August  2020, 25(8): 3005-3031. doi: 10.3934/dcdsb.2020049

Periodic traveling wave solutions of periodic integrodifference systems

1. 

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

2. 

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received  March 2019 Revised  September 2019 Published  February 2020

This paper is concerned with the periodic traveling wave solutions of integrodifference systems with periodic parameters. Without the assumptions on monotonicity, the existence of periodic traveling wave solutions is deduced to the existence of generalized upper and lower solutions by fixed point theorem and an operator with multi steps. The asymptotic behavior of periodic traveling wave solutions is investigated by the stability of periodic solutions in the corresponding initial value problem or the corresponding difference systems. To illustrate our conclusions, we study the periodic traveling wave solutions of two models including a scalar equation and a competitive type system, which do not generate monotone semiflows. The existence or nonexistence of periodic traveling wave solutions with all positive wave speeds is presented, which implies the minimal wave speeds of these models.

Citation: Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
References:
[1]

W. J. BoG. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189.  Google Scholar

[2]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.  doi: 10.1017/S0305004100053494.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

W. DingX. Liang and B. Xu, Spreading speeds of $N$-season spatially periodic integro-difference models, Discrete Contin. Dyn. Syst., 33 (2013), 3443-3472.  doi: 10.3934/dcds.2013.33.3443.  Google Scholar

[5]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[6]

J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.  doi: 10.1007/BF00173295.  Google Scholar

[11]

M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[12]

B. LiM. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[13]

L. LiS. KangL. Kong and H. Chen, Minimal wave speed of a competition integrodifference system, J. Difference Equ. Appl., 24 (2018), 941-954.  doi: 10.1080/10236198.2018.1442446.  Google Scholar

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X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019). doi: 10.3390/math7030269.  Google Scholar

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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

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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

[17]

G. Lin, Traveling wave solutions for integro-difference systems, J. Differential Equations, 258 (2015), 2908-2940.  doi: 10.1016/j.jde.2014.12.030.  Google Scholar

[18]

G. Lin, Spreading speeds and traveling wave solutions for a delayed diffusion equation without quasimonotonicity, J. Dynam. Differential Equations, 31 (2019), 2275-2292.  doi: 10.1007/s10884-018-9707-6.  Google Scholar

[19]

G. LinW.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201.  doi: 10.1007/s00285-010-0334-z.  Google Scholar

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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

[21]

G. Lin and T. Su, Asymptotic speeds of spread and traveling wave solutions of a second order integrodifference equation without monotonicity, J. Difference Equ. Appl., 22 (2016), 542-557.  doi: 10.1080/10236198.2015.1112383.  Google Scholar

[22]

G. Lin and H. Wang, Traveling wave solutions of a reaction-diffusion equation with state-dependent delay, Commun. Pure Appl. Anal., 15 (2016), 319-334.  doi: 10.3934/cpaa.2016.15.319.  Google Scholar

[23]

X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019). doi: 10.3390/math7030291.  Google Scholar

[24]

R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.  doi: 10.1137/0513064.  Google Scholar

[25]

R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953.  doi: 10.1137/0513065.  Google Scholar

[26]

R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206.  doi: 10.1137/0516087.  Google Scholar

[27]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar

[28]

R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168.  doi: 10.1137/0517015.  Google Scholar

[29]

W. M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.  doi: 10.3934/nhm.2013.8.379.  Google Scholar

[30]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[31]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 535-544.  doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar

[32]

S. PanG. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.  Google Scholar

[33]

R. J. Sacker, Global stability in a multi-species periodic Leslie-Gower model, J. Biol. Dyn., 5 (2011), 549-562.  doi: 10.1080/17513758.2011.554891.  Google Scholar

[34]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.   Google Scholar

[35]

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

[36]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266.  doi: 10.3934/dcdsb.2012.17.2243.  Google Scholar

[37]

Z.-C. WangW.-T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[38]

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

[39]

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

[40]

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

[41]

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

[42]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[43]

R. Wu and X. Q. Zhao, Propagation dynamics for a spatially periodic integrodifference competition model, J. Differential Equations, 264 (2018), 6507-6534.  doi: 10.1016/j.jde.2018.01.039.  Google Scholar

[44]

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

[45]

L. Zhang and S. Pan, Entire solutions of integrodifference equations, J. Difference Equ. Appl., 25 (2019), 505-514.  doi: 10.1080/10236198.2019.1583748.  Google Scholar

[46]

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. (9), 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[47]

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

[48]

X. Q. Zhao, Dynamincal Systems in Population Biology, CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[49]

Z. Zhou and X. Zou, Stable periodic solutions in a discrete periodic logistic equation, Appl. Math. Lett., 16 (2003), 165-171.  doi: 10.1016/S0893-9659(03)80027-7.  Google Scholar

show all references

References:
[1]

W. J. BoG. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189.  Google Scholar

[2]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.  doi: 10.1017/S0305004100053494.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

W. DingX. Liang and B. Xu, Spreading speeds of $N$-season spatially periodic integro-difference models, Discrete Contin. Dyn. Syst., 33 (2013), 3443-3472.  doi: 10.3934/dcds.2013.33.3443.  Google Scholar

[5]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[6]

J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.  doi: 10.1007/BF00173295.  Google Scholar

[11]

M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[12]

B. LiM. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[13]

L. LiS. KangL. Kong and H. Chen, Minimal wave speed of a competition integrodifference system, J. Difference Equ. Appl., 24 (2018), 941-954.  doi: 10.1080/10236198.2018.1442446.  Google Scholar

[14]

X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019). doi: 10.3390/math7030269.  Google Scholar

[15]

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

[16]

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

[17]

G. Lin, Traveling wave solutions for integro-difference systems, J. Differential Equations, 258 (2015), 2908-2940.  doi: 10.1016/j.jde.2014.12.030.  Google Scholar

[18]

G. Lin, Spreading speeds and traveling wave solutions for a delayed diffusion equation without quasimonotonicity, J. Dynam. Differential Equations, 31 (2019), 2275-2292.  doi: 10.1007/s10884-018-9707-6.  Google Scholar

[19]

G. LinW.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201.  doi: 10.1007/s00285-010-0334-z.  Google Scholar

[20]

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

[21]

G. Lin and T. Su, Asymptotic speeds of spread and traveling wave solutions of a second order integrodifference equation without monotonicity, J. Difference Equ. Appl., 22 (2016), 542-557.  doi: 10.1080/10236198.2015.1112383.  Google Scholar

[22]

G. Lin and H. Wang, Traveling wave solutions of a reaction-diffusion equation with state-dependent delay, Commun. Pure Appl. Anal., 15 (2016), 319-334.  doi: 10.3934/cpaa.2016.15.319.  Google Scholar

[23]

X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019). doi: 10.3390/math7030291.  Google Scholar

[24]

R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.  doi: 10.1137/0513064.  Google Scholar

[25]

R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953.  doi: 10.1137/0513065.  Google Scholar

[26]

R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206.  doi: 10.1137/0516087.  Google Scholar

[27]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar

[28]

R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168.  doi: 10.1137/0517015.  Google Scholar

[29]

W. M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.  doi: 10.3934/nhm.2013.8.379.  Google Scholar

[30]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[31]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 535-544.  doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar

[32]

S. PanG. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.  Google Scholar

[33]

R. J. Sacker, Global stability in a multi-species periodic Leslie-Gower model, J. Biol. Dyn., 5 (2011), 549-562.  doi: 10.1080/17513758.2011.554891.  Google Scholar

[34]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.   Google Scholar

[35]

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

[36]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266.  doi: 10.3934/dcdsb.2012.17.2243.  Google Scholar

[37]

Z.-C. WangW.-T. Li and S. Ruan, Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[38]

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

[39]

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

[40]

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

[41]

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

[42]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[43]

R. Wu and X. Q. Zhao, Propagation dynamics for a spatially periodic integrodifference competition model, J. Differential Equations, 264 (2018), 6507-6534.  doi: 10.1016/j.jde.2018.01.039.  Google Scholar

[44]

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

[45]

L. Zhang and S. Pan, Entire solutions of integrodifference equations, J. Difference Equ. Appl., 25 (2019), 505-514.  doi: 10.1080/10236198.2019.1583748.  Google Scholar

[46]

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. (9), 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[47]

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

[48]

X. Q. Zhao, Dynamincal Systems in Population Biology, CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[49]

Z. Zhou and X. Zou, Stable periodic solutions in a discrete periodic logistic equation, Appl. Math. Lett., 16 (2003), 165-171.  doi: 10.1016/S0893-9659(03)80027-7.  Google Scholar

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