Geodesic
Spatial Dynamics of A Reaction-Diffusion Model with Distributed Delay
Y. Zhang1 ; X.-Q. Zhao1
1 Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7, Canada
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 3, pp. 60-77

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This paper is devoted to the study of spreading speeds and traveling waves for a class of reaction-diffusion equations with distributed delay. Such an equation describes growth and diffusion in a population where the individuals enter a quiescent phase exponentially and stay 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 of monostable traveling waves are established via the finite-delay approximation approach. We also prove the existence of bistable traveling waves in the case where the associated reaction system admits a bistable structure. Moreover, the global stability and uniqueness of the bistable waves are obtained in the case where the density function has zero tail
DOI : 10.1051/mmnp/20138306

Y. Zhang 1 ; X.-Q. Zhao 1

1 Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL A1C 5S7, Canada 
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Y. Zhang; X.-Q. Zhao. Spatial Dynamics of A Reaction-Diffusion Model with Distributed Delay. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 3, pp. 60-77. doi: 10.1051/mmnp/20138306

[1] X. Chen Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations Adv. Diff. Eq. 1997 125 160

[2] J. Fang, J. Wei, X.-Q. Zhao Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system J. Diff. Eq. 2008 2749 2770

[3] J. Fang, X.-Q. Zhao Monotone wavefronts for partially degenerate reaction-diffusion systems J. Dyn. Diff. Eq. 2009 663 680

[4] J. Fang, X.-Q. Zhao. Bistable traveling waves for monotone semiflows with applications. Journal of European Math. Soc., in press.

[5] M. Gyllenberg, G. F. Webb A nonlinear structured population model of tumor growth with quiescence J. Math. Biol. 1990 671 694

[6] K. P. Hadeler Lin. Alg. Appl. 2008 1620 1627

[7] K. P. Hadeler Math. Models Natur. Phenom. 2008 115 125

[8] K. P. Hadeler, M. A. Lewis Can. Appl. Math. Q. 2002 473 499

[9] K. P. Hadeler, F. Lutscher Disc. Cont. Dyn. Sys.(Ser. B) 2012 849 869

[10] W. Jäger, S. Krömker, B. Tang Math. Biosci. 1994 225 239

[11] X. Liang, X.-Q. Zhao 2007 1 40

[12] S. Ma, J. Wu J. Dyn. Diff. Eq. 2007 391 436

[13] A. Maler, F. Lutscher Mathematics in Medicine and Biology 2010 313 342

[14] T. Malik, H. Smith J. Math. Biol. 2006 231 252

[15] R. H. Martin, H. L. Smith Trans. Amer. Math. Soc. 1990 1 44

[16] K. W. Schaaf Trans. Amer. Math. Soc. 1987 587 615

[17] 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.

[18] H. L. Smith, X.-Q. Zhao SIAM J. Math. Anal. 2000 514 534

[19] H. R. Thieme, X.-Q. Zhao J. Diff. Eq 2003 430 470

[20] Z.-C. Wang, W.-T. Li, S. Ruan J. Diff. Eq 2006 185 232

[21] J. Wu, X. Zou J. Dyn. Diff. Eq. 2001 651 687

[22] X.-Q. Zhao, D. Xiao J. Dyn. Diff. Eq. 2006 1001 1019

[23] X.-Q. Zhao. Dynamical Systems in Population Biology. Springer-Verlag, New York, 2003.

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