American Institute of Mathematical Sciences

Advanced
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  July 2019 Early access  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

[1]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[2]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[3]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[4]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
[5]

Guo Lin, Wan-Tong Li, Mingju Ma. Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 393-414. doi: 10.3934/dcdsb.2010.13.393
[6]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[7]

Jianhua Huang, Xingfu Zou. Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 925-936. doi: 10.3934/dcds.2003.9.925
[8]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258
[9]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[10]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[11]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[12]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[13]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819
[14]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[15]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[16]

Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697
[17]

Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054
[18]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[19]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[20]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (263)
  • HTML views (679)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences