Current Topics in Reaction-Diffusion Systems

  • Paul C. Fife
Part of the NATO ASI Series book series (NSSB, volume 116)


In these lectures, I want to outline various topics of high current interest in the theory of reaction-diffusion systems. I say “theory”, but in fact they all reside in the applied side of the theory, as in all cases they are motivated by applications in the natural sciences. RD systems, in fact, provide an extremely fertile source of models in all these sciences. In accordance with the theme of the conference, the applications will by and large be related to dissipative structures of one kind or another.


Singular Perturbation Dissipative Structure Left Branch Solution Branch Target Pattern 


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  1. D. G. Aronson and H. F. Weinberger 1978, Adv. in Math. 30, 33–76.MathSciNetMATHCrossRefGoogle Scholar
  2. D. S. Cohen, F. C. Hoppensteadt, and R. M. Miura (1977), SIAM J. Appl. Math. 33, 217–229.MathSciNetMATHCrossRefGoogle Scholar
  3. D. S. Cohen, J. C. Neu, and R. R. Rosales 1978, SIAM J. Appl. Math., 35, 536.MathSciNetMATHCrossRefGoogle Scholar
  4. G. B. Ermentrout 1981a, Quart. Appl. Math., Apr. 1981, 61–86. 1981b, preprints on stationary homogeneous media.MathSciNetGoogle Scholar
  5. R.J. Field, E. Koros, and R. M. Noyes 1972, J. Amer. Chem. Soc. 94, 8649.CrossRefGoogle Scholar
  6. P. C. Fife 1974, J. Differential Equations 15, 77–105.MathSciNetMATHCrossRefGoogle Scholar
  7. P. C. Fife 1976a, J. Chem. Phys. 64, 854–864.CrossRefGoogle Scholar
  8. P. C. Fife 1976b, J. Math. Anal, and Appls. 54, 497–521.MathSciNetMATHCrossRefGoogle Scholar
  9. P. C. Fife 1976c, SIAM-AMS Proceedings, Symp. on Asymptotic Methods and Singular Perturbations, New York, 23–49.Google Scholar
  10. P. C. Fife 1977 J. Math. Biol. 4, 358–362.MathSciNetCrossRefGoogle Scholar
  11. P. C. Fife 1978, Bull. Amer. Math. Soc. 84, 693–726.MathSciNetMATHCrossRefGoogle Scholar
  12. P. C. Fife 1979, pp. 143–160 of Applied Nonlinear Analysis, V. Lakshmikantham, ed., Academic Press, New York.Google Scholar
  13. P. C. Fife 1980, Math. Research Center TSR No. 2110.Google Scholar
  14. P. C. Fife 1981, pp. 45–56 in Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Axelsson, Frank, and Van der Sluis, ed., Mathematics Studies 47, North-Holland, Amsterdam.CrossRefGoogle Scholar
  15. P. C. Fife and J. B. McLeod 1977, Arch. Rational Mech. Anal. 65, 335–361.MathSciNetADSMATHCrossRefGoogle Scholar
  16. P. C. Fife and J. B. McLeod 1981, Arch. Rational Mech. Anal. 75, 281–314.MathSciNetADSMATHCrossRefGoogle Scholar
  17. R. A. Fisher 1937, Ann. of Eugenics 7, 355–369.CrossRefGoogle Scholar
  18. H. Fujii, M. Mimura and Y. Nishiura 1982, Physica D-Nonlinear Phenomena 5D, 1–42.MathSciNetADSCrossRefGoogle Scholar
  19. H. Fujii and Y. Nishiura 1982, Proceedings, U.S.-Japan Seminar on Nonlinear Partial Diff. Eqns. July 1982, Tokyo.Google Scholar
  20. A. Gierer and H. Meinhardt 1972, Kybernetika 12, 30–39.CrossRefGoogle Scholar
  21. J. M. Greenberg 1978, SIAM J. Appl. Math. 34, 391–397.MathSciNetMATHCrossRefGoogle Scholar
  22. J. M. Greenberg 1980 SIAM J. Appl. Math 39, 301–309.MathSciNetMATHCrossRefGoogle Scholar
  23. P. Hagan 1980, manuscript on spiral waves. V. G. Jahno 1975, Biofizika 20, 669–674.Google Scholar
  24. Ya. I. Kanel’ 1962, Mat. Sbornik 59, 245–288.MathSciNetGoogle Scholar
  25. J. P. Keener 1980, SIAM J. Appl. Math. 39, 528–548.MathSciNetMATHCrossRefGoogle Scholar
  26. S. Koga and Y. Kuramoto 1980, Prog. Theor. Phys. 63, 106–121.ADSCrossRefGoogle Scholar
  27. N. Kopell and L. N. Howard 1973, Stud. Appl. Math. 52, 291–328.MathSciNetMATHGoogle Scholar
  28. N. Kopell and L. N. Howard 1979, Stud. Appl. Math. 64 (8), 1–56.MathSciNetGoogle Scholar
  29. Y. Kuramoto 1984, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics #19.MATHCrossRefGoogle Scholar
  30. Y. Kuramoto and S. Koga 1981, Prog. Theor. Phys. 66, 1081–1085.ADSCrossRefGoogle Scholar
  31. Y. Kuramoto and T. Tsuzuki 1975, Prog. Theor. Phys. 54, 687.ADSCrossRefGoogle Scholar
  32. M. Mimura, M. Tabata and Y. Hosono 1980, SIAM J. Math. Anal. 11, 613–631.MathSciNetMATHCrossRefGoogle Scholar
  33. A. C. Newell 1974 Lectures in Appl. Mathematics, Vol. 15, Nonlinear Wave Motion, A. C. Newell, ed., Amer. Math. Soc, Providence.Google Scholar
  34. A. C. Newell and J. A. Whitehead 1969, J. Fluid Mech. 38, 279.ADSMATHCrossRefGoogle Scholar
  35. Y. Nishiura 1982, SIAM J. Math. Anal. 13, 555–593.MathSciNetADSMATHCrossRefGoogle Scholar
  36. Y. Nishiura 1983, Proc, Workshop on modelling of patterns in space and time, Heidelberg, July, 1983.Google Scholar
  37. P. Ortoleva and J. Ross 1975, J. Chem. Phys. 63, 3398–3408.ADSCrossRefGoogle Scholar
  38. L. A. Ostrovskii and V. G. Yahno 1975, Biofizika 20, 489–493.Google Scholar
  39. O. E. Rossler and C. Kahlert 1979, Z. Naturforsch. 34a, 565–570.ADSGoogle Scholar
  40. M.-L. Smoes 1980, pp. 80–96 in Dynamics of Synergetic Systems, H. Haken, ed., Springer-Verlag, Berlin.CrossRefGoogle Scholar
  41. J. Smoller 1983, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  42. J. Tyson 1979, Ann. N. Y. Acad. Sci. 36, 279–295.MathSciNetADSCrossRefGoogle Scholar
  43. J. Tyson 1982, preprint on reducing the F-K-N mechanism.Google Scholar
  44. J. Tyson and P. C. Fife 1980, J. Chem. Phys. 73, 2224–2237.MathSciNetADSCrossRefGoogle Scholar
  45. A. T. Winfree 1972, Science 175, 634–636.ADSCrossRefGoogle Scholar
  46. A. T. Winfree 1978, Theor. Chem. Vol. 4, Academic Press, New York, pp. 1–51.Google Scholar
  47. A. N. Zaikin and A. M. Zhabotinsky 1970, Nature 225, 535–537.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • Paul C. Fife
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaUSA

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