# American Institute of Mathematical Sciences

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  August 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
##### References:
 [1] W. J. Bo, G. 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. [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. [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. [4] W. Ding, X. 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. [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. [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. [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. [8] J. Fang, X. 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. [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. [10] M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.  doi: 10.1007/BF00173295. [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. [12] B. Li, M. 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. [13] L. Li, S. Kang, L. 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. [14] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019). doi: 10.3390/math7030269. [15] X. Liang, Y. 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. [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. [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. [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. [19] G. Lin, W.-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. [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. [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. [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. [23] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019). doi: 10.3390/math7030291. [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. [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. [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. [27] F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2. [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. [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. [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. [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. [32] S. Pan, G. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597. [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. [34] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756. [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. [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. [37] Z.-C. Wang, W.-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. [38] Z. C. Wang, L. 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. [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. [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. [41] H. F. Weinberger, M. 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. [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. [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. [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. [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. [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. [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. [48] X. Q. Zhao, Dynamincal Systems in Population Biology, CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3. [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.

show all references

##### References:
 [1] W. J. Bo, G. 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. [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. [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. [4] W. Ding, X. 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. [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. [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. [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. [8] J. Fang, X. 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. [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. [10] M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.  doi: 10.1007/BF00173295. [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. [12] B. Li, M. 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. [13] L. Li, S. Kang, L. 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. [14] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019). doi: 10.3390/math7030269. [15] X. Liang, Y. 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. [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. [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. [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. [19] G. Lin, W.-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. [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. [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. [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. [23] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019). doi: 10.3390/math7030291. [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. [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. [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. [27] F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2. [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. [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. [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. [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. [32] S. Pan, G. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597. [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. [34] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756. [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. [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. [37] Z.-C. Wang, W.-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. [38] Z. C. Wang, L. 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. [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. [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. [41] H. F. Weinberger, M. 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. [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. [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. [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. [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. [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. [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. [48] X. Q. Zhao, Dynamincal Systems in Population Biology, CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3. [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.
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