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

Weihua Jiang; Xun Cao; Chuncheng Wang

doi:10.3934/dcdsb.2021085

Article Contents

2022, Volume 27, Issue 2: 1163-1178. Doi: 10.3934/dcdsb.2021085

This issue Previous Article Mixed Hegselmann-Krause dynamics Next Article Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations

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
^* Corresponding author: Weihua Jiang

Received: December 2020

Revised: January 2021

Early access: March 2021

Published: February 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.

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Figure 1. 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

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Figure 2. The first Turing bifurcation curve

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Figure 3. $ \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 $

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Figure 4. Turing patterns for system (18) with different values of $ (d,\varepsilon) $ given in Table 2

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Figure 5. (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) $

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Figure 6. (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)) $

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Figure 7. (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) $

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Table 1. 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

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Table 2. 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)

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Table 3. 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

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