# American Institute of Mathematical Sciences

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

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

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

Received  April 2016 Revised  July 2017 Published  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.

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. [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. [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. [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 [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. [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. [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. [8] P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. [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. [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. [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 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] J. Murray, Mathematical Biology, 3 $^{nd}$, Springer, Berlin-Heidelberg, New York, 2003. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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 [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. [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. [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. [8] P. Fife, Some nonclassical trends in parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. [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. [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. [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 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] J. Murray, Mathematical Biology, 3 $^{nd}$, Springer, Berlin-Heidelberg, New York, 2003. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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