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Volumn 13, Issue 2, 2003, Pages 209-264

Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional gierer-meinhardt model

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EID: 84867997764     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-002-0531-z     Document Type: Article
Times cited : (109)

References (33)
  • 1
    • 84968466504 scopus 로고
    • Collocation software for boundary value ODE's
    • U. Ascher, R. Christiansen, R. Russell, Collocation Software for Boundary Value ODE's, Math. Comp., 33, (1979), pp. 659-679.
    • (1979) Math. Comp. , vol.33 , pp. 659-679
    • Ascher, U.1    Christiansen, R.2    Russell, R.3
  • 2
    • 12044252828 scopus 로고
    • Pattern formation outside of equilibrium
    • M. Cross, P. Hohenburg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys., 65, (1993), pp. 851-1112.
    • (1993) Rev. Mod. Phys. , vol.65 , pp. 851-1112
    • Cross, M.1    Hohenburg, P.2
  • 3
    • 0013238621 scopus 로고    scopus 로고
    • On Stability and Hopf Bifurcations for Chemotaxis Systems
    • E. N. Dancer, On Stability and Hopf Bifurcations for Chemotaxis Systems, Meth. Appl. Anal., 8(2), (2001), pp. 245-256. (Pubitemid 36289753)
    • (2001) Methods and Applications of Analysis , vol.8 , Issue.2 , pp. 245-256
    • Dancer, E.N.1
  • 4
    • 0000706319 scopus 로고    scopus 로고
    • Pattern formation in the one-dimensional Gray-Scott model
    • PII S0951771597775662
    • A. Doelman, T. J. Kaper, P. Zegeling, Pattern Formation in the One-Dimensional Gray-Scott Model, Nonlinearity, 10, (1997), pp. 523-563. (Pubitemid 127430235)
    • (1997) Nonlinearity , vol.10 , Issue.2 , pp. 523-563
    • Doelman, A.1    Kaper, T.J.2    Zegeling, P.A.3
  • 5
    • 0011517497 scopus 로고    scopus 로고
    • Spatially Periodic and Aperiodic Multi-pulse Patterns in the One-dimensional Gierer-Meinhardt Equation
    • A. Doelman, T. Kaper, H. van der Ploeg, Spatially Periodic and Aperiodic Multi-Pulse Patterns in the One-Dimensional Gierer-Meinhardt Model, Meth. Appl. Anal., 8(3), (2001), pp. 387-414. (Pubitemid 36299751)
    • (2001) Methods and Applications of Analysis , vol.8 , Issue.3 , pp. 387-414
    • Doelman, A.1    Kaper, T.J.2    Van Der Ploeg, H.3
  • 6
    • 0005468518 scopus 로고    scopus 로고
    • Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach
    • PII S0167278998001808
    • A. Doelman, R. A. Gardner, T. J. Kaper, Stability Analysis of Singular Patterns in the 1-D Gray-Scott Model: A Matched Asymptotics Approach, Physica D, 122, (1998), pp. 1-36. (Pubitemid 128636761)
    • (1998) Physica D: Nonlinear Phenomena , vol.122 , Issue.1-4 , pp. 1-36
    • Doelman, A.1    Gardner, R.A.2    Kaper, T.J.3
  • 7
    • 85087535076 scopus 로고    scopus 로고
    • A stability index analysis of 1-D patterns of the gray scott model
    • A. Doelman, R. A. Gardner, T. Kaper, A Stability Index Analysis of 1-D Patterns of the Gray Scott Model, Mem. AMS, 155, no. 737, (2002).
    • (2002) Mem. AMS , vol.155 , Issue.737
    • Doelman, A.1    Gardner, R.A.2    Kaper, T.3
  • 10
    • 0006030342 scopus 로고    scopus 로고
    • The motion of weakly interacting pulses in reaction-diffusion systems
    • S. Ei, The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems, J. Dyn. Diff. Eq., 14, no. 1, (2002), pp. 85-137.
    • (2002) J. Dyn. Diff. Eq. , vol.14 , Issue.1 , pp. 85-137
    • Ei, S.1
  • 11
    • 0015454185 scopus 로고
    • A theory of biological pattern formation
    • A. Gierer, H. Meinhardt, A Theory of Biological Pattern Formation, Kybernetik, 12, (1972), pp. 30-39.
    • (1972) Kybernetik , vol.12 , pp. 30-39
    • Gierer, A.1    Meinhardt, H.2
  • 12
    • 0001067122 scopus 로고
    • Order and localization in reaction-diffusion pattern
    • L. Harrison, D. Holloway, Order and Localization in Reaction-Diffusion Pattern, Physica A, 222, (1995), pp. 210-233.
    • (1995) Physica A , vol.222 , pp. 210-233
    • Harrison, L.1    Holloway, D.2
  • 13
    • 0036665085 scopus 로고    scopus 로고
    • The dynamics of multi-spike solutions to the one-dimensional gierer-meinhardt model
    • electronic
    • D. Iron, M. J.Ward, The Dynamics of Multi-Spike Solutions to the One-Dimensional Gierer-Meinhardt Model, SIAM J. Appl. Math., 62(6), (2002), pp. 1924-1951 (electronic).
    • (2002) SIAM J. Appl. Math. , vol.62 , Issue.6 , pp. 1924-1951
    • Iron, D.1    Ward, M.J.2
  • 14
    • 0035277877 scopus 로고    scopus 로고
    • The stability of spike solutions to the one-dimensional gierer-meinhardt model
    • D. Iron, M. J.Ward, J.Wei, The Stability of Spike Solutions to the One-Dimensional Gierer-Meinhardt Model, Physica D, 150(1-2), (2001), pp. 25-62.
    • (2001) Physica D , vol.150 , Issue.1-2 , pp. 25-62
    • Iron, D.1    Ward, M.J.2    Wei, J.3
  • 15
    • 0018003001 scopus 로고
    • Activators and inhibitors in pattern formation
    • J. Keener, Activators and Inhibitors in Pattern Formation, Stud. Appl. Math., 59, (1978), pp. 1-23.
    • (1978) Stud. Appl. Math. , vol.59 , pp. 1-23
    • Keener, J.1
  • 16
    • 0002542873 scopus 로고
    • Large amplitude stationary solutions to a chemotaxis system
    • C. S. Lin,W. M. Ni, I. Takagi, Large Amplitude Stationary Solutions to a Chemotaxis System, J. Diff. Eq., 72, (1988), pp. 1-27.
    • (1988) J. Diff. Eq. , vol.72 , pp. 1-27
    • Lin, C.S.1    Ni, W.M.2    Takagi, I.3
  • 19
    • 84867986083 scopus 로고    scopus 로고
    • NAG Fortran Library Mark 17, routine D03PCF,Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995)
    • NAG Fortran Library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).
  • 20
    • 0003002827 scopus 로고    scopus 로고
    • Diffusion cross-diffusion and their spike-layer steady-states
    • W. Ni, Diffusion, Cross-Diffusion, and Their Spike-Layer Steady-States, Not. AMS, 45(1), (1998), pp. 9-18.
    • (1998) Not. AMS , vol.45 , Issue.1
    • Ni, W.1
  • 22
    • 0001347301 scopus 로고
    • Global structure of bifurcating solutions of some reaction-diffusion systems
    • Y. Nishiura, Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems, SIAM J. Math. Anal., 13(4), (1982), pp. 555-593.
    • (1982) SIAM J. Math. Anal. , vol.13 , Issue.4 , pp. 555-593
    • Nishiura, Y.1
  • 23
    • 0002882425 scopus 로고
    • Coexistence of infinitely many stable solutions to reaction-diffusion equations in the singular limit
    • editors: C. K. R. T. Jones, U. Kirchgraber), Springer-Verlag, New York
    • Y. Nishiura, Coexistence of Infinitely Many Stable Solutions to Reaction-Diffusion Equations in the Singular Limit, in Dynamics Reported: Expositions in Dynamical Systems, vol. 3 (editors: C. K. R. T. Jones, U. Kirchgraber), Springer-Verlag, New York (1995).
    • (1995) Dynamics Reported: Expositions in Dynamical Systems , vol.3
    • Nishiura, Y.1
  • 24
    • 0035311814 scopus 로고    scopus 로고
    • Spatio-temporal chaos for the Gray-Scott model
    • DOI 10.1016/S0167-2789(00)00214-1
    • Y. Nishiura, D. Ueyama, Spatio-Temporal Chaos for the Gray-Scott Model, Physica D, 150, (2001), pp. 137-162. (Pubitemid 32265308)
    • (2001) Physica D: Nonlinear Phenomena , vol.150 , Issue.3-4 , pp. 137-162
    • Nishiura, Y.1    Ueyama, D.2
  • 25
    • 1642548932 scopus 로고    scopus 로고
    • Far-from-equilibrium dynamics
    • AMS Publications, Providence, RI
    • Y. Nishiura, Far-from-Equilibrium Dynamics, Translations of Mathematical Monographs, vol. 209, AMS Publications, Providence, RI (2002).
    • (2002) Translations of Mathematical Monographs , vol.209
    • Nishiura, Y.1
  • 26
    • 0018784247 scopus 로고
    • Simple chemical reaction systems with limit cycle behavior
    • J. Schnakenberg, Simple Chemical Reaction Systems with Limit Cycle Behavior, J. Theor. Biol., 81, (1979), pp. 389-400.
    • (1979) J. Theor. Biol. , vol.81 , pp. 389-400
    • Schnakenberg, J.1
  • 27
    • 84867984485 scopus 로고    scopus 로고
    • Numerical challenges for resolving spike dynamics for two reaction-diffusion systems
    • W. Sun, T. Tang, M. J. Ward, J. Wei, Numerical Challenges for Resolving Spike Dynamics for Two Reaction-Diffusion Systems, accepted, Stud. Appl. Math., (2002).
    • (2002) Stud. Appl. Math.
    • Sun, W.1    Tang, T.2    Ward, M.J.3    Wei, J.4
  • 28
    • 0000864203 scopus 로고
    • Point-condensation for a reaction-diffusion system
    • I. Takagi, Point-Condensation for a Reaction-Diffusion System, J. Diff. Eq., 61, (1986), pp. 208-249.
    • (1986) J. Diff. Eq. , vol.61 , pp. 208-249
    • Takagi, I.1
  • 29
    • 0002011401 scopus 로고
    • The chemical basis of morphogenesis
    • A. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. B, 327, (1952), pp. 37-72.
    • (1952) Phil. Trans. Roy. Soc. B , vol.327 , pp. 37-72
    • Turing, A.1
  • 30
    • 0036067340 scopus 로고    scopus 로고
    • Asymmetric spike patterns for the one-dimensional gierer-meinhardt model: Equilibria and stability
    • M. J. Ward, J. Wei, Asymmetric Spike Patterns for the One-Dimensional Gierer-Meinhardt Model: Equilibria and Stability, Eur. J. Appl. Math., 13(3), (2002), pp. 283-320.
    • (2002) Eur. J. Appl. Math. , vol.13 , Issue.3 , pp. 283-320
    • Ward, M.J.1    Wei, J.2
  • 31
    • 84867986086 scopus 로고    scopus 로고
    • Hopf bifurcation of spike solutions for the shadow gierer-meinhardt model
    • M. J. Ward, J. Wei, Hopf Bifurcation of Spike Solutions for the Shadow Gierer-Meinhardt Model, submitted, Eur. J. Appl. Math., (2002).
    • (2002) Eur. J. Appl. Math.
    • Ward, M.J.1    Wei, J.2
  • 32
    • 0033414750 scopus 로고    scopus 로고
    • On single interior spike solutions for the gierer-meinhardt system: Uniqueness and stability estimates
    • J.Wei, On Single Interior Spike Solutions for the Gierer-Meinhardt System: Uniqueness and Stability Estimates, Eur. J. Appl. Math., 10(4), (1999), pp. 353-378.
    • (1999) Eur. J. Appl. Math. , vol.10 , Issue.4 , pp. 353-378
    • Wei, J.1
  • 33
    • 30244514666 scopus 로고    scopus 로고
    • The gierer-meinhardt system: Existence and stability of n-peaked Solutions
    • J. Wei, M. Winter, The Gierer-Meinhardt System: Existence and Stability of N-Peaked Solutions, submitted, Trans. Am. Math Soc., (2001).
    • (2001) Trans. Am. Math Soc.
    • Wei, J.1    Winter, M.2


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