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January  2011, 29(1): 343-366. doi: 10.3934/dcds.2011.29.343

Spatial dynamics of a nonlocal and delayed population model in a periodic habitat

Peixuan Weng 1, and  Xiao-Qiang Zhao 2,

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

Received  January 2010 Revised  July 2010 Published  September 2010

We derived an age-structured population model with nonlocal effects and time delay in a periodic habitat. The spatial dynamics of the model including the comparison principle, the global attractivity of spatially periodic equilibrium, spreading speeds, and spatially periodic traveling wavefronts is investigated. It turns out that the spreading speed coincides with the minimal wave speed for spatially periodic travel waves.
Keywords: nonlocal effect, periodic habitat, spreading speeds, Age-structured population model, spatially periodic traveling wavefronts..
Mathematics Subject Classification: Primary: 35K57, 34K30; Secondary: 92D2.
Citation: Peixuan Weng, Xiao-Qiang Zhao. Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 343-366. doi: 10.3934/dcds.2011.29.343
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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