# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 1611-1633. 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), 139-153. 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), 855-869. 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), 2063-2086. 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), 13-33. 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), 391-407. [7] H. Andrewartha and L. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954, p. 370. [8] H. Barclay and P. driessche, A model for a single species with two life history stages and added mortality, Ecol. Model, 11 (1980), 157-166. [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), Art. ID 863561, 7 pp. [10] J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Develop., 17 (1973), 307-313. 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), 69-76. doi: 10.1016/j.ecolmodel.2008.02.019. [12] R. Gambell, Birds and mammals-Antarctic whales, in Antarctica (eds. W. Bonner and D. Walton), Pergamon Press, New York, 1985, 223-241. [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), 1563-1579. 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), 479-485. doi: 10.2307/4567. [15] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 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, P. Drabek and A. Fonda), Elsevier Science B. V., North-Holland, Amsterdam, 2006, 435-545. 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), 83-103. 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), 133-146. 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), 2829-2833. 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), 993-1028. 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, November 30, 2014, arXiv:1412.0219. [22] Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Academic, New York, 1993. [23] H. Landahl and B. Hanson, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17. [24] X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40; Comm. Pure Appl. Math., 61 (2008), 137-138 (erratum). doi: 10.1002/cpa.20221. [25] M. Memory, Stable and unstable manifolds for partial functional differential equations, Nonlinear Anal., 16 (1991), 131-142. 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), 99-113. 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), 587-615. 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), 2365-2386. doi: 10.1098/rspa.2000.0616. [31] K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204. 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), 395-418. 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), 887-905. 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), 47-68. doi: 10.1093/imammb/6.1.47. [35] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 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), 228-235. 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, China Science Publishing Group, 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), 185-194. 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), 1021-1029. 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), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007.

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##### 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), 1611-1633. 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), 139-153. 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), 855-869. 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), 2063-2086. 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), 13-33. 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), 391-407. [7] H. Andrewartha and L. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954, p. 370. [8] H. Barclay and P. driessche, A model for a single species with two life history stages and added mortality, Ecol. Model, 11 (1980), 157-166. [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), Art. ID 863561, 7 pp. [10] J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Develop., 17 (1973), 307-313. 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), 69-76. doi: 10.1016/j.ecolmodel.2008.02.019. [12] R. Gambell, Birds and mammals-Antarctic whales, in Antarctica (eds. W. Bonner and D. Walton), Pergamon Press, New York, 1985, 223-241. [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), 1563-1579. 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), 479-485. doi: 10.2307/4567. [15] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 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, P. Drabek and A. Fonda), Elsevier Science B. V., North-Holland, Amsterdam, 2006, 435-545. 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), 83-103. 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), 133-146. 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), 2829-2833. 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), 993-1028. 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, November 30, 2014, arXiv:1412.0219. [22] Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Academic, New York, 1993. [23] H. Landahl and B. Hanson, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17. [24] X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40; Comm. Pure Appl. Math., 61 (2008), 137-138 (erratum). doi: 10.1002/cpa.20221. [25] M. Memory, Stable and unstable manifolds for partial functional differential equations, Nonlinear Anal., 16 (1991), 131-142. 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), 99-113. 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), 587-615. 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), 2365-2386. doi: 10.1098/rspa.2000.0616. [31] K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204. 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), 395-418. 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), 887-905. 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), 47-68. doi: 10.1093/imammb/6.1.47. [35] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 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), 228-235. 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, China Science Publishing Group, 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), 185-194. 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), 1021-1029. 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), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007.
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