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doi: 10.3934/dcdss.2022061
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Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel

Department of Mathematics, University of Louisiana at Laffayette, Lafayette, LA 70504-1010, USA

Received  July 2021 Revised  December 2021 Early access March 2022

This paper deals with front propagation for nonlocal monostable equations in spatially periodic habitats. In the authors' earlier works, assuming the existence of principal eigenvalue, it is shown that there are periodic traveling wave solutions to a spatially periodic nonlocal monostable equation with symmetric and compact kernel connecting its unique positive stationary solution and the trivial solution in every direction with all propagating speeds greater than the spreading speed in that direction. In this paper, first assuming the existence of principal eigenvalue, we extend the results to the case that the kernel is asymmetric and supported on a non-compact region. In addition, without the assumption of the existence of principal eigenvalue, we explore the existence of semicontinuous traveling wave solutions.

Citation: Aijun Zhang. Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022061
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partail Differential Equations and Related Topics" (J. Goldstein, Ed.), Lecture Notes in Math., Springer-Verlag, New York, 446 (1975), 5-49.

[2]

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

[3]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.

[9]

C. CortazarM. Elgueta and and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. of Math., 170 (2009), 53-60.  doi: 10.1007/s11856-009-0019-8.

[10]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.

[11]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., (185) (2006), 461–485. doi: 10.1007/s10231-005-0163-7.

[12]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

[13]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[14]

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.

[15]

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

[16]

P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.

[17]

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

[18]

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

[19]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.  doi: 10.1016/j.jde.2008.04.015.

[20]

M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. 

[21]

F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.  doi: 10.1016/j.matpur.2007.12.005.

[22]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[23]

G. HetzerT. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.

[24]

G. HetzerW. 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, 43 (2013), 489-513.  doi: 10.1216/RMJ-2013-43-2-489.

[25]

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

[26]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, World Sci. Publ., River Edge, NJ, (1995), 187–199.

[27]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[28]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[29]

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

[30]

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

[31]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.

[32]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. 

[33]

W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[34]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eq., 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[35]

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.

[36]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅱ. Biological theory, Math. Biosciences, 93 (1989), 269-312.  doi: 10.1016/0025-5564(89)90027-8.

[37]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations, Journal of Mathematical Analysis and Applications, 385 (2012), 1094-1106.  doi: 10.1016/j.jmaa.2011.07.033.

[38]

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.

[39]

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.

[40]

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.

[41]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis, 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[43]

N. Popovic and T. J. Kaper, Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations, J. Dynam. Diff. Eq., 18 (2006), 103-139.  doi: 10.1007/s10884-005-9002-1.

[44]

L. Ryzhik and A Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci., 5 (2007), 575-593.  doi: 10.4310/CMS.2007.v5.n3.a4.

[45]

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.

[46]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), 262-297.  doi: 10.1016/j.jde.2006.12.015.

[47]

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

[48]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proceedings of the American Mathematical Society, 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.

[49]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101. 

[50]

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.

[51]

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

[52]

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: 10.1007/s00285-002-0169-3.

[53]

A. Zhang, Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 815–824. doi: 10.3934/proc.2013.2013.815.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partail Differential Equations and Related Topics" (J. Goldstein, Ed.), Lecture Notes in Math., Springer-Verlag, New York, 446 (1975), 5-49.

[2]

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

[3]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.

[9]

C. CortazarM. Elgueta and and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. of Math., 170 (2009), 53-60.  doi: 10.1007/s11856-009-0019-8.

[10]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.

[11]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., (185) (2006), 461–485. doi: 10.1007/s10231-005-0163-7.

[12]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

[13]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.

[14]

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.

[15]

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

[16]

P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.

[17]

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

[18]

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

[19]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.  doi: 10.1016/j.jde.2008.04.015.

[20]

M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. 

[21]

F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399.  doi: 10.1016/j.matpur.2007.12.005.

[22]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[23]

G. HetzerT. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.

[24]

G. HetzerW. 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, 43 (2013), 489-513.  doi: 10.1216/RMJ-2013-43-2-489.

[25]

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

[26]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, World Sci. Publ., River Edge, NJ, (1995), 187–199.

[27]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[28]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[29]

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

[30]

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

[31]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.

[32]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. 

[33]

W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[34]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eq., 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[35]

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.

[36]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅱ. Biological theory, Math. Biosciences, 93 (1989), 269-312.  doi: 10.1016/0025-5564(89)90027-8.

[37]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations, Journal of Mathematical Analysis and Applications, 385 (2012), 1094-1106.  doi: 10.1016/j.jmaa.2011.07.033.

[38]

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.

[39]

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.

[40]

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.

[41]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis, 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[43]

N. Popovic and T. J. Kaper, Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations, J. Dynam. Diff. Eq., 18 (2006), 103-139.  doi: 10.1007/s10884-005-9002-1.

[44]

L. Ryzhik and A Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci., 5 (2007), 575-593.  doi: 10.4310/CMS.2007.v5.n3.a4.

[45]

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.

[46]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), 262-297.  doi: 10.1016/j.jde.2006.12.015.

[47]

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

[48]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proceedings of the American Mathematical Society, 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.

[49]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101. 

[50]

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.

[51]

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

[52]

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: 10.1007/s00285-002-0169-3.

[53]

A. Zhang, Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 815–824. doi: 10.3934/proc.2013.2013.815.

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