(u_{\ast},v_{\ast})\) of system (1.3) is spatially unstable when \(b>b^{c}\), and the critical wave number is given by \(k_{c}^{2}\), where \(b^{c}\) and \(k_{c}^{2}\) can be found in the proof of the theorem.

Proof

From Theorem 3.1 and Theorem 3.2, we know that system (1.3) still remains stable without diffusion or in the presence of self-diffusion alone. Therefore, it follows that the only potential destabilizing mechanism is the presence of the cross-diffusion terms. Then for the Turing instability to be realized and spatial patterns to form, there must be some \(k\neq0\) such that the real part of at least one root of the characteristic equation (3.9) is greater than zero. Since \(g(k^{2})>0\) for \(\forall k\neq0\), instability can only be obtained in the case of \(h(k^{2})<0\) for some \(k\neq0\). Since \(\operatorname {det}(D^{b})>0\), we get the condition for the marginal stability as follows:

$$ \min\bigl(h\bigl(k^{2}\bigr)\bigr)=h\bigl(k_{m}^{2} \bigr)=0, $$

(3.12)

where

$$ k_{m}^{2}=-\frac{q}{2\operatorname {det}(D^{b})}, $$

(3.13)

which requires that \(q<0\). From \(q<0\), we know that the necessary condition of Turing instability is as follows:

$$ d_{1}u_{\ast}-\beta_{1}v_{\ast}+ \frac{\alpha_{12}c_{2}v_{\ast }}{(c_{2}+v_{\ast})^{2}}< 0, $$

(3.14)

which is equivalent to the assumption (3.3).

In what follows, we shall choose the cross-diffusion coefficient b as bifurcation parameter. Let us now set \(q=-\gamma b+\delta\), where the positive quantities γ and δ are defined as:

$$\begin{aligned} \begin{aligned} \gamma=-u_{\ast} \biggl( d_{1}u_{\ast}- \beta_{1}v_{\ast}+\frac {\alpha_{12}c_{2}v_{\ast}}{(c_{2}+v_{\ast})^{2}} \biggr) ,\quad \quad \delta=a_{1}d_{2}v_{\ast}+a_{2}d_{1}u_{\ast}. \end{aligned} \end{aligned}$$

Let \(b=\frac{\delta}{\gamma}+\eta\). At bifurcation, \(\min (h(k^{2}))=0\). Then substituting b into the second equality of (3.12) yields the following equation for η:

$$ \frac{\gamma^{2}\eta^{2}}{4\operatorname {det}(K)}-a_{1}u_{\ast}\eta - \biggl(a_{1}a_{2}+a_{1}u_{\ast} \frac{\delta}{\gamma}\biggr)=0. $$

(3.15)

Clearly, equation (3.15) has a unique positive root. Denote the unique positive root of equation (3.15) by \(\eta^{+}\). Set

$$ b^{c}=\frac{\delta}{\gamma}+\eta^{+}. $$

(3.16)

Since

$$ q=0\quad \Leftrightarrow \quad b^{\ast}=\frac{\delta}{\gamma}, $$

we can see that \(0< b^{\ast}< b^{c}\) and \(q<0\), \(\min(h(k^{2}))<0\) when \(b>b^{c}\). Therefore, the unique steady state \(E^{\ast}\) is unstable. This indicates that a Turing instability occurs and the critical value for bifurcation is \(b=b^{c}\). The critical wave number \(k_{c}\) is then given (using (3.13)) by

$$ k_{c}^{2}=\frac{\gamma\eta^{+}}{2\operatorname {det}(D^{b^{c}})}. $$

(3.17)

This completes the proof. □

Remark 1

When \(b>b^{c}\), there exists a range \((k_{1}^{2}, k_{2}^{2})\) of unstable wave numbers \(k^{2}\) such that \(h(k^{2})<0\), and correspondingly the real part of the eigenvalue of equation (3.9) \(\operatorname {Re}(\lambda)>0\); see on the right of Figure 2.

In what follows, we shall consider only the case where there is one unstable eigenvalue, admissible for the Neumann boundary conditions, which falls within the band \((k_{1}^{2}, k_{2}^{2})\) in the sense of \(k_{1}^{2}< k^{2}=\phi^{2}+\psi^{2}< k_{2}^{2}\) with \(\phi=\frac{l\pi }{L_{x}}\) and \(\psi=\frac{n\pi}{L_{y}}\), where l, n are integers. We denote by \(\hat{k}_{c}^{2}\) the unstable admissible eigenvalue to distinguish it from the critical value \(k_{c}^{2}\). As an example, we take the parameters of system (1.3) as \(r_{1}=0.6\), \(r_{2}=0.3\), \(c_{1}=c_{2}=0.1\), \(d_{1}=d_{2}=0.01\), \(\alpha_{12}=0.4\), \(\beta _{1}=0.02\), \(\alpha_{21}=0.3\), \(a_{1}=0.1\), \(a_{2}=0.2\), \(b=0.8\). The spatial domain is taken as \(L_{x}=9\pi\), \(L_{y}=8\pi\). Using Shengjin’s formula [44] and Matlab algorithm software, we can calculate that system (1.3) has a unique positive steady state \(E_{\ast }=(0.1959, 49.8614)\). By (3.16), we compute that \(b^{c}=0.7571< b=0.8\) and that \(k_{1}^{2}=0.9430\), \(k_{2}^{2}=2.0173\). Then there exists only a pair of \((l,n)=(9,8)\) such that the only unstable model allowed by the boundary condition is \(\hat {k}_{c}^{2}=2\), which falls within the band of the unstable modes. According to Theorem 3.3, the unique positive spatially homogeneous steady state \(E_{\ast}\) is spatially unstable and the spatial inhomogenous distribution of the two species are dipicted in Figure 3.

In this section, we shall employ the multiple scale method to derive the amplitude equations describing the dynamics close to onset \(b=b^{c}\) based on the weakly nonlinear analysis theory. For more details as regards the weakly nonlinear analysis, we refer the reader to [45, 46] and some articles which use similar techniques [47, 48]. In what follows, we will suppose that there exists only one unstable eigenvalue \(\lambda(\hat {k}_{c}^{2})\), which lies in the instability band.

4.1 Simple eigenvalue case

In this case, \(m=1\), that is, given \(\hat{k}_{c}^{2}\in[k_{1}^{2},k_{2}^{2}]\), there exists only one pair of integers \((l,n)\) such that the following condition holds:

$$ \phi_{1}^{2}+\psi_{1}^{2}=\biggl( \frac{l\pi}{L_{x}}\biggr)^{2}+\biggl(\frac{n\pi }{L_{y}} \biggr)^{2}=\hat{k}_{c}^{2}. $$

(4.17)

Since the eigenvalue λ is simple, the solution of the linear problem (4.9) satisfying the Neumann boundary conditions is given by

$$ {\mathbf {w}}_{1}=A_{1}(T_{1},T_{2}){ \boldsymbol{\varrho}}\cos(\phi_{1} x)\cos(\psi_{1} y),\quad \text{with }\phi_{1}=\frac{l\pi }{L_{x}}, \psi_{1}= \frac{n\pi}{L_{y}}. $$

(4.18)

From (4.10), we get the following form of the vector G:

$$ \begin{aligned} {\mathbf {G}}&=\left ( \frac{\partial A_{1}}{\partial T_{1}} \boldsymbol {\varrho}+b^{(1)}\hat{k}_{c}^{2}A_{1} \left ( \begin{matrix}0\\ u_{\ast}M+v_{\ast} \end{matrix} \right ) \right ) \cos(\phi_{1}x)\cos(\psi_{1}y) \\ &\quad{}- \frac{1}{8}A_{1}^{2}\sum _{i,j=0,2}\mathcal {M}_{ij}^{1}(\boldsymbol{ \varrho},\boldsymbol{\varrho}) \cos(i\phi_{1}x)\cos(j \psi_{1}y), \end{aligned} $$

with \(\mathcal{M}_{ij}^{l}==\Phi_{K}-(i^{2}\phi_{l}^{2}+j^{2}\psi _{l}^{2})\Phi_{D}^{b^{c}}\), \(l=1,2\).

By imposing the solvability condition at \(O(\varepsilon^{2})\) to equation (4.10), we obtain the quintic Stuart-Landau equations as follows:

$$ \frac{\partial A_{1}}{\partial T_{1}}=\alpha A_{1},\quad\quad\alpha =- \frac{b^{(1)}\hat{k}_{c}^{2}(u_{\ast}M+v_{\ast})}{1+MM^{\ast}}. $$

It follows from the above equation that \(A_{1}\to0\) as \(t\to\infty\), which implies that the pattern amplitude dies out at this order and there is no information that may be helpful at this stage and we should push the weakly nonlinear analysis to a higher order to obtain some qualitative results as regards the amplitude. Hence we impose \(T_{1}=0\) and \(b^{(1)}=0\) to suppress the secular terms at this order. Then the compatibility condition is automatically satisfied and the solution of linear problem (4.10) satisfying the Neumann boundary conditions is then calculated as follows:

$$ {\mathbf {w}}_{2}=A_{1}^{2}\sum _{i,j=0,2}{\mathbf {w}}_{2ij}\cos(i\phi _{1} x) \cos(j\psi_{1} y), $$

(4.19)

where the vectors \({\mathbf {w}}_{2ij}\) are the solutions to the following linear systems:

$$ \mathfrak{L}_{ij}^{l}{\mathbf {w}}_{2ij}=- \frac{1}{8}\mathcal {M}_{ij}^{l}({\boldsymbol{ \varrho}}, {\boldsymbol{\varrho}}),\quad i,j=0,2, $$

(4.20)

with \(\mathfrak{L}_{ij}^{l}=K-(i^{2}\phi_{l}^{2}+j^{2}\psi _{l}^{2})D^{b^{c}}\), \(l=1,2\).

According to (4.11), the vector H is given by

$$ {\mathbf {H}}=\biggl(\frac{dA_{1}}{dT_{2}}{\boldsymbol{\varrho}}+A_{1}{ \mathbf {H}}_{11}^{(1)}+A_{1}^{3}{\mathbf {H}}_{11}^{(3)}\biggr)\cos(\phi_{1} x)\cos(\psi _{1} y)+A_{1}^{3}{\mathbf {H}}^{\ast}, $$

(4.21)

where \({\mathbf {H}}_{11}^{(j)}\), \(j=1,3\), and \({\mathbf {H}}^{\ast}\) depend on parameters of the original system (1.3), given in Appendix A.1.1. Applying the solvability condition \(\langle {\mathbf {H}}, {\boldsymbol{\varphi}}\rangle=0\) to equation (4.11), we obtain the Stuart-Landau equation corresponding to the amplitude \(A_{1}(T_{2})\) as follows:

$$ \frac{dA_{1}}{dT_{2}}=\sigma A_{1}-LA_{1}^{3}, $$

(4.22)

where σ and L are given by

$$ \sigma=-\frac{\langle{\mathbf {H}}_{11}^{(1)}\cos(\phi_{1} x)\cos(\psi _{1} y),{\boldsymbol{\varphi}}\rangle}{\langle{\boldsymbol{\varrho }}\cos(\phi_{1} x)\cos(\psi_{1} y),{\boldsymbol{\varphi}}\rangle }, \quad\quad L=\frac{\langle{\mathbf {H}}_{11}^{(3)}\cos(\phi_{1} x)\cos(\psi_{1} y),{\boldsymbol{\varphi}}\rangle}{\langle {\boldsymbol{\varrho}\cos(\phi_{1} x)\cos(\psi_{1} y)},{\boldsymbol{\varphi}}\rangle}. $$

Since the coefficient σ in equation (4.22) is always positive in the pattern-forming region, we distinguish two cases for the qualitative dynamics of the Stuart-Landau equation (4.22) according to the sign of the Landau constant L: (i) \(L>0\), the supercritical bifurcation case; (ii) \(L<0\), the subcritical bifurcation case.

In the following two subsections, we will concentrate ourselves to the discussion of the dynamics of the Stuart-Landau equation (4.22) according to the sign of Landau constant L.

4.1.1 The supercritical bifurcation case

In this case, since σ and L are both positive, for equation (4.22) there exists the stable stationary state \(\sqrt{\frac {\sigma}{L}}\). Then summarizing the above analysis yields the following proposition.

Proposition 4.1

Assume that

1. (i)

   \(\varepsilon^{2}=\frac{b-b^{c}}{b^{c}}\) is small enough so that the positive constant steady state \((u_{\ast},v_{\ast})\) of system (1.3) is unstable to modes corresponding only to the eigenvalue \(\hat{k}_{c}^{2}\), which is defined in (4.17);

    
2. (ii)

   there exists only one pair of integers \((l,n)\) in (4.17);

    
3. (iii)

   the Landau coefficient L in equation (4.22) is positive.

    

Then, according to (4.7), system (1.3) has a stationary pattern as follows:

$$ \begin{aligned}[b] \left ( \begin{matrix}u(x,y)\\ v(x,y) \end{matrix} \right ) &=\left ( \begin{matrix}u_{\ast}\\ v_{\ast} \end{matrix} \right ) +\varepsilon \sqrt{\frac{\sigma}{L}} {\boldsymbol{\varrho }}\cos(\phi_{1} x)\cos( \psi_{1} y) \\ &\quad{} +\varepsilon^{2} \frac{\sigma }{L}\sum _{i,j=0,2}{\mathbf {w}}_{2ij}\cos(i\phi_{1} x) \cos(j\psi _{1} y)+O\bigl(\varepsilon^{3}\bigr), \end{aligned} $$

(4.23)

where ϱ is given by (4.16), and \({\mathbf {w}}_{2ij}\) is given by (4.20).

As an illustrated example, from Figure 4, we calculate the band of the unstable modes is \([1.1849, 1.6372]\). Thus the most unstable model is chosen as \(\hat{k}_{c}^{2}=1.25\), which falls within the band of the unstable modes. In this rectangular domain, there exists only the pair \((l,n)=(9,4)\) such that the equality (4.17) is satisfied. By calculation, we get \(\sigma=0.2576\) and \(L=1.3353>0\). In view of Proposition 4.1, a supercritical bifurcation occurs in this case. By (4.23), we get the approximation solution of the first order to the stationary pattern by weakly nonlinear analysis to read

$$ {\mathbf {w}}=0.4392\varepsilon{\boldsymbol{\rho}}\cos(0.5x)\cos (y)+O\bigl( \varepsilon^{2}\bigr), $$

which shows a good qualitative agreement with the numerical solution of system (1.3). Through numerical computation we know that the error between the approximation solution and the simulation solution is \(O(\varepsilon^{2})\) with \(\varepsilon=0.1\).

4.1.2 The subcritical bifurcation case

For certain values of the parameters appearing in system (1.3), the Landau coefficient L in equation (4.22) has a negative value. In this case equation (4.22) cannot capture the amplitude of the pattern. In order to predict the amplitude of the pattern, one needs to extend weakly nonlinear expansion to higher orders as suggested by [49] and references therein.

Pushing the weakly nonlinear analysis up to \(O(\varepsilon^{5})\), one obtains the quintic Stuart-Landau equation for the amplitude \(A_{1}\) at the time \(T(T_{2}, T_{4})\) as follows:

$$ \frac{dA_{1}}{dT}=\overline{\sigma}A_{1}-\overline{L} A_{1}^{3}+\overline{R} A_{1}^{5}, $$

(4.24)

with

$$ \overline{\sigma}=\sigma+\varepsilon^{2}\widehat{\sigma},\quad \quad \overline{L}=L+\varepsilon^{2}\widehat{L},\quad \quad\overline {R}= \varepsilon^{2}\widehat{R}. $$

Here the details of the derivation and the explicit expression of the coefficients σ̂, L̂, R̂ are given in Appendix A.1.2.

Since \(\sigma>0\), \(L<0\), there exists \(\vert \varepsilon \vert \ll1\) such that \(\overline{\sigma}>0\), \(\overline{L}<0\). Then when \(\overline{R}<0\), there exists one stable stationary state

$$ A_{1\infty}=\sqrt{\frac{\overline{L}-\sqrt{\overline {L}^{2}-4\overline{\sigma}\overline{R}}}{2\overline{R}}}. $$

(4.25)

Then the above analysis implies the following proposition.

Proposition 4.2

Assume that the hypotheses (i) and (ii) of Proposition  4.1 hold and that

1. (i)

   the control parameter ε is small enough so that the Landau coefficient L in (4.22) is negative;

    
2. (ii)

   the coefficient R̅ or R̂ is negative.

    

Then the asymptotic solution of the reaction-diffusion system (1.3) can be expressed as

$$\begin{aligned} & \left ( \begin{matrix}u(x,y)\\ v(x,y) \end{matrix} \right ) \\ &\quad =\left ( \begin{matrix}u_{\ast}\\ v_{\ast} \end{matrix} \right ) +\varepsilon A_{1\infty}{\boldsymbol{\varrho}} \cos(\phi _{1} x)\cos(\psi_{1} y)+\varepsilon^{2} A_{1\infty}^{2}\sum_{i,j=0,2}{\mathbf {w}}_{2ij}\cos(i\phi_{1} x)\cos(j\psi_{1} y) \\ &\quad \quad{}+ \varepsilon^{3} \biggl( A_{1\infty}{\mathbf {w}}_{311}^{(1)} \cos(\phi_{1} x)\cos(\psi_{1} y)+A_{1\infty}^{3} \sum_{i,j=1,3}{\mathbf {w}}_{3ij}\cos(i \phi_{1} x)\cos(j\psi_{1} y) \biggr) \\ &\quad \quad{}+\varepsilon^{4} \biggl( \sum_{i,j=0,2}A_{1\infty}^{2}{ \mathbf {w}}_{4ij}^{(1)}\cos(i\phi_{1} x)\cos(j \psi_{1} y)+\sum_{i,j=0,2,4}A_{1\infty}^{4}{ \mathbf {w}}_{4ij}\cos(i\phi_{1} x)\cos(j\psi _{1} y) \biggr) \\ &\quad \quad{} +O\bigl(\varepsilon^{5}\bigr), \end{aligned}$$

(4.26)

where ϱ is given by (4.16), \({\mathbf {w}}_{2ij}\) (\(i,j=0,2\)) are given by (4.20), \({\mathbf {w}}_{311}^{(1)}\), \({\mathbf {w}}_{3ij}\) (\(i,j=1,3\)) are given by (A.1), \({\mathbf {w}}_{4ij}^{(1)}\) (\(i,j=0,2\)), \({\mathbf {w}}_{4ij}\) (\(i,j=0,2,4\)) are given by (A.2) and \(A_{1\infty}\) is given by (4.25).

Figure 5 shows that the resource supply rate has an important effect on the Turing bifurcation direction.

As an example, we take the parameters of system (1.3) as \(r_{1}=0.6\), \(r_{2}=0.3\), \(c_{1}=c_{2}=0.1\), \(d_{1}=d_{2}=0.01\), \(\alpha _{12}=0.6\), \(\beta_{1}=0.02\), \(\alpha_{21}=0.3\), \(a_{1}=0.1\), \(a_{2}=0.2\). Here \(b^{c}=0.045\), \(\varepsilon=0.1\), \(b=1.0101\), \(b^{c}=0.0455\). Solving equations (2.1)-(2.2) yields a unique positive steady state \(E_{\ast\ast}=(2.3485, 58.7748)\) and that the critical value of the cross-diffusion coefficient is \(b^{c}=0.045\). Set \(\varepsilon =0.1\), \(b=(1+\varepsilon^{2}+\varepsilon^{4})\), \(b^{c}=0.0455\). Then the band of the unstable modes is \([0.7748, 1.1615]\). The spatial domain is taken as \(L_{x}=9\pi\), \(L_{y}=5\pi\) and the only unstable model is chosen as \(\hat{k}_{c}^{2}=1.04\in[0.7748,1.1645]\). In this rectangular domain, there exists only the pair \((l,n)=(9,1)\) such that equality (4.17) is satisfied. According to (4.22), we get the Landau coefficient \(L=-0.0926\), which is less than zero. By a calculation, we obtain \(\overline{\sigma}=0.2013\), \(\overline {L}=-0.0214\) and \(\overline{R}=-0.046\) for \(\varepsilon=0.1\). Thus a subcritical bifurcation occurs according to Proposition 4.2 in this case. It can be seen from Figure 6 that both the approximation solution at \(O(\varepsilon^{3})\) and the approximation at \(O(\varepsilon^{5})\) have only a subtle difference with \(\varepsilon=0.1\).

Figure 7 presents a complete bifurcation diagram for the bifurcation parameter b. It also shows that system (1.3) is a stable stationary pattern with a large amplitude branch which coexists in the range \(b_{s}< b< b_{c}\), which is known as a hysteresis cycle. Varying the bifurcation parameter b following the direction of the arrows in Figure 7, the corresponding numerical solution of the full system shows the hysteresis phenomenon.

4.2 Double and non-resonant eigenvalue

When \(m=2\), the two pairs of allowed spatial modes \((\phi_{1},\psi _{1})\) and \((\phi_{2}, \psi_{2})\) satisfy the following no-resonance condition [38]:

$$\begin{aligned}& \begin{aligned} &\phi_{i}+\phi_{j}\neq \phi_{j} \quad\text{or} \quad\psi_{i}-\psi_{j} \neq\psi_{j} \\ &\text{and} \\ &\phi_{i}-\phi_{j}\neq\phi_{j} \quad\text{or} \quad\psi_{i}+\psi_{j}\neq\psi_{j},\end{aligned} \end{aligned}$$

(4.27)

with \(i,j=1,2\) and \(i\neq j\).

Pushing weakly nonlinear analysis up to \(O(\varepsilon^{3})\), we get the following Stuart-Landau equations for the amplitudes \(A_{1}\) and \(A_{2}\) at the time \(T(T_{2})\):

$$ \begin{aligned} &\frac{dA_{1}}{dT}=\sigma A_{1}-L_{1}A_{1}^{3}+S_{1}A_{1}A_{2}^{2}, \\ &\frac{dA_{2}}{dT}=\sigma A_{2}-L_{2}A_{2}^{3}+S_{2}A_{1}^{2}A_{2}, \end{aligned} $$

(4.28)

where the coefficients σ, \(L_{l}\), \(S_{l}\), with \(l=1,2\), are given in Appendix A.2.1. For the existence and stability of the equilibria of equations (4.28), we have the following propositions.

Proposition 4.3

1. (i)

   The trivial equilibrium \(E_{0}=(0,0)\) always exists.

    
2. (ii)

   The boundary equilibria \(E_{1}^{\pm}=(\pm\sqrt{\frac{\sigma }{L_{1}}},0)\) exist if and only if \(L_{1}>0\).

    
3. (iii)

   The boundary equilibria \(E_{2}^{\pm}=(0,\pm\sqrt{\frac{\sigma }{L_{2}}})\) exist if and only if \(L_{2}>0\).

    
4. (iv)

   The interior equilibria \(E_{3}^{\pm}= ( \pm\sqrt{\frac{\sigma (L_{2}+S_{1})}{L_{1}L_{2}-S_{1}S_{2}}},\pm\sqrt{\frac{\sigma (L_{1}+S_{2})}{L_{1}L_{2}-S_{1}S_{2}}} ) \) exist if and only if either \(L_{1}L_{2}-S_{1}S_{2}<0\), \(L_{1}+S_{2}<0\), \(L_{2}+S_{1}<0\) or \(L_{1}L_{2}-S_{1}S_{2}>0\), \(L_{1}+S_{2}>0\), \(L_{2}+S_{1}>0\).

    

By a linear stability analysis, we have the following results.

Proposition 4.4

1. (i)

   The trivial equilibrium \(E_{0}\) is an unstable node.

    
2. (ii)

   The boundary equilibria \(E_{1}^{\pm}\) are stable nodes if \(L_{1}+S_{2}<0\).

    
3. (iii)

   The boundary equilibria \(E_{2}^{\pm}\) are stable nodes if \(L_{2}+S_{1}<0\).

    
4. (iv)

   The interior equilibria \(E_{3}^{\pm}\) are locally asymptotically stable if \(L_{1}L_{2}-S_{1}S_{2}>0\), \(L_{1}+S_{2}>0\), \(L_{2}+S_{1}>0\) and \(L_{1}S_{1}+L_{2}S_{2}+2L_{1}L_{2}>0\), while they are saddle points if \(L_{1}L_{2}-S_{1}S_{2}<0\), \(L_{1}+S_{2}<0\) and \(L_{2}+S_{1}<0\).

    

Proof

(i), (ii) and (iii) are easily checked; here we omit their proofs. For (iv), we only give the proof for the positive equilibrium \(E_{3}^{+}= ( \sqrt{\frac{\sigma (L_{2}+S_{1})}{L_{1}L_{2}-S_{1}S_{2}}},\sqrt{\frac{\sigma (L_{1}+S_{2})}{L_{1}L_{2}-S_{1}S_{2}}} ) \) as the other cases can be proved similarly. Linearizing system (4.28) about \(E_{3}^{+}\) yields the Jacobian matrix as follows:

$$ J=\left ( \begin{matrix}{} \frac{-2\sigma L_{1}L_{2}-2\sigma L_{1}S_{1}}{L_{1}L_{2}-S_{1}S_{2}} & 2S_{1}\sqrt{\frac{\sigma (L_{2}+S_{1})}{L_{1}L_{2}-S_{1}S_{2}}}\sqrt{\frac{\sigma (L_{1}+S_{2})}{L_{1}L_{2}-S_{1}S_{2}}}\\ 2S_{2}\sqrt{\frac{\sigma(L_{2}+S_{1})}{L_{1}L_{2}-S_{1}S_{2}}}\sqrt {\frac{\sigma(L_{1}+S_{2})}{L_{1}L_{2}-S_{1}S_{2}}}&\frac{-2\sigma L_{1}L_{2}-2\sigma L_{2}S_{2}}{L_{1}L_{2}-S_{1}S_{2}} \end{matrix} \right ) . $$

Then the characteristic equation of the linearized system can be calculated as follows:

$$ \lambda^{2}+\frac{2\sigma (L_{1}S_{1}+L_{2}S_{2}+2L_{1}L_{2})}{L_{1}L_{2}-S_{1}S_{2}}\lambda +\frac{4\sigma^{2}(L_{1}+S_{2})(L_{2}+S_{1})}{L_{1}L_{2}-S_{1}S_{2}}=0. $$

(4.29)

When \(L_{1}L_{2}-S_{1}S_{2}>0\), \(L_{1}+S_{2}>0\), \(L_{2}+S_{1}>0\) and \(L_{1}S_{1}+L_{2}S_{2}+2L_{1}L_{2}>0\), it is easy to know that all the zeros of the characteristic equation (4.29) have negative real parts. However, the characteristic equation (4.29) has at least one positive real zero when \(L_{1}L_{2}-S_{1}S_{2}<0\), \(L_{1}+S_{2}<0\) and \(L_{2}+S_{1}<0\). This completes the proof. □

Remark 2

From the above analysis, we see that the boundary equilibria \(E_{1}^{\pm}\) and \(E_{2}^{\pm}\) are unstable when any equilibrium in \(E_{3}^{\pm}\) exists and is stable. When one of the two boundary equilibria \(E_{1}\), \(E_{2}\) is stable, the interior equilibria \(E_{3}^{\pm}\) are all unstable. Thus, we shall distinguish two kinds of stationary patterns: (i): a single mode stationary pattern when any of the equilibria \(E_{1}^{\pm}\) or \(E_{2}^{\pm}\) is stable; (ii): a mixed-mode stationary pattern when any of the equilibria \(E_{3}^{\pm}\) is stable.

Remark 3

It should be pointed out that the stability condition of the interior equilibria \(E_{3}^{\pm}\) requires that \(L_{1}S_{1}+L_{2}S_{2}+2L_{1}L_{2}>0\) but not \(L_{1}S_{1}+L_{2}S_{2}+2L_{1}L_{2}<0\), emerging in [38]. Here we give a detailed analysis and the corrected proof.

Suppose that system (4.28) admits at least one stable equilibrium \((A_{1\infty}, A_{2\infty})\). Then we have the following conclusion.

Proposition 4.5

Assume that the hypothesis (i) of Proposition  4.1 and the no-resonance condition (4.27) hold. Then system (1.3) has an asymptotic solution,

$$ \left ( \begin{matrix}u(x,y)\\ v(x,y) \end{matrix} \right ) =\left ( \begin{matrix}u_{\ast}\\ v_{\ast} \end{matrix} \right ) +\varepsilon{\boldsymbol{\varrho}} \sum _{i=1}^{2}A_{i\infty}\cos(\phi_{i} x) \cos(\psi_{i} y)+O\bigl(\varepsilon^{2}\bigr), $$

(4.30)

where \((A_{1\infty}, A_{2\infty})\) is a stable equilibrium of system (4.28).

In Figure 8, we show the pattern formation, starting from initial conditions, which reveals random periodic perturbations about the equilibrium \((0.167, 4.8763)\). Here we consider the rectangular domain with \(L_{x}=8\pi\) and \(L_{y}=4\pi\). With these parameter values the only admitted unstable mode is chosen as \(\hat{k}_{c}^{2}=0.8125\in [0.8021,2.4506]\), which is the band of the unstable modes. Then there exist two mode pairs \((4,3)\) and \((6,2)\) which satisfy the condition (4.17) and the non-resonant condition (4.27). By a calculation, we have \(L_{1}=L_{2}=3.6765\), \(S_{1}=S_{2}=-51.6425\). Then we get \(L_{1}+S_{2}=L_{2}+S_{1}=-47.9662<0\), \(L_{1}L_{2}-S_{1}S_{2}=-2653.4<0\). Therefore, the equilibria \(E_{1}^{\pm }\) and \(E_{2}^{\pm}\) are all stable and \(E_{3}^{\pm}\) are all unstable. Therefore, the predicted asymptotic solution is the following single mode pattern:

$$ {\mathbf {w}}=\varepsilon{\boldsymbol{\varrho}} \bigl( A_{1\infty}\cos (0.5x) \cos(0.75y) \bigr) +O\bigl(\epsilon^{2}\bigr) $$

(4.31)

or

$$ {\mathbf {w}}=\varepsilon{\boldsymbol{\varrho}} \bigl( A_{2\infty}\cos (0.75x)\cos(0.5y) \bigr) +O\bigl(\epsilon^{2}\bigr), $$

(4.32)

where \(A_{1\infty}\) and \(A_{2\infty}\) are the coordinates of the equilibria \(E_{1}^{\pm}\) and \(E_{2}^{\pm}\), respectively. In Figure 8, we see a good qualitative agreement of the approximation (4.31) or (4.32) with the numerical solution of the full system (1.3).

According to the parameter values of Figure 9, the only admitted unstable mode is chosen as \(\hat{k}_{c}^{2}=1.25\), which belongs to the band of unstable modes allowed by the boundary conditions. Then the two mode pairs \((8,2)\) and \((4,4)\) satisfy the conditions (4.17) and (4.27). From (4.28), we obtain \(\sigma _{1}=\sigma_{2}=0.2324\), \(L_{1}=L_{2}=0.6442\), \(S_{1}=S_{2}=-0.1545\). Thus the only stable positive equilibrium is \(E_{3}^{+}\) according to Proposition 4.4. Therefore, the predicted asymptotic solution given by the positive equilibrium \(E_{3}^{+}\) is the following mixed-mode pattern:

$$ {\mathbf {w}}=\varepsilon{\boldsymbol{\varrho}} \bigl( A_{1\infty}\cos (x)\cos(0.5y)+A_{2\infty}\cos(0.5x)\cos(y) \bigr) +O \bigl(\epsilon^{2}\bigr), $$

(4.33)

where \(A_{1\infty}\) and \(A_{2\infty}\) are the coordinates of the positive equilibrium \(E_{3}^{+}\). Figure 9 provides the comparison between the numerical solution of system (1.3) and the weakly nonlinear approximation solution with \(L_{x}=8\pi\) and \(L_{y}=4\pi\) at the first order.

According to the parameter values of Figure 10, the only unstable model is chosen as \(\hat{k}_{c}^{2}=1.25\), which falls within the band of admitted unstable modes. Then there exist the two mode pairs \((8,4)\) and \((4,8)\) satisfying the conditions (4.17) and (4.27). The predicted asymptotic solution has the same form as (4.33). From Figures 9 and 10, we see a significant quantitative discrepancy between the numerical solution of the original system and the approximated solution. The fact that the weakly nonlinear analysis has poorer performances in predicting the mixed-mode pattern was also reported in [38].

4.3 Double and resonant eigenvalue

When \(m=2\), the two pairs of allowed spatial modes \((\phi_{1},\psi _{1})\) and \((\phi_{2}, \psi_{2})\) satisfy the following resonance condition [38]:

$$\begin{aligned} \begin{aligned} &\phi_{i}+\phi_{j}= \phi_{j} \quad\text{and}\quad \psi_{i}-\psi_{j}= \psi_{j} \\ &\text{or} \\ &\phi_{i}-\phi_{j}= \phi_{j} \quad\text{and} \quad \psi_{i}+\psi_{j}= \psi_{j}, \end{aligned} \end{aligned}$$

(4.34)

with \(i,j=1,2\) and \(i\neq j\). In what follows, without of loss generality, we shall assume that the first conditions in (4.34) hold with \(i=2\), \(j=1\). Thus it follows that \(\phi_{2}=0\), \(\psi _{2}=\hat{k}_{c}\), \(\psi_{1}=\frac{\hat{k}_{c}}{2}\), \(\phi_{1}=\sqrt {3}\psi_{1}\) and \(\psi_{2}=2\psi_{1}\).

When the resonance condition (4.34) holds, the solution (4.14) at the order ε reads

$$ {\mathbf {w}}_{1}=A_{1}\boldsymbol{\varrho}\cos(\sqrt{3} \psi_{1} x)\cos (\psi_{1} y)+A_{2}\boldsymbol{ \varrho}\cos(2\psi_{1} y). $$

(4.35)

By imposing the solvability condition at \(O(\varepsilon^{2})\) to equation (4.10), we obtain the Stuart-Landau equations as follows:

$$ \begin{aligned} &\frac{\partial A_{1}}{\partial T_{1}}=\tilde{\sigma}A_{1}- \tilde {L}A_{1}A_{2}, \\ &\frac{\partial A_{2}}{\partial T_{1}}=\tilde{\sigma}A_{2}- \frac {\tilde{L}}{4}A_{1}^{2}. \end{aligned} $$

(4.36)

Here the coefficients σ̃ and L̃ are given in Appendix A.3.1.

It is easy to see that the steady states of equation (4.36) are the trivial equilibrium \((0,0)\) and \(E^{\pm}=(\pm\frac{2\tilde {\sigma}}{\tilde{L}}, \frac{\tilde{\sigma}}{\tilde{L}})\). By computing the eigenvalues of the Jacobian matrix evaluated at \(E^{\pm }\), we obtain the stationary solutions \(E^{\pm}\) are always unstable, and it is not possible to predict the amplitude of the pattern at this order. Therefore the weakly nonlinear analysis as regards the amplitude has to be pushed to a higher order to obtain some qualitatively results. Pushing the weakly nonlinear analysis to \(O(\varepsilon ^{3})\), we obtain the following Stuart-Landau equations for the amplitudes \(A_{1}\) and \(A_{2}\) at the time \(T(T_{1},T_{2})\):

$$ \begin{aligned} &\frac{dA_{1}}{dT}=\bar{\sigma}_{1}A_{1}- \bar{L}_{1} A_{1}A_{2}+\bar{\gamma}_{1} A_{1}^{3}+\bar{\delta }_{1}A_{1}A_{2}^{2}, \\ &\frac{dA_{2}}{dT}=\bar{\sigma}_{2}A_{2}- \bar{L}_{2} A_{1}^{2}+\bar {\gamma}_{2} A_{2}^{3}+\bar{\delta}_{2}A_{1}^{2}A_{2}, \end{aligned} $$

(4.37)

where the coefficients in (4.37) are given in Appendix A.3.2. Then we can draw a conclusion from the weakly nonlinear analysis about the resonant case as follows.

Proposition 4.6

Assume that the hypothesis (i) of Proposition  4.1 and the resonance condition (4.34) hold. Then system (1.3) has an asymptotic solution

$$ \left ( \begin{matrix}u(x,y)\\ v(x,y) \end{matrix} \right ) =\left ( \begin{matrix}u_{\ast}\\ v_{\ast} \end{matrix} \right ) +\varepsilon{\boldsymbol{\varrho}} \sum _{i=1}^{2}A_{i\infty}\cos(\phi_{i} x) \cos(\psi_{i} y)+O\bigl(\varepsilon^{2}\bigr), $$

(4.38)

where \((A_{1\infty}, A_{2\infty})\) is a stable equilibrium of system (4.37).

Clearly, system (4.37) has a trivial equilibrium \(E_{0}=(0,0)\). The boundary equilibrium \(R^{\pm}=(0,\pm\sqrt{\frac {-\bar{\sigma}_{2}}{\bar{\gamma}_{2}}})\) exists if and only if \(\bar{\gamma}_{2}<0\). System (4.37) admits an interior equilibrium \(H=(A_{1}, A_{2})\) if H is a solution to the following two equations:

$$\begin{aligned} \textstyle\begin{cases} A_{2}^{3}(\bar{\gamma}_{1}\bar{\gamma}_{2}-\bar{\delta}_{1}\bar {\delta}_{2})+A_{2}^{2}(\bar{L}_{1}\bar{\delta}_{2}+\bar {L}_{2}\bar{\delta}_{1})+A_{2}(\bar{\gamma}_{1}\bar{\sigma }_{2}-\bar{L}_{1}\bar{L}_{2}-\bar{\sigma}_{1}\bar{\delta }_{2})+\bar{\sigma}_{1}\bar{L}_{2}=0,\\ A_{1}^{2}=\frac{-\bar{\sigma}_{1}+\bar{L}_{1}A_{2}-\bar{\delta }_{1}A_{2}^{2}}{\bar{\gamma}_{1}}. \end{cases}\displaystyle \end{aligned}$$

(4.39)

Without loss of generality, we assume the first equation of (4.39) has three real roots, denoted by \(A_{2i}\), \(i=1,2,3\). Substituting it into the second equation of (4.39) yields the existence of interior equilibria as follows.

Proposition 4.7

System (4.37) has six interior equilibria \(H_{i}^{\pm}= ( \pm\sqrt{\frac{-\bar{\sigma}_{1}+\bar{L}_{1}A_{2i}-\bar{\delta }_{1}A_{2i}^{2}}{\bar{\gamma}_{1}}}, A_{2i} ) \) if and only if \(\bar{\gamma}_{1}(-\bar{\sigma}_{1}+\bar{L}_{1}A_{2i}-\bar{\delta }_{1}A_{2i}^{2})>0\) for \(i=1,2,3\).

To investigate the stability of the interior equilibrium \((A_{1i}, A_{2i})\), linearizing system (4.37) yields the Jacobian matrix about it as follows:

$$ \bar{J}=\left ( \begin{matrix}{} \bar{\sigma}_{1}-\bar{L}_{1}A_{2i}+3\bar{\gamma }_{1} A_{1i}^{2}+\bar{\delta}_{1}A_{2i}^{2}& -\bar {L}_{1}A_{1i}+2\bar{\delta}_{1}A_{1i}A_{2i}\\ -2\bar{L}_{2}A_{1i}+2\bar{\delta}_{2}A_{1i}A_{2i}&\bar{\sigma }_{2}+3\bar{\gamma}_{2}A_{2i}^{2}+\bar{\delta}_{2}A_{1i}^{2} \end{matrix} \right ) . $$

Denote

$$\begin{aligned} &\operatorname {tr}(\bar{J}_{i})=\bar{\sigma}_{1}+\bar{\sigma }_{2}+(3\bar{\gamma}_{1}+\bar{\delta}_{2})A_{1i}^{2}+(3 \bar{\gamma }_{2}+\bar{\delta}_{1})A_{2i}^{2}- \bar{L}_{1}A_{2i}, \\ &\begin{aligned} \operatorname {det}(\bar{J}_{i})&=3\bar{\gamma}_{1}\bar{\delta }_{2}A_{1i}^{4}+3\bar{\delta}_{1}\bar{ \gamma}_{2}A_{2i}^{4}+ (9\bar{\gamma}_{1} \bar{\gamma}_{2}-3\bar{\delta}_{1}\bar{\delta }_{2})A_{1i}^{2}A_{2i}^{2}-3 \bar{L}_{1}\bar{\gamma }_{2}A_{2i}^{3}+( \bar{L}_{1}\bar{\delta}_{2}+4\bar{L}_{2}\bar { \delta}_{1})A_{1i}^{2}A_{2i} \\ &\quad{}+(\bar{\sigma}_{1}\bar{\delta}_{2}-2\bar {L}_{1}\bar{L}_{2}+3\bar{\sigma}_{2}\bar{\gamma }_{1})A_{1i}^{2}+(3\bar{\sigma}_{1}\bar{ \gamma}_{2}+\bar{\sigma }_{2}\bar{\delta}_{1})A_{2i}^{2} -\bar{\sigma}_{2}\bar{L}_{1}A_{2i}+\bar{ \sigma}_{1}\bar{\sigma}_{2}, \end{aligned} \end{aligned}$$

where \(A_{1i}^{2}=\frac{-\bar{\sigma}_{1}+\bar{L}_{1}A_{2i}-\bar {\delta}_{1}A_{2i}^{2}}{\bar{\gamma}_{1}}\).

Then we have the following stability results about the equilibria of system (4.37).

Proposition 4.8

1. (i)

   The trivial equilibrium \(E_{0}\) is an unstable node.

    
2. (ii)

   The boundary equilibrium \(R^{+}\) is a stable node if \(\bar {\sigma}_{1}<\bar{L}_{1}\sqrt{\frac{-\bar{\sigma}_{2}}{\bar {\gamma}_{2}}}+\frac{\bar{\delta}_{1}\bar{\sigma}_{2}}{\bar {\gamma}_{2}}\), while unstable if \(\bar{\sigma}_{1}>\bar {L}_{1}\sqrt{\frac{-\bar{\sigma}_{2}}{\bar{\gamma}_{2}}}+\frac {\bar{\delta}_{1}\bar{\sigma}_{2}}{\bar{\gamma}_{2}}\). The boundary equilibrium \(R^{-}\) is a stable node if \(\bar{\sigma }_{1}<-\bar{L}_{1}\sqrt{\frac{-\bar{\sigma}_{2}}{\bar{\gamma }_{2}}}+\frac{\bar{\delta}_{1}\bar{\sigma}_{2}}{\bar{\gamma }_{2}}\), while unstable if \(\bar{\sigma}_{1}>-\bar{L}_{1}\sqrt {\frac{-\bar{\sigma}_{2}}{\bar{\gamma}_{2}}}+\frac{\bar{\delta }_{1}\bar{\sigma}_{2}}{\bar{\gamma}_{2}}\).

    
3. (iii)

   The equilibria \(H_{i}^{\pm}\) are locally asymptotically stable if \(\operatorname {tr}(\bar{J}_{i})<0\) and \(\operatorname {det}(\bar{J}_{i})>0\), while unstable if \(\operatorname {det}(\bar{J}_{i})<0\).

    

Proof

The proof is trivial, we omit it. □

According to the parameter values of Figure 11, we get the critical value \(b^{c}=1.0769\), letting \(\varepsilon=0.03\) be a very small positive constant. The only admitted discrete unstable mode is chosen as \(\hat{k}_{c}^{2}=2.25\in[0.8428, 2.9863]\), which is the band of unstable modes allowed by the boundary conditions. Then there exist two mode pairs \((18,3)\) and \((0,6)\) satisfying the conditions (4.15) and (4.34). By (A.8), we get \(\bar{\sigma }_{1}=\bar{\sigma}_{2}=0.8156\), \(\bar{L}_{1}=0.8037\), \(\bar {L}_{2}=0.2009\), \(\bar{\gamma}_{1}=0.0808\), \(\bar{\gamma}_{2}=-0.2169\), \(\bar{\delta}_{1}=0.1631\), \(\bar{\delta}_{2}=0.0816\). Thus according to Proposition 4.8, the only stable equilibrium of system (4.37) is \(R^{+}=(0, 1.9391)\). Therefore the predicted asymptotic solution of system (1.3) via the weakly nonlinear analysis is the following single mode pattern:

$$ {\mathbf {w}}=1.9391\varepsilon{\boldsymbol{\varrho}}\cos (1.5y)+O\bigl( \varepsilon^{2}\bigr). $$

(4.40)

By performing 20 thousand simulations, which start from a random periodic perturbation of the equilibrium of \((0.3236,5.2918)\), we observe that the solution eventually evolves to the hexagonal pattern on the left of Figure 11. However, by equation (4.40), we obtain the roll pattern on the right of Figure 11.

According to the parameter values of Figure 12, the only admitted allowed unstable mode is chosen as \(\hat{k}_{c}^{2}=1\), which falls within the band of unstable modes allowed by the boundary conditions. Then the two pair modes \((12,2)\) and \((0,4)\) satisfy the conditions (4.15) and the resonance conditions (4.34). By (A.8), we get \(\bar{\sigma}_{1}=\bar{\sigma}_{2}=0.2642\), \(\bar {L}_{1}=0.083\), \(\bar{L}_{2}=0.0207\), \(\bar{\gamma}_{1}=-0.0517\), \(\bar {\gamma}_{2}=0.4712\), \(\bar{\delta}_{1}=-0.8454\), \(\bar{\delta }_{2}=-0.4712\). Since \(\bar{\gamma}_{2}>0\), the boundary equilibria \(R^{\pm}\) do not exist. By a calculation, we have two interior equilibria \(H_{1}^{\pm}=(\pm2.2682, -0.0559)\) and \(\operatorname {tr}(\bar {J}_{1})=-2.4383<0\) and \(\operatorname {det}(\bar{J}_{1})=1.0144>0\). Thus the two interior equilibria are stable according to Proposition 4.8. Then the predicted asymptotic solution by the two stable states \(H_{1}^{\pm}\) can be expressed as

$$ {\mathbf {w}}=\varepsilon{\boldsymbol{\varrho}}\bigl(2.2682\cos(\sqrt {3}x/{2}) \cos(0.5y)-0.0559\cos(y)\bigr)+O\bigl(\epsilon^{2}\bigr) $$

or

$$ {\mathbf {w}}=\varepsilon{\boldsymbol{\varrho}}\bigl(-2.2682\cos(\sqrt {3}x/{2}) \cos(0.5y)-0.0559\cos(y)\bigr)+O\bigl(\epsilon^{2}\bigr). $$

From Figure 12, we also observe a qualitative agreement of the approximation with the numerical solution of the original system.

In addition, in our 2D numerical experiments, the time step is always chosen as 0.001, which was always small enough to ensure the numerical stability of the method, and we have always used the same spatial step for the two spatial directions and the number of Chebyshey points was always chosen as \(N=64\) to ensure the spectral method accuracy.

In this paper, we have investigated the Turing mechanism and the phenomena of pattern formation induced by nonlinear cross-diffusion in a uni-directional C-R system. We discuss in detail the asymptotic behavior of patterns by virtue of amplitude equation in three cases: simple eigenvalue, double and non-resonant eigenvalue, double and resonant eigenvalue. In every case, we give a detailed analysis and make a comparison between a numerical solution of system (1.3) and its weakly nonlinear approximation solution by numerical simulations. In our numerical simulations, we show that the system exhibits patterns like rhombic pattern, mixed-mode pattern, super-squares pattern, roll, and hexagonal pattern. We also tested the influence of the resource supply rate \(\beta_{1}\) on the bifurcation direction (see Figure 5), and found that the bifurcation direction varied when we changed the parameter \(\beta_{1}\). We have also found that there exists a hysteresis cycle as the bifurcation parameter b starts from the value larger that its critical value \(b^{c}\), then decreases to become less than another threshold value, \(b^{s}\), and next increases to become larger than \(b^{c}\) (see Figure 7). Clearly, comparing our results with those in [8], the effect caused by the nonlinear cross-diffusion was presented.