September  2017, 37(9): 4697-4727. doi: 10.3934/dcds.2017202

Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China

2. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

Received  July 2016 Revised  April 2017 Published  June 2017

Fund Project: The first author was supported by the Fundamental Research Funds for the Central Universities No. NZ2014104

The current paper is devoted to the study of spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. We first prove the existence, uniqueness, and stability of spatially homogeneous entire positive solutions. Next, we establish lower and upper bounds of the (generalized) spreading speed intervals. Then, by constructing appropriate sub-solutions and super-solutions, we show the existence and continuity of transition fronts with given front position functions. Also, we prove the existence of some kind of critical front.

Citation: 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
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Lecture Notes in Math., Springer, Berlin, 446 (1974), 5-49.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations}, Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.  Google Scholar

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H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[5]

H. BerestyckiF. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Func. Anal., 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.  Google Scholar

[7]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.  Google Scholar

[8]

H. BerestyckiF. Hamel and L. Roques, Analysis of periodically fragmented environment model: Ⅱ -Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[9]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23 pp. doi: 10.1063/1.4764932.  Google Scholar

[10]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[11]

X. ChenS.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

[12]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[13]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[14]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[15]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

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R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[17]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic analysis and applications, ed. M. Pinsky, Advances in Probability and related topics, 7 (1984), 147-166.   Google Scholar

[18]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

[19]

S.-C. FuJ.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Anal., 48 (2002), 1137-1149.  doi: 10.1016/S0362-546X(00)00242-X.  Google Scholar

[20]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[21]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833.  doi: 10.1016/j.jde.2009.03.010.  Google Scholar

[22]

D. Hankerson and B. Zinner, Wavefronts for a cooperative tridiagonal system of differential equations, J. Dynam. Differential Equations, 5 (1993), 359-373.  doi: 10.1007/BF01053165.  Google Scholar

[23]

J.H. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J Appl. Dyn. Syst., 8 (2009), 790-821.  doi: 10.1137/080723259.  Google Scholar

[24]

W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Anal., 1 (1994), 23-46.   Google Scholar

[25]

V. HutsonW. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

[26]

Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math., 13 (1976), 11-66.   Google Scholar

[27]

A. KolmogorovI. Petrovsky and N. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105-130.  doi: 10.1016/B978-0-08-092523-3.50014-9.  Google Scholar

[28]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.  Google Scholar

[29]

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.  doi: 10.1002/cpa.20154.  Google Scholar

[30]

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: 10.1016/j.jfa.2010.04.018.  Google Scholar

[31]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602.  Google Scholar

[32]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[33]

G. Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 841-873.  doi: 10.1016/j.anihpc.2014.03.007.  Google Scholar

[34]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.  Google Scholar

[35]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[36]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.  Google Scholar

[37]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[38]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[39]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[40]

W. Shen, Spreading and generalized propagating speeds of discrete KPP models in time varying environments, Front. Math. China, 4 (2009), 523-562.  doi: 10.1007/s11464-009-0032-6.  Google Scholar

[41]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[42]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[43]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44.  doi: 10.1007/s10884-010-9200-3.  Google Scholar

[44]

W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations, Journal of Dynamics and Differential Equations, (2015), 1-32, arXiv: 1504.02525. doi: 10.1007/s10884-016-9528-4.  Google Scholar

[45]

W. Shen and Z. Shen, Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media, Commun. Pure Appl. Anal., 15 (2016), 1193-1213.  doi: 10.3934/cpaa.2016.15.1193.  Google Scholar

[46]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[47]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[48]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[49]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997. Google Scholar

[50]

B. Shorrocks and I. R. Swingland, Living in a Patch Environment Oxford University Press, New York, 1990. Google Scholar

[51]

T. TaoB. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416.  doi: 10.1088/0951-7715/27/9/2409.  Google Scholar

[52]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[53]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[54]

H. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[55]

J. Wu and X. Zou, Asymptotic and periodic boundary values problems of mixed PDEs and wave solutions of lattice differential equations, J. Differential Equations, 135 (1997), 315-357.  doi: 10.1006/jdeq.1996.3232.  Google Scholar

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B. ZinnerG. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.  doi: 10.1006/jdeq.1993.1082.  Google Scholar

[57]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Lecture Notes in Math., Springer, Berlin, 446 (1974), 5-49.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations}, Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.  Google Scholar

[4]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[5]

H. BerestyckiF. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Func. Anal., 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.  Google Scholar

[7]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.  Google Scholar

[8]

H. BerestyckiF. Hamel and L. Roques, Analysis of periodically fragmented environment model: Ⅱ -Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[9]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23 pp. doi: 10.1063/1.4764932.  Google Scholar

[10]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[11]

X. ChenS.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

[12]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[13]

X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[14]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[15]

J. CovilleJ. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[16]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[17]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic analysis and applications, ed. M. Pinsky, Advances in Probability and related topics, 7 (1984), 147-166.   Google Scholar

[18]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), 1282-1286.   Google Scholar

[19]

S.-C. FuJ.-S. Guo and S.-Y Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations, Nonlinear Anal., 48 (2002), 1137-1149.  doi: 10.1016/S0362-546X(00)00242-X.  Google Scholar

[20]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[21]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833.  doi: 10.1016/j.jde.2009.03.010.  Google Scholar

[22]

D. Hankerson and B. Zinner, Wavefronts for a cooperative tridiagonal system of differential equations, J. Dynam. Differential Equations, 5 (1993), 359-373.  doi: 10.1007/BF01053165.  Google Scholar

[23]

J.H. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J Appl. Dyn. Syst., 8 (2009), 790-821.  doi: 10.1137/080723259.  Google Scholar

[24]

W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Comm. Appl. Nonlinear Anal., 1 (1994), 23-46.   Google Scholar

[25]

V. HutsonW. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

[26]

Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math., 13 (1976), 11-66.   Google Scholar

[27]

A. KolmogorovI. Petrovsky and N. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105-130.  doi: 10.1016/B978-0-08-092523-3.50014-9.  Google Scholar

[28]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.  Google Scholar

[29]

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.  doi: 10.1002/cpa.20154.  Google Scholar

[30]

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: 10.1016/j.jfa.2010.04.018.  Google Scholar

[31]

T. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602.  Google Scholar

[32]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[33]

G. Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 841-873.  doi: 10.1016/j.anihpc.2014.03.007.  Google Scholar

[34]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.  Google Scholar

[35]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.  Google Scholar

[36]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.  Google Scholar

[37]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[38]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[39]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[40]

W. Shen, Spreading and generalized propagating speeds of discrete KPP models in time varying environments, Front. Math. China, 4 (2009), 523-562.  doi: 10.1007/s11464-009-0032-6.  Google Scholar

[41]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[42]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[43]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44.  doi: 10.1007/s10884-010-9200-3.  Google Scholar

[44]

W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations, Journal of Dynamics and Differential Equations, (2015), 1-32, arXiv: 1504.02525. doi: 10.1007/s10884-016-9528-4.  Google Scholar

[45]

W. Shen and Z. Shen, Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media, Commun. Pure Appl. Anal., 15 (2016), 1193-1213.  doi: 10.3934/cpaa.2016.15.1193.  Google Scholar

[46]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[47]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[48]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[49]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997. Google Scholar

[50]

B. Shorrocks and I. R. Swingland, Living in a Patch Environment Oxford University Press, New York, 1990. Google Scholar

[51]

T. TaoB. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416.  doi: 10.1088/0951-7715/27/9/2409.  Google Scholar

[52]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[53]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[54]

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