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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[47]

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

[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.

[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.

[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.

[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.

[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.

[53]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[47]

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

[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.

[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.

[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.

[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.

[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.

[53]

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

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