April  2019, 24(4): 1511-1541. doi: 10.3934/dcdsb.2018218

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, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218
References:
[1]

M. AguerreaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. ChasseigneM. 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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F.-D. DongW.-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.  Google Scholar

[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.  Google Scholar

[10]

P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[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.  Google Scholar

[12]

J.-S. GuoY. WangC.-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.  Google Scholar

[13]

X. HouB. 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.  Google Scholar

[14]

R. HuangM. 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.  Google Scholar

[15]

R. HuangM. MeiK. 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.  Google Scholar

[16]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biology, 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. LiJ. 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.  Google Scholar

[20]

Y. LiW.-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.  Google Scholar

[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.  Google Scholar

[22]

W.-T. LiL. 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.  Google Scholar

[23]

C. K. LinC. T. LinY. 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.  Google Scholar

[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.  Google Scholar

[25]

M. MeiC. K. LinC. 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.  Google Scholar

[26]

M. MeiC. K. LinC. 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.  Google Scholar

[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.  Google Scholar

[28]

M. MeiJ.W.H. SoM.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.  Google Scholar

[29]

M. MeiC. 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.  Google Scholar

[30]

S. PanW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Y.-R. YangW.-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.  Google Scholar

[34]

Y.-R. YangW.-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.  Google Scholar

[35]

Z. YuF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

G.-B. ZhangW.-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.  Google Scholar

[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.  Google Scholar

[41]

G.-B. ZhangR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. AguerreaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. ChasseigneM. 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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F.-D. DongW.-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.  Google Scholar

[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.  Google Scholar

[10]

P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[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.  Google Scholar

[12]

J.-S. GuoY. WangC.-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.  Google Scholar

[13]

X. HouB. 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.  Google Scholar

[14]

R. HuangM. 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.  Google Scholar

[15]

R. HuangM. MeiK. 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.  Google Scholar

[16]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biology, 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. LiJ. 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.  Google Scholar

[20]

Y. LiW.-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.  Google Scholar

[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.  Google Scholar

[22]

W.-T. LiL. 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.  Google Scholar

[23]

C. K. LinC. T. LinY. 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.  Google Scholar

[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.  Google Scholar

[25]

M. MeiC. K. LinC. 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.  Google Scholar

[26]

M. MeiC. K. LinC. 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.  Google Scholar

[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.  Google Scholar

[28]

M. MeiJ.W.H. SoM.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.  Google Scholar

[29]

M. MeiC. 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.  Google Scholar

[30]

S. PanW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Y.-R. YangW.-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.  Google Scholar

[34]

Y.-R. YangW.-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.  Google Scholar

[35]

Z. YuF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

G.-B. ZhangW.-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.  Google Scholar

[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.  Google Scholar

[41]

G.-B. ZhangR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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