# American Institute of Mathematical Sciences

April  2015, 35(4): 1531-1560. doi: 10.3934/dcds.2015.35.1531

## Invasion entire solutions in a competition system with nonlocal dispersal

 1 School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 3 School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received  July 2013 Revised  December 2013 Published  November 2014

This paper is concerned with invasion entire solutions of a Lotka-Volterra competition system with nonlocal dispersal, which formulate a new invasion way of the stronger species to the weaker one. We first give the asymptotic behavior of traveling wave solutions at infinity. Then by the comparison principle and sub-super solutions method, we establish the existence of invasion entire solutions which behave as two monotone waves with different speeds and coming from both sides of $x$-axis.
Citation: Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar [2] P. W. Bates, On some nonlocal evolution equations arising in materials science,, in H. Brunner, 48 (2006), 13. Google Scholar [3] P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar [4] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [5] X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation,, J. Differential Equations, 212 (2005), 62. doi: 10.1016/j.jde.2004.10.028. Google Scholar [6] X. Chen, J.-S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearity,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207. doi: 10.1017/S0308210500004959. Google Scholar [7] Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation,, Taiwanese J. Math., 8 (2004), 15. Google Scholar [8] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153. Google Scholar [9] 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. doi: 10.3934/dcds.2005.12.193. Google Scholar [10] J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku. Math. J., 62 (2010), 17. doi: 10.2748/tmj/1270041024. 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. 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 Contin. Dyn. Syst. Ser. B, 17 (2012), 2713. doi: 10.3934/dcdsb.2012.17.2713. Google Scholar [13] F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution,, Comm. Pure Appl. Math., 52 (1999), 1255. doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. Google Scholar [14] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^N$,, Arch. Rational Mech. Anal., 157 (2001), 91. doi: 10.1007/PL00004238. Google Scholar [15] Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, Numerical and Applied Mathematics, (1989), 687. Google Scholar [16] M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233. doi: 10.1007/BF03167402. Google Scholar [17] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. Google Scholar [18] C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar [19] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar [20] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar [21] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar [22] 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. doi: 10.1016/j.jde.2008.03.023. Google Scholar [23] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,, J. Differential Equations, 244 (2008), 487. doi: 10.1016/j.jde.2007.10.019. Google Scholar [24] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [25] Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations,, J. Dynam. Differential Equations, 18 (2006), 841. doi: 10.1007/s10884-006-9046-x. Google Scholar [26] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. doi: 10.1137/080723715. Google Scholar [27] J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,, Third edition. Interdisciplinary Applied Mathematics, 18 (2003). Google Scholar [28] 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. doi: 10.1007/s00033-007-7005-y. Google Scholar [29] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012), 1. doi: 10.1186/1687-2770-2012-120. Google Scholar [30] 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. doi: 10.1016/j.jde.2011.04.020. Google Scholar [31] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems,, Translation of Mathematical Monographs, (1994). Google Scholar [32] M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed,, Nonlinearity, 23 (2010), 1609. doi: 10.1088/0951-7715/23/7/005. Google Scholar [33] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047. doi: 10.1090/S0002-9947-08-04694-1. Google Scholar [34] 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. doi: 10.1137/080727312. Google Scholar [35] S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics,, Nonlinear Anal. Real World Appl., 13 (2012), 1991. doi: 10.1016/j.nonrwa.2011.12.020. Google Scholar [36] S.-L. Wu, Y.-J. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921. doi: 10.3934/dcds.2013.33.921. Google Scholar [37] H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts,, Publ. Res. Inst. Math. Sci., 39 (2003), 117. doi: 10.2977/prims/1145476150. Google Scholar [38] Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications,, ANZIAM. J., 51 (2009), 49. doi: 10.1017/S1446181109000406. Google Scholar [39] Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems,, Chaos, 45 (2012), 1361. doi: 10.1016/j.chaos.2012.07.002. Google Scholar [40] G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021. Google Scholar [41] 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. doi: 10.1016/j.jde.2012.01.014. Google Scholar

show all references

##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar [2] P. W. Bates, On some nonlocal evolution equations arising in materials science,, in H. Brunner, 48 (2006), 13. Google Scholar [3] P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar [4] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [5] X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation,, J. Differential Equations, 212 (2005), 62. doi: 10.1016/j.jde.2004.10.028. Google Scholar [6] X. Chen, J.-S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearity,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207. doi: 10.1017/S0308210500004959. Google Scholar [7] Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation,, Taiwanese J. Math., 8 (2004), 15. Google Scholar [8] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153. Google Scholar [9] 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. doi: 10.3934/dcds.2005.12.193. Google Scholar [10] J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku. Math. J., 62 (2010), 17. doi: 10.2748/tmj/1270041024. 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. 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 Contin. Dyn. Syst. Ser. B, 17 (2012), 2713. doi: 10.3934/dcdsb.2012.17.2713. Google Scholar [13] F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution,, Comm. Pure Appl. Math., 52 (1999), 1255. doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W. Google Scholar [14] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^N$,, Arch. Rational Mech. Anal., 157 (2001), 91. doi: 10.1007/PL00004238. Google Scholar [15] Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, Numerical and Applied Mathematics, (1989), 687. Google Scholar [16] M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233. doi: 10.1007/BF03167402. Google Scholar [17] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. Google Scholar [18] C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar [19] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar [20] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar [21] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar [22] 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. doi: 10.1016/j.jde.2008.03.023. Google Scholar [23] G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,, J. Differential Equations, 244 (2008), 487. doi: 10.1016/j.jde.2007.10.019. Google Scholar [24] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [25] Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations,, J. Dynam. Differential Equations, 18 (2006), 841. doi: 10.1007/s10884-006-9046-x. Google Scholar [26] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. doi: 10.1137/080723715. Google Scholar [27] J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,, Third edition. Interdisciplinary Applied Mathematics, 18 (2003). Google Scholar [28] 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. doi: 10.1007/s00033-007-7005-y. Google Scholar [29] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012), 1. doi: 10.1186/1687-2770-2012-120. Google Scholar [30] 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. doi: 10.1016/j.jde.2011.04.020. Google Scholar [31] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems,, Translation of Mathematical Monographs, (1994). Google Scholar [32] M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed,, Nonlinearity, 23 (2010), 1609. doi: 10.1088/0951-7715/23/7/005. Google Scholar [33] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047. doi: 10.1090/S0002-9947-08-04694-1. Google Scholar [34] 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. doi: 10.1137/080727312. Google Scholar [35] S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics,, Nonlinear Anal. Real World Appl., 13 (2012), 1991. doi: 10.1016/j.nonrwa.2011.12.020. Google Scholar [36] S.-L. Wu, Y.-J. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921. doi: 10.3934/dcds.2013.33.921. Google Scholar [37] H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts,, Publ. Res. Inst. Math. Sci., 39 (2003), 117. doi: 10.2977/prims/1145476150. Google Scholar [38] Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications,, ANZIAM. J., 51 (2009), 49. doi: 10.1017/S1446181109000406. Google Scholar [39] Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems,, Chaos, 45 (2012), 1361. doi: 10.1016/j.chaos.2012.07.002. Google Scholar [40] G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021. Google Scholar [41] 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. doi: 10.1016/j.jde.2012.01.014. Google Scholar
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