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September  2018, 38(9): 4329-4351. doi: 10.3934/dcds.2018189

Traveling wave solutions for time periodic reaction-diffusion systems

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

* Corresponding author: Guo Lin

Received  July 2017 Revised  April 2018 Published  June 2018

This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super- and sub-solutions, which reduces the existence of traveling wave solutions to the existence of super- and sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.

Citation: Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805. doi: 10.1090/S0002-9947-99-02134-0. Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Mathematics, 446. Springer, Berlin, 1975. Google Scholar

[3]

X. BaoW. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[4]

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

[5]

P. Bates and F. Chen, Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 1999 (1999), no. 26, 19 pp. Google Scholar

[6]

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

[7]

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

[8]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. Google Scholar

[9]

A. Ducrot, M. Marion and V. Volpert, Reaction-diffusion Waves (with the Lewis Number Different from 1), Mathematics and Mathematical Modelling, Publibook, Paris, 2009. Google Scholar

[10]

J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. Google Scholar

[11]

J. Fang and X. Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. Google Scholar

[12]

J. Fang and X. Q. Zhao, Bistable traveling waves for monotone semiflows with application, J. Eur. Math. Soc., 17 (2015), 2243-2288. doi: 10.4171/JEMS/556. Google Scholar

[13]

T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. doi: 10.1016/j.jde.2006.05.006. Google Scholar

[14]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, 1979. Google Scholar

[15]

P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185. doi: 10.1016/0022-0396(81)90016-4. Google Scholar

[16]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[17]

B. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhauser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4. Google Scholar

[18]

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

[19]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. vol. 247, Longman Scientific and Technical, Wiley, Harlow, Essex, 1991. Google Scholar

[20]

S. B. Hsu and X. Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[21]

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

[22]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. Google Scholar

[27]

B. Lisena, Global stability in periodic competitive systems, Nonlinear Anal. RWA, 5 (2004), 613-627. doi: 10.1016/j.nonrwa.2004.01.002. Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Boston, 1995. Google Scholar

[29]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. Google Scholar

[30]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[31]

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

[32]

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

[33]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992. Google Scholar

[34]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅰ and Ⅱ, J. Differential Equations, 159 (1999), 1-101. doi: 10.1006/jdeq.1999.3652. Google Scholar

[35]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548. doi: 10.1006/jdeq.2000.3906. Google Scholar

[36]

W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339. doi: 10.1016/S0362-546X(03)00065-8. Google Scholar

[37]

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

[38]

W. J. Sheng and M. L. Cao, Entire solutions of the Fisher-KPP equation in time periodic media, Dyn. Partial Differ. Equ., 9 (2012), 133-145. doi: 10.4310/DPDE.2012.v9.n2.a3. Google Scholar

[39]

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Second Edition), Springer-Verlag, NewYork, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[40]

M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[41]

Z. Teng and L. Chen, Global asymptotic stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095. doi: 10.1016/S0362-546X(99)00441-1. Google Scholar

[42]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs 140, AMS, Providence, RI, 1994. Google Scholar

[43]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9. Google Scholar

[44]

Z. C. WangL. Zhang and X. Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differential Equations, 30 (2018), 379-403. doi: 10.1007/s10884-016-9546-2. Google Scholar

[45]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar

[46]

X. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations, 3 (1991), 541-573. doi: 10.1007/BF01049099. Google Scholar

[47]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations (The Second Edition), Science Press, Beijing, 2011.Google Scholar

[48]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572. doi: 10.1016/j.jde.2013.01.031. Google Scholar

[49]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611. doi: 10.1016/j.jde.2011.04.027. Google Scholar

[50]

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

[51]

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

[52]

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

[53]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

show all references

References:
[1]

N. D. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805. doi: 10.1090/S0002-9947-99-02134-0. Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Mathematics, 446. Springer, Berlin, 1975. Google Scholar

[3]

X. BaoW. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[4]

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

[5]

P. Bates and F. Chen, Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 1999 (1999), no. 26, 19 pp. Google Scholar

[6]

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

[7]

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

[8]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. Google Scholar

[9]

A. Ducrot, M. Marion and V. Volpert, Reaction-diffusion Waves (with the Lewis Number Different from 1), Mathematics and Mathematical Modelling, Publibook, Paris, 2009. Google Scholar

[10]

J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. Google Scholar

[11]

J. Fang and X. Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002. Google Scholar

[12]

J. Fang and X. Q. Zhao, Bistable traveling waves for monotone semiflows with application, J. Eur. Math. Soc., 17 (2015), 2243-2288. doi: 10.4171/JEMS/556. Google Scholar

[13]

T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. doi: 10.1016/j.jde.2006.05.006. Google Scholar

[14]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, 1979. Google Scholar

[15]

P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185. doi: 10.1016/0022-0396(81)90016-4. Google Scholar

[16]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[17]

B. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhauser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4. Google Scholar

[18]

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

[19]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. vol. 247, Longman Scientific and Technical, Wiley, Harlow, Essex, 1991. Google Scholar

[20]

S. B. Hsu and X. Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[21]

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

[22]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. Google Scholar

[27]

B. Lisena, Global stability in periodic competitive systems, Nonlinear Anal. RWA, 5 (2004), 613-627. doi: 10.1016/j.nonrwa.2004.01.002. Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Boston, 1995. Google Scholar

[29]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. Google Scholar

[30]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[31]

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

[32]

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

[33]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992. Google Scholar

[34]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅰ and Ⅱ, J. Differential Equations, 159 (1999), 1-101. doi: 10.1006/jdeq.1999.3652. Google Scholar

[35]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548. doi: 10.1006/jdeq.2000.3906. Google Scholar

[36]

W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003), 319-339. doi: 10.1016/S0362-546X(03)00065-8. Google Scholar

[37]

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

[38]

W. J. Sheng and M. L. Cao, Entire solutions of the Fisher-KPP equation in time periodic media, Dyn. Partial Differ. Equ., 9 (2012), 133-145. doi: 10.4310/DPDE.2012.v9.n2.a3. Google Scholar

[39]

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Second Edition), Springer-Verlag, NewYork, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[40]

M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[41]

Z. Teng and L. Chen, Global asymptotic stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095. doi: 10.1016/S0362-546X(99)00441-1. Google Scholar

[42]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs 140, AMS, Providence, RI, 1994. Google Scholar

[43]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates Z. Angew. Math. Phys., 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9. Google Scholar

[44]

Z. C. WangL. Zhang and X. Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differential Equations, 30 (2018), 379-403. doi: 10.1007/s10884-016-9546-2. Google Scholar

[45]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar

[46]

X. Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations, 3 (1991), 541-573. doi: 10.1007/BF01049099. Google Scholar

[47]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations (The Second Edition), Science Press, Beijing, 2011.Google Scholar

[48]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572. doi: 10.1016/j.jde.2013.01.031. Google Scholar

[49]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611. doi: 10.1016/j.jde.2011.04.027. Google Scholar

[50]

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

[51]

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

[52]

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

[53]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

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