# 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

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

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. 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

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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. 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
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