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 - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931
References:
[1]

M. Adimy, F. Crauste, M. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay,, SIAM J. Appl. Math., 70 (2010), 1611. doi: 10.1137/080742713. Google Scholar

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J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay,, Nonlinear Anal. Real World Appl., 6 (2005), 13. doi: 10.1016/j.nonrwa.2004.04.002. Google Scholar

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J. Al-Omari and A. Tallafha, Modelling and analysis of stage-structured population model with state-dependent maturation delay and harvesting,, Int. J. Math. Analysis, 1 (2007), 391. Google Scholar

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M. Benchohra, I. Medjadj, J. Nieto and P. Prakash, Global existence for functional differential equations with state-dependent delay,, J. Funct. Space Appl., (2013). Google Scholar

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J. Canosa, On a nonlinear diffusion equation describing population growth,, IBM J. Res. Develop., 17 (1973), 307. doi: 10.1147/rd.174.0307. Google Scholar

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K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting,, Nonlinear Anal. Real World Appl., 14 (2013), 83. doi: 10.1016/j.nonrwa.2012.05.004. Google Scholar

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Q. Hu and X. Zhao, Global dynamics of a state-dependent delay model with unimodal feedback,, J. Math. Anal. Appl., 399 (2013), 133. doi: 10.1016/j.jmaa.2012.09.058. Google Scholar

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T. Krisztin, A local unstable manifold for differential equations with state-dependent delay,, Discret. Contin. Dyn. S., 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. Google Scholar

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Y. Kuang, Delay Differential Equation with Applications in Population Dynamics,, Academic, (1993). Google Scholar

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X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20221. Google Scholar

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M. Memory, Stable and unstable manifolds for partial functional differential equations,, Nonlinear Anal., 16 (1991), 131. doi: 10.1016/0362-546X(91)90164-V. Google Scholar

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J. Murray, Mathematical Biology,, Springer, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

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K. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. doi: 10.2307/2000859. Google Scholar

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J. Sherratt, Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations,, Proc. R. Soc. Lond. A, 456 (2000), 2365. doi: 10.1098/rspa.2000.0616. Google Scholar

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K. Tognetti, The two stage stochastic model,, Math. Biosci., 25 (1975), 195. doi: 10.1016/0025-5564(75)90002-4. Google Scholar

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C. Travis and G. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3. Google Scholar

[33]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887. doi: 10.1016/j.jde.2009.04.002. Google Scholar

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S. Wood, S. Blythe, W. Gurney and R. Nisbet, Instability in mortality estimation schemes related to stage-structure population models,, IMA J. Math. Appl. in Medicine and Biology, 6 (1989), 47. doi: 10.1093/imammb/6.1.47. Google Scholar

[35]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[36]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay,, Appl. Math. Comput., 214 (2009), 228. doi: 10.1016/j.amc.2009.03.078. Google Scholar

[37]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Second edition, (2011). Google Scholar

[38]

A. Zaghrout and S. Attalah, Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay,, Appl. Math. Comput., 77 (1996), 185. doi: 10.1016/S0096-3003(95)00212-X. Google Scholar

[39]

G. Zhang, W. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure,, Math. Comput. Model., 49 (2009), 1021. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

[40]

L. Zhang, B. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure,, Nonlinear Anal. Real World Appl., 13 (2012), 1429. doi: 10.1016/j.nonrwa.2011.11.007. Google Scholar

show all references

References:
[1]

M. Adimy, F. Crauste, M. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay,, SIAM J. Appl. Math., 70 (2010), 1611. doi: 10.1137/080742713. Google Scholar

[2]

W. Aiello and H. Freedman, A time-delay model of a single species growth with stage structure,, Math. Biosci, 101 (1990), 139. doi: 10.1016/0025-5564(90)90019-U. Google Scholar

[3]

W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay,, SIAM J. Appl. Math, 52 (1992), 855. doi: 10.1137/0152048. Google Scholar

[4]

J. Al-omari and S. Gourley, Stability and traveling fronts in Lotka-Volterra competition models with stage structure,, SIAM J. Appl. Math., 63 (2003), 2063. doi: 10.1137/S0036139902416500. Google Scholar

[5]

J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay,, Nonlinear Anal. Real World Appl., 6 (2005), 13. doi: 10.1016/j.nonrwa.2004.04.002. Google Scholar

[6]

J. Al-Omari and A. Tallafha, Modelling and analysis of stage-structured population model with state-dependent maturation delay and harvesting,, Int. J. Math. Analysis, 1 (2007), 391. Google Scholar

[7]

H. Andrewartha and L. Birch, The Distribution and Abundance of Animals,, University of Chicago Press, (1954). Google Scholar

[8]

H. Barclay and P. driessche, A model for a single species with two life history stages and added mortality,, Ecol. Model, 11 (1980), 157. Google Scholar

[9]

M. Benchohra, I. Medjadj, J. Nieto and P. Prakash, Global existence for functional differential equations with state-dependent delay,, J. Funct. Space Appl., (2013). Google Scholar

[10]

J. Canosa, On a nonlinear diffusion equation describing population growth,, IBM J. Res. Develop., 17 (1973), 307. doi: 10.1147/rd.174.0307. Google Scholar

[11]

K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system,, Ecol. Model., 215 (2008), 69. doi: 10.1016/j.ecolmodel.2008.02.019. Google Scholar

[12]

R. Gambell, Birds and mammals-Antarctic whales,, in Antarctica (eds. W. Bonner and D. Walton), (1985), 223. Google Scholar

[13]

S. Gourley and Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure,, Proc. R. Soc. Lond. A, 459 (2003), 1563. doi: 10.1098/rspa.2002.1094. Google Scholar

[14]

W. Gurney, R. Nisbet and J. Lawton, The systematic formulation of tractible single species population models incorporating age structure,, J. Animal Ecol., 52 (1983), 479. doi: 10.2307/4567. Google Scholar

[15]

J. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977). Google Scholar

[16]

F. Hartung, T. Krisztin, H. Walther and J. Wu, Functional differential equations with state-dependent delay: Theory and applications,, in Handbook of Differential Equations: Ordinary Differential Equations. Vol. III (eds. A. Canada, (2006), 435. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar

[17]

K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting,, Nonlinear Anal. Real World Appl., 14 (2013), 83. doi: 10.1016/j.nonrwa.2012.05.004. Google Scholar

[18]

Q. Hu and X. Zhao, Global dynamics of a state-dependent delay model with unimodal feedback,, J. Math. Anal. Appl., 399 (2013), 133. doi: 10.1016/j.jmaa.2012.09.058. Google Scholar

[19]

D. Jones and C. Walters, Catastrophe theory and fisheries regulation,, J. Fish. Res. Bd. Can., 33 (1976), 2829. doi: 10.1139/f76-338. Google Scholar

[20]

T. Krisztin, A local unstable manifold for differential equations with state-dependent delay,, Discret. Contin. Dyn. S., 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. Google Scholar

[21]

T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold,, preprint, (2014). Google Scholar

[22]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics,, Academic, (1993). Google Scholar

[23]

H. Landahl and B. Hanson, A three stage population model with cannibalism,, Bull. Math. Biol., 37 (1975), 11. Google Scholar

[24]

X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20221. Google Scholar

[25]

M. Memory, Stable and unstable manifolds for partial functional differential equations,, Nonlinear Anal., 16 (1991), 131. doi: 10.1016/0362-546X(91)90164-V. Google Scholar

[26]

J. Murray, Mathematical Biology,, Springer, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

[27]

A. Rezounenko and J. Wu, A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors,, J. Comput. Appl. Math., 190 (2006), 99. doi: 10.1016/j.cam.2005.01.047. Google Scholar

[28]

W. Rudin, Functional Analysis,, McGraw-Hill, (1991). Google Scholar

[29]

K. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. doi: 10.2307/2000859. Google Scholar

[30]

J. Sherratt, Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations,, Proc. R. Soc. Lond. A, 456 (2000), 2365. doi: 10.1098/rspa.2000.0616. Google Scholar

[31]

K. Tognetti, The two stage stochastic model,, Math. Biosci., 25 (1975), 195. doi: 10.1016/0025-5564(75)90002-4. Google Scholar

[32]

C. Travis and G. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3. Google Scholar

[33]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887. doi: 10.1016/j.jde.2009.04.002. Google Scholar

[34]

S. Wood, S. Blythe, W. Gurney and R. Nisbet, Instability in mortality estimation schemes related to stage-structure population models,, IMA J. Math. Appl. in Medicine and Biology, 6 (1989), 47. doi: 10.1093/imammb/6.1.47. Google Scholar

[35]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[36]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay,, Appl. Math. Comput., 214 (2009), 228. doi: 10.1016/j.amc.2009.03.078. Google Scholar

[37]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Second edition, (2011). Google Scholar

[38]

A. Zaghrout and S. Attalah, Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay,, Appl. Math. Comput., 77 (1996), 185. doi: 10.1016/S0096-3003(95)00212-X. Google Scholar

[39]

G. Zhang, W. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure,, Math. Comput. Model., 49 (2009), 1021. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

[40]

L. Zhang, B. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure,, Nonlinear Anal. Real World Appl., 13 (2012), 1429. doi: 10.1016/j.nonrwa.2011.11.007. Google Scholar

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