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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[18]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[48]

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

[49]

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

[50]

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

[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.

[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.

[53]

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

[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.

[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.

[56]

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.

[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.

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[18]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[48]

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

[49]

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

[50]

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

[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.

[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.

[53]

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

[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.

[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.

[56]

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.

[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.

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