Abstract
A model nonlinear network involving chemical reactions and diffusion is studied. The time evolution and bounds on the steady state solutions are analyzed. Spatially ordered solutions of the equations of the dissipative structure type are found by bifurcation theory. These solutions are calculated analytically and their qualitative properties are discussed.
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Literature
Babloyantz, A. and J. Hiernaux. 1974. “Models for Positional Information and Positional Differentiation.”Proc. Nat. Acad. Sci., U.S.A.,71, 1530–1533.
Babloyantz, A. and J. Hiernaux. 1975. “Models for Cell Differentiation and Generation of Polarity in Diffusion Governed Morphogenetic Fields.”Bull. Math. Biology (in press).
Boa, J. 1974. “A Model Biochemical Reaction.” Ph.D. Thesis, California Inst. Tech.
Eigen, M. 1971. “Self Organization of Matter and the Evolution of Biological Macromolecules.”Naturwissenschaften,58, 465–523.
Glansdorff, P. and I. Prigogine. 1971.Thermodynamics of Structure, Stability and Fluctuations. New York: Wiley-Interscience.
Goldbeter, A. 1973. “Patterns of Spatiotemporal Organization in an Allosteric Enzyme Model.”Proc. Nat. Acad. Sci., U.S.A.,70, 3255–3259.
Hanson, M. P. 1974. “Spatial Structures in Dissipative Systems.”J. Chem. Phys.,60, 3210–3214.
Hanusse, P. 1972. “De l'Existence d'un Cycle Limite dans l'Évolution des Systèmes Chimiques Ouverts.”C.R. Acad. Sci.,274, 1245–1247.
Herschkowitz-Kaufman, M. 1975. “Bifurcation Analysis of Nonlinear Reaction-Diffusion Equations II.”Bull. Math. Biology (in press).
Kopell, N. and L. N. Howard. 1973. “Horizontal Bands in the Belousov Reaction.”Science,180, 1171–1173.
Lions, J. L. 1969.Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires. Paris: Dunod.
Nicolis, G. and J. F. G. Auchmuty. 1974. “Dissipative Structures, Catastrophes and Pattern Formation: A Bifurcation Analysis.”Proc. Nat. Acad. Sci., U.S.A.,71, 2748–2751.
— and J. Portnow. 1973. “Chemical Oscillations.”Chem. Rev.,73, 365–384.
Prigogine, I., R. Lefever, A. Goldbeter and M. Herschkowitz-Kaufman. 1969. “Symmetry Breaking Instabilities in Biological Systems.”Nature,223, 913–916.
— and R. Lefever. 1974. “Stability and Self-Organization in Open Systems.” InMembranes, Dissipative Structures and Evolution, Nicolis, G. and Lefever, R., eds. New York: Wiley.
—, G. Nicolis and A. Babloyantz. 1972. “Thermodynamics of Evolution.”Physics Today,11, 23–28 and12, 38–44.
Protter, M. H. and H. F. Weinberger. 1967.Maximum Principles in Differential Equations. Englewood Cliffs: Prentice Hall.
Sattinger, D. 1973.Lecture Notes in Mathematics, Vol. 309. Berlin: Springer.
—, I. Stakgold and D. Joseph. 1973.Lecture Notes in Mathematics, Vol. 322. Berlin: Springer.
Stakgold, I. 1971. “Branching of Solutions of Nonlinear Equations.”SIAM Rev.,13, 289–332.
Turing, A. M. 1952. “The Chemical Basis of Morphogenesis.”Phil. Trans. Roy. Soc., London,B 237, 37–72.
Tyson, J. 1973. “Some Further Studies of Nonlinear Oscillations in Chemical Systems.”J. Chem. Phys.,58, 3919–3930.
— and J. Light. 1973. “Properties of Two-component Bimolecular and Trimolecular Chemical Reaction Systems.”J. Chem. Phys.,59, 4164–4173.
Winfree, A. M. 1974. “Rotating Solutions of Reaction-Diffusion Equations in Simply Connected Media.” InProc. Symp. in Appl. Math., Vol. 8. Providence: American Mathematical Society.
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Auchmuty, J.F.G., Nicolis, G. Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions. Bltn Mathcal Biology 37, 323–365 (1975). https://doi.org/10.1007/BF02459519
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DOI: https://doi.org/10.1007/BF02459519