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Volumn 57, Issue 1, 2008, Pages 53-89

Stationary multiple spots for reaction-diffusion systems

Author keywords

Multiple spots; Pattern formation; Reaction diffusion systems

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; BIOPHYSICS; CHEMICAL MODEL; COMPUTER SIMULATION; DIFFUSION; HUMAN; KINETICS; MATHEMATICS; METHODOLOGY; MORPHOGENESIS; NONLINEAR SYSTEM; STATISTICAL MODEL;

EID: 42149174297     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-007-0146-y     Document Type: Article
Times cited : (66)

References (77)
  • 1
    • 0000578162 scopus 로고    scopus 로고
    • Unravelling the Turing bifurcation using spatially varying diffusion coefficients
    • Benson D.L., Maini P.K. and Sherratt J.A. (1998). Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37: 381-417
    • (1998) J. Math. Biol. , vol.37 , pp. 381-417
    • Benson, D.L.1    Maini, P.K.2    Sherratt, J.A.3
  • 2
    • 5544250703 scopus 로고
    • Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern
    • Castets V., Dulos E., Boissonade J. and De Kepper P. (1990). Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64: 2953-2956
    • (1990) Phys. Rev. Lett. , vol.64 , pp. 2953-2956
    • Castets, V.1    Dulos, E.2    Boissonade, J.3    De Kepper, P.4
  • 3
    • 0032701813 scopus 로고    scopus 로고
    • Reaction and diffusion on growing domains: Scenarios for robust pattern formation
    • Crampin E.J., Gaffney E.A. and Maini P.K. (1999). Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61: 1093-1120
    • (1999) Bull. Math. Biol. , vol.61 , pp. 1093-1120
    • Crampin, E.J.1    Gaffney, E.A.2    Maini, P.K.3
  • 4
    • 0036485212 scopus 로고    scopus 로고
    • Mode doubling and tripling in reaction-diffusion patterns on growing domains: A piece-wise linear model
    • 1093-1120
    • Crampin, E.J., Gaffney, E.A., Maini, P.K.: Mode doubling and tripling in reaction-diffusion patterns on growing domains: a piece-wise linear model. J. Math. Biol. 44, 107-128, 1093-1120 (1999)
    • (1999) J. Math. Biol. , vol.44 , pp. 107-128
    • Crampin, E.J.1    Gaffney, E.A.2    Maini, P.K.3
  • 5
    • 0013238621 scopus 로고    scopus 로고
    • On stability and Hopf bifurcations for chemotaxis systems
    • Dancer E.N. (2001). On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8: 245-256
    • (2001) Methods Appl. Anal. , vol.8 , pp. 245-256
    • Dancer, E.N.1
  • 6
    • 44949270147 scopus 로고
    • Turing-type chemical pattern in the chlorite-iodide-malonic acid reaction
    • De Kepper P., Castets V., Dulos E. and Boissonade J. (1991). Turing-type chemical pattern in the chlorite-iodide-malonic acid reaction. Phys. D 49: 161-169
    • (1991) Phys. D , vol.49 , pp. 161-169
    • De Kepper, P.1    Castets, V.2    Dulos, E.3    Boissonade, J.4
  • 7
    • 0000132086 scopus 로고    scopus 로고
    • Large stable pulse solutions in reaction-diffusion equations
    • Doelman A., Gardner R.A. and Kaper T.J. (2001). Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50: 443-507
    • (2001) Indiana Univ. Math. J. , vol.50 , pp. 443-507
    • Doelman, A.1    Gardner, R.A.2    Kaper, T.J.3
  • 8
    • 0005468518 scopus 로고    scopus 로고
    • Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach
    • Doelman A., Gardner A. and Kaper T.J. (1998). Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach. Phys. D 122: 1-36
    • (1998) Phys. D , vol.122 , pp. 1-36
    • Doelman, A.1    Gardner, A.2    Kaper, T.J.3
  • 9
    • 85087535076 scopus 로고    scopus 로고
    • A stability index analysis of 1-D patterns of the Gray-Scott model
    • Doelman, A., Gardner, A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray-Scott model. Mem. Am. Math. Soc. 155(737), xii+64 (2002)
    • (2002) Mem. Am. Math. Soc. , vol.155 , Issue.737
    • Doelman, A.1    Gardner, A.2    Kaper, T.J.3
  • 10
    • 0000706319 scopus 로고    scopus 로고
    • Pattern formation in the one-dimensional Gray-Scott model
    • Doelman A., Kaper T. and Zegeling P.A. (1997). Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10: 523-563
    • (1997) Nonlinearity , vol.10 , pp. 523-563
    • Doelman, A.1    Kaper, T.2    Zegeling, P.A.3
  • 11
    • 36449007762 scopus 로고
    • Conventional and unconventional Turing patterns
    • Dufiet V. and Boissonade J. (1992). Conventional and unconventional Turing patterns. J. Chem. Phys. 96: 664-673
    • (1992) J. Chem. Phys. , vol.96 , pp. 664-673
    • Dufiet, V.1    Boissonade, J.2
  • 12
    • 0006030342 scopus 로고    scopus 로고
    • The motion of weakly interacting pulses in reaction-diffusion systems
    • Ei S. (2002). The motion of weakly interacting pulses in reaction-diffusion systems. J. Dyn. Diff. Equ. 14: 85-87
    • (2002) J. Dyn. Diff. Equ. , vol.14 , pp. 85-87
    • Ei, S.1
  • 13
    • 0347670293 scopus 로고    scopus 로고
    • n splitting or edge splitting: A manner of splitting in dissipative systems
    • n splitting or edge splitting: a manner of splitting in dissipative systems. Jpn J. Ind. Appl. Math. 18: 181-205
    • (2001) Jpn J. Ind. Appl. Math. , vol.18 , pp. 181-205
    • Ei, S.1    Nishiura, Y.2    Ueda, K.3
  • 14
    • 3042943807 scopus 로고
    • Stationary patterns for reaction-diffusion systems
    • Nonlinear Diffusion Pitman, London
    • Fife, P.C.: Stationary patterns for reaction-diffusion systems. In: Nonlinear Diffusion. Research Notes in Math., vol. 14, pp. 81-121. Pitman, London (1977)
    • (1977) Research Notes in Math. , vol.14 , pp. 81-121
    • Fife, P.C.1
  • 15
    • 0002910353 scopus 로고
    • Large time behaviour of solutions of bistable nonlinear diffusion equations
    • Fife P.C. (1979). Large time behaviour of solutions of bistable nonlinear diffusion equations. Arch. Rat. Mech. Anal. 70: 31-46
    • (1979) Arch. Rat. Mech. Anal. , vol.70 , pp. 31-46
    • Fife, P.C.1
  • 17
    • 0015454185 scopus 로고
    • A theory of biological pattern formation
    • Gierer A. and Meinhardt H. (1972). A theory of biological pattern formation. Kybernetik (Berlin) 12: 30-39
    • (1972) Kybernetik (Berlin) , vol.12 , pp. 30-39
    • Gierer, A.1    Meinhardt, H.2
  • 18
    • 0020607165 scopus 로고
    • Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability
    • Gray P. and Scott S.K. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38: 29-43
    • (1983) Chem. Eng. Sci. , vol.38 , pp. 29-43
    • Gray, P.1    Scott, S.K.2
  • 19
    • 0021310291 scopus 로고
    • Autocatalytic reactions in the isothermal. continuous stirred tank reactor: Oscillations and instabilites to the system A + 2B → 3B, B → C
    • Gray P. and Scott S.K. (1984). Autocatalytic reactions in the isothermal. continuous stirred tank reactor: oscillations and instabilites to the system A + 2B → 3B, B → C. Chem. Eng. Sci. 39: 1087-1097
    • (1984) Chem. Eng. Sci. , vol.39 , pp. 1087-1097
    • Gray, P.1    Scott, S.K.2
  • 20
    • 0034458192 scopus 로고    scopus 로고
    • Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis
    • Hale J.K., Peletier L.A. and Troy W.C. (2000). Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. SIAM J. Appl. Math. 61: 102-130
    • (2000) SIAM J. Appl. Math. , vol.61 , pp. 102-130
    • Hale, J.K.1    Peletier, L.A.2    Troy, W.C.3
  • 21
    • 0003126227 scopus 로고    scopus 로고
    • Stability and instability of the Gray-Scott model: The case of equal diffusion constants
    • Hale J.K., Peletier L.A. and Troy W.C. (1999). Stability and instability of the Gray-Scott model: the case of equal diffusion constants. Appl. Math. Lett. 12: 59-65
    • (1999) Appl. Math. Lett. , vol.12 , pp. 59-65
    • Hale, J.K.1    Peletier, L.A.2    Troy, W.C.3
  • 22
    • 12944263637 scopus 로고    scopus 로고
    • Stability analysis of Turing patterns generated by the Schnakenberg model
    • Iron D., Wei J. and Winter M. (2004). Stability analysis of Turing patterns generated by the Schnakenberg model. J. Math. Biol. 49: 358-390
    • (2004) J. Math. Biol. , vol.49 , pp. 358-390
    • Iron, D.1    Wei, J.2    Winter, M.3
  • 23
    • 0242626885 scopus 로고    scopus 로고
    • Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model
    • Kolokolnikov T. and Ward M.J. (2003). Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model. Eur. J. Appl. Math. 14: 513-545
    • (2003) Eur. J. Appl. Math. , vol.14 , pp. 513-545
    • Kolokolnikov, T.1    Ward, M.J.2
  • 24
    • 4944224628 scopus 로고    scopus 로고
    • Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model
    • Kolokolnikov T. and Ward M.J. (2004). Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discret. Contin. Dyn. Syst. Ser. B 4: 1033-1064
    • (2004) Discret. Contin. Dyn. Syst. Ser. B , vol.4 , pp. 1033-1064
    • Kolokolnikov, T.1    Ward, M.J.2
  • 25
    • 19044393401 scopus 로고    scopus 로고
    • The existence and stability of spike equilibria in the one- dimensional Gray-Scott model: The low-feed regime
    • Kolokolnikov T., Ward M.J. and Wei J. (2005). The existence and stability of spike equilibria in the one- dimensional Gray-Scott model: the low-feed regime. Stud. Appl. Math. 115: 21-71
    • (2005) Stud. Appl. Math. , vol.115 , pp. 21-71
    • Kolokolnikov, T.1    Ward, M.J.2    Wei, J.3
  • 26
    • 16344362507 scopus 로고    scopus 로고
    • The existence and stability of spike equilibria in the one- dimensional Gray-Scott model: The pulse-splitting regime
    • Kolokolnikov T., Ward M.J. and Wei J. (2005). The existence and stability of spike equilibria in the one- dimensional Gray-Scott model: the pulse-splitting regime. Phys. D 202: 258-293
    • (2005) Phys. D , vol.202 , pp. 258-293
    • Kolokolnikov, T.1    Ward, M.J.2    Wei, J.3
  • 27
    • 0029163545 scopus 로고
    • A viable reaction-diffusion wave on the skin of Pomacanthus, a marine Angelfish
    • Kondo S. and Asai R. (1995). A viable reaction-diffusion wave on the skin of Pomacanthus, a marine Angelfish. Nature 376: 765-768
    • (1995) Nature , vol.376 , pp. 765-768
    • Kondo, S.1    Asai, R.2
  • 28
    • 12044250363 scopus 로고
    • Biological pattern formation from basic mechanisms to complex structures
    • Koch A.J. and Meinhardt H. (1994). Biological pattern formation from basic mechanisms to complex structures. Rev. Mod. Phys. 66: 1481-1507
    • (1994) Rev. Mod. Phys. , vol.66 , pp. 1481-1507
    • Koch, A.J.1    Meinhardt, H.2
  • 29
    • 84972508309 scopus 로고
    • Uniqueness of positive solutions of Δu + f(u) = 0 in an annulus
    • Kwong M.K. and Zhang L. (1991). Uniqueness of positive solutions of Δu + f(u) = 0 in an annulus. Diff. Integral Equ. 4: 583-599
    • (1991) Diff. Integral Equ. , vol.4 , pp. 583-599
    • Kwong, M.K.1    Zhang, L.2
  • 30
    • 12044256206 scopus 로고
    • Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system
    • Lengyel I. and Epstein I.R. (1991). Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science 251: 650-652
    • (1991) Science , vol.251 , pp. 650-652
    • Lengyel, I.1    Epstein, I.R.2
  • 31
    • 0028242433 scopus 로고
    • Experimental observation of self- replicating spots in a reaction-diffusion system
    • Lee K.J., McCormick W.D., Pearson J.E. and Swinney H.L. (1994). Experimental observation of self- replicating spots in a reaction-diffusion system. Nature 369: 215-218
    • (1994) Nature , vol.369 , pp. 215-218
    • Lee, K.J.1    McCormick, W.D.2    Pearson, J.E.3    Swinney, H.L.4
  • 32
    • 0000037738 scopus 로고
    • Pattern formation by interacting chemical fronts
    • Lee K.J., McCormick W.D., Ouyang Q. and Swinney H.L. (1993). Pattern formation by interacting chemical fronts. Science 261: 192-194
    • (1993) Science , vol.261 , pp. 192-194
    • Lee, K.J.1    McCormick, W.D.2    Ouyang, Q.3    Swinney, H.L.4
  • 33
    • 0027101175 scopus 로고
    • The problem of pattern and scale in ecology
    • Levin S.A. (1992). The problem of pattern and scale in ecology. Ecology 73: 1943-1967
    • (1992) Ecology , vol.73 , pp. 1943-1967
    • Levin, S.A.1
  • 34
    • 23944439965 scopus 로고    scopus 로고
    • A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains
    • Madzvamuse A., Maini P.K. and Wathen A.J. (2005). A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput. 24: 247-262
    • (2005) J. Sci. Comput. , vol.24 , pp. 247-262
    • Madzvamuse, A.1    Maini, P.K.2    Wathen, A.J.3
  • 35
    • 0141515669 scopus 로고    scopus 로고
    • A moving grid finite element method applied to a model biological pattern generator
    • Madzvamuse A., Wathen A.J. and Maini P.K. (2003). A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190: 478-500
    • (2003) J. Comput. Phys. , vol.190 , pp. 478-500
    • Madzvamuse, A.1    Wathen, A.J.2    Maini, P.K.3
  • 36
    • 33845326442 scopus 로고    scopus 로고
    • The Turing model comes of molecular age
    • Maini P.K., Baker R.E. and Chuong C.M. (2006). The Turing model comes of molecular age. Science 314: 1397-1398
    • (2006) Science , vol.314 , pp. 1397-1398
    • Maini, P.K.1    Baker, R.E.2    Chuong, C.M.3
  • 37
    • 33748737797 scopus 로고    scopus 로고
    • Spatial pattern formation in chemical and biological systems
    • Maini P.K., Painter K.J. and Chau H. (1997). Spatial pattern formation in chemical and biological systems. J. Chem. Soc. Faraday Trans. 93: 3601-3610
    • (1997) J. Chem. Soc. Faraday Trans. , vol.93 , pp. 3601-3610
    • Maini, P.K.1    Painter, K.J.2    Chau, H.3
  • 40
    • 33746873060 scopus 로고    scopus 로고
    • Reaction-diffusion systems arising in biological and chemical systems: Applications of singular limit procedures
    • Mathematical Aspects of Evolving Interfaces (Funchal, 2000) Springer, Berlin
    • Mimura, M.: Reaction-diffusion systems arising in biological and chemical systems: applications of singular limit procedures. In: Mathematical Aspects of Evolving Interfaces (Funchal, 2000). Lecture Notes in Mathematics, vol. 1812. Springer, Berlin (2003)
    • (2003) Lecture Notes in Mathematics , vol.1812
    • Mimura, M.1
  • 41
    • 0034623753 scopus 로고    scopus 로고
    • Static spike autosolitons in the Gray-Scott model
    • Muratov C.B. and Osipov V.V. (2000). Static spike autosolitons in the Gray-Scott model. J. Phys. A Math. Gen. 33: 8893-8916
    • (2000) J. Phys. A Math. Gen. , vol.33 , pp. 8893-8916
    • Muratov, C.B.1    Osipov, V.V.2
  • 42
    • 0036588860 scopus 로고    scopus 로고
    • Stability of the static spike autosolitons in the Gray-Scott model
    • Muratov C.B. and Osipov V.V. (2002). Stability of the static spike autosolitons in the Gray-Scott model. SIAM J. Appl. Math. 62: 1463-1487
    • (2002) SIAM J. Appl. Math. , vol.62 , pp. 1463-1487
    • Muratov, C.B.1    Osipov, V.V.2
  • 44
    • 0003002827 scopus 로고    scopus 로고
    • Diffusion, cross-diffusion, and their spike-layer steady-states
    • Ni W.-M. (1998). Diffusion, cross-diffusion and their spike-layer steady-states. Not. Am. Math. Soc. 45: 9-18
    • (1998) Not. Am. Math. Soc. , vol.45 , pp. 9-18
    • Ni, W.-M.1
  • 45
    • 1642548932 scopus 로고    scopus 로고
    • Far-From-Equilibrium-Dynamics
    • AMS publications, Providence, Rhode Island
    • Nishiura, Y.: Far-From-Equilibrium-Dynamics, Translations of Mathematical Monographs, vol. 209. AMS publications, Providence, Rhode Island (2002)
    • (2002) Translations of Mathematical Monographs , vol.209
    • Nishiura, Y.1
  • 46
    • 0001347301 scopus 로고
    • Global structure of bifurcating solutions of some reaction-diffusion systems
    • Nishiura Y. (1982). Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13: 555-593
    • (1982) SIAM J. Math. Anal. , vol.13 , pp. 555-593
    • Nishiura, Y.1
  • 47
    • 0000988067 scopus 로고
    • Stability of singularly perturbed solutions to systems of reaction-diffusion equations
    • Nishiura Y. and Fujii H. (1987). Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18: 1726-1770
    • (1987) SIAM J. Math. Anal. , vol.18 , pp. 1726-1770
    • Nishiura, Y.1    Fujii, H.2
  • 48
    • 42749108966 scopus 로고    scopus 로고
    • Scattering and separators in dissipative systems
    • 5
    • Nishiura Y., Teramoto T. and Ueda K. (2003). Scattering and separators in dissipative systems. Phys. Rev. E 67(5): 56210
    • (2003) Phys. Rev. e , vol.67 , pp. 56210
    • Nishiura, Y.1    Teramoto, T.2    Ueda, K.3
  • 49
    • 0347038753 scopus 로고    scopus 로고
    • A skeleton structure of self-replicating dynamics
    • Nishiura Y. and Ueyama D. (1999). A skeleton structure of self-replicating dynamics. Phys. D 130: 73-104
    • (1999) Phys. D , vol.130 , pp. 73-104
    • Nishiura, Y.1    Ueyama, D.2
  • 50
    • 0035311814 scopus 로고    scopus 로고
    • Spatio-temporal chaos for the Gray-Scott model
    • Nishiura Y. and Ueyama D. (2001). Spatio-temporal chaos for the Gray-Scott model. Phys. D 150: 137-162
    • (2001) Phys. D , vol.150 , pp. 137-162
    • Nishiura, Y.1    Ueyama, D.2
  • 51
    • 34047187677 scopus 로고
    • Transition from a uniform state to hexagonal and striped Turing patterns
    • Ouyang Q. and Swinney H.L. (1991). Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610-612
    • (1991) Nature , vol.352 , pp. 610-612
    • Ouyang, Q.1    Swinney, H.L.2
  • 52
    • 0000494953 scopus 로고
    • Transition to chemical turbulence
    • Ouyang Q. and Swinney H.L. (1991). Transition to chemical turbulence. Chaos 1: 411-420
    • (1991) Chaos , vol.1 , pp. 411-420
    • Ouyang, Q.1    Swinney, H.L.2
  • 53
    • 0033545988 scopus 로고    scopus 로고
    • Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis
    • Painter K.J., Maini P.K. and Othmer H.G. (1999). Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Nat. Acad. Sci. USA Dev. Biol. 96: 5549-5554
    • (1999) Proc. Nat. Acad. Sci. USA Dev. Biol. , vol.96 , pp. 5549-5554
    • Painter, K.J.1    Maini, P.K.2    Othmer, H.G.3
  • 54
    • 0001102631 scopus 로고
    • Complex patterns in a simple system
    • Pearson J.E. (1993). Complex patterns in a simple system. Science 261: 189-192
    • (1993) Science , vol.261 , pp. 189-192
    • Pearson, J.E.1
  • 55
    • 0001692679 scopus 로고
    • Turing instabilities with nearly equal diffusion constants
    • Pearson J.E. and Horsthemke W. (1989). Turing instabilities with nearly equal diffusion constants. J. Chem. Phys. 90: 1588-1599
    • (1989) J. Chem. Phys. , vol.90 , pp. 1588-1599
    • Pearson, J.E.1    Horsthemke, W.2
  • 56
    • 0001360757 scopus 로고
    • Dynamics of self-replicating patterns in reaction diffusion systems
    • Reynolds J., Pearson J. and Ponce-Dawson S. (1994). Dynamics of self-replicating patterns in reaction diffusion systems. Phys. Rev. Lett. 72: 2797-2800
    • (1994) Phys. Rev. Lett. , vol.72 , pp. 2797-2800
    • Reynolds, J.1    Pearson, J.2    Ponce-Dawson, S.3
  • 57
    • 0000838693 scopus 로고    scopus 로고
    • Dynamics of self-replicating spots in reaction-diffusion systems
    • Reynolds J., Pearson J. and Ponce-Dawson S. (1997). Dynamics of self-replicating spots in reaction-diffusion systems. Phys. Rev. E 56: 185-198
    • (1997) Phys. Rev. e , vol.56 , pp. 185-198
    • Reynolds, J.1    Pearson, J.2    Ponce-Dawson, S.3
  • 58
    • 11844279687 scopus 로고    scopus 로고
    • Absolute inequalities of standing pulses
    • Sandstede B. and Scheel A. (2005). Absolute inequalities of standing pulses. Nonlinearity 18: 331-378
    • (2005) Nonlinearity , vol.18 , pp. 331-378
    • Sandstede, B.1    Scheel, A.2
  • 59
    • 0018784247 scopus 로고
    • Simple chemical reaction systems with limit cycle behaviour
    • Schnakenberg J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81: 389-400
    • (1979) J. Theor. Biol. , vol.81 , pp. 389-400
    • Schnakenberg, J.1
  • 60
    • 0001371672 scopus 로고
    • Appliations of nonlinear stability theory to the study of the effects of dispersion on predator-prey interactions
    • Piccirelli, R. (ed.). Conference Proceedings No. 27. American Inst. Physics, New York
    • Segel, L.A., Levin, S.A.: Appliations of nonlinear stability theory to the study of the effects of dispersion on predator-prey interactions. In: Piccirelli, R. (ed.) Selected Topics in Statistical Mechanics and Biophysics. Conference Proceedings no. 27. American Inst. Physics, New York (1976)
    • (1976) Selected Topics in Statistical Mechanics and Biophysics
    • Segel, L.A.1    Levin, S.A.2
  • 61
    • 33845301220 scopus 로고    scopus 로고
    • WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism
    • Sick S., Reinker S., Timmer J. and Schlake T. (2006). WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science 314: 1447-1450
    • (2006) Science , vol.314 , pp. 1447-1450
    • Sick, S.1    Reinker, S.2    Timmer, J.3    Schlake, T.4
  • 62
    • 20044376798 scopus 로고    scopus 로고
    • The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: Competition and oscillatory instabilities
    • Sun W., Ward M.J. and Russell R. (2005). The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Sys. 4: 904-953
    • (2005) SIAM J. Appl. Dyn. Sys. , vol.4 , pp. 904-953
    • Sun, W.1    Ward, M.J.2    Russell, R.3
  • 63
    • 0000864203 scopus 로고
    • Point-condensation for a reaction-diffusion system
    • Takagi I. (1986). Point-condensation for a reaction-diffusion system. J. Diff. Equ. 61: 208-249
    • (1986) J. Diff. Equ. , vol.61 , pp. 208-249
    • Takagi, I.1
  • 64
    • 0002011401 scopus 로고
    • The chemical basis of morphogenesis
    • Turing A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237: 37-72
    • (1952) Phil. Trans. R. Soc. Lond. B , vol.237 , pp. 37-72
    • Turing, A.M.1
  • 67
    • 33746911928 scopus 로고    scopus 로고
    • Asymptotic methods for reaction-diffusion systems: Past and present
    • Ward M.J. (2006). Asymptotic methods for reaction-diffusion systems: past and present. Bull. Math. Biol. 68: 1151-1167
    • (2006) Bull. Math. Biol. , vol.68 , pp. 1151-1167
    • Ward, M.J.1
  • 68
    • 0033414750 scopus 로고    scopus 로고
    • On single interior spike solutions of the Gierer-Meinhardt system: Uniqueness and spectrum estimates
    • Wei J. (1999). On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10: 353-378
    • (1999) Eur. J. Appl. Math. , vol.10 , pp. 353-378
    • Wei, J.1
  • 69
    • 0033130430 scopus 로고    scopus 로고
    • Existence, stability and metastability of point condensation patterns generated by Gray-Scott system
    • Wei J. (1999). Existence, stability and metastability of point condensation patterns generated by Gray-Scott system. Nonlinearity 12: 593-616
    • (1999) Nonlinearity , vol.12 , pp. 593-616
    • Wei, J.1
  • 70
    • 0003077491 scopus 로고    scopus 로고
    • Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability
    • Wei J. (2001). Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability. Phys. D 148: 20-48
    • (2001) Phys. D , vol.148 , pp. 20-48
    • Wei, J.1
  • 71
    • 0036809985 scopus 로고    scopus 로고
    • The existence and stability of asymmetric spike patterns for the Schnakenberg model
    • Ward M.J. and Wei J. (2002). The existence and stability of asymmetric spike patterns for the Schnakenberg model. Stud. Appl. Math. 109: 229-264
    • (2002) Stud. Appl. Math. , vol.109 , pp. 229-264
    • Ward, M.J.1    Wei, J.2
  • 72
    • 84867997764 scopus 로고    scopus 로고
    • Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer-Meinhardt model
    • Ward M.J. and Wei J. (2003). Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13: 209-264
    • (2003) J. Nonlinear Sci. , vol.13 , pp. 209-264
    • Ward, M.J.1    Wei, J.2
  • 73
    • 0033249122 scopus 로고    scopus 로고
    • On the two-dimensional Gierer-Meinhardt system with strong coupling
    • Wei J. and Winter M. (1999). On the two-dimensional Gierer-Meinhardt system with strong coupling. SIAM J. Math. Anal. 30: 1241-1263
    • (1999) SIAM J. Math. Anal. , vol.30 , pp. 1241-1263
    • Wei, J.1    Winter, M.2
  • 74
    • 0037137989 scopus 로고    scopus 로고
    • Spikes for the two-dimensional Gierer-Meinhardt system: The strong coupling case
    • Wei J. and Winter M. (2000). Spikes for the two-dimensional Gierer-Meinhardt system: the strong coupling case. J. Diff. Equ. 178: 478-518
    • (2000) J. Diff. Equ. , vol.178 , pp. 478-518
    • Wei, J.1    Winter, M.2
  • 75
    • 0013236903 scopus 로고    scopus 로고
    • Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case
    • Wei J. and Winter M. (2001). Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. J. Nonlinear Sci. 11: 415-458
    • (2001) J. Nonlinear Sci. , vol.11 , pp. 415-458
    • Wei, J.1    Winter, M.2


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