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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 |
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. |
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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