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

[1]

Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243

[2]

Liang Kong, Tung Nguyen, Wenxian Shen. Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077

[3]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663

[4]

Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187

[5]

Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026

[6]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268

[7]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019

[8]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202

[9]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[10]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537

[11]

Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1565-1583. doi: 10.3934/mbe.2017081

[12]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[13]

Nar Rawal, Wenxian Shen, Aijun Zhang. Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1609-1640. doi: 10.3934/dcds.2015.35.1609

[14]

Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087

[15]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69

[16]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

[17]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[18]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[19]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[20]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]