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January  2019, 18(1): 361-396. doi: 10.3934/cpaa.2019019

Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems

1. 

School of Science, Chang'an University, Xi'an, Shaanxi 710064, China

2. 

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

3. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

* Corresponding author

Received  January 2018 Revised  January 2018 Published  August 2018

Fund Project: X. Bao was partially supported by Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1014) and NSF of China (11701041). Z. Shen was supported by a start-up grant from the University of Alberta.

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution ${\bf u} = {\bf 0}$ of such a system is unstable and the system has a stable space-time periodic positive solution ${\bf u^*}(t,x)$. We first show that in any direction $ξ∈ \mathbb{S}^{N-1}$, such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed $c^{*}(ξ)$ in the direction of $ξ$. Next, we show that for any $c>c^{*}(ξ)$, there are space-time periodic traveling wave solutions of the form ${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$ connecting ${\bf u^*}$ and ${\bf 0}$, and propagating in the direction of $ξ$ with speed $c$, where $Φ(x,t,y)$ is periodic in $t$ and $y$, and there is no such solution for $c<c^{*}(ξ)$. We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

Citation: 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
References:
[1]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.  Google Scholar

[2]

X. BaoW.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition sys tems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.  Google Scholar

[3]

X. Bao and W. Shen, Criteria for the existence of principal eigenvalue of time periodic cooper ative linear system with nonlocal dispersal, Proc. Amer. Math. Soc., 145 (2017), 2881-2894.  doi: 10.1090/proc/13602.  Google Scholar

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.7018.  Google Scholar

[5]

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

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

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W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar

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S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[9]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.   Google Scholar

[10]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Functional Analysis, 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.  Google Scholar

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J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Eqnuations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[13]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for Lotka-volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[14]

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

[15]

G. HetzerW. Shen and A. Zhang, Effects of spatical variatious 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.  Google Scholar

[16]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448.   Google Scholar

[17]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction diffusion compe tition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[18]

C. HuY. KuangB. Li and H. Liu, Spreading speeds and traveling wave solutions in coop erative intergal-differential system, Discrete Contin. Dyn. Syst. B, 20 (2015), 1663-1684.  doi: 10.3934/dcdsb.2015.30.1663.  Google Scholar

[19]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[20]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Non linear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[21] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg, 1976.   Google Scholar
[22]

L. KongNar Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.  Google Scholar

[23]

M. LewisH. Weinberger and B. Li, Spreading speed and linear determincay for two species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[24]

B. LiH. F. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative system, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[25]

W.-T. LiG. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction diffusion systems with applications to diffusion competition system, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[26]

W.-T. LiL Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[27]

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

[28]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Analysis, 259 (2012), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[29]

R. Lui, Biological growth and spread modeled by system of recursions. Ⅰ. Mathematical theory, Math. Biosci., 0 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[30]

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

[31]

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

[32]

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 Contin. Dyn. Syst., 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.617.  Google Scholar

[33]

S. PanW.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffsion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.  Google Scholar

[34]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonliner Anal. Theory, Methods, Applications, 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[35]

S. Pan and G. Lin, Invasion traveling wave solutions of a competition system with dispersal, Boundary Value Problems 2012, (2012), 120.  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[36] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, Berlin Heidelberg Tokyo, 1983.  doi: 10.1007/978-1-4612-5561-1.  Google Scholar
[37]

Nar Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, Dynamics J., Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[38]

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

[39]

W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

[40]

W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101.   Google Scholar

[41]

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

[42]

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

[43]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction advection diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.  Google Scholar

[44]

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

[45]

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.  Google Scholar

[46]

H. F. WeinbergerM. Lewis and B. Li, Analysis of linear determinacy for speed in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[47]

X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics, Ser. 16, Springer-Verlag, NewYork, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[48]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[49]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection reaction diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

show all references

References:
[1]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.  Google Scholar

[2]

X. BaoW.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition sys tems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.  Google Scholar

[3]

X. Bao and W. Shen, Criteria for the existence of principal eigenvalue of time periodic cooper ative linear system with nonlocal dispersal, Proc. Amer. Math. Soc., 145 (2017), 2881-2894.  doi: 10.1090/proc/13602.  Google Scholar

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.7018.  Google Scholar

[5]

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

[6]

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

[7]

W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar

[8]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[9]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.   Google Scholar

[10]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Functional Analysis, 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.  Google Scholar

[11]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Eqnuations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[13]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for Lotka-volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[14]

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

[15]

G. HetzerW. Shen and A. Zhang, Effects of spatical variatious 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.  Google Scholar

[16]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448.   Google Scholar

[17]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction diffusion compe tition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[18]

C. HuY. KuangB. Li and H. Liu, Spreading speeds and traveling wave solutions in coop erative intergal-differential system, Discrete Contin. Dyn. Syst. B, 20 (2015), 1663-1684.  doi: 10.3934/dcdsb.2015.30.1663.  Google Scholar

[19]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[20]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Non linear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[21] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg, 1976.   Google Scholar
[22]

L. KongNar Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.  Google Scholar

[23]

M. LewisH. Weinberger and B. Li, Spreading speed and linear determincay for two species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[24]

B. LiH. F. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative system, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[25]

W.-T. LiG. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction diffusion systems with applications to diffusion competition system, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[26]

W.-T. LiL Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[27]

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

[28]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Analysis, 259 (2012), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[29]

R. Lui, Biological growth and spread modeled by system of recursions. Ⅰ. Mathematical theory, Math. Biosci., 0 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[30]

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

[31]

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

[32]

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 Contin. Dyn. Syst., 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.617.  Google Scholar

[33]

S. PanW.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffsion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.  Google Scholar

[34]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonliner Anal. Theory, Methods, Applications, 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[35]

S. Pan and G. Lin, Invasion traveling wave solutions of a competition system with dispersal, Boundary Value Problems 2012, (2012), 120.  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[36] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, Berlin Heidelberg Tokyo, 1983.  doi: 10.1007/978-1-4612-5561-1.  Google Scholar
[37]

Nar Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, Dynamics J., Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[38]

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

[39]

W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

[40]

W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101.   Google Scholar

[41]

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

[42]

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

[43]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction advection diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.  Google Scholar

[44]

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

[45]

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.  Google Scholar

[46]

H. F. WeinbergerM. Lewis and B. Li, Analysis of linear determinacy for speed in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[47]

X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics, Ser. 16, Springer-Verlag, NewYork, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[48]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[49]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection reaction diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

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