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Spatial dynamics of a nonlocal and delayed population model in a periodic habitat
1. | School of Mathematics, South China Normal University, Guangzhou 510631, China |
2. | Department of Mathematics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada |
References:
[1] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: doi:10.1007/s00285-004-0313-3. |
[2] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
doi: doi:10.1016/j.matpur.2004.10.006. |
[3] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.
doi: doi:10.1007/BF02450783. |
[4] |
N. Dunford and J. T. Schwartz, "Linear Operators, Part I: General Theory," 1st edition, Interscience, New York, 1958, 1994. |
[5] |
M. I. Friedlin, On wavefront propagation in periodic media, stochastic analysis and applications, in "Adv. Probab. Related Topics, 7", Dekker, New York, 1984, 147-166. |
[6] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[7] |
J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), 1282-1286. |
[8] |
J. Guo and F. Hamel, Front propagation for discrete periodic monostable equations Mathematische Annalen, 335 (2006), 489-525.
doi: doi:10.1007/s00208-005-0729-0. |
[9] |
S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563-1579.
doi: doi:10.1098/rspa.2002.1094. |
[10] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. |
[11] |
P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, New York, 1991. |
[12] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: doi:10.1137/070703016. |
[13] |
Y. Jin and X.-Q. Zhao, Spacial dynamics of a discrete-time population model in a periodic lattics habitat, J. Dynam. Differential Equations, 21 (2009), 501-525.
doi: doi:10.1007/s10884-009-9138-5. |
[14] |
W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Applicandae Math., 2 (1984), 297-309.
doi: doi:10.1007/BF02280856. |
[15] |
N. Kinezaki, K. Kawasaki, F. Takasu and N. Shigesada, Modelling biological invasions into periodically fragmented environments, Theor. Popul. Biol., 64 (2003), 291-302.
doi: doi:10.1016/S0040-5809(03)00091-1. |
[16] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in the Banach space, Amer. Math. Soc. Translations, 26 (1950), 3-128. |
[17] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. (Erratum: 61 (2008),137-138). |
[18] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: doi:10.1016/j.jfa.2010.04.018. |
[19] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: doi:10.1016/0025-5564(89)90026-6. |
[20] |
R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-331.
doi: doi:10.1016/0025-5564(89)90027-8. |
[21] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: doi:10.2307/2001590. |
[22] |
R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. reine angew. Math., 413 (1991), 1-35. |
[23] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. |
[24] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Mathematical Surveys and Monographs, 102, American Mathematical Society, Providence, RI, 2003. |
[25] |
K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[26] |
N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theoretical Population Biol., 30 (1986), 143-160.
doi: doi:10.1016/0040-5809(86)90029-8. |
[27] |
J. W.-H. So and J. Wu, X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on the unbounded domains, Proc. R. Soc. Lond. Ser. A., 457 (2001), 1841-1853.
doi: doi:10.1098/rspa.2001.0789. |
[28] |
H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: doi:10.1137/S0036141098346785. |
[29] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biology, 8 (1979), 173-187.
doi: doi:10.1007/BF00279720. |
[30] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: doi:10.1016/S0022-0396(03)00175-X. |
[31] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Application, 2 (2001), 145-160.
doi: doi:10.1016/S0362-546X(00)00112-7. |
[32] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translation of Mathematical Monographs, 140, Amer. Math. Soc. 1994. |
[33] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: doi:10.1137/0513028. |
[34] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: doi:10.1007/s00285-002-0169-3. |
[35] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: doi:10.1007/s002850200145. |
[36] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.
doi: doi:10.1007/s00285-008-0168-0. |
[37] |
P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.
doi: doi:10.1093/imamat/68.4.409. |
[38] |
P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.
doi: doi:10.1016/j.jde.2006.01.020. |
[39] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: doi:10.1137/S0036144599364296. |
[40] |
X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003. |
show all references
References:
[1] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: doi:10.1007/s00285-004-0313-3. |
[2] |
H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
doi: doi:10.1016/j.matpur.2004.10.006. |
[3] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.
doi: doi:10.1007/BF02450783. |
[4] |
N. Dunford and J. T. Schwartz, "Linear Operators, Part I: General Theory," 1st edition, Interscience, New York, 1958, 1994. |
[5] |
M. I. Friedlin, On wavefront propagation in periodic media, stochastic analysis and applications, in "Adv. Probab. Related Topics, 7", Dekker, New York, 1984, 147-166. |
[6] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[7] |
J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), 1282-1286. |
[8] |
J. Guo and F. Hamel, Front propagation for discrete periodic monostable equations Mathematische Annalen, 335 (2006), 489-525.
doi: doi:10.1007/s00208-005-0729-0. |
[9] |
S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563-1579.
doi: doi:10.1098/rspa.2002.1094. |
[10] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. |
[11] |
P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, New York, 1991. |
[12] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: doi:10.1137/070703016. |
[13] |
Y. Jin and X.-Q. Zhao, Spacial dynamics of a discrete-time population model in a periodic lattics habitat, J. Dynam. Differential Equations, 21 (2009), 501-525.
doi: doi:10.1007/s10884-009-9138-5. |
[14] |
W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Applicandae Math., 2 (1984), 297-309.
doi: doi:10.1007/BF02280856. |
[15] |
N. Kinezaki, K. Kawasaki, F. Takasu and N. Shigesada, Modelling biological invasions into periodically fragmented environments, Theor. Popul. Biol., 64 (2003), 291-302.
doi: doi:10.1016/S0040-5809(03)00091-1. |
[16] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in the Banach space, Amer. Math. Soc. Translations, 26 (1950), 3-128. |
[17] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. (Erratum: 61 (2008),137-138). |
[18] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: doi:10.1016/j.jfa.2010.04.018. |
[19] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: doi:10.1016/0025-5564(89)90026-6. |
[20] |
R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-331.
doi: doi:10.1016/0025-5564(89)90027-8. |
[21] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: doi:10.2307/2001590. |
[22] |
R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. reine angew. Math., 413 (1991), 1-35. |
[23] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. |
[24] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Mathematical Surveys and Monographs, 102, American Mathematical Society, Providence, RI, 2003. |
[25] |
K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. |
[26] |
N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theoretical Population Biol., 30 (1986), 143-160.
doi: doi:10.1016/0040-5809(86)90029-8. |
[27] |
J. W.-H. So and J. Wu, X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on the unbounded domains, Proc. R. Soc. Lond. Ser. A., 457 (2001), 1841-1853.
doi: doi:10.1098/rspa.2001.0789. |
[28] |
H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: doi:10.1137/S0036141098346785. |
[29] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biology, 8 (1979), 173-187.
doi: doi:10.1007/BF00279720. |
[30] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: doi:10.1016/S0022-0396(03)00175-X. |
[31] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Application, 2 (2001), 145-160.
doi: doi:10.1016/S0362-546X(00)00112-7. |
[32] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translation of Mathematical Monographs, 140, Amer. Math. Soc. 1994. |
[33] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: doi:10.1137/0513028. |
[34] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: doi:10.1007/s00285-002-0169-3. |
[35] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: doi:10.1007/s002850200145. |
[36] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.
doi: doi:10.1007/s00285-008-0168-0. |
[37] |
P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.
doi: doi:10.1093/imamat/68.4.409. |
[38] |
P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.
doi: doi:10.1016/j.jde.2006.01.020. |
[39] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: doi:10.1137/S0036144599364296. |
[40] |
X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003. |
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