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 and Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085
References:
[1]

Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[2]

Yu. I. BalkareiA. V. Grigor'yantsYu. A. Rzhanov and M. I. Elinson, Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors, Opt. Commun., 66 (1988), 161-166.  doi: 10.1016/0030-4018(88)90054-5.

[3]

X. Cao and W. Jiang, On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns, Submitted.

[4]

V. Dufiet and J. Boissonade, Conventional and unconventional Turing patterns, J. Chem. Phys., 96 (1992), 664-673.  doi: 10.1063/1.462450.

[5]

L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005. doi: 10.1137/1.9780898719147.

[6]

E. A. Gaffney and N. A. L. x Monk, Gene expression time delays and Turing pattern formation systems, Bull. Math. Biol., 68 (2006), 99-130.  doi: 10.1007/s11538-006-9066-z.

[7]

G. H. Gunaratne, Complex spatial patterns on planar continua, Phys. Rev. Lett., 71 (1993), 1367-1370.  doi: 10.1103/PhysRevLett.71.1367.

[8]

Z.-G. Guo, L.-P. Song, G.-Q. Sun, C. Li and Z. Jin, Pattern dynamics of an SIS epidemic model with nonlocal delay, Internat. J. Bifur. Chaos, 29 (2019), 1950027, 12 pp. doi: 10.1142/S0218127419500275.

[9]

K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 95-105.  doi: 10.3934/dcdsb.2007.8.95.

[10]

W. JiangH. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay, J. Dynam. Differential Equations, 31 (2019), 2223-2247.  doi: 10.1007/s10884-018-9702-y.

[11]

S. L. Judd and M. Silber, Simple and superlattice Turing patterns in reaction-diffusion systems: bifurcation, bistability, and parameter collapse, Phys. D, 136 (2000), 45-65.  doi: 10.1016/S0167-2789(99)00154-2.

[12]

I. Lengyel and I. R. Epsten, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[13]

S. LiJ. Wu and Y. Doug, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[14]

P. LiuJ. ShiY. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.  doi: 10.1007/s10910-013-0196-x.

[15]

P. K. MainiK. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.  doi: 10.1039/a702602a.

[16]

J. D. Murray, Parameter space for Turing instability in reaction diffusion mechanisms: A comparison of models, J. Theoret. Biol., 98 (1982), 143-163.  doi: 10.1016/0022-5193(82)90063-7.

[17]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[18]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003

[19]

W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[20]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[21]

T. Nozakura and S. Ikeuchi, Formation of dissipative structures in galaxies, Astrophys. J., 279 (1984), 40-52.  doi: 10.1086/161863.

[22]

M.R. Richard and S. Mischler, Turing instabilities at Hopf bifurcation, J. Nonlinear Sci., 19 (2009), 467-496.  doi: 10.1007/s00332-009-9041-6.

[23]

R. A. SatnoianuM. Menzinger and P. K. Maini, Turing instabilities in general system, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.

[25]

L. A. Segel and J. L. Jackson, Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.

[26]

L. Seirin LeeE.A. Gaffney and R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Biol., 73 (2011), 2527-2551.  doi: 10.1007/s11538-011-9634-8.

[27]

G.-Q. SunJ. ZhangL.-P. SongZ. Jin and B.-L. Li, Pattern formation of a spatial predator-prey system, Appl. Math. Comput., 218 (2012), 11151-11162.  doi: 10.1016/j.amc.2012.04.071.

[28]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[29]

W. WangX. GaoY. CaiH. Shi and S. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.  doi: 10.1016/j.jfranklin.2018.07.014.

[30]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.

[31]

M. WeiJ. Wu and G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction, Nonlinear Anal. Real World Appl., 22 (2015), 155-175.  doi: 10.1016/j.nonrwa.2014.08.003.

[32]

P. W. Williams, Geomorphology and hydrology of karst terrains, Nature, 336 (1988), 322-322.  doi: 10.1038/336322b0.

[33] L. Wolpert and T. Jessell, Principles of Development, Oxford University Press, 1998. 
[34]

T. E. Woolley, R. E. Baker and P. K. Maini, Turing's theory of morphogenesis: Where we started, where we are and where we want to go, in The Incomputable, in Theory Appl. Comput., Springer, Cham, 2017,219–235.

[35]

C. Xu and J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.

[36]

F. YiE. A. Gaffney and S. Seirin-Lee, The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 647-668.  doi: 10.3934/dcdsb.2017031.

[37]

F. YiJ. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.

[38]

J.-F. ZhangW.-T. Li and Y.-T. Wang, Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Anal., 74 (2011), 847-858.  doi: 10.1016/j.na.2010.09.035.

show all references

References:
[1]

Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510.  doi: 10.3934/dcdsb.2018183.

[2]

Yu. I. BalkareiA. V. Grigor'yantsYu. A. Rzhanov and M. I. Elinson, Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors, Opt. Commun., 66 (1988), 161-166.  doi: 10.1016/0030-4018(88)90054-5.

[3]

X. Cao and W. Jiang, On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns, Submitted.

[4]

V. Dufiet and J. Boissonade, Conventional and unconventional Turing patterns, J. Chem. Phys., 96 (1992), 664-673.  doi: 10.1063/1.462450.

[5]

L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005. doi: 10.1137/1.9780898719147.

[6]

E. A. Gaffney and N. A. L. x Monk, Gene expression time delays and Turing pattern formation systems, Bull. Math. Biol., 68 (2006), 99-130.  doi: 10.1007/s11538-006-9066-z.

[7]

G. H. Gunaratne, Complex spatial patterns on planar continua, Phys. Rev. Lett., 71 (1993), 1367-1370.  doi: 10.1103/PhysRevLett.71.1367.

[8]

Z.-G. Guo, L.-P. Song, G.-Q. Sun, C. Li and Z. Jin, Pattern dynamics of an SIS epidemic model with nonlocal delay, Internat. J. Bifur. Chaos, 29 (2019), 1950027, 12 pp. doi: 10.1142/S0218127419500275.

[9]

K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 95-105.  doi: 10.3934/dcdsb.2007.8.95.

[10]

W. JiangH. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay, J. Dynam. Differential Equations, 31 (2019), 2223-2247.  doi: 10.1007/s10884-018-9702-y.

[11]

S. L. Judd and M. Silber, Simple and superlattice Turing patterns in reaction-diffusion systems: bifurcation, bistability, and parameter collapse, Phys. D, 136 (2000), 45-65.  doi: 10.1016/S0167-2789(99)00154-2.

[12]

I. Lengyel and I. R. Epsten, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652.  doi: 10.1126/science.251.4994.650.

[13]

S. LiJ. Wu and Y. Doug, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[14]

P. LiuJ. ShiY. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.  doi: 10.1007/s10910-013-0196-x.

[15]

P. K. MainiK. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610.  doi: 10.1039/a702602a.

[16]

J. D. Murray, Parameter space for Turing instability in reaction diffusion mechanisms: A comparison of models, J. Theoret. Biol., 98 (1982), 143-163.  doi: 10.1016/0022-5193(82)90063-7.

[17]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[18]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003

[19]

W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9.

[20]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[21]

T. Nozakura and S. Ikeuchi, Formation of dissipative structures in galaxies, Astrophys. J., 279 (1984), 40-52.  doi: 10.1086/161863.

[22]

M.R. Richard and S. Mischler, Turing instabilities at Hopf bifurcation, J. Nonlinear Sci., 19 (2009), 467-496.  doi: 10.1007/s00332-009-9041-6.

[23]

R. A. SatnoianuM. Menzinger and P. K. Maini, Turing instabilities in general system, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.

[25]

L. A. Segel and J. L. Jackson, Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.  doi: 10.1016/0022-5193(72)90090-2.

[26]

L. Seirin LeeE.A. Gaffney and R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Biol., 73 (2011), 2527-2551.  doi: 10.1007/s11538-011-9634-8.

[27]

G.-Q. SunJ. ZhangL.-P. SongZ. Jin and B.-L. Li, Pattern formation of a spatial predator-prey system, Appl. Math. Comput., 218 (2012), 11151-11162.  doi: 10.1016/j.amc.2012.04.071.

[28]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[29]

W. WangX. GaoY. CaiH. Shi and S. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.  doi: 10.1016/j.jfranklin.2018.07.014.

[30]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.

[31]

M. WeiJ. Wu and G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction, Nonlinear Anal. Real World Appl., 22 (2015), 155-175.  doi: 10.1016/j.nonrwa.2014.08.003.

[32]

P. W. Williams, Geomorphology and hydrology of karst terrains, Nature, 336 (1988), 322-322.  doi: 10.1038/336322b0.

[33] L. Wolpert and T. Jessell, Principles of Development, Oxford University Press, 1998. 
[34]

T. E. Woolley, R. E. Baker and P. K. Maini, Turing's theory of morphogenesis: Where we started, where we are and where we want to go, in The Incomputable, in Theory Appl. Comput., Springer, Cham, 2017,219–235.

[35]

C. Xu and J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.  doi: 10.1016/j.nonrwa.2012.01.001.

[36]

F. YiE. A. Gaffney and S. Seirin-Lee, The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 647-668.  doi: 10.3934/dcdsb.2017031.

[37]

F. YiJ. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 9 (2008), 1038-1051.  doi: 10.1016/j.nonrwa.2007.02.005.

[38]

J.-F. ZhangW.-T. Li and Y.-T. Wang, Turing patterns of a strongly coupled predator-prey system with diffusion effects, Nonlinear Anal., 74 (2011), 847-858.  doi: 10.1016/j.na.2010.09.035.

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