American Institute of Mathematical Sciences

Advanced
September  2021, 14(9): 3097-3111. doi: 10.3934/dcdss.2021010

Traveling wave fronts in a diffusive and competitive Lotka-Volterra system

Zengji Du , , Shuling Yan and  Kaige Zhuang

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

* The corresponding author

Received  March 2019 Revised  December 2020 Published  September 2021 Early access  January 2021

Fund Project: This work is partially supported by the Natural Science Foundation of China (Grant Nos. 11871251, 12090011 and 11801231)

In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.

Keywords: Lotka-Volterra model, geometrical singular perturbation, traveling wave solution, bistable, competition.
Mathematics Subject Classification: Primary: 35K57, 34D15, 92D25.
Citation: Zengji Du, Shuling Yan, Kaige Zhuang. Traveling wave fronts in a diffusive and competitive Lotka-Volterra system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3097-3111. doi: 10.3934/dcdss.2021010
References:
[1]

J. Al-Omari and S. A. Gourley, Monotone traveling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159.  Google Scholar

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[3]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.  doi: 10.1090/S0002-9947-05-03839-0.  Google Scholar

[4]

Z. Du, J. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

Z. Du and Q. Qiao, The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differential Equations, 269 (2020) 7214–7230. doi: 10.1016/j.jde.2020.05.033.  Google Scholar

[6]

F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Liénard equations, J. Differential Equations, 174 (2001), 1-29.  doi: 10.1006/jdeq.2000.3947.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.  Google Scholar

[9]

S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[10]

B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959-1983.  doi: 10.3934/dcdsb.2020011.  Google Scholar

[11]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[12]

J. Huang and X. Zou, Travelling wave fronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[13]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, in: R Johnson(Ed.), Dynamical Systems, Lecture Notes in Math., Springer, New York, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[14]

Y. Kan-On, Parameter dependence of propagation speed of travelling waves for competition diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[15]

X. Li and X. Miao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.  doi: 10.3934/dcds.2009.24.523.  Google Scholar

[16]

W.-T. Li and Z.-C. Wang, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591.  doi: 10.1007/s00033-006-5125-4.  Google Scholar

[17]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar

[18]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[19]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[20]

G. Lv and M. X. Wang, Travelling wave fronts in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[21]

M. B. A. Mansour, A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation, J. Geom. Phys., 69 (2013), 116-122.  doi: 10.1016/j.geomphys.2013.03.004.  Google Scholar

[22]

R. H. Martin Jr. and H. L. Smith, Reaction-diffusion systems with the time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.  Google Scholar

[23]

A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monoger., vol. 140, AMS, Providence, RI, 1994. doi: 10.1090/mmono/140.  Google Scholar

[24]

Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[25]

Y. Xu, Z. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation, Nonlinear Dynam., 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

[26]

Z. Zhao and Y. Xu, Solitary waves for Korteweg-de Vries equation with small delay, J. Math. Anal. Appl., 368 (2010), 43-53.  doi: 10.1016/j.jmaa.2010.02.014.  Google Scholar

show all references

References:
[1]

J. Al-Omari and S. A. Gourley, Monotone traveling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159.  Google Scholar

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[3]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.  doi: 10.1090/S0002-9947-05-03839-0.  Google Scholar

[4]

Z. Du, J. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

Z. Du and Q. Qiao, The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differential Equations, 269 (2020) 7214–7230. doi: 10.1016/j.jde.2020.05.033.  Google Scholar

[6]

F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Liénard equations, J. Differential Equations, 174 (2001), 1-29.  doi: 10.1006/jdeq.2000.3947.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.  Google Scholar

[9]

S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[10]

B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959-1983.  doi: 10.3934/dcdsb.2020011.  Google Scholar

[11]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[12]

J. Huang and X. Zou, Travelling wave fronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[13]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, in: R Johnson(Ed.), Dynamical Systems, Lecture Notes in Math., Springer, New York, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[14]

Y. Kan-On, Parameter dependence of propagation speed of travelling waves for competition diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[15]

X. Li and X. Miao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.  doi: 10.3934/dcds.2009.24.523.  Google Scholar

[16]

W.-T. Li and Z.-C. Wang, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591.  doi: 10.1007/s00033-006-5125-4.  Google Scholar

[17]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar

[18]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[19]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[20]

G. Lv and M. X. Wang, Travelling wave fronts in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[21]

M. B. A. Mansour, A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation, J. Geom. Phys., 69 (2013), 116-122.  doi: 10.1016/j.geomphys.2013.03.004.  Google Scholar

[22]

R. H. Martin Jr. and H. L. Smith, Reaction-diffusion systems with the time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.  Google Scholar

[23]

A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monoger., vol. 140, AMS, Providence, RI, 1994. doi: 10.1090/mmono/140.  Google Scholar

[24]

Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[25]

Y. Xu, Z. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation, Nonlinear Dynam., 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

[26]

Z. Zhao and Y. Xu, Solitary waves for Korteweg-de Vries equation with small delay, J. Math. Anal. Appl., 368 (2010), 43-53.  doi: 10.1016/j.jmaa.2010.02.014.  Google Scholar

[1]

Yuzo Hosono. Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 79-95. doi: 10.3934/dcdsb.2003.3.79
[2]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[3]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[4]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[5]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[6]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[7]

Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014
[8]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[9]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[10]

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
[11]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[12]

Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451
[13]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[14]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 859-875. doi: 10.3934/dcdsb.2019193
[15]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[16]

De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, 2021, 15 (5) : 951-974. doi: 10.3934/ipi.2021023
[17]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
[18]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[19]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[20]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (179)
  • HTML views (277)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences