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 and 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.

[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.

[1]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[15]

Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697
[16]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[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 and 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 and 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 and 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 and 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 (300)
  • HTML views (682)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy