American Institute of Mathematical Sciences

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July  2019, 24(7): 3067-3075. doi: 10.3934/dcdsb.2018300

Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case

Yang Wang 1, and  Xiong Li 2,,

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.

Keywords: Uniqueness, traveling front solutions, reaction diffusion systems.
Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 92D25.
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, 2019, 24 (7) : 3067-3075. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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