# American Institute of Mathematical Sciences

February  2022, 27(2): 1163-1178. doi: 10.3934/dcdsb.2021085

## Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang

Received  December 2020 Revised  January 2021 Published  February 2022 Early access  March 2021

Fund Project: The work was supported in part by the National Natural Science Foundation of China (No. 11871176, 11671110) and the Fundamental Research Funds for the Central Universities

In this article, Turing instability and the formations of spatial patterns for a general two-component reaction-diffusion system defined on 2D bounded domain, are investigated. By analyzing characteristic equation at positive constant steady states and further selecting diffusion rate $d$ and diffusion ratio $\varepsilon$ as bifurcation parameters, sufficient and necessary conditions for the occurrence of Turing instability are established, which is called the first Turing bifurcation curve. Furthermore, parameter regions in which single-mode Turing patterns arise and multiple-mode (or superposition) Turing patterns coexist when bifurcations parameters are chosen, are described. Especially, the boundary of parameter region for the emergence of single-mode Turing patterns, consists of the first and the second Turing bifurcation curves which are given in explicit formulas. Finally, by taking diffusive Schnakenberg system as an example, parameter regions for the emergence of various kinds of spatially inhomogeneous patterns with different spatial frequencies and superposition Turing patterns, are estimated theoretically and shown numerically.

Citation: Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085
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The graphs of functions $\varepsilon = \varepsilon_1$ and $\varepsilon = \varepsilon_*(\mathbf{k}(i),d),d\ge d_i,\; i = 1,2,3\cdots$ in $d \rm{-}\varepsilon$ plane
The first Turing bifurcation curve
$\varepsilon = \varepsilon_{**}(d)$ is the second Turing bifurcation curve. The grey area enclosed by the first and the second Turing bifurcation curves, represents $\mathfrak{D}^1$. The region below the second Turing bifurcation curve $\varepsilon = \varepsilon_{**}(d)$ is $\mathfrak{D}^2$, and the blue area denoted by $D_{\mathbf{k}(3), \mathbf{k}(7)}$ is one component of $\mathfrak{D}^2$
">Figure 4.  Turing patterns for system (18) with different values of $(d,\varepsilon)$ given in Table 2
(a), (b): For $\mathbf{k}(i) = (4,0)$ and $\mathbf{k}(j) = (0,4)$, there exists superposition pattern of (18) when $(d,\varepsilon) = (0.09,0.025)$; (c): The graph of function $z(x,y) = 0.9+1.8\cos(4\cdot\pi x)\cos(0\cdot\frac{\pi^2}{3} y)+\cos(0\cdot\pi x)\cos(4\cdot\frac{\pi^2}{3} y)$
(a), (b): Superposition pattern of (18) for $\mathbf{k}(i) = (3,0)$ and $\mathbf{k}(j) = (0,3)$; (c): The graph of function $z(x,y) = 1+1.8\cos(3\cdot\pi x)\cos(0\cdot\frac{\pi^2}{3} (\frac{\pi}{3}-y))+\cos(0\cdot\pi x)\cos(3\cdot\frac{\pi^2}{3} (\frac{\pi}{3}-y))$
(a), (b): Superposition of three kinds of spatial patterns with spatial wave numbers $(2,0)$, $(0,2)$ and $(2,2)$ for system (18); (c): The graph of function $z(x,y) = 0.9+\cos(2\cdot\pi x)\cos(0\cdot\frac{\pi^2}{3} y)+5\cos(0\cdot\pi x)\cos(2\cdot\frac{\pi^2}{3} y)+2\cos(2\cdot\pi x)\cos(2\cdot\frac{\pi^2}{3} y)$
These values of $\mathbf{k}(i),\; \mu_i/\pi^2$ and $d_{i,i+1}$ when parameters are chosen as (21), $i = 1,2,\ldots,28$
 $i$ $\mathbf{k}(i)$ $\mu_i/\pi^2$ $d_{i,i+1}$ $i$ $\mathbf{k}(i)$ $\mu_i/\pi^2$ $d_{i,i+1}$ 1 (1, 0) 1 0.2834 15 (4, 0) 16 0.0179 2 (0, 1) 1.0966 0.2034 16 (4, 1) 17.0966 0.0171 3 (1, 1) 2.0966 0.1064 17 (0, 4) 17.5460 0.0164 4 (2, 0) 4 0.0709 18 (1, 4) 18.5460 0.0160 5 (0, 2) 4.3865 0.0629 19 (3, 3) 18.8696 0.0151 6 (2, 1) 5.0966 0.0566 20 (4, 2) 20.3865 0.0142 7 (1, 2) 5.3865 0.0449 21 (2, 4) 21.5460 0.0128 8 (2, 2) 8.3865 0.0342 22 (5, 0) 25 0.0117 9 (3, 0) 9 0.0315 23 (4, 3) 25.8696 0.0114 10 (0, 3) 9.8696 0.0297 24 (5, 1) 26.0966 0.0113 11 (3, 1) 10.0966 0.0283 25 (3, 4) 26.5460 0.0110 12 (1, 3) 10.8696 0.0247 26 (0, 5) 27.4156 0.0106 13 (3, 2) 13.3865 0.0218 27 (1, 5) 28.4156 0.0103 14 (2, 3) 13.8696 0.0199 28 (5, 2) 29.3865 0.0098
 $i$ $\mathbf{k}(i)$ $\mu_i/\pi^2$ $d_{i,i+1}$ $i$ $\mathbf{k}(i)$ $\mu_i/\pi^2$ $d_{i,i+1}$ 1 (1, 0) 1 0.2834 15 (4, 0) 16 0.0179 2 (0, 1) 1.0966 0.2034 16 (4, 1) 17.0966 0.0171 3 (1, 1) 2.0966 0.1064 17 (0, 4) 17.5460 0.0164 4 (2, 0) 4 0.0709 18 (1, 4) 18.5460 0.0160 5 (0, 2) 4.3865 0.0629 19 (3, 3) 18.8696 0.0151 6 (2, 1) 5.0966 0.0566 20 (4, 2) 20.3865 0.0142 7 (1, 2) 5.3865 0.0449 21 (2, 4) 21.5460 0.0128 8 (2, 2) 8.3865 0.0342 22 (5, 0) 25 0.0117 9 (3, 0) 9 0.0315 23 (4, 3) 25.8696 0.0114 10 (0, 3) 9.8696 0.0297 24 (5, 1) 26.0966 0.0113 11 (3, 1) 10.0966 0.0283 25 (3, 4) 26.5460 0.0110 12 (1, 3) 10.8696 0.0247 26 (0, 5) 27.4156 0.0106 13 (3, 2) 13.3865 0.0218 27 (1, 5) 28.4156 0.0103 14 (2, 3) 13.8696 0.0199 28 (5, 2) 29.3865 0.0098
Parameter values of $(d,\varepsilon)$ in $\mathfrak{D}^1$ satisfying that (18) has $\mathbf{k}(i)-$mode Turing patterns
 $i$ $\mathbf{k}(i)$ $\varepsilon$ $(d_i^-(\varepsilon), d_i^+(\varepsilon))$ $d$ $(d,\varepsilon)\in$ Figure 3 (1, 1) 0.09 (0.08870, 0.2925) 0.1300 $D_{\mathbf{k}(3), \mathbf{k}(5)}$ 4(a) 6 (2, 1) 0.09 (0.0365, 0.12034) 0.0660 $D_{\mathbf{k}(4), \mathbf{k}(9)}$ 4(b) 8 (2, 2) 0.09 (0.02218, 0.0731) 0.0434 $D_{\mathbf{k}(5), \mathbf{k}(14)}$ 4(c) 9 (3, 0) 0.09 (0.0207, 0.0681) 0.0340 $D_{\mathbf{k}(8), \mathbf{k}(17)}$ 4(d) 12 (1, 3) 0.08 (0.0171, 0.0564) 0.0270 $D_{\mathbf{k}(8), \mathbf{k}(25)}$ 4(e) 13 (3, 2) 0.09 (0.0139, 0.0458) 0.0260 $D_{\mathbf{k}(8), \mathbf{k}(21)}$ 4(f)
 $i$ $\mathbf{k}(i)$ $\varepsilon$ $(d_i^-(\varepsilon), d_i^+(\varepsilon))$ $d$ $(d,\varepsilon)\in$ Figure 3 (1, 1) 0.09 (0.08870, 0.2925) 0.1300 $D_{\mathbf{k}(3), \mathbf{k}(5)}$ 4(a) 6 (2, 1) 0.09 (0.0365, 0.12034) 0.0660 $D_{\mathbf{k}(4), \mathbf{k}(9)}$ 4(b) 8 (2, 2) 0.09 (0.02218, 0.0731) 0.0434 $D_{\mathbf{k}(5), \mathbf{k}(14)}$ 4(c) 9 (3, 0) 0.09 (0.0207, 0.0681) 0.0340 $D_{\mathbf{k}(8), \mathbf{k}(17)}$ 4(d) 12 (1, 3) 0.08 (0.0171, 0.0564) 0.0270 $D_{\mathbf{k}(8), \mathbf{k}(25)}$ 4(e) 13 (3, 2) 0.09 (0.0139, 0.0458) 0.0260 $D_{\mathbf{k}(8), \mathbf{k}(21)}$ 4(f)
Parameter values of $(d,\varepsilon)$ in $\mathfrak{D}^2$ satisfying that (18) has superposition patterns
 $(d,\varepsilon)$ $\mathbf{k}(i)$ $\mathbf{k}(j)$ $\mathbf{k}(l)$ Figure (0.025, 0.09) $\in D_{\mathbf{k}(8), \mathbf{k}(21)}$ (4, 0) (0, 4) --- 5 (0.044, 0.09) $\in D_{\mathbf{k}(5), \mathbf{k}(14)}$ (3, 0) (0, 3) --- 6 (0.094, 0.09) $\in D_{\mathbf{k}(3), \mathbf{k}(7)}$ (2, 0) (0, 2) (2, 2) 7
 $(d,\varepsilon)$ $\mathbf{k}(i)$ $\mathbf{k}(j)$ $\mathbf{k}(l)$ Figure (0.025, 0.09) $\in D_{\mathbf{k}(8), \mathbf{k}(21)}$ (4, 0) (0, 4) --- 5 (0.044, 0.09) $\in D_{\mathbf{k}(5), \mathbf{k}(14)}$ (3, 0) (0, 3) --- 6 (0.094, 0.09) $\in D_{\mathbf{k}(3), \mathbf{k}(7)}$ (2, 0) (0, 2) (2, 2) 7
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