American Institute of Mathematical Sciences

Advanced
March  2018, 23(2): 587-608. doi: 10.3934/dcdsb.2018035

Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal

Guo-Bao Zhang 1,, , Ruyun Ma 1, and  Xue-Shi Li 2,

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
2. 

School of Mathematics, Lanzhou City University, Lanzhou, Gansu 730070, China

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

Received  April 2016 Revised  July 2017 Published  March 2018 Early access  December 2017

This paper is concerned with the traveling waves of a nonlocal dispersal Lotka-Volterra strong competition model with bistable nonlinearity. We first establish the asymptotic behavior of traveling waves at infinity. Then by applying the stronger comparison principle and the sliding method, we prove that the traveling waves with nonzero speed are strictly monotone. Moreover, the uniqueness of wave speeds is also obtained.

Keywords: Nonlocal dispersal, strong competition system, traveling waves, monotonicity, uniqueness, wave speed.
Mathematics Subject Classification: Primary: 35K57, 34K60; Secondary: 92D25.
Citation: Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
References:
[1]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect.A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[6]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

[7]

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

[8]

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

[9]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  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. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Continuous Dynam. Systems -B, 17 (2012), 2713-2724.  doi: 10.3934/dcdsb.2012.17.2713.  Google Scholar

[13]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterrra-Lotka competition model with nonlocal dispersal, Commu. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[14]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, in "Numerical and Applied Mathematic, Part Ⅱ" (Paris, 1988), IMACS Ann. Comput. Appl. Math., 1. 2, Baltzer, Basel, (1989), 687–692.  Google Scholar

[15]

Y. Hosono, The minimal speed of traveling fronts for a diffusion Lotka-Volterra competition model, Bulletin of Math. Biology, 60 (1998), 435-448.  doi: 10.1006/bulm.1997.0008.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[19]

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

[20]

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.  Google Scholar

[21]

G. Lin and W. T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 224 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[22]

J. Murray, Mathematical Biology, 3 $^{nd}$, Springer, Berlin-Heidelberg, New York, 2003.  Google Scholar

[23]

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.  Google Scholar

[24]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 2012 (2012), 1-11.  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[25]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[26]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[27]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems Translation of Mathematical Monographs, Vol. 140, Amer. Math. Soc., Priovidence, 1994.  Google Scholar

[28]

C.-C. Wu, Existence of traveling wavefront for discrete bistable competition model, Discrete Continuous Dynam. Systems -B, 16 (2011), 973-984.  doi: 10.3934/dcdsb.2011.16.973.  Google Scholar

[29]

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

[30]

G.-B. Zhang, W.-T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar

[31]

G.-B. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Anal., 74 (2011), 5030-5047.  doi: 10.1016/j.na.2011.04.069.  Google Scholar

[32]

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.  Google Scholar

[33]

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]

P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[4]

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

[5]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect.A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[6]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

[7]

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

[8]

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

[9]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  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. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Continuous Dynam. Systems -B, 17 (2012), 2713-2724.  doi: 10.3934/dcdsb.2012.17.2713.  Google Scholar

[13]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterrra-Lotka competition model with nonlocal dispersal, Commu. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[14]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, in "Numerical and Applied Mathematic, Part Ⅱ" (Paris, 1988), IMACS Ann. Comput. Appl. Math., 1. 2, Baltzer, Basel, (1989), 687–692.  Google Scholar

[15]

Y. Hosono, The minimal speed of traveling fronts for a diffusion Lotka-Volterra competition model, Bulletin of Math. Biology, 60 (1998), 435-448.  doi: 10.1006/bulm.1997.0008.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[19]

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

[20]

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.  Google Scholar

[21]

G. Lin and W. T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 224 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[22]

J. Murray, Mathematical Biology, 3 $^{nd}$, Springer, Berlin-Heidelberg, New York, 2003.  Google Scholar

[23]

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.  Google Scholar

[24]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 2012 (2012), 1-11.  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[25]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[26]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[27]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems Translation of Mathematical Monographs, Vol. 140, Amer. Math. Soc., Priovidence, 1994.  Google Scholar

[28]

C.-C. Wu, Existence of traveling wavefront for discrete bistable competition model, Discrete Continuous Dynam. Systems -B, 16 (2011), 973-984.  doi: 10.3934/dcdsb.2011.16.973.  Google Scholar

[29]

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

[30]

G.-B. Zhang, W.-T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar

[31]

G.-B. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Anal., 74 (2011), 5030-5047.  doi: 10.1016/j.na.2011.04.069.  Google Scholar

[32]

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.  Google Scholar

[33]

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

[1]

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
[2]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
[3]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
[4]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[5]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[6]

Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
[7]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
[8]

Fang-Di Dong, Wan-Tong Li, Jia-Bing Wang. Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272
[9]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[10]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
[11]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
[12]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
[13]

Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001
[14]

Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340
[15]

Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121
[16]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[17]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[18]

Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053
[19]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
[20]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (305)
  • HTML views (300)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences