Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations


We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].


Reaction-diffusion equations FKPP equation traveling waves critical wave speeds geometric desingularization blow-up technique 

Mathematics Subject Classification, 1991

35K57 34E15 34E05 


  1. 1.
    Abramowitz, M. A., and Stegun, I. A. (eds.), (1974). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, Inc., New York, Ninth printing.Google Scholar
  2. 2.
    Bender C.M., Orszag S.A. (1978). Advanced Mathematical Methods for Scientists and Engineers. Mc Graw-Hill, Inc., New YorkMATHGoogle Scholar
  3. 3.
    Billingham J., Needham D.J. (1991). A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis. Dynam. Stabil. Sys. 6(1): 33–49MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Britton N.F. (1986). Reaction-Diffusion Equations and Their Applications to Biology. Academic Press Inc., LondonMATHGoogle Scholar
  5. 5.
    Carr, J. (1981). Applications of Centre Manifold Theory, volume 35, in Applied Mathematical Sciences. Springer-Verlag, New York.Google Scholar
  6. 6.
    Casten R.G., Cohen H., Lagerstrom P.A. (1975). Perturbation analysis of an approximation to the Hodgkin-Huxley theory. Quart. Appl. Math. 32, 365–402MATHMathSciNetGoogle Scholar
  7. 7.
    Chow S.N., Lin X.B. (1990). Bifurcation of a homoclinic orbit with a saddle-node equilibrium. Diff. Int. Equ. 3(3): 435–466MATHMathSciNetGoogle Scholar
  8. 8.
    Chow S.N., Li C., Wang D. (1994). Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, CambridgeMATHGoogle Scholar
  9. 9.
    Denkowska Z., Roussarie R. (1991). A method of desingularization for analytic two-dimensional vector field families. Bol. Soc. Bras. Mat. 22(1): 93–126MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Diener M. (1994). The canard unchained or how fast/slow dynamical systems bifurcate. Math. Intelligencer 6(3): 38–49MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dumortier, F. (1993). Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, In Schlomiuk, D. (ed.), Bifurcations and Periodic Orbits of Vector Fields, number 408 in NATO ASI Series C, Mathematical and Physical Sciences, Dordrecht, Kluwer Academic Publishers.Google Scholar
  12. 12.
    Fenichel N. (1971). Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fenichel N. (1979). Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equs. 31(1): 53–98MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fisher R.A. (2000). The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369Google Scholar
  15. 15.
    Guckenheimer J., Hoffman K., Weckesser W. (2000). Numerical computation of canards. Int. J. Bifur. Chaos Appl. Sci. Engrg. 10(12): 2669–2687MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jones, C. K. R. T. (1995). Geometric singular perturbation theory. In Dynamical Systems, volume 1609 of Springer Lecture Notes in Mathematics, Springer-Verlag, New York.Google Scholar
  17. 17.
    Kolmogorov A.N., Petrowskii I.G., Piscounov N. (1997). Etude de l’équation de la diffusion avec croissance de la quantité de matiére et son application à un problème biologique. Moscow Univ. Math. Bull. 1, 1–25Google Scholar
  18. 18.
    Krupa M., Szmolyan P. (2001). Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2): 286–314MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Merkin J.H., Needham D.J. (1993). Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. J. Appl. Math. Phys. (ZAMP) A 44(4): 707–721MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Murray J.D. (2002). Mathematical Biology, I: An Introduction, volume 17 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin Heidelberg, third editionGoogle Scholar
  21. 21.
    Needham D.J., Barnes A.N. (1999). Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations. Nonlinearity 12(1): 41–58MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Robinson C. (1983). Sustained resonance for a nonlinear system with slowly varying coefficients. SIAM J. Math. Anal. 14(5): 847–860MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Salam F.M.A. (1987). The Mel’nikov technique for highly dissipative systems. SIAM J. Appl. Math. 47(2): 232–243MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sherratt J.A., Marchant B.P. (1996). Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation. IMA J. Appl. Math. 56(3): 289–302MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Witelski T.P., Ono K., Kaper T.J. (2001). Critical wave speeds for a family of scalar reaction-diffusion equations. Appl. Math. Lett. 14(1): 65–73MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Center for BioDynamics and Department of Mathematics and StatisticsBoston UniversityBostonUSA

Personalised recommendations