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Keener, James P.

Activators and inhibitors in pattern formation. (English) Zbl 0407.92023

Studies Appl. Math. 59, 1-23 (1978).
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Cited in 46 Documents

MSC:

92Exx Chemistry
92B05 General biology and biomathematics
93C20 Control/observation systems governed by partial differential equations
92Cxx Physiological, cellular and medical topics
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35K45 Initial value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations

Keywords:

Activator-Inhibitor Reaction Dynamics; Pattern Formation; Nonlinear Diffusion; Reacting Chemical System; Gierer Meinhardt Model; Stability; Bifurcation; Initial Value Problem
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References:

[1] Aronson, Lecture Notes in Mathematics, #446, in: Proceedings of the Tulane Program in Partial Differential Equations and Related Topics (1975)
[2] Fife, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Bull. Amer. Math. Soc. 81 pp 1075– (1975) · Zbl 0318.35046 · doi:10.1090/S0002-9904-1975-13922-X
[3] Fisher, The wave of advances of advantageous genes, Ann. Eugenics 1 pp 355– (1936)
[4] McKean, Nagumo’s equation, Advances in Math. 4 pp 209– (1970) · Zbl 0202.16203 · doi:10.1016/0001-8708(70)90023-X
[5] Segal, Dissipative structure: an explanation and an ecological example, J. Theor. Biol 37 pp 545– (1972) · doi:10.1016/0022-5193(72)90090-2
[6] Gierer, A theory of biological pattern formation, Kybernetik 12 pp 30– (1972) · doi:10.1007/BF00289234
[7] Child, Patterns and Problems of Development (1941) · doi:10.5962/bhl.title.6415
[8] Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theor. Biol. 25 pp 1– (1969) · doi:10.1016/S0022-5193(69)80016-0
[9] Waddington, Principles of Embryology (1962)
[10] Nicolis, Dissipative structures, catastrophes and pattern formation: a bifurcation analysis, Proc. Nat. Acad. Sci 71 pp 2748– (1974) · Zbl 0303.76045 · doi:10.1073/pnas.71.7.2748
[11] Boa, Bifurcation of localized disturbances on a model biochemical reaction, SIAM J. Appl. Math. 30 pp 123– (1976) · Zbl 0328.76065 · doi:10.1137/0130015
[12] Fife, Pattern formation in reacting and diffusing systems, J. Chem. Phys. 64 pp 554– (1976)
[13] Proc. AMS-SIAM Symposium on Asymtotic Method and singular Perturbations (1976)
[14] Keener, Pattern formation in reacting and diffusing systems, J. Chem. Phys. 64 pp 554– (1976) · doi:10.1063/1.432246
[15] Turing, The chemical basis of morphogenisis, Philos. Trans. Roy. Soc. London Ser. B 237 pp 32– (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
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