메뉴 건너뛰기




Volumn 366, Issue 2, 2010, Pages 679-693

Pattern formation of a coupled two-cell Brusselator model

Author keywords

A priori estimates; Bifurcation; Brusselator; Index; Leray Schauder degree; Pattern formation; Two cell model

Indexed keywords


EID: 76749157436     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2009.12.021     Document Type: Article
Times cited : (28)

References (49)
  • 1
    • 0016754941 scopus 로고
    • Bifurcation analysis of nonlinear reaction-diffusion equations I: Evolution equations and the steady state solutions
    • Auchmuty J.F.G., and Nicolis G. Bifurcation analysis of nonlinear reaction-diffusion equations I: Evolution equations and the steady state solutions. Bull. Math. Biol. 37 (1975) 323-365
    • (1975) Bull. Math. Biol. , vol.37 , pp. 323-365
    • Auchmuty, J.F.G.1    Nicolis, G.2
  • 2
    • 0038778726 scopus 로고
    • Global bifurcation in the Brusselator system
    • Brown K.J., and Davidson F.A. Global bifurcation in the Brusselator system. Nonlinear Anal. 12 (1995) 1713-1725
    • (1995) Nonlinear Anal. , vol.12 , pp. 1713-1725
    • Brown, K.J.1    Davidson, F.A.2
  • 3
    • 0346072299 scopus 로고    scopus 로고
    • Pattern formation in three-dimensional reaction-diffusion systems
    • Callahan T.K., and Knobloch E. Pattern formation in three-dimensional reaction-diffusion systems. Phys. D 132 (1999) 339-362
    • (1999) Phys. D , vol.132 , pp. 339-362
    • Callahan, T.K.1    Knobloch, E.2
  • 4
    • 84972537529 scopus 로고
    • Existence and uniqueness of coexistence states for a predator-prey model with diffusion
    • Casal A., Eilbeck J.C., and López-Gómez J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion. Differential Integral Equations 7 (1994) 411-439
    • (1994) Differential Integral Equations , vol.7 , pp. 411-439
    • Casal, A.1    Eilbeck, J.C.2    López-Gómez, J.3
  • 5
    • 1542720931 scopus 로고    scopus 로고
    • Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model
    • Chen W.Y., and Peng R. Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model. J. Math. Anal. Appl. 291 (2004) 550-564
    • (2004) J. Math. Anal. Appl. , vol.291 , pp. 550-564
    • Chen, W.Y.1    Peng, R.2
  • 6
    • 41449113551 scopus 로고    scopus 로고
    • Sharp estimates on minimum traveling wave speed of reaction-diffusion systems modeling autocatalysis
    • Chen X., and Qi Y. Sharp estimates on minimum traveling wave speed of reaction-diffusion systems modeling autocatalysis. SIAM J. Math. Anal. 39 (2007) 437-448
    • (2007) SIAM J. Math. Anal. , vol.39 , pp. 437-448
    • Chen, X.1    Qi, Y.2
  • 7
    • 0000814522 scopus 로고    scopus 로고
    • A counter example of competing species equations
    • Dancer E.N. A counter example of competing species equations. Differential Integral Equations 9 (1996) 239-246
    • (1996) Differential Integral Equations , vol.9 , pp. 239-246
    • Dancer, E.N.1
  • 8
    • 0033634578 scopus 로고    scopus 로고
    • On the symbiotic Lotka-Volterra model with diffusion and transport effects
    • Delgado M., López-Gómez J., and Suarez A. On the symbiotic Lotka-Volterra model with diffusion and transport effects. J. Differential Equations 160 (2000) 175-262
    • (2000) J. Differential Equations , vol.160 , pp. 175-262
    • Delgado, M.1    López-Gómez, J.2    Suarez, A.3
  • 9
    • 21744459836 scopus 로고    scopus 로고
    • Some uniqueness and exact multiplicity results for a predator-prey model
    • Du Y.H., and Lou Y. Some uniqueness and exact multiplicity results for a predator-prey model. Trans. Amer. Math. Soc. 349 (1997) 2443-2475
    • (1997) Trans. Amer. Math. Soc. , vol.349 , pp. 2443-2475
    • Du, Y.H.1    Lou, Y.2
  • 10
    • 23044527512 scopus 로고    scopus 로고
    • Qualitative behavior of positive solutions of a predator-prey model: effects of saturation
    • Du Y.H., and Lou Y. Qualitative behavior of positive solutions of a predator-prey model: effects of saturation. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 321-349
    • (2001) Proc. Roy. Soc. Edinburgh Sect. A , vol.131 , pp. 321-349
    • Du, Y.H.1    Lou, Y.2
  • 11
    • 0000002470 scopus 로고
    • Strips or spots? Nonlinear effects in bifurcation of reaction-diffusion equation on the square
    • Ermentrout B. Strips or spots? Nonlinear effects in bifurcation of reaction-diffusion equation on the square. Proc. R. Soc. Lond. 434 (1991) 413-417
    • (1991) Proc. R. Soc. Lond. , vol.434 , pp. 413-417
    • Ermentrout, B.1
  • 15
    • 55149099624 scopus 로고    scopus 로고
    • Turing pattern formation in the Brusselator model with superdiffusion
    • Golovin A.A., Matkowsky B.J., and Volpert V.A. Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69 (2009) 251-272
    • (2009) SIAM J. Appl. Math. , vol.69 , pp. 251-272
    • Golovin, A.A.1    Matkowsky, B.J.2    Volpert, V.A.3
  • 16
    • 41349095866 scopus 로고    scopus 로고
    • Modeling molecular interaction pathways using a two-stage identification algorithm
    • Gormley P., Li K., and Irwin G.W. Modeling molecular interaction pathways using a two-stage identification algorithm. Systems Synthetic Biol. 1 (2007) 145-160
    • (2007) Systems Synthetic Biol. , vol.1 , pp. 145-160
    • Gormley, P.1    Li, K.2    Irwin, G.W.3
  • 17
    • 0039978016 scopus 로고
    • Geometric Theory of Semilinear Parabolic Equation
    • Springer-Verlag, Berlin
    • Henry D. Geometric Theory of Semilinear Parabolic Equation. Lecture Notes in Math. vol. 840 (1993), Springer-Verlag, Berlin
    • (1993) Lecture Notes in Math. , vol.840
    • Henry, D.1
  • 18
    • 0018305813 scopus 로고
    • Multiple stable steady states and chemical hysteresis in a two-box model of the Brusselator
    • Jetschke G. Multiple stable steady states and chemical hysteresis in a two-box model of the Brusselator. J. Non-Equilib. Thermodyn. 4 2 (1979)
    • (1979) J. Non-Equilib. Thermodyn. , vol.4 , Issue.2
    • Jetschke, G.1
  • 19
    • 0030688319 scopus 로고    scopus 로고
    • Dynamical analysis of a reaction-diffusion system with Brusselator kinetics under feedback control
    • Albuquerque, NM, June
    • I. Karafyllis, P.D. Christofides, P. Daoutidis, Dynamical analysis of a reaction-diffusion system with Brusselator kinetics under feedback control, in: Proc. Amer. Control Conference, Albuquerque, NM, June 1997, pp. 2213-2217
    • (1997) Proc. Amer. Control Conference , pp. 2213-2217
    • Karafyllis, I.1    Christofides, P.D.2    Daoutidis, P.3
  • 20
    • 0034652878 scopus 로고    scopus 로고
    • Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion
    • Kan-on Y. Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion. J. Math. Anal. Appl. 243 (2000) 357-372
    • (2000) J. Math. Anal. Appl. , vol.243 , pp. 357-372
    • Kan-on, Y.1
  • 21
    • 0032389543 scopus 로고    scopus 로고
    • Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics
    • Kan-on Y., and Mimura M. Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics. SIAM J. Math. Anal. 29 (1998) 1519-1536
    • (1998) SIAM J. Math. Anal. , vol.29 , pp. 1519-1536
    • Kan-on, Y.1    Mimura, M.2
  • 22
    • 0019197080 scopus 로고
    • Two coupled neural oscillators as a model of the circadian pacemaker
    • Kawato M., and Suzuki R. Two coupled neural oscillators as a model of the circadian pacemaker. J. Theoret. Biol. 86 (1980) 547-575
    • (1980) J. Theoret. Biol. , vol.86 , pp. 547-575
    • Kawato, M.1    Suzuki, R.2
  • 23
    • 0036500993 scopus 로고    scopus 로고
    • Systems biology: a brief overview
    • Kitano H. Systems biology: a brief overview. Science 295 (2002) 1662-1664
    • (2002) Science , vol.295 , pp. 1662-1664
    • Kitano, H.1
  • 24
    • 0011958476 scopus 로고
    • Period doubling bifurcations in a two-box model of the Brusselator
    • Klič. Period doubling bifurcations in a two-box model of the Brusselator. Aplikace Matematiky 28 (1983) 335-343
    • (1983) Aplikace Matematiky , vol.28 , pp. 335-343
    • Klič1
  • 25
    • 31144461544 scopus 로고    scopus 로고
    • Mesa-type patterns in one-dimensional Brusselator and their stability
    • Kolokolnikov T., Erneux T., and Wei J. Mesa-type patterns in one-dimensional Brusselator and their stability. Phys. D 214 1 (2006) 63-77
    • (2006) Phys. D , vol.214 , Issue.1 , pp. 63-77
    • Kolokolnikov, T.1    Erneux, T.2    Wei, J.3
  • 26
    • 18244407182 scopus 로고    scopus 로고
    • Spectral Theory and Nonlinear Functional Analysis
    • Chapman and Hall/CRC, Boca Raton, FL
    • López-Gómez J. Spectral Theory and Nonlinear Functional Analysis. Res. Notes Math. vol. 426 (2001), Chapman and Hall/CRC, Boca Raton, FL
    • (2001) Res. Notes Math. , vol.426
    • López-Gómez, J.1
  • 27
    • 55049131713 scopus 로고    scopus 로고
    • Algebraic Multiplicity of Eigenvalues of Linear Operators
    • Birkhäuser, Springer, Basel, Boston, Berlin
    • López-Gómez J., and Mora-Corral C. Algebraic Multiplicity of Eigenvalues of Linear Operators. Oper. Theory Adv. Appl. vol. 177 (2007), Birkhäuser, Springer, Basel, Boston, Berlin
    • (2007) Oper. Theory Adv. Appl. , vol.177
    • López-Gómez, J.1    Mora-Corral, C.2
  • 28
    • 0030579038 scopus 로고    scopus 로고
    • Diffusion, self-diffusion and cross-diffusion
    • Lou Y., and Ni W.M. Diffusion, self-diffusion and cross-diffusion. J. Differential Equations 131 (1996) 79-131
    • (1996) J. Differential Equations , vol.131 , pp. 79-131
    • Lou, Y.1    Ni, W.M.2
  • 29
    • 0033130048 scopus 로고    scopus 로고
    • Diffusion vs. cross-diffusion: an elliptic approach
    • Lou Y., and Ni W.M. Diffusion vs. cross-diffusion: an elliptic approach. J. Differential Equations 154 (1999) 157-190
    • (1999) J. Differential Equations , vol.154 , pp. 157-190
    • Lou, Y.1    Ni, W.M.2
  • 30
    • 0347055376 scopus 로고    scopus 로고
    • On 3 × 3 Lotka-Volterra competition systems with cross-diffusion
    • Lou Y., Martinez S., and Ni W.M. On 3 × 3 Lotka-Volterra competition systems with cross-diffusion. Discrete Contin. Dyn. Syst. 6 (2000) 175-190
    • (2000) Discrete Contin. Dyn. Syst. , vol.6 , pp. 175-190
    • Lou, Y.1    Martinez, S.2    Ni, W.M.3
  • 31
    • 1442279582 scopus 로고    scopus 로고
    • Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion
    • Pang P.Y.H., and Wang M.X. Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion. Proc. London Math. Soc. 88 (2004) 135-157
    • (2004) Proc. London Math. Soc. , vol.88 , pp. 135-157
    • Pang, P.Y.H.1    Wang, M.X.2
  • 32
    • 0141957171 scopus 로고    scopus 로고
    • Qualitative analysis of a ratio-dependent predator-prey system with diffusion
    • Pang P.Y.H., and Wang M.X. Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 919-942
    • (2003) Proc. Roy. Soc. Edinburgh Sect. A , vol.133 , pp. 919-942
    • Pang, P.Y.H.1    Wang, M.X.2
  • 33
    • 0346340796 scopus 로고    scopus 로고
    • Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction
    • Peng R., and Wang M.X. Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction. Nonlinear Anal. 56 (2004) 451-464
    • (2004) Nonlinear Anal. , vol.56 , pp. 451-464
    • Peng, R.1    Wang, M.X.2
  • 34
    • 22144487744 scopus 로고    scopus 로고
    • Pattern formation in the Brusselator system
    • Peng R., and Wang M.X. Pattern formation in the Brusselator system. J. Math. Anal. Appl. 309 (2005) 151-166
    • (2005) J. Math. Anal. Appl. , vol.309 , pp. 151-166
    • Peng, R.1    Wang, M.X.2
  • 35
    • 36849102494 scopus 로고
    • Symmetry-breaking instabilities in dissipative systems
    • Prigogine I., and Lefever R. Symmetry-breaking instabilities in dissipative systems. J. Chem. Phys. 48 (1968) 1665-1700
    • (1968) J. Chem. Phys. , vol.48 , pp. 1665-1700
    • Prigogine, I.1    Lefever, R.2
  • 36
    • 33846590412 scopus 로고    scopus 로고
    • The development of traveling waves in cubic auto-catalysis with different rates of diffusion
    • Qi Y. The development of traveling waves in cubic auto-catalysis with different rates of diffusion. Phys. D 226 (2007) 129-135
    • (2007) Phys. D , vol.226 , pp. 129-135
    • Qi, Y.1
  • 37
    • 49049139221 scopus 로고
    • Strange attractors in coupled reaction-diffusion cells
    • Schreiber I., and Marek M. Strange attractors in coupled reaction-diffusion cells. Phys. D 5 (1982) 258-272
    • (1982) Phys. D , vol.5 , pp. 258-272
    • Schreiber, I.1    Marek, M.2
  • 38
    • 0002268213 scopus 로고
    • Simple and complex reaction-diffusion fronts
    • Kapral R., and Showalter K. (Eds), Kluwer Acad. Publ., Dordrecht
    • Scott S.K., and Showalter K. Simple and complex reaction-diffusion fronts. In: Kapral R., and Showalter K. (Eds). Chemical Waves and Patterns (1995), Kluwer Acad. Publ., Dordrecht 485-516
    • (1995) Chemical Waves and Patterns , pp. 485-516
    • Scott, S.K.1    Showalter, K.2
  • 39
    • 0007393460 scopus 로고
    • Transition to chaos via two-torus in coupled reaction-diffusion cells
    • Schreiber L., and Marek M. Transition to chaos via two-torus in coupled reaction-diffusion cells. Phys. Lett. 91 6 (1982) 263
    • (1982) Phys. Lett. , vol.91 , Issue.6 , pp. 263
    • Schreiber, L.1    Marek, M.2
  • 41
    • 0002011401 scopus 로고
    • The chemical basis of morphogenesis
    • Turing A. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Ser. B 237 (1952) 37-72
    • (1952) Philos. Trans. R. Soc. Ser. B , vol.237 , pp. 37-72
    • Turing, A.1
  • 42
    • 4244101701 scopus 로고
    • Some further studies of nonlinear oscillations in chemical systems
    • Tyson J.J. Some further studies of nonlinear oscillations in chemical systems. J. Chem. Phys. 18 9 (1973)
    • (1973) J. Chem. Phys. , vol.18 , Issue.9
    • Tyson, J.J.1
  • 44
    • 0037845132 scopus 로고    scopus 로고
    • Non-constant positive steady-states of the Sel'kov model
    • Wang M.X. Non-constant positive steady-states of the Sel'kov model. J. Differential Equations 190 (2003) 600-620
    • (2003) J. Differential Equations , vol.190 , pp. 600-620
    • Wang, M.X.1
  • 45
    • 0035918674 scopus 로고    scopus 로고
    • A system of resource-based growth models with two resources in the unstirred chemostat
    • Wu J.H., and Wolkowicz G.S.K. A system of resource-based growth models with two resources in the unstirred chemostat. J. Differential Equations 172 (2001) 300-332
    • (2001) J. Differential Equations , vol.172 , pp. 300-332
    • Wu, J.H.1    Wolkowicz, G.S.K.2
  • 46
    • 3042513690 scopus 로고    scopus 로고
    • Stable squares and other oscillatory Turing patterns in a reaction-diffusion model
    • Yang L., Zhabotinsky A.M., and Epstein I.R. Stable squares and other oscillatory Turing patterns in a reaction-diffusion model. Phys. Rev. Lett. 92 198303 (2004) 1-4
    • (2004) Phys. Rev. Lett. , vol.92 , Issue.198303 , pp. 1-4
    • Yang, L.1    Zhabotinsky, A.M.2    Epstein, I.R.3
  • 48
    • 44949096501 scopus 로고    scopus 로고
    • Global dynamics of the Brusselator equations
    • You Y.C. Global dynamics of the Brusselator equations. Dyn. Partial Differ. Equ. 4 (2007) 167-196
    • (2007) Dyn. Partial Differ. Equ. , vol.4 , pp. 167-196
    • You, Y.C.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.