American Institute of Mathematical Sciences

December  2017, 37(12): 6291-6318. doi: 10.3934/dcds.2017272

Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application

 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  January 2017 Revised  June 2017 Published  August 2017

In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of $x$-axis for a three-species Lotka-Volterra competition system.

Citation: 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 and Continuous Dynamical Systems, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272
References:
 [1] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037. [2] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5. [3] 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. [4] X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028. [5] X. Chen, J. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959. [6] X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824. [7] A. De Masi, T. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506. [8] A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.  doi: 10.1017/S0308210500022472. [9] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9. [10] B. Ermentrout and J. Mcleod, Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X. [11] P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191. [12] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979. [13] Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454. [14] 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. [15] 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. [16] J. S. Guo and Y. C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629. [17] J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024. [18] J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193. [19] F. Hamel and N. Nadirashvili, Entire solutions of the KPP equations, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. [20] F. Hamel and N. Nadirashvili, Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [21] W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005. [22] W. T. Li, J. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006. [23] W. T. Li, Z. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023. [24] W. T. Li, L. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531. [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [26] Y. Morita and H. Ninomiya, Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x. [27] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715. [28] 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. [29] M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005. [30] Z. C. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8. [31] Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1. [32] Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312. [33] C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8. [34] S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6. [35] H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150. [36] 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. [37] L. Zhang, W. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.

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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. Ration. Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037. [2] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5. [3] 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. [4] X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028. [5] X. Chen, J. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959. [6] X. Chen, S. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824. [7] A. De Masi, T. Gobron and E. Presutti, Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506. [8] A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.  doi: 10.1017/S0308210500022472. [9] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9. [10] B. Ermentrout and J. Mcleod, Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X. [11] P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191. [12] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979. [13] Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.  doi: 10.11650/twjm/1500558454. [14] 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. [15] 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. [16] J. S. Guo and Y. C. Lin, Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629. [17] J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024. [18] J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193. [19] F. Hamel and N. Nadirashvili, Entire solutions of the KPP equations, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. [20] F. Hamel and N. Nadirashvili, Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [21] W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005. [22] W. T. Li, J. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006. [23] W. T. Li, Z. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023. [24] W. T. Li, L. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531. [25] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [26] Y. Morita and H. Ninomiya, Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x. [27] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715. [28] 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. [29] M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005. [30] Z. C. Wang, W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8. [31] Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1. [32] Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312. [33] C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8. [34] S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6. [35] H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150. [36] 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. [37] L. Zhang, W. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.
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