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

Previous Article
Wave extension problem for the fractional Laplacian
DCDS Home
This Issue
Next Article
Regions of stability for a linear differential equation with two rationally dependent delays

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

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

Yunfei Lv ^1, , Rong Yuan ^2, and Yuan He ^3,

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

Abstract
Related Papers

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.

Keywords: Schauder's fixed point theorem., traveling wave fronts, stage structured model, State--dependent delay, stability.

Mathematics Subject Classification: 92D25, 34D23, 34K20, 35K5.

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.
[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.*
[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.
[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.
[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.
[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.
[7]	H. Andrewartha and L. Birch, The Distribution and Abundance of Animals,, University of Chicago Press, (1954).
[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.
[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).
[10]	J. Canosa, On a nonlinear diffusion equation describing population growth,, IBM J. Res. Develop., 17 (1973), 307. doi: 10.1147/rd.174.0307.
[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.*
[12]	R. Gambell, Birds and mammals-Antarctic whales,, in Antarctica* (eds. W. Bonner and D. Walton)*, (1985), 223.
[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.*
[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.
[15]	J. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977).
[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.
[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.
[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.*
[19]	D. Jones and C. Walters, Catastrophe theory and fisheries regulation,, J. Fish. Res. Bd. Can., 33 (1976), 2829. doi: 10.1139/f76-338.
[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.
[21]	T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold,, preprint, (2014).
[22]	Y. Kuang, Delay Differential Equation with Applications in Population Dynamics,, Academic, (1993).
[23]	H. Landahl and B. Hanson, A three stage population model with cannibalism,, Bull. Math. Biol., 37 (1975), 11.
[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.
[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.
[26]	J. Murray, Mathematical Biology,, Springer, (1989). doi: 10.1007/978-3-662-08539-4.
[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.*
[28]	W. Rudin, Functional Analysis,, McGraw-Hill, (1991).
[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.*
[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.*
[31]	K. Tognetti, The two stage stochastic model,, Math. Biosci., 25 (1975), 195. doi: 10.1016/0025-5564(75)90002-4.
[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.*
[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.*
[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.
[35]	J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1.
[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.*
[37]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Second edition, (2011).
[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.
[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.
[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.

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.
[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.*
[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.
[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.
[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.
[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.
[7]	H. Andrewartha and L. Birch, The Distribution and Abundance of Animals,, University of Chicago Press, (1954).
[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.
[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).
[10]	J. Canosa, On a nonlinear diffusion equation describing population growth,, IBM J. Res. Develop., 17 (1973), 307. doi: 10.1147/rd.174.0307.
[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.*
[12]	R. Gambell, Birds and mammals-Antarctic whales,, in Antarctica* (eds. W. Bonner and D. Walton)*, (1985), 223.
[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.*
[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.
[15]	J. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977).
[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.
[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.
[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.*
[19]	D. Jones and C. Walters, Catastrophe theory and fisheries regulation,, J. Fish. Res. Bd. Can., 33 (1976), 2829. doi: 10.1139/f76-338.
[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.
[21]	T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold,, preprint, (2014).
[22]	Y. Kuang, Delay Differential Equation with Applications in Population Dynamics,, Academic, (1993).
[23]	H. Landahl and B. Hanson, A three stage population model with cannibalism,, Bull. Math. Biol., 37 (1975), 11.
[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.
[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.
[26]	J. Murray, Mathematical Biology,, Springer, (1989). doi: 10.1007/978-3-662-08539-4.
[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.*
[28]	W. Rudin, Functional Analysis,, McGraw-Hill, (1991).
[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.*
[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.*
[31]	K. Tognetti, The two stage stochastic model,, Math. Biosci., 25 (1975), 195. doi: 10.1016/0025-5564(75)90002-4.
[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.*
[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.*
[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.
[35]	J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1.
[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.*
[37]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations,, Second edition, (2011).
[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.
[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.
[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.

[1]	Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[2]	Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
[3]	Guangying Lv, Mingxin Wang. Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 415-433. doi: 10.3934/dcdsb.2010.13.415
[4]	Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
[5]	Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365
[6]	István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[7]	Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074
[8]	Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143
[9]	M. B. A. Mansour. Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 83-91. doi: 10.3934/mbe.2009.6.83
[10]	Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[11]	Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[12]	Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[13]	Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[14]	Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[15]	Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039
[16]	Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[17]	Wenjie Zuo, Junping Shi. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1179-1200. doi: 10.3934/cpaa.2018057
[18]	Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559
[19]	Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753
[20]	G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences