March 2017, 22(2): 647-668. doi: 10.3934/dcdsb.2017031

The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

2. 

Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK

3. 

Department of Mathematical and Life Sciences, Hiroshima University, Kagamiyama 1-3-1, Higashi-hiroshima 739-0046, Japan and JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

1Corresponding Author

Received  January 2016 Revised  June 2016 Published  December 2016

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

Citation: Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
References:
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B. Alberts, A. Johnson, P. Walter, J. Lewis, M. Raff and K. Roberts, Molecular Biology of The Cell 5th ed. Garland Science, New York, 2002.

[2]

E. Beretta and Y. Kuang, Geometry stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer, New York, 1992.

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K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8.

[5]

Y. Chen and A. F. Schier, Lefty proteins are long-range inhibitors of squint-mediated nodal signaling, Curr. Biol., 12 (2002), 2124-2128.

[6]

S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonl. Anal. RWA, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004.

[7]

S. ChenJ. Shi and J. Wei, A note on Hopf bifurcation in delayed diffusive Lotka-Volterra predator-prey system, Comput. Math. Appl., 62 (2011), 2240-2245. doi: 10.1016/j.camwa.2011.07.011.

[8]

S. ChenJ. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commu. Pure. Appl. Anal., 12 (2013), 481-501. doi: 10.3934/cpaa.2013.12.481.

[9]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38. doi: 10.1007/s00332-012-9138-1.

[10]

C. F. Drew, C. M. Lin, T. X. Jiang, G. Blunt, C. Mou, C. M. Chuong and D. J. Headon, The Edar subfamily in feather placode formation, Developmental Biology, 305 (2007), 232-245.

[11]

T. Faria and L. Magalhses, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Diff. Equa, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144.

[12]

T. Faria and L. Magalhses, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Diff. Equa., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145.

[13]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7.

[14]

T. Faria, On the study of singularities for a planar system with two delays, Dyn. Contin. Discrete Implus. Syst., 10 (2003), 357-371.

[15]

E. A. Gaffney and N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Trans. Amer. Math. Soc, 12 (1972), 30-39. doi: 10.1007/s11538-006-9066-z.

[16]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.

[17]

K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Dyn. Con. Dis. Sys. Series B., 8 (2007), 95-105. doi: 10.3934/dcdsb.2007.8.95.

[18]

M. P. HarrisS. WilliamsonJ. F. FallonH. Meinhardt and R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci., 102 (2005), 11734-11739.

[19]

M. P. HarrisS. WilliamsonJ. F. FallonH. Meinhardt and R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci. USA, 102 (2005), 11734-11739.

[20]

H. Juan and H. Hamada, Roles of nodal-lefty regulatory loops in embryonic patterning of vertebrates, Genes Cells., 6 (2001), 923-930.

[21]

J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol., 13 (2003), 1398-1408.

[22]

T. Miura and K. Shiota, Extracellular matrix environment influences chondrogenic pattern formation in limb bud micromass culture: Experimental verification of theoretical models, Anat. Rec., 258 (2000), 100-107.

[23]

T. Miura and K. Shiota, TGFβ2 acts as an activator molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture, Dev. Dyn., 217 (2000), 241-249.

[24]

T. MiuraK. ShiotaG. Morriss-Kay and P. K. Maini, Mixed-mode pattern in doublefoot mutant mouse limb-Turing reaction-diffusion model on a growing domain during limb development, J. Theor. Biol., 240 (2006), 562-573. doi: 10.1016/j.jtbi.2005.10.016.

[25]

Y. Morita, Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions, Japan J. Appl. Math., 1 (1984), 39-65. doi: 10.1007/BF03167861.

[26]

C. MouB. JacksonP. SchneiderP. A. Overbeek and D. J. Headon, Generation of the primary hair follicle pattern, Proceedings Of The National Academy Of Sciences Of The United States Of America, 103 (2006), 9075-9080.

[27]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.

[28]

L. A. F. de Oliveira, Instability of homogeneous periodic solutions of parabolic-delay equations, J. Diff. Equa., 109 (1994), 42-76. doi: 10.1006/jdeq.1994.1044.

[29]

S. Ruan, Absolute stability, conditional stability an bifurcation in Kolmogorov-type predator-prey systems with discrete delay, Quart. Appl. Math., 59 (2001), 159-176.

[30]

S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 61 (1998), 15-32. doi: 10.1093/imamat/61.1.15.

[31]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Con. Dis. Impul. Sys. Series A: Math. Anal., 10 (2003), 863-874.

[32]

R. SakumaY. OhnishiC. MenoH. FujiiH. JuanJ. TakeuchiT. OguraE. LiK. Miyazono and H. Hamada, Inhibition of nodal signalling by lefty mediated through interaction with common receptors and efficient diffusion, Genes Cells., 7 (2002), 401-412.

[33]

L. A. Segel and J. L. Jackson, Dissipative structure. an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.

[34]

L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr. Biol. , 13 (2003), R7-R9,401-412.

[35]

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

[36]

S. Seirin Lee and E. A. Gaffney, Aberrant behaviours of reaction diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-010-9533-4.

[37]

S. 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.

[38]

S. Seirin LeeE. A. Gaffney and N. A. M. Monk, The influence of gene expression time delays on Gierer-Meihardt pattern formation systems, Bull. Math. Biol., 72 (2010), 2139-2160. doi: 10.1007/s11538-010-9532-5.

[39]

S. SickS. ReinkerJ. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450. doi: 10.1126/science.1130088.

[40]

A. Sorkin and M. von Zastrow, Signal transduction and endocytosis: Close encounters of many kinds, Nature Reviews Molecular Cell Biol., 3 (2002), 600-614.

[41]

C. N. TennysonH. J. Klamut and R. G. Worton, The human dystrophin gene requires 16 hr to be transcribed and is contranscriptionally spliced, Nat. Gen., 9 (1995), 184-190.

[42]

A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72.

[43]

J. XiaZ. LiuR. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type Ⅱ functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200. doi: 10.1137/080728512.

show all references

References:
[1]

B. Alberts, A. Johnson, P. Walter, J. Lewis, M. Raff and K. Roberts, Molecular Biology of The Cell 5th ed. Garland Science, New York, 2002.

[2]

E. Beretta and Y. Kuang, Geometry stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer, New York, 1992.

[4]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8.

[5]

Y. Chen and A. F. Schier, Lefty proteins are long-range inhibitors of squint-mediated nodal signaling, Curr. Biol., 12 (2002), 2124-2128.

[6]

S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonl. Anal. RWA, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004.

[7]

S. ChenJ. Shi and J. Wei, A note on Hopf bifurcation in delayed diffusive Lotka-Volterra predator-prey system, Comput. Math. Appl., 62 (2011), 2240-2245. doi: 10.1016/j.camwa.2011.07.011.

[8]

S. ChenJ. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commu. Pure. Appl. Anal., 12 (2013), 481-501. doi: 10.3934/cpaa.2013.12.481.

[9]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38. doi: 10.1007/s00332-012-9138-1.

[10]

C. F. Drew, C. M. Lin, T. X. Jiang, G. Blunt, C. Mou, C. M. Chuong and D. J. Headon, The Edar subfamily in feather placode formation, Developmental Biology, 305 (2007), 232-245.

[11]

T. Faria and L. Magalhses, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Diff. Equa, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144.

[12]

T. Faria and L. Magalhses, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Diff. Equa., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145.

[13]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7.

[14]

T. Faria, On the study of singularities for a planar system with two delays, Dyn. Contin. Discrete Implus. Syst., 10 (2003), 357-371.

[15]

E. A. Gaffney and N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Trans. Amer. Math. Soc, 12 (1972), 30-39. doi: 10.1007/s11538-006-9066-z.

[16]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.

[17]

K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Dyn. Con. Dis. Sys. Series B., 8 (2007), 95-105. doi: 10.3934/dcdsb.2007.8.95.

[18]

M. P. HarrisS. WilliamsonJ. F. FallonH. Meinhardt and R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci., 102 (2005), 11734-11739.

[19]

M. P. HarrisS. WilliamsonJ. F. FallonH. Meinhardt and R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci. USA, 102 (2005), 11734-11739.

[20]

H. Juan and H. Hamada, Roles of nodal-lefty regulatory loops in embryonic patterning of vertebrates, Genes Cells., 6 (2001), 923-930.

[21]

J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol., 13 (2003), 1398-1408.

[22]

T. Miura and K. Shiota, Extracellular matrix environment influences chondrogenic pattern formation in limb bud micromass culture: Experimental verification of theoretical models, Anat. Rec., 258 (2000), 100-107.

[23]

T. Miura and K. Shiota, TGFβ2 acts as an activator molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture, Dev. Dyn., 217 (2000), 241-249.

[24]

T. MiuraK. ShiotaG. Morriss-Kay and P. K. Maini, Mixed-mode pattern in doublefoot mutant mouse limb-Turing reaction-diffusion model on a growing domain during limb development, J. Theor. Biol., 240 (2006), 562-573. doi: 10.1016/j.jtbi.2005.10.016.

[25]

Y. Morita, Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions, Japan J. Appl. Math., 1 (1984), 39-65. doi: 10.1007/BF03167861.

[26]

C. MouB. JacksonP. SchneiderP. A. Overbeek and D. J. Headon, Generation of the primary hair follicle pattern, Proceedings Of The National Academy Of Sciences Of The United States Of America, 103 (2006), 9075-9080.

[27]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.

[28]

L. A. F. de Oliveira, Instability of homogeneous periodic solutions of parabolic-delay equations, J. Diff. Equa., 109 (1994), 42-76. doi: 10.1006/jdeq.1994.1044.

[29]

S. Ruan, Absolute stability, conditional stability an bifurcation in Kolmogorov-type predator-prey systems with discrete delay, Quart. Appl. Math., 59 (2001), 159-176.

[30]

S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 61 (1998), 15-32. doi: 10.1093/imamat/61.1.15.

[31]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Con. Dis. Impul. Sys. Series A: Math. Anal., 10 (2003), 863-874.

[32]

R. SakumaY. OhnishiC. MenoH. FujiiH. JuanJ. TakeuchiT. OguraE. LiK. Miyazono and H. Hamada, Inhibition of nodal signalling by lefty mediated through interaction with common receptors and efficient diffusion, Genes Cells., 7 (2002), 401-412.

[33]

L. A. Segel and J. L. Jackson, Dissipative structure. an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559.

[34]

L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr. Biol. , 13 (2003), R7-R9,401-412.

[35]

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

[36]

S. Seirin Lee and E. A. Gaffney, Aberrant behaviours of reaction diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-010-9533-4.

[37]

S. 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.

[38]

S. Seirin LeeE. A. Gaffney and N. A. M. Monk, The influence of gene expression time delays on Gierer-Meihardt pattern formation systems, Bull. Math. Biol., 72 (2010), 2139-2160. doi: 10.1007/s11538-010-9532-5.

[39]

S. SickS. ReinkerJ. Timmer and T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450. doi: 10.1126/science.1130088.

[40]

A. Sorkin and M. von Zastrow, Signal transduction and endocytosis: Close encounters of many kinds, Nature Reviews Molecular Cell Biol., 3 (2002), 600-614.

[41]

C. N. TennysonH. J. Klamut and R. G. Worton, The human dystrophin gene requires 16 hr to be transcribed and is contranscriptionally spliced, Nat. Gen., 9 (1995), 184-190.

[42]

A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72.

[43]

J. XiaZ. LiuR. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type Ⅱ functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200. doi: 10.1137/080728512.

Figure 1.  A: The parameter space of $(a, b)$ which satisfies Theorem 2.4. In the white region, we will have a stable periodic solution for the FDEs. B: A bifurcation diagram for $\tau$ for the FDEs. This has been plotted using the value of $u(t)$ at which $v(t)=v_*$. The parameter values $(a, b)=(0.1, 0.9)$ have been chosen and the system exhibits the first Hopf bifurcation at approximately $\tau_0=0.2171$. For $\tau\in[0, \tau_0)$, $E_*=(1.0, 0.9)$ is always asymptotically stable, bifurcating first at $\tau_0$ then exhibiting subsequent bifurcations with frequency doubling prior to chaotic dynamics
Figure 2.  A homogeneous periodic solution for $u$ with $0< \varepsilon<1$. The parameter values for the simulations are given by $a=0.1, b=0.9, \varepsilon=0.1$. A: A plot of $u$ for $d=0.5$, when equation (43) is satisfied. B: A plot of $u$ for $d=5\times 10^{-5}$, when equation (43) is not satisfied
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