Journal of Dynamics and Differential Equations

, Volume 21, Issue 4, pp 663–680

Monotone Wavefronts for Partially Degenerate Reaction-Diffusion Systems



This paper is devoted to the study of monotone wavefronts for cooperative and partially degenerate reaction-diffusion systems. The existence of monostable wavefronts is established via the vector-valued upper and lower solutions method. It turns out that the minimal wave speed of monostable wavefronts coincides with the spreading speed. The existence of bistable wavefronts is obtained by the vanishing viscosity approach combined with the properties of spreading speeds in the monostable case.


Degenerate reaction-diffusion systems Spreading speeds Monostable and bistable wavefronts Minimal wave speeds 

Mathematics Subject Classification (2000)

35K57 35B40 35B20 


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  1. 1.
    Capasso V., Maddalena L.: Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173–184 (1981)MATHMathSciNetGoogle Scholar
  2. 2.
    Chen X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)MATHGoogle Scholar
  3. 3.
    Hadeler K.P., Lewis M.A.: Spatial dynamics of the diffusive logistic equation with a sedentary compartment. Can. Appl. Math. Q. 10, 473–499 (2002)MATHMathSciNetGoogle Scholar
  4. 4.
    Jin Y., Zhao X.-Q.: Bistable waves for a class of cooperative reaction-diffusion systems. J. Biol. Dyn. 2, 196–207 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kazmierczak B., Volpert V.: Calcium aves in systems with immobile buffers as a limit of waves for systems with nonzero diffusion. Nonlinearity 21, 71–96 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Li B., Weinberger H.F., Lewis M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Liang X., Yi Y., Zhao X.-Q.: Spreading speeds and traveling waves for periodic evolution systems. J. Differ. Equ. 231, 57–77 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (Erratum: 61(2008), 137–138)Google Scholar
  9. 9.
    Ma S.: Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem. J. Differ Equ. 171, 294–314 (2001)MATHCrossRefGoogle Scholar
  10. 10.
    Murray J.D.: Mathematical Biology, I & II, Interdisciplinary Applied Mathematics. Springer, New York (2003)Google Scholar
  11. 11.
    Lewis M.A., Schmitz G.: Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis. Forma 11, 1–25 (1996)MATHMathSciNetGoogle Scholar
  12. 12.
    Shen W.: Traveling waves in time periodic lattice differential equations. Nonl. Anal. 54, 319–339 (2003)MATHCrossRefGoogle Scholar
  13. 13.
    Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math Surveys and Monographs, vol. 41. American Mathematical Society, Providence, RI (1995)Google Scholar
  14. 14.
    Smoller J.: Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften 258. Springer, New York (1994)Google Scholar
  15. 15.
    Thieme H.R.: Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306, 94–121 (1979)MATHMathSciNetGoogle Scholar
  16. 16.
    Thieme H.R., Zhao X.-Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J. Differ. Equ. 195, 430–470 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Tsai J.-C.: Asymptotic stability of traveling wave fronts in the buffered bistable system. SIAM J. Math. Anal. 39, 138–159 (2007)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Tsai J.-C.: Global exponential stability of traveling waves in monotone bistable systems. Discrete Contin. Dyn. Syst. A 21, 601–623 (2008)MATHGoogle Scholar
  19. 19.
    Tsai J.-C., Sneyd J.: Existence and stability of traveling waves in buffered systems. SIAM J. Appl. Math. 66, 237–265 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, vol. 140. American Mathematical Society, Providence, RI (1994)Google Scholar
  21. 21.
    Wang Q., Zhao X.-Q.: Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment. Dyn. Contin. Discrete Impuls. Syst. A 13, 231–246 (2006)MATHGoogle Scholar
  22. 22.
    Weinberger H.F.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Weinberger H.F., Lewis M.A., Li B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Weng P., Zhao X.-Q.: Spreading speed and traveling waves for a multi-type SIS epidemic model. J. Differ. Equ. 229, 270–296 (2006)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Xu D., Zhao X.-Q.: Bistable waves in an epidemic model. J. Dyn. Differ. Equ. 16, 679–707 (2004)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Zhang F., Zhao X.-Q.: Asymptotic behavior of a reaction-diffusion model with a quiescent stage. Proc. R. Soc. Lond. Ser. A 463, 1029–1043 (2007)MATHCrossRefGoogle Scholar
  27. 27.
    Zhao X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)MATHGoogle Scholar
  28. 28.
    Zhao X.-Q., Wang W.: Fisher waves in an epidemic model. Disc. Cont. Dyn. Syst. B 4, 1117–1128 (2004)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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