Existence, Uniqueness and Asymptotic Stability of Traveling Wavefronts in A Non-Local Delayed Diffusion Equation


In this paper, we study the existence, uniqueness, and global asymptotic stability of traveling wave fronts in a non-local reaction–diffusion model for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. Under realistic assumptions on the birth function, we construct various pairs of super and sub solutions and utilize the comparison and squeezing technique to prove that the equation has exactly one non-decreasing traveling wavefront (up to a translation) which is monotonically increasing and globally asymptotic stable with phase shift.


Non-local reaction-diffusion equation traveling wave front existence uniqueness asymptotic stability comparison principle 

AMS (1991) Subject Classification

34K30 35B40 35R10 58D25 


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  1. 1.
    Chen X. (1997). Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eq. 2, 125–160MATHGoogle Scholar
  2. 2.
    Faria T., Huang W., Wu J. (2006). Traveling waves for delayed reactiondiffusion equations with non-local response. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462, 229–261MathSciNetMATHGoogle Scholar
  3. 3.
    Gourley S.A., So J.W.-H., Wu J.H. (2004). Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124, 5119–5153CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ma S.W. (2001).Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem. J. Diff. Eq. 171, 294–314MATHCrossRefGoogle Scholar
  5. 5.
    Martin R.H., Smith H.L. (1990) Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321, 1–44MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Smith H.L., Thieme H. (1991). Strongly order preserving semiflows generated by functional differential equations. J. Diff. Eq. 93, 332–363MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Smith H.L., Zhao X.Q. (2000). Global asymptotic stability of traveling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal. 31, 514–534MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    So J.W.-H., Wu J.H., Zou X.F. (2001) A reaction-diffusion model for a single species with age structure. I Traveling wavefronts on unbounded domains. Proc. R. Soc. Lond. A, 457, 1841–1853MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Wu J.H. (1996). Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Science, Vol. 119. Spring-Verlag, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.School of Mathematical SciencesNankai UniversityTianjinPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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