Turing Pattern Formation in a Semiarid Vegetation Model with Fractional-in-Space Diffusion

Abstract

A fractional power of the Laplacian is introduced to a reaction–diffusion system to describe water’s anomalous diffusion in a semiarid vegetation model. Our linear stability analysis shows that the wavenumber of Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations, and the analysis of these amplitude equations shows that the spatial patterns are asymptotic stable due to the supercritical Turing bifurcation. Numerical simulations exhibit a bistable regime composed of hexagons and stripes, which confirm our analytical results. Moreover, the characteristic length of the emergent spatial pattern is consistent with the scale of vegetation patterns observed in field studies.

Appendix: Derivation of the Amplitude Equation

In this appendix, we sketch a derivation of the amplitude Eq. (18). Note that in Zhang and Tian (2014), the authors study a fractional system with the same fractional order. However, in the system (4), the two fractional orders are different.

Since $\mathbf {L_c}$ is the linear operator of the system at the Turing instability threshold, $(u_1,v_1)^T$ is the eigenvector corresponding to the eigenvalue 0. Therefore, at $O(\varepsilon )$ the solution is given in the form

$$\begin{aligned} (u_1,v_1)^T=\mathbf {\rho }A(T) \cos (k_cx),~\text { with } \mathbf {\rho }\equiv (\rho _1, \rho _2)^T\in Ker\left( \mathbf {L}-\mathbf {D}\mathrm{diag}\left( {k_c}^{\gamma }, {k_c}^2\right) \right) , \end{aligned}$$

where A(T) is the amplitude of the pattern and is still arbitrary at this level. Its form is determined by the perturbational term of the higher order. The vector $\mathbf {\rho }$ is defined up to a constant, and we shall make the normalization in the following way:

$$\begin{aligned} \mathbf {\rho }=(1, \rho _2)^T, \text { with } \rho _2=-(1+v_s^2+dx)/(2u_sv_s). \end{aligned}$$

(22)

Next, we turn to $O(\varepsilon ^2)$. The equation is written in the form

$$\begin{aligned} \mathbf {L_c}\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) =A^2(1+\cos (2k_c x))(\rho _2v_s+\rho _2^2u_s/2)\left( \begin{array}{c} 1\\ -1\\ \end{array} \right) . \end{aligned}$$

(23)

Since the right-hand side does not have the resonance, the Fredholm alternative is automatically satisfied. The solution of system (23) is then explicitly computed in terms of the parameters of the full system

$$\begin{aligned} \left( \begin{array}{c} u_2 \\ v_2 \end{array} \right) =A^2\mathbf {(w_{20}}+\mathbf {w_{22}}\cos (2k_cx)), \end{aligned}$$

where $\mathbf {w_{20}}$ and $\mathbf {w_{22}}$ are, respectively, the solutions of the following linear systems

$$\begin{aligned} (\mathbf {L}-\mathbf {D}\mathrm{diag}\left( {0k_c}^{\gamma }, {0k_c}^2\right) )\mathbf {w_{20}}=(\rho _2v_s+\rho _2^2u_s/2)(1, -1)^T, \end{aligned}$$

(24)

$$\begin{aligned} (\mathbf {L}-\mathbf {D}\mathrm{diag}({(2k_c)}^{\gamma }, {(2k_c)}^2))\mathbf {w_{22}}=(\rho _2v_s+\rho _2^2u_s/2)(1, -1)^T. \end{aligned}$$

(25)

Now $O(\varepsilon ^3)$. The equation is written in the form

$$\begin{aligned} \mathbf {L_c}\left( \begin{array}{c} u_3\\ v_3\\ \end{array} \right)= & {} \left( \begin{array}{c} 1\\ -1\\ \end{array} \right) \left[ \cos (k_c x)\left( \frac{\mathrm{d}A}{\mathrm{d} T}+A^3\left( \frac{3}{4}\rho _2^2+2v_s\rho _2w_{20}^1 +(2v_s+2u_s\rho _2)w_{20}^2\right. \right. \right. \nonumber \\&+\,\left. \left. \left. v_s\rho _2w_{22}^1+(v_s+u_s\rho _2)w_{22}^2\right) \right) +\cos (3k_c x)\left( \frac{1}{4}\rho _2^2+v_s\rho _2w_{22}^1\right. \right. \nonumber \\&+\,\left. \left. (v_2+u_s\rho _2)w_{22}^2\right) \right] -\left( \begin{array}{c} 0\\ b_c\rho _2A\cos (k_cx)\\ \end{array} \right) . \end{aligned}$$

(26)

According to the Fredholm solubility condition, the vector function of the right-hand side must be orthogonal with the zero eigenvalues of the operator $\mathbf {L_c^+}$ to ensure the existence of the nontrivial solution to this equation, where $\mathbf {L_c^+}$ is the adjoint operator of $\mathbf {L_c}$. The nontrivial kernel of the operator $\mathbf {L_c^+}$ is

$$\begin{aligned} \left( \begin{array}{c} 1 \\ (v_s^2+1+dx)/u_s^2\\ \end{array} \right) \cos (k_c x). \end{aligned}$$

(27)

In order to ensure the Fredholm solubility condition, we only consider the resonance of the system (26). Thus, we have the amplitude equation

$$\begin{aligned} \frac{dA}{dt}=\sigma A-LA^3, \end{aligned}$$

(28)

where

$$\begin{aligned} \begin{array}{c} \displaystyle \sigma =\frac{\rho _2(v_s^2+a+dx)}{u_s^2-v_s^2-1-dx}(b-b_c), \nonumber \\ \displaystyle L=\frac{3}{4}\rho _2^2+2v_s\rho _2w_{20}^1 +(2v_s+2u_s\rho _2)w_{20}^2 +v_s\rho _2w_{22}^1+(v_s+u_s\rho _2)w_{22}^2.\nonumber \\ \end{array}\\ \end{aligned}$$

(29)