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Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case
1. | School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China |
2. | School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China |
In this paper, we will prove the uniqueness of traveling front solutions with critical and noncritical speeds, connecting the origin and the positive equilibrium, for the classical competitive Lotka-Volterra system with diffusion in the weak competition, which partially answers the open problem presented by Tang and Fife in [
References:
[1] |
S. Ahmad and A. C. Lazer,
An elementary approach to traveling front solutions to a system of N competition-diffusion equations, Nonlinear Anal., 16 (1991), 893-901.
doi: 10.1016/0362-546X(91)90152-Q. |
[2] |
S. Ahmad, A. C. Lazer and A. Tineo,
Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912.
doi: 10.1016/j.na.2007.04.029. |
[3] |
Q. Bian, W. Zhang and Z. X. Yu,
Temporally discrete three-species Lotka-Volterra competitive systems with time delays, Taiwanese J. Math., 20 (2016), 49-75.
doi: 10.11650/tjm.20.2016.5597. |
[4] |
P de Mottoni,
Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79.
|
[5] |
J. Fang and J. H. Wu,
Monotone traveling waves for delayed Lotka-Volterra competition systems, Discret. Contin. Dyn. Syst., 32 (2012), 3043-3058.
doi: 10.3934/dcds.2012.32.3043. |
[6] |
W. Feng, W. H. Ruan and X. Lu,
On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discret. Contin. Dyn. Syst. Ser. B, 21 (2016), 815-836.
doi: 10.3934/dcdsb.2016.21.815. |
[7] |
A. W. Leung, X. J. Hou and W. Feng,
Traveling wave solutions for Lotka-Volterra system re-visited, Discret. Contin. Dyn. Syst. Ser. B, 15 (2011), 171-196.
doi: 10.3934/dcdsb.2011.15.171. |
[8] |
A. W. Leung, X. J. Hou and Y. Li,
Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924.
doi: 10.1016/j.jmaa.2007.05.066. |
[9] |
K. Li and X. Li,
Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497.
doi: 10.1016/j.jmaa.2011.11.055. |
[10] |
W. T. Li, G. Lin and S. G. Ruan,
Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[11] |
G. Lin,
Minimal wave speed of competitive diffusive systems with time delays, Appl. Math. Lett., 76 (2018), 164-169.
doi: 10.1016/j.aml.2017.08.018. |
[12] |
G. Lin and W. T. Li,
Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.
doi: 10.1017/S0956792512000198. |
[13] |
G. Lin, W. T. Li and M. Ma,
Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discret. Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.
doi: 10.3934/dcdsb.2010.13.393. |
[14] |
G. Lin and S. G. Ruan,
Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605.
doi: 10.1007/s10884-014-9355-4. |
[15] |
Z. G. Lin, M. Pedersen and C. R. Tian,
Traveling wave solutions for reaction-diffusion systems, Nonlinear Anal., 73 (2010), 3303-3313.
doi: 10.1016/j.na.2010.07.010. |
[16] |
W. H. Ruan, W. Feng and X. Lu,
On traveling wave solutions in general reaction-diffusion systems with time delays, J. Math. Anal. Appl., 448 (2017), 376-400.
doi: 10.1016/j.jmaa.2016.10.070. |
[17] |
M. M. Tang and P. C. Fife,
Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[18] |
J. H. V. Vuuren,
The existence of travelling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148.
doi: 10.1093/imamat/55.2.135. |
[19] |
Y. Wang and X. Li,
Entire solutions for the classical competitive Lotka-Volterra system with diffusion in the weak competition case, Nonlinear Anal. Real World Appl., 42 (2018), 1-23.
doi: 10.1016/j.nonrwa.2017.12.002. |
[20] |
C. H. Wu,
Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discret. Cont. Dyn. Syst. B, 18 (2013), 2441-2455.
doi: 10.3934/dcdsb.2013.18.2441. |
[21] |
J. Xia, Z. X. Yu, Y. C. Dong and H. Y. Li,
Traveling waves for n-species competitive system with nonlocal dispersals and delays, Appl. Math. Comput., 287/288 (2016), 201-213.
doi: 10.1016/j.amc.2016.04.025. |
[22] |
Z. X. Yu and R. Yuan,
Traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66.
doi: 10.1017/S1446181109000406. |
[23] |
Z. X. Yu and R. Yuan,
Traveling waves for a Lotka-Volterra competition system with diffusion, Math. Comput. Model, 53 (2011), 1035-1043.
doi: 10.1016/j.mcm.2010.11.061. |
[24] |
Z. X. Yu and H. K. Zhao,
Traveling waves for competitive Lotka-Volterra systems with spatial diffusions and spatio-temporal delays, Appl. Math. Comput., 242 (2014), 669-678.
doi: 10.1016/j.amc.2014.06.058. |
show all references
References:
[1] |
S. Ahmad and A. C. Lazer,
An elementary approach to traveling front solutions to a system of N competition-diffusion equations, Nonlinear Anal., 16 (1991), 893-901.
doi: 10.1016/0362-546X(91)90152-Q. |
[2] |
S. Ahmad, A. C. Lazer and A. Tineo,
Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912.
doi: 10.1016/j.na.2007.04.029. |
[3] |
Q. Bian, W. Zhang and Z. X. Yu,
Temporally discrete three-species Lotka-Volterra competitive systems with time delays, Taiwanese J. Math., 20 (2016), 49-75.
doi: 10.11650/tjm.20.2016.5597. |
[4] |
P de Mottoni,
Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79.
|
[5] |
J. Fang and J. H. Wu,
Monotone traveling waves for delayed Lotka-Volterra competition systems, Discret. Contin. Dyn. Syst., 32 (2012), 3043-3058.
doi: 10.3934/dcds.2012.32.3043. |
[6] |
W. Feng, W. H. Ruan and X. Lu,
On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discret. Contin. Dyn. Syst. Ser. B, 21 (2016), 815-836.
doi: 10.3934/dcdsb.2016.21.815. |
[7] |
A. W. Leung, X. J. Hou and W. Feng,
Traveling wave solutions for Lotka-Volterra system re-visited, Discret. Contin. Dyn. Syst. Ser. B, 15 (2011), 171-196.
doi: 10.3934/dcdsb.2011.15.171. |
[8] |
A. W. Leung, X. J. Hou and Y. Li,
Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924.
doi: 10.1016/j.jmaa.2007.05.066. |
[9] |
K. Li and X. Li,
Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497.
doi: 10.1016/j.jmaa.2011.11.055. |
[10] |
W. T. Li, G. Lin and S. G. Ruan,
Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[11] |
G. Lin,
Minimal wave speed of competitive diffusive systems with time delays, Appl. Math. Lett., 76 (2018), 164-169.
doi: 10.1016/j.aml.2017.08.018. |
[12] |
G. Lin and W. T. Li,
Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.
doi: 10.1017/S0956792512000198. |
[13] |
G. Lin, W. T. Li and M. Ma,
Travelling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discret. Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.
doi: 10.3934/dcdsb.2010.13.393. |
[14] |
G. Lin and S. G. Ruan,
Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605.
doi: 10.1007/s10884-014-9355-4. |
[15] |
Z. G. Lin, M. Pedersen and C. R. Tian,
Traveling wave solutions for reaction-diffusion systems, Nonlinear Anal., 73 (2010), 3303-3313.
doi: 10.1016/j.na.2010.07.010. |
[16] |
W. H. Ruan, W. Feng and X. Lu,
On traveling wave solutions in general reaction-diffusion systems with time delays, J. Math. Anal. Appl., 448 (2017), 376-400.
doi: 10.1016/j.jmaa.2016.10.070. |
[17] |
M. M. Tang and P. C. Fife,
Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[18] |
J. H. V. Vuuren,
The existence of travelling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148.
doi: 10.1093/imamat/55.2.135. |
[19] |
Y. Wang and X. Li,
Entire solutions for the classical competitive Lotka-Volterra system with diffusion in the weak competition case, Nonlinear Anal. Real World Appl., 42 (2018), 1-23.
doi: 10.1016/j.nonrwa.2017.12.002. |
[20] |
C. H. Wu,
Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discret. Cont. Dyn. Syst. B, 18 (2013), 2441-2455.
doi: 10.3934/dcdsb.2013.18.2441. |
[21] |
J. Xia, Z. X. Yu, Y. C. Dong and H. Y. Li,
Traveling waves for n-species competitive system with nonlocal dispersals and delays, Appl. Math. Comput., 287/288 (2016), 201-213.
doi: 10.1016/j.amc.2016.04.025. |
[22] |
Z. X. Yu and R. Yuan,
Traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66.
doi: 10.1017/S1446181109000406. |
[23] |
Z. X. Yu and R. Yuan,
Traveling waves for a Lotka-Volterra competition system with diffusion, Math. Comput. Model, 53 (2011), 1035-1043.
doi: 10.1016/j.mcm.2010.11.061. |
[24] |
Z. X. Yu and H. K. Zhao,
Traveling waves for competitive Lotka-Volterra systems with spatial diffusions and spatio-temporal delays, Appl. Math. Comput., 242 (2014), 669-678.
doi: 10.1016/j.amc.2014.06.058. |
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