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Pattern formation and wave propagation in the s-a system

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Bifurcation and Nonlinear Eigenvalue Problems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 782))

Abstract

A system of two coupled reaction-diffusion equations involving substrate inhibited enzyme kinetics is studied with a view to describing and explaining stable non uniform steady state solutions and propagating wave front solutions which they admit. The pattern formation phenomenon, reminiscent of morphogenesis, is compared to the predictions of Kauffman for sequential compartment formation in Drosophila imaginal disks. A modified perturbation technique is used to obtain the emerging bifurcation branches.

Numerical analysis of pattern formation needs methods to follow branches of solutions including turning points and bifurcation points. A simple dissipative structure is given in order to test such algorithms.

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References

  1. ASHKENAZI M., OTHMER H.G., Spatial patterns in coupled biochemical oscillators, J. Math. Biol., Vol. 5, 1978, pp. 305–350.

    MathSciNet  MATH  Google Scholar 

  2. BATHE K.J., WILSON E.L., Numerical methods in finite element analysis, Englewood Cliffs, Prentice Hall, 1976.

    MATH  Google Scholar 

  3. BOA J.A., COHEN D.S., Bifurcation of localized disturbances in a model biochemical reaction, Siam. J. Appl. Math., Vol. 30, No 1, 1976.

    Google Scholar 

  4. BRAUNER C.M., NICOLAENKO B., Singular perturbation, multiple solutions, and hysteresis in a nonlinear problem, Lect. Notes in Math., No594, Springer Verlag, 1977, pp. 50–76.

    Google Scholar 

  5. BRITTON N.F., MURRAY J.D., Threshold wave and cell-cell avalanche behavior in a class of substrate inhibition oscillators, J. Theor. Biol. (in press).

    Google Scholar 

  6. BUNOW B., COLTON C.K., Substrate inhibition kinetics in assemblages of cells, Biosystems, 7, 1975, 160–171.

    Article  Google Scholar 

  7. FIFE P.C., Pattern formation in reacting and diffusing systems, J. Chem. Phys. Vol. 64, 1976, pp. 554–564.

    Article  ADS  Google Scholar 

  8. FIFE P.C., Singular perturbations and wave front techniques in reaction-diffusion problems, Siam-AMS Proceedings, Vol. 10, 1976, pp. 23–50.

    MathSciNet  Google Scholar 

  9. FIFE P.C., Stationary patterns for reaction-diffusion equations, MRC Technical Summary Report, 1976, pp. 1–50.

    Google Scholar 

  10. FIFE P.C., Asymptotic states for equations of reaction and diffusion, Bull. Am. Math. Soc., Vol. 84, No5, 1978, pp. 693–726.

    Article  MathSciNet  MATH  Google Scholar 

  11. HERSCHKOWITZ-KAUFFMAN M., NICOLIS G., Localized spatial structures and non-linear chemical waves in dissipative systems, J. Chem. Phys., Vol. 56, 1972, pp. 1890–1895.

    Article  ADS  Google Scholar 

  12. IOOSS G., Bifurcation et stabilité, Cours de 3ème Cycle, Université de Paris XI, 1972.

    Google Scholar 

  13. KAUFFMAN S.A., SHYMKO R.M., TRABERT K., Control of sequential compartment formation in Drosophila, Science, Vol. 199, 1978, pp. 259–270.

    Article  ADS  MATH  Google Scholar 

  14. KERNEVEZ J.P., THOMAS D., Numerical analysis and control of some biochemical systems, Appl. Math. and Opt., Vol. 1, No3, 1975.

    Google Scholar 

  15. KERNEVEZ J.P., DUBAN M.C., JOLY G., THOMAS D., Hysteresis, oscillations and morphogenesis in immobilized enzyme systems in "The significance of nonlinearity in the natural sciences", Ed. Perlmutter and Scott, Plenum, 1977, pp.327–353.

    Google Scholar 

  16. KERNEVEZ J.P., JOLY G., DUBAN M.C., BUNOW B., THOMAS D., Hysteresis, oscillations and pattern formation in realistic immobilized enzyme systems, J. Math. Biol., 7,1979,41–56.

    Article  MathSciNet  MATH  Google Scholar 

  17. KERNEVEZ J.P., MURRAY J.D., JOLY G., DUBAN M.C., THOMAS D., Propagation d'onde dans un système à enzyme immobilisée, C.R.A.S., 287-A, 1978, 961–964.

    MathSciNet  MATH  Google Scholar 

  18. KOGELMAN S., KELLER J.B., Transient behavior of unstable nonlinear systems with applications to the Benard and Taylor problems, Siam. J. Appl. Math., Vol. 20, No4, 1971, pp. 619–637.

    Article  MathSciNet  MATH  Google Scholar 

  19. MATKOWSKY B.J., A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., Vol. 76, 1970, pp. 620–625.

    Article  MathSciNet  MATH  Google Scholar 

  20. MARSDEN J.E., MAC CRACKEN M., The Hopf bifurcation and its applications, Applied Mathematical Sciences 19, Springer Verlag, New York, 1976.

    Google Scholar 

  21. MEURANT G., SAUT J.C., Bifurcation and stability in a chemical system, J. Math. Anal. and Appl., Vol. 59, 1, 1977, pp. 69–92.

    Article  MathSciNet  MATH  Google Scholar 

  22. MIMURA M., MURRAY J.D., Spatial structures in a model substrate-inhibition reaction diffusion system, Z. für Natürfosch, 33C, 1978, 580–586.

    Google Scholar 

  23. MURRAY J.D., Nonlinear differential equation models in biology, Clarendon, Oxford, 1977.

    MATH  Google Scholar 

  24. NAPARSTEK A., ROMETTE J.L., KERNEVEZ J.P., THOMAS D., Memory in enzyme membranes, Nature, Vol. 249, 1974, p. 490.

    Article  ADS  Google Scholar 

  25. NICOLIS G., PRIGOGINE I., Self-organization in nonequilibrium systems, Wiley Interscience, 1977.

    Google Scholar 

  26. ORTOVELA P., ROSS J., Theory of propagation of discontinuities in kinetic systems with multiple time scales: front, front multiplicity, and pulses, J. Chem. Phys., Vol. 63, No8, 1975, pp. 3398–3408.

    Article  ADS  Google Scholar 

  27. OTHMER H.G., Nonlinear wave propagation in reacting systems, J. Math. Biol., Vol. 2, 1975, pp. 133–163.

    Article  MathSciNet  MATH  Google Scholar 

  28. OTHMER H.G., SCRIVEN L.E., Instability and dynamic pattern in cellular networks J. Theor. Biol., Vol. 32, 1971, pp. 507–537.

    Article  Google Scholar 

  29. SATTINGER D.H., Topics in stability and bifurcation theory, Lect. Notes in Math., No 309, Springer Verlag, 1973.

    Google Scholar 

  30. SPREY T.H., Aldehyde oxidase distribution in the imaginal disks of some diptera, Wilhelm Roux's Archives 183, 1–15 (1977).

    Article  Google Scholar 

  31. THOMAS D., BARBOTIN J.N., DAVID A., HERVAGAULT J.F., ROMETTE J.L., Experimental evidence for a kinetic and electrochemical memory in enzyme membranes, Proc. Natl. Sci. USA, Vol. 74, No12, 1977, pp. 5314–5317.

    Article  ADS  Google Scholar 

  32. THOMAS D., BROUN G., Artificial enzyme membranes, Methods in Enzymology, Vol. 44, 1976, pp. 901–929.

    Article  Google Scholar 

  33. TURING A.M., The chemical basis of morphogenesis, Phil. Trans. Roy. Soc., Vol. B237, 1952, pp. 37–72.

    Article  ADS  MathSciNet  Google Scholar 

  34. KELLER H.B., Perturbation theory, Notes on a series of six lectures presented at the Department of Mathematics, Michigan State University, East Lansing, Michigan, 1968, pp. 1–64.

    Google Scholar 

  35. OTHMER H.G., Current problems in pattern formation, S.A. Levin, ed., Lectures on Mathematics in the Life Sciences, Vol. 9: Some Mathematical Questions in Biology VIII (1976), pp. 57–85.

    Google Scholar 

  36. IOOSS, personal communication.

    Google Scholar 

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C. Bardos J. M. Lasry M. Schatzman

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© 1980 Springer-Verlag

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Kernevez, J.P., Joly, G., Thomas, D., Bunow, B. (1980). Pattern formation and wave propagation in the s-a system. In: Bardos, C., Lasry, J.M., Schatzman, M. (eds) Bifurcation and Nonlinear Eigenvalue Problems. Lecture Notes in Mathematics, vol 782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090434

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  • DOI: https://doi.org/10.1007/BFb0090434

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  • Print ISBN: 978-3-540-09758-7

  • Online ISBN: 978-3-540-38637-7

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