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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 |
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] | |
[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. |
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), 1611-1633.
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] | |
[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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