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Spatial Dynamics of A Reaction-Diffusion Model with DistributedDelay

Published online by Cambridge University Press:  12 June 2013

Y. Zhang  and
X.-Q. Zhao
Y. Zhang
Affiliation:
Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7, Canada
X.-Q. Zhao*
Affiliation:
Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7, Canada
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

This paper is devoted to the study of spreading speeds and traveling waves for a class ofreaction-diffusion equations with distributed delay. Such an equation describes growth anddiffusion in a population where the individuals enter a quiescent phase exponentially andstay quiescent for some arbitrary time that is given by a probability density function.The existence of the spreading speed and its coincidence with the minimum wave speed ofmonostable traveling waves are established via the finite-delay approximation approach. Wealso prove the existence of bistable traveling waves in the case where the associatedreaction system admits a bistable structure. Moreover, the global stability and uniquenessof the bistable waves are obtained in the case where the density function has zerotail

Keywords

Distributed delayspreading speedstraveling wavesglobal stability
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 60 - 77
Copyright
© EDP Sciences, 2013

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References

Chen, X.. Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eq., 2(1997), 125–160. Google Scholar
Fang, J., Wei, J., Zhao, X.-Q.. Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system. J. Diff. Eq., 245(2008), 2749–2770. CrossRefGoogle Scholar
Fang, J., Zhao, X.-Q.. Monotone wavefronts for partially degenerate reaction-diffusion systems. J. Dyn. Diff. Eq., 21(2009), 663–680. CrossRefGoogle Scholar
J. Fang, X.-Q. Zhao. Bistable traveling waves for monotone semiflows with applications. Journal of European Math. Soc., in press.
Gyllenberg, M., Webb, G. F.. A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol., 28(1990), 671–694. CrossRefGoogle ScholarPubMed
Hadeler, K. P.. Quiescent phases and stability. Lin. Alg. Appl., 428(2008), 1620-1627. CrossRefGoogle Scholar
Hadeler, K. P.. Homogenerous systems with a quiescent phase. Math. Models Natur. Phenom., 3(2008), 115–125. CrossRefGoogle Scholar
Hadeler, K. P., Lewis, M. A.. Spatial dynamics of the diffusive logistic equation with a sedentary compartment. Can. Appl. Math. Q., 10(2002), 473-499. Google Scholar
Hadeler, K. P., Lutscher, F.. Quiescent phases with distributed exit times. Disc. Cont. Dyn. Sys.(Ser. B), 17(2012), 849–869. CrossRefGoogle Scholar
Jäger, W., Krömker, S., Tang, B.. Quiescence and transient growth dynamics in chemostat models. Math. Biosci., 119(1994), 225–239. CrossRefGoogle ScholarPubMed
Liang, X., Zhao, X.-Q.. Asymptotic speeds of spread and traveling waves for monotone semiflow with applications. Comm. Pure Appl. Math., 60(2007), 1–40, Erratum: Comm. Pure Appl. Math., 61(2008), 137–138. Google Scholar
Ma, S., Wu, J.. Existence, uniqueness and asymptotic stability of traveling wavefronts in A non-local delayed diffusion equation. J. Dyn. Diff. Eq., 19(2007), 391–436. CrossRefGoogle Scholar
Maler, A., Lutscher, F.. Cell cycle times and the tumor control probability. Mathematics in Medicine and Biology, 27(2010), 313–342. CrossRefGoogle Scholar
Malik, T., Smith, H.. A resource-based model of microbial quiescence. J. Math. Biol., 53(2006), 231–252. CrossRefGoogle Scholar
Martin, R. H., Smith, H. L.. Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc., 321(1990), 1–44. Google Scholar
Schaaf, K. W.. Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations Trans. Amer. Math. Soc., 302(1987), 587–615. Google Scholar
H. L. Smith. Monotone dynamical systems: An introduction to the Theory of Competitive and Cooperative Systems. Math. Surveys Monogr.,vol. 41. Amer. Math. Soc., Providence, RI, 1995.
Smith, H. L., Zhao, X.-Q.. Global asymptotic stability of traveling waves in delayed reaction-diffusion equations SIAM J. Math. Anal., 31(2000), 514–534. CrossRefGoogle Scholar
Thieme, H. R., Zhao, X.-Q.. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J. Diff. Eq, 195(2003), 430–470. CrossRefGoogle Scholar
Wang, Z.-C., Li, W.-T., Ruan, S.. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. J. Diff. Eq, 222(2006), 185–232. CrossRefGoogle Scholar
Wu, J., Zou, X.. Traveling wave fronts of reaction-diffusion systems with delay. J. Dyn. Diff. Eq., 13(2001), 651–687. Erratum: J. Dyn. Diff. Eq., 20(2008), 531–533. CrossRefGoogle Scholar
Zhao, X.-Q., Xiao, D.. The asymptotic speed of spread and traveling waves for a vector disease model. J. Dyn. Diff. Eq., 18(2006), 1001–1019, Erratum: J. Dyn. Diff. Eq., 20(2008), 277–279. CrossRefGoogle Scholar
X.-Q. Zhao. Dynamical Systems in Population Biology. Springer-Verlag, New York, 2003.