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:
[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.Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

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.Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

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
[1]

Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975

[2]

Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784

[3]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[4]

Reinhard Racke. Instability of coupled systems with delay. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753

[5]

Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029

[6]

Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403

[7]

Karl Peter Hadeler, Shigui Ruan. Interaction of diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 95-105. doi: 10.3934/dcdsb.2007.8.95

[8]

Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861

[9]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[10]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[11]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[12]

Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042

[13]

Hong-Ming Yin. A free boundary problem arising from a stress-driven diffusion in polymers. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 191-202. doi: 10.3934/dcds.1996.2.191

[14]

Renhai Wang, Yangrong Li, Bixiang Wang. Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4091-4126. doi: 10.3934/dcds.2019165

[15]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[16]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085

[17]

Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717

[18]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[19]

Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053

[20]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (42)
  • HTML views (78)
  • Cited by (0)

[Back to Top]