American Institute of Mathematical Sciences

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

Invasion entire solutions in a competition system with nonlocal dispersal

Wan-Tong Li 1, , Li Zhang 2, and  Guo-Bao Zhang 3,

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.
Keywords: asymptotic behavior., traveling wave solution, Entire solution, nonlocal dispersal.
Mathematics Subject Classification: Primary: 35K57, 37C65; Secondary: 92D3.
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

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[6]

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[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

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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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