Invasion entire solutions in a competition system with nonlocal dispersal

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531

References:

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[2]	P. W. Bates, On some nonlocal evolution equations arising in materials science, in H. Brunner, X.Q. Zhao and X. Zou (Eds.), Nonlinear dynamics and evolution equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52.
[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), 105-136. doi: 10.1007/s002050050037.
[4]	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.
[5]	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.
[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), 1207-1237. doi: 10.1017/S0308210500004959.
[7]	Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
[8]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
[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), 193-212. doi: 10.3934/dcds.2005.12.193.
[10]	J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28. doi: 10.2748/tmj/1270041024.
[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 Contin. Dyn. Syst. Ser. B, 17 (2012), 2713-2724. doi: 10.3934/dcdsb.2012.17.2713.
[13]	F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution, 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.
[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), 91-163. doi: 10.1007/PL00004238.
[15]	Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, Numerical and Applied Mathematics, Part II (Paris,1988), Baltzer,Basel, (1989), 687-692.
[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), 233-252. doi: 10.1007/BF03167402.
[17]	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.
[18]	C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.
[19]	C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047.
[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), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.
[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), 102-129. doi: 10.1016/j.jde.2008.03.023.
[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), 487-513. doi: 10.1016/j.jde.2007.10.019.
[24]	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.
[25]	Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861. doi: 10.1007/s10884-006-9046-x.
[26]	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.
[27]	J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.
[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), 377-392. doi: 10.1007/s00033-007-7005-y.
[29]	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.
[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), 551-581. doi: 10.1016/j.jde.2011.04.020.
[31]	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).
[32]	M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed, Nonlinearity, 23 (2010), 1609-1630. doi: 10.1088/0951-7715/23/7/005.
[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), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1.
[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), 2392-2420. doi: 10.1137/080727312.
[35]	S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. Real World Appl., 13 (2012), 1991-2005. doi: 10.1016/j.nonrwa.2011.12.020.
[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), 921-946. doi: 10.3934/dcds.2013.33.921.
[37]	H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164. doi: 10.2977/prims/1145476150.
[38]	Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM. J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406.
[39]	Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems, Chaos, Solitons Fractals, 45 (2012), 1361-1367. doi: 10.1016/j.chaos.2012.07.002.
[40]	G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations, Nonlinear Anal., 72 (2010), 4466-4474. doi: 10.1016/j.na.2010.02.021.
[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), 5096-5124. doi: 10.1016/j.jde.2012.01.014.

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, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165.
[2]	P. W. Bates, On some nonlocal evolution equations arising in materials science, in H. Brunner, X.Q. Zhao and X. Zou (Eds.), Nonlinear dynamics and evolution equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52.
[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), 105-136. doi: 10.1007/s002050050037.
[4]	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.
[5]	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.
[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), 1207-1237. doi: 10.1017/S0308210500004959.
[7]	Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.
[8]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
[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), 193-212. doi: 10.3934/dcds.2005.12.193.
[10]	J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28. doi: 10.2748/tmj/1270041024.
[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 Contin. Dyn. Syst. Ser. B, 17 (2012), 2713-2724. doi: 10.3934/dcdsb.2012.17.2713.
[13]	F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution, 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.
[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), 91-163. doi: 10.1007/PL00004238.
[15]	Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, Numerical and Applied Mathematics, Part II (Paris,1988), Baltzer,Basel, (1989), 687-692.
[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), 233-252. doi: 10.1007/BF03167402.
[17]	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.
[18]	C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.
[19]	C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047.
[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), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.
[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), 102-129. doi: 10.1016/j.jde.2008.03.023.
[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), 487-513. doi: 10.1016/j.jde.2007.10.019.
[24]	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.
[25]	Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861. doi: 10.1007/s10884-006-9046-x.
[26]	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.
[27]	J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.
[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), 377-392. doi: 10.1007/s00033-007-7005-y.
[29]	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.
[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), 551-581. doi: 10.1016/j.jde.2011.04.020.
[31]	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).
[32]	M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed, Nonlinearity, 23 (2010), 1609-1630. doi: 10.1088/0951-7715/23/7/005.
[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), 2047-2084. doi: 10.1090/S0002-9947-08-04694-1.
[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), 2392-2420. doi: 10.1137/080727312.
[35]	S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. Real World Appl., 13 (2012), 1991-2005. doi: 10.1016/j.nonrwa.2011.12.020.
[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), 921-946. doi: 10.3934/dcds.2013.33.921.
[37]	H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164. doi: 10.2977/prims/1145476150.
[38]	Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM. J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406.
[39]	Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems, Chaos, Solitons Fractals, 45 (2012), 1361-1367. doi: 10.1016/j.chaos.2012.07.002.
[40]	G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations, Nonlinear Anal., 72 (2010), 4466-4474. doi: 10.1016/j.na.2010.02.021.
[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), 5096-5124. doi: 10.1016/j.jde.2012.01.014.

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