Abstract
The purpose of this paper is to investigate properties of traveling wave fronts for three species Lotka–Volterra system: the asymptotic behavior and uniqueness. Applying the Ikehara’s theorem, we determine the exponential rates of traveling wave fronts at the negative infinity. We further investigate the uniqueness of traveling wave fronts with the help of the sliding method.
Similar content being viewed by others
References
Chen, X., Fu, S.C., Guo, J.S.: Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. SIAM J. Math. Anal. 38, 233–258 (2006)
Chen, X.: Existence, uniqueness and asymptotic stability Of traveling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)
Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (1998)
Conley, C., Gardner, R.: An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model. Indiana Univ. Math. J. 33, 319–345 (1984)
Ducrot, A., Nadin, G.: Asymptotic behaviour of travelling waves for the delayed fisher-KPP equation. J. Differ. Equ. 256, 3115–3140 (2014)
Ellison, W., Ellison, F.: Prime Numbers, A Wiley-Interscience Publication. Wiley, New York (1985)
Fisher, R.A.: The Genetical Theory of Natural Selection: A Complete Variorum Edtion. Oxford University Press, Oxford (1999)
Fisher, R.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)
Gardner, R.: Existence of traveling wave solutions of competing models, A degree theoretic approach. J. Differ. Equ. 44, 343–364 (1982)
Guo, J.S., Wu, C.H.: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 252, 4357–4391 (2012)
Hung, L.C.: Traveling wave solutions of competitive-cooperative Lotka–Volterra systems of three species. Nonlinear Anal. Real Word Appl. 12, 3691–3700 (2011)
Huang, J., Zou, X.: traveling wavefronts in diffusive and cooperative Lotka–Voltera system with delays. J. Math. Anal. Appl. 271, 455–466 (2002)
Hou, X., Leung, A.W.: Traveling wave solutions for a competitive reaction-diffusion sysrem and their asymptotics. Nonlinear Anal. Real Word Appl. 9, 2196–2213 (2008)
Hsu, C.H., Yang, T.S.: Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for a epidemic model, Nonlinearity, 26 (2013), 121-139. Corrigendum: 26 (2013), 2925–2928
Kan-on, Y.: Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Math. Anal. 26(2), 340–363 (1995)
Kan-on, Y.: Fisher wave fronts for the Lotka–Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)
Li, K., Huang, J.H., Li, X.: Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Commu. Pure Appl. Anal. 16, 131–150 (2017)
Li, K., Li, X.: Asymptotic behavior and uniqueness of traveiling wave solutions in Ricker competiton system. J. Math. Anal. Appl. 389, 486–497 (2012)
Li, K., Li, X.: Traveling wave solutions in a delayed diffusive competition system. Nonlinear Anal. TMA. 75, 3705–3722 (2012)
Li, B., Weinberger, H., Lewis, M.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)
Lv, G.Y., Wang, M.X.: Traveling wave front in diffusive and competitive Lotka–Volterra system with delays. Nonlinear Anal. Real Word Appl. 11, 1323–1329 (2010)
Li, W.T., Lin, G., Ruan, S.G.: Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion-competition systems. Nonlinearity 19, 1253–1273 (2006)
Ma, S.H., Wu, X., Yuan, R.: Nonlinear stability of traveling wavefronts for competitive-cooperative Lotka–Volterra systems of three species. Appl. Math. Comput. 351, 331–346 (2017)
Schaaf, K.: Asymptotic behavier and traveling wave solutions for parabolic functional-differential equations. Trans. Am. Math. Soc. 302, 587–615 (1987)
Tang, M.M., Fife, P.C.: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 73, 69–77 (1980)
Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, (1994), Translated from the Russian manuscript by James F. Heyda
Widder, D.V.: The Laplace Tranform. Princeton University Press, Princeton (1941)
Yu, Z.X., Mei, M.: Uniqueness and stability of traveling waves for cellular neural networks with multiple delays. J. Differ. Equ. 260, 241–267 (2016)
Yu, Z.X., Yuan, R.: Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response. Taiwanese J. Math. 17, 2163–2190 (2013)
Yu, Z.X., Yuan, R.: Traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications. ANZIAM J. 51, 49–66 (2009)
Zhao, G., Ruan, S.: Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion. J. Math. Pure. Appl. 96, 627–671 (2011)
Acknowledgements
The research of Weiguo Zhang was partially supported by National Natural Science Foundation of China (No. 11471215), by Shanghai Leading Academic Discipline Project (No. XTKX2012) and by the Hujiang Foundation of China (B14005). Yanling Meng is supported by Natural Science Foundation of Shanghai (No.18ZR1426500). The authors would like to thank the anonymous referees for their valuable comments and suggestions which have led to an improvement of the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Meng, Y., Zhang, W. Properties of Traveling Wave Fronts for Three Species Lotka–Volterra System. Qual. Theory Dyn. Syst. 19, 67 (2020). https://doi.org/10.1007/s12346-020-00404-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-020-00404-2
Keywords
- Traveling wave fronts
- Competitive-cooperative Lotka–Volterra system
- Asymptotic behavior
- Uniqueness
- The sliding method