August  2013, 33(8): 3443-3472. doi: 10.3934/dcds.2013.33.3443

Spreading speeds of $N$-season spatially periodic integro-difference models

1. 

Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui 230026, China, China, China

Received  May 2012 Revised  August 2012 Published  January 2013

In this paper, the spreading speeds of $N$-season spatially periodic integro-difference models are investigated. The variational formula of the spreading speeds is given via the principal eigenvalues of the respective positive linear operators. The effects of the spatial and temporal distribution of the intrinsic growth rates on the spreading speeds are considered.
Citation: Weiwei Ding, Xing Liang, Bin Xu. Spreading speeds of $N$-season spatially periodic integro-difference models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3443-3472. doi: 10.3934/dcds.2013.33.3443
References:
[1]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[2]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework,, J. Eur. Math. Soc., 7 (2005), 173.  doi: 10.4171/JEMS/26.  Google Scholar

[3]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts,, J. Math. Pures Appl. (9), 84 (2005), 1101.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[4]

M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model,, Disc. Cont. Dyn. Systems, 25 (2009), 321.  doi: 10.3934/dcds.2009.25.321.  Google Scholar

[5]

M. I. Freidlin, On wavefront propagation in periodic media,, in, (1984), 147.   Google Scholar

[6]

J. Gärtner and M. I. Freidlin, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 20 (1979), 1282.   Google Scholar

[7]

J. S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, Math. Ann., 335 (2006), 489.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[8]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., (2010).   Google Scholar

[9]

F. Hamel, L. Roques and J. Fayard, Spreading speeds in slowly oscillating environments,, Bull. Math. Biol., 72 (2010), 1166.  doi: 10.1007/s11538-009-9486-7.  Google Scholar

[10]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations,, Rocky Mountain Journal of Mathematics, ().   Google Scholar

[11]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[12]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a discrete-time population model in a periodic lattice habitat,, J. Dyn. Diff. Equat., 21 (2009), 501.  doi: 10.1007/s10884-009-9138-5.  Google Scholar

[13]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.   Google Scholar

[14]

K. Kawasaki and N. Shigesada, An integrodifference model for biological invasions in a periodically fragmented environment,, Japan J. Indust. Appl. Math., 34 (2007), 3.   Google Scholar

[15]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[16]

X. Liang, X. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations,, Trans. Am. Math. Soc., 362 (2010), 5605.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[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.  doi: 10.1002/cpa.20154.  Google Scholar

[18]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[19]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 4 (2009), 2388.  doi: 10.1137/080743597.  Google Scholar

[20]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator,, Ann. Mat. Pura Appl., 188 (2009), 269.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[21]

G. Nadin, Some dependence results between the spreading speed and the coeffcients of the space-time periodic Fisher-KPP equation,, European J. Appl. Math., 22 (2011), 169.  doi: 10.1017/S0956792511000027.  Google Scholar

[22]

W. E. Ricker, Stock and recruitment,, J. Fisheries Research Board of Canada, 11 (1954), 559.   Google Scholar

[23]

L. Ryzhik and A. Zlatoš, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.   Google Scholar

[24]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[25]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997).   Google Scholar

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Popul. Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[27]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[28]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173.  doi: 10.1007/BF00279720.  Google Scholar

[29]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[30]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[31]

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.  doi: 10.1007/s00285-008-0168-0.  Google Scholar

[32]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.  doi: 10.1137/S0036144599364296.  Google Scholar

[33]

A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

show all references

References:
[1]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[2]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework,, J. Eur. Math. Soc., 7 (2005), 173.  doi: 10.4171/JEMS/26.  Google Scholar

[3]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating traveling fronts,, J. Math. Pures Appl. (9), 84 (2005), 1101.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[4]

M. El Smaily, F. Hamel and L. Roques, Homogenization and influence of fragmentation in a biological invasion model,, Disc. Cont. Dyn. Systems, 25 (2009), 321.  doi: 10.3934/dcds.2009.25.321.  Google Scholar

[5]

M. I. Freidlin, On wavefront propagation in periodic media,, in, (1984), 147.   Google Scholar

[6]

J. Gärtner and M. I. Freidlin, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 20 (1979), 1282.   Google Scholar

[7]

J. S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, Math. Ann., 335 (2006), 489.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[8]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., (2010).   Google Scholar

[9]

F. Hamel, L. Roques and J. Fayard, Spreading speeds in slowly oscillating environments,, Bull. Math. Biol., 72 (2010), 1166.  doi: 10.1007/s11538-009-9486-7.  Google Scholar

[10]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations,, Rocky Mountain Journal of Mathematics, ().   Google Scholar

[11]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[12]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a discrete-time population model in a periodic lattice habitat,, J. Dyn. Diff. Equat., 21 (2009), 501.  doi: 10.1007/s10884-009-9138-5.  Google Scholar

[13]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.   Google Scholar

[14]

K. Kawasaki and N. Shigesada, An integrodifference model for biological invasions in a periodically fragmented environment,, Japan J. Indust. Appl. Math., 34 (2007), 3.   Google Scholar

[15]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[16]

X. Liang, X. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations,, Trans. Am. Math. Soc., 362 (2010), 5605.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[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.  doi: 10.1002/cpa.20154.  Google Scholar

[18]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[19]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 4 (2009), 2388.  doi: 10.1137/080743597.  Google Scholar

[20]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator,, Ann. Mat. Pura Appl., 188 (2009), 269.  doi: 10.1007/s10231-008-0075-4.  Google Scholar

[21]

G. Nadin, Some dependence results between the spreading speed and the coeffcients of the space-time periodic Fisher-KPP equation,, European J. Appl. Math., 22 (2011), 169.  doi: 10.1017/S0956792511000027.  Google Scholar

[22]

W. E. Ricker, Stock and recruitment,, J. Fisheries Research Board of Canada, 11 (1954), 559.   Google Scholar

[23]

L. Ryzhik and A. Zlatoš, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.   Google Scholar

[24]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[25]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997).   Google Scholar

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Popul. Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[27]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[28]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173.  doi: 10.1007/BF00279720.  Google Scholar

[29]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[30]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[31]

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.  doi: 10.1007/s00285-008-0168-0.  Google Scholar

[32]

J. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.  doi: 10.1137/S0036144599364296.  Google Scholar

[33]

A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

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