Journal of Dynamics and Differential Equations

, Volume 26, Issue 3, pp 583–605 | Cite as

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

  • Guo LinEmail author
  • Shigui Ruan


This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.


Nonmonotone traveling wave solutions Contracting rectangle Invariant region Generalized upper and lower solutions 

Mathematics Subject Classification

35C07 37C65 35K57 



The authors would like to thank an anonymous reviewer for his/her helpful comments and Yanli Huang for her valuable suggestions. This research was partially supported by the the National Natural Science Foundation of China (11101194) and the National Science Foundation (DMS-1022728).


  1. 1.
    Ahmad, S., Lazer, A.C.: An elementary approach to traveling front solutions to a system of \(N\) competition–diffusion equations. Nonlinear Anal. TMA 16, 893–901 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ai, S.: Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differ. Equ. 232, 104–133 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J.A. (ed.) Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, New York (1975)CrossRefGoogle Scholar
  4. 4.
    Gourley, S.A., Ruan, S.: Convergence and traveling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J. Math. Anal. 35, 806–822 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fang, J., Wu, J.: Monotone traveling waves for delayed Lotka–Volterra competition systems. Discrete Contin. Dyn. Syst. Ser. A 32, 3043–3058 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Faria, T., Huang, W., Wu, J.: Traveling waves for delayed reaction–diffusion equations with global response. Proc. R. Soc. Lond. 462A, 229–261 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Faria, T., Trofimchuk, S.: Nonmonotone travelling waves in a single species reaction–diffusion equation with delay. J. Differ. Equ. 228, 357–376 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Faria, T., Trofimchuk, S.: Positive travelling fronts for reaction–diffusion systems with distributed delay. Nonlinearity 23, 2457–2481 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Springer-Verlag, Berlin (1979)CrossRefzbMATHGoogle Scholar
  10. 10.
    Guo, J.S., Liang, X.: The minimal speed of traveling fronts for the Lotka–Volterra competition system. J. Dyn. Differ. Equ. 23, 353–363 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional–Differential Equations. Springer-Verlag, New York (1993)zbMATHGoogle Scholar
  12. 12.
    Huang, J., Zou, X.: Existence of traveling wavefronts of delayed reaction–diffusion systems without monotonicity. Discrete Cont. Dyn. Sys. Ser. B 9, 925–936 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Huang, W.: Problem on minimum wave speed for a Lotka–Volterra reaction–diffusion competition model. J. Dyn. Differ. Equ. 22, 285–297 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kwong, M.K., Ou, C.: Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J. Differ. Equ. 249, 728–745 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Li, W.T., Lin, G., Ruan, S.: Existence of traveling wave solutions in delayed reaction–diffusion systems with applications to diffusion-competition systems. Nonlinearity 19, 1253–1273 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Liang, X., Zhao, X.Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. 60, 1–40 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lin, G., Li, W.T., Ma, M.: Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin. Dyn. Syst. Ser. B 19, 393–414 (2010)MathSciNetGoogle Scholar
  18. 18.
    Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Ma, S.: Traveling waves for non-local delayed diffusion equations via auxiliary equations. J. Differ. Equ. 237, 259–277 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ma, S., Wu, J.: Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation. J. Dyn. Differ. Equ. 19, 391–436 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Martin, R.H., Smith, H.L.: Reaction–diffusion systems with the time delay: monotonicity, invariance, comparison and convergence. J. Reine. Angew. Math. 413, 1–35 (1991)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Mei, M., Lin, C.-K., Lin, C.-T., So, J.W.-H.: Traveling wavefronts for time-delayed reaction–diffusion equation: (I) local nonlinearity. J. Differ. Equ. 247, 495–510 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ou, C., Wu, J.: Persistence of wavefronts in delayed nonlocal reaction–diffusion equations. J. Differ. Equ. 235, 219–261 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ruan, S.: Delay differential equations. In: Arino, O., Hbid, M., Ait Dads, E. (eds.) Delay Differential Equations with Applications. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 205, pp. 477–517. Springer-Verlag, Berlin (2006)CrossRefGoogle Scholar
  25. 25.
    Ruan, S., Wu, J.: Reaction–diffusion equations with infinite delay. Canad. Appl. Math. Quart. 2, 485–550 (1994)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Schaaf, K.W.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Am. Math. Soc. 302, 587–615 (1987)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997)Google Scholar
  28. 28.
    Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. AMS, Providence, RI (1995)zbMATHGoogle Scholar
  29. 29.
    Smith, H.L., Zhao, X.Q.: Global asymptotic stability of traveling waves in delayed reaction–diffusion equations. SIAM J. Math. Anal. 31, 514–534 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Tang, M.M., Fife, P.: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 73, 69–77 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. AMS, Providence, RI (1994)Google Scholar
  33. 33.
    Wang, H.Y.: On the existence of traveling waves for delayed reaction–diffusion equations. J. Differ. Equ. 247, 887–905 (2009)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Z.C., Li, W.T., Ruan, S.: Traveling wave fronts of reaction–diffusion systems with spatio-temporal delays. J. Differ. Equ. 222, 185–232 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Wang, Z.C., Li, W.T., Ruan, S.: Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Differ. Equ. 238, 153–200 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Wang, Z.C., Li, W.T., Ruan, S.: Traveling fronts in monostable equations with nonlocal delayed effects. J. Dyn. Differ. Equ. 20, 573–603 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer-Verlag, New York (1996)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wu, J., Zou, X.: Traveling wave fronts of reaction–diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Ye, Q., Li, Z., Wang, M.X., Wu, Y.: Introduction to Reaction–Diffusion Equations, 2nd edn. Science Press, Beijing (2011)Google Scholar
  40. 40.
    Yi, T., Chen, Y., Wu, J.: Unimodal dynamical systems: comparison principles, spreading speeds and travelling waves. J. Differ. Equ. 254, 3538–3572 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Zou, X.: Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type. J. Comput. Appl. Math. 146, 309–321 (2002)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations