October  2015, 35(10): 4931-4954. doi: 10.3934/dcds.2015.35.4931

Wavefronts of a stage structured model with state--dependent delay

1. 

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  July 2013 Revised  January 2015 Published  April 2015

This paper deals with a diffusive stage structured model with state-dependent delay which is assumed to be an increasing function of the population density. Compared with the constant delay, the state--dependent delay makes the dynamic behavior more complex. For the state--dependent delay system, the dynamic behavior is dependent of the diffusion coefficients, while the equilibrium state of constant delay system is not destabilized by diffusion. Through calculating the minimum wave speed, we find that the wave is slowed down by the state-dependent delay. Then, the existence of traveling waves is obtained by constructing a pair of upper--lower solutions and using Schauder's fixed point theorem. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.
Citation: Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931
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show all references

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SIAM J. Appl. Math., 70 (2010), 1611-1633. doi: 10.1137/080742713.  Google Scholar

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Math. Biosci, 101 (1990), 139-153. doi: 10.1016/0025-5564(90)90019-U.  Google Scholar

[3]

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SIAM J. Appl. Math., 63 (2003), 2063-2086. doi: 10.1137/S0036139902416500.  Google Scholar

[5]

Nonlinear Anal. Real World Appl., 6 (2005), 13-33. doi: 10.1016/j.nonrwa.2004.04.002.  Google Scholar

[6]

Int. J. Math. Analysis, 1 (2007), 391-407.  Google Scholar

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University of Chicago Press, Chicago, IL, 1954, p. 370. Google Scholar

[8]

Ecol. Model, 11 (1980), 157-166. Google Scholar

[9]

J. Funct. Space Appl., (2013), Art. ID 863561, 7 pp.  Google Scholar

[10]

IBM J. Res. Develop., 17 (1973), 307-313. doi: 10.1147/rd.174.0307.  Google Scholar

[11]

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[12]

in Antarctica (eds. W. Bonner and D. Walton), Pergamon Press, New York, 1985, 223-241. Google Scholar

[13]

Proc. R. Soc. Lond. A, 459 (2003), 1563-1579. doi: 10.1098/rspa.2002.1094.  Google Scholar

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J. Animal Ecol., 52 (1983), 479-485. doi: 10.2307/4567.  Google Scholar

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Springer-Verlag, New York, 1977.  Google Scholar

[16]

in Handbook of Differential Equations: Ordinary Differential Equations. Vol. III (eds. A. Canada, P. Drabek and A. Fonda), Elsevier Science B. V., North-Holland, Amsterdam, 2006, 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

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[18]

J. Math. Anal. Appl., 399 (2013), 133-146. doi: 10.1016/j.jmaa.2012.09.058.  Google Scholar

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Nonlinear Anal. Real World Appl., 13 (2012), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007.  Google Scholar

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