doi: 10.3934/dcdsb.2018300

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

* Corresponding author: Xiong Li

Received  March 2018 Revised  July 2018 Published  October 2018

Fund Project: The second author is supported by NSF grant 11571041 and the Fundamental Research Funds for the Central Universities

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 [17]. In fact, once these traveling front solutions have the same wave speed and the same asymptotic behavior at $ξ = ±∞$, they are unique up to translation.

Citation: Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018300
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. AhmadA. 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. BianW. 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. FengW. 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. LeungX. 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. LeungX. 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. LiG. 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. LinW. 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. LinM. 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. RuanW. 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. XiaZ. X. YuY. 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. AhmadA. 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. BianW. 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. FengW. 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. LeungX. 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. LeungX. 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. LiG. 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. LinW. 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. LinM. 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. RuanW. 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. XiaZ. X. YuY. 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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