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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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