# American Institute of Mathematical Sciences

## Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

E-mail address: zhanggb2011@nwnu.edu.cn (G.-B. Zhang)(Corresponding author)

Received  November 2017 Revised  February 2018 Published  June 2018

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → -∞, but can be arbitrarily large in other locations.

Citation: Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018218
##### References:
 [1] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited), Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincare Anal. Non Lineaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3. [3] J. Carr and A. Chmaj, Uniqueness of travelling waves of nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [4] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [5] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. http://projecteuclid.org/euclid.ade/1366809230 [6] X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0. [7] J. Coville, On uniqueness and monotonicity of solutions of nonlocal reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [8] F.-D. Dong, W.-T. Li and J.-B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Continuous Dynam. Systems, 37 (2017), 6291-6318. doi: 10.3934/dcds.2017272. [9] 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. [10] P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191. [11] J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391. doi: 10.1016/j.jde.2012.01.009. [12] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829. doi: 10.11650/tjm.19.2015.5373. [13] X. Hou, B. Wang and Z. C. Zhang, The mutual inclusion in a nonlocal competitive Lotka Volterra system, Japan J. Indust. Appl. Math., 31 (2014), 87-110. doi: 10.1007/s13160-013-0126-0. [14] R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Continuous Dynam. Systems, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. [15] R. Huang, M. Mei, K. J. Zhang and Q. F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Continuous Dynam. Systems, 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331. [16] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biology, 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [17] K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal., 75 (2012), 3705-3722. doi: 10.1016/j.na.2012.01.024. [18] K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497. doi: 10.1016/j.jmaa.2011.11.055. [19] K. Li, J. Huang and X. Li, Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal competitive system, Commu. Pure Appl. Anal., 16 (2017), 131-150. doi: 10.3934/cpaa.2017006. [20] Y. Li, W.-T. Li and G.-B. Zhang, Stability and uniqueness of traveling waves of a nonlocal dispersal SIR epidemic model, Dynam. Part. Differential Equations, 14 (2017), 87-123. doi: 10.4310/DPDE.2017.v14.n2.a1. [21] X.-S. Li and G. Lin, Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays, Appl. Math. Comput., 204 (2008), 738-744. doi: 10.1016/j.amc.2008.07.016. [22] W.-T. Li, L. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Continuous Dynam. Systems, 35 (2015), 1531-1560. doi: 10.3934/dcds.2015.35.1531. [23] C. K. Lin, C. T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. [24] G. Lv and X. Wang, Stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, Z. Anal. Anwend. J. Anal. Appl., 33 (2014), 463-480. doi: 10.4171/ZAA/1523. [25] M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (Ⅰ) local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [26] M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (Ⅱ) nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [27] M. Mei and J. W. H. So, Stability of strong travelling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. [28] M. Mei, J.W.H. So, M.Y. Li and S.S.P. Shen, Asymptotic stability of travelling waves for the Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinbourgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. [29] M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790. doi: 10.1137/090776342. [30] S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. [31] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 120 (2012), 1-11. doi: 10.1186/1687-2770-2012-120. [32] H. L. Smith and X.-Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [33] Y.-R. Yang, W.-T. Li and S.-L. Wu, Exponential stability of traveling fronts in a diffusion epidemic system with delay, Nonlinear Anal. RWA, 12 (2011), 1223-1234. doi: 10.1016/j.nonrwa.2010.09.017. [34] Y.-R. Yang, W.-T. Li and S.-L. Wu, Stability of traveling waves in a monostable delayed system without quasi-monotonicity, Nonlinear Anal. RWA, 14 (2013), 1511-1526. doi: 10.1016/j.nonrwa.2012.10.015. [35] Z. Yu, F. Xu and W. G. Zhang, Stability of invasion traveling waves for a competition system with nonlocal dispersals, Appl. Anal., 96 (2017), 1107-1125. doi: 10.1080/00036811.2016.1178242. [36] Z.-X. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. [37] Z. Yu and R. Yuan, Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwan. J. Math., 17 (2013), 2163-2190. doi: 10.11650/tjm.17.2013.3794. [38] G.-B. Zhang, Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation, Appl. Anal., 96 (2017), 1830-1866. doi: 10.1080/00036811.2016.1197913. [39] G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124. doi: 10.1016/j.jde.2012.01.014. [40] G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844. doi: 10.1007/s00033-013-0353-x. [41] G.-B. Zhang, R. Ma and X.-S. Li, Traveling waves for a Lotka-Volterra strong competition system with nonlocal dispersal, Discrete Continuous Dynam. Systems - B, 23 (2018), 587-608. doi: 10.3934/dcdsb.2018035. [42] G.-B. Zhang, Y. Li and Z.-S. Feng, Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay, J. Comput. Appl. Math., (2018), in press. doi: 10.1016/j.cam.2018.05.018. [43] L. Zhang and B. Li, Traveling wave solutions in an integro-differential competition model, Discrete Continuous Dynam. Systems - B, 17 (2012), 417-428. doi: 10.3934/dcdsb.2012.17.417.

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##### References:
 [1] M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited), Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8. [2] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincare Anal. Non Lineaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3. [3] J. Carr and A. Chmaj, Uniqueness of travelling waves of nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. [4] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [5] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. http://projecteuclid.org/euclid.ade/1366809230 [6] X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146. doi: 10.1007/s00208-003-0414-0. [7] J. Coville, On uniqueness and monotonicity of solutions of nonlocal reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [8] F.-D. Dong, W.-T. Li and J.-B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Continuous Dynam. Systems, 37 (2017), 6291-6318. doi: 10.3934/dcds.2017272. [9] 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. [10] P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191. [11] J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391. doi: 10.1016/j.jde.2012.01.009. [12] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829. doi: 10.11650/tjm.19.2015.5373. [13] X. Hou, B. Wang and Z. C. Zhang, The mutual inclusion in a nonlocal competitive Lotka Volterra system, Japan J. Indust. Appl. Math., 31 (2014), 87-110. doi: 10.1007/s13160-013-0126-0. [14] R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Continuous Dynam. Systems, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. [15] R. Huang, M. Mei, K. J. Zhang and Q. F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Continuous Dynam. Systems, 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331. [16] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biology, 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [17] K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal., 75 (2012), 3705-3722. doi: 10.1016/j.na.2012.01.024. [18] K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497. doi: 10.1016/j.jmaa.2011.11.055. [19] K. Li, J. Huang and X. Li, Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal competitive system, Commu. Pure Appl. Anal., 16 (2017), 131-150. doi: 10.3934/cpaa.2017006. [20] Y. Li, W.-T. Li and G.-B. Zhang, Stability and uniqueness of traveling waves of a nonlocal dispersal SIR epidemic model, Dynam. Part. Differential Equations, 14 (2017), 87-123. doi: 10.4310/DPDE.2017.v14.n2.a1. [21] X.-S. Li and G. Lin, Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays, Appl. Math. Comput., 204 (2008), 738-744. doi: 10.1016/j.amc.2008.07.016. [22] W.-T. Li, L. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Continuous Dynam. Systems, 35 (2015), 1531-1560. doi: 10.3934/dcds.2015.35.1531. [23] C. K. Lin, C. T. Lin, Y. P. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. [24] G. Lv and X. Wang, Stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, Z. Anal. Anwend. J. Anal. Appl., 33 (2014), 463-480. doi: 10.4171/ZAA/1523. [25] M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (Ⅰ) local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026. [26] M. Mei, C. K. Lin, C. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: (Ⅱ) nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020. [27] M. Mei and J. W. H. So, Stability of strong travelling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Royal Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. [28] M. Mei, J.W.H. So, M.Y. Li and S.S.P. Shen, Asymptotic stability of travelling waves for the Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinbourgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. [29] M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790. doi: 10.1137/090776342. [30] S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. [31] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 120 (2012), 1-11. doi: 10.1186/1687-2770-2012-120. [32] H. L. Smith and X.-Q. Zhao, Global asymptotical stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [33] Y.-R. Yang, W.-T. Li and S.-L. Wu, Exponential stability of traveling fronts in a diffusion epidemic system with delay, Nonlinear Anal. RWA, 12 (2011), 1223-1234. doi: 10.1016/j.nonrwa.2010.09.017. [34] Y.-R. Yang, W.-T. Li and S.-L. Wu, Stability of traveling waves in a monostable delayed system without quasi-monotonicity, Nonlinear Anal. RWA, 14 (2013), 1511-1526. doi: 10.1016/j.nonrwa.2012.10.015. [35] Z. Yu, F. Xu and W. G. Zhang, Stability of invasion traveling waves for a competition system with nonlocal dispersals, Appl. Anal., 96 (2017), 1107-1125. doi: 10.1080/00036811.2016.1178242. [36] Z.-X. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. [37] Z. Yu and R. Yuan, Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwan. J. Math., 17 (2013), 2163-2190. doi: 10.11650/tjm.17.2013.3794. [38] G.-B. Zhang, Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation, Appl. Anal., 96 (2017), 1830-1866. doi: 10.1080/00036811.2016.1197913. [39] G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124. doi: 10.1016/j.jde.2012.01.014. [40] G.-B. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844. doi: 10.1007/s00033-013-0353-x. [41] G.-B. Zhang, R. Ma and X.-S. Li, Traveling waves for a Lotka-Volterra strong competition system with nonlocal dispersal, Discrete Continuous Dynam. Systems - B, 23 (2018), 587-608. doi: 10.3934/dcdsb.2018035. [42] G.-B. Zhang, Y. Li and Z.-S. Feng, Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay, J. Comput. Appl. Math., (2018), in press. doi: 10.1016/j.cam.2018.05.018. [43] L. Zhang and B. Li, Traveling wave solutions in an integro-differential competition model, Discrete Continuous Dynam. Systems - B, 17 (2012), 417-428. doi: 10.3934/dcdsb.2012.17.417.
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