Abstract
We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: I-Species persistence. J. Math. Biol. 51, 75–113 (2005)
Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating traveling fronts. J. Math. Pures Appl. 84, 1101–1146 (2005)
Du, L.J., Li, W.T., Wu, S.L.: Pulsating fronts and front-like entire solutions for a reaction-advection-diffusion competition model in a periodic habitat. J. Differ. Equ. 266, 8419–8458 (2019)
Du, L.J., Li, W.T., Wu, S.L.: Propagation phenomena for a bistable Lotka–Volterra competition system with advection in a periodic habitat. Z. Angew. Math. Phys. 71, 11 (2020)
Ducrot, A.: A multi-dimensional bistable nonlinear diffusion equation in a periodic medium. Math. Ann. 366, 783–818 (2016)
Fang, J., Zhao, X.Q.: Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc. 17, 2243–2288 (2015)
Fang, J., Yu, X., Zhao, X.Q.: Traveling waves and spreading speeds for time-space periodic monotone systems. J. Funct. Anal. 272, 4222–4262 (2017)
Fife, P.C., Tang, M.: Comparison principles for reaction–diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances. J. Differ. Equ. 40, 168–185 (1981)
Furter, J., López-Gómez, J.: On the existence and uniqueness of coexistence states for the Lotka–Volterra competition model with diffusion and spatially dependent coefficients. Nonlinear Anal. 25, 363–398 (1995)
Girardin, L., Nadin, G.: Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed. Eur. J. Appl. Math. 26, 521–534 (2015)
He, X., Ni, W.M.: Global dynamics of the Lotka–Volterra competition-diffusion system: diffusion and spatial heterogeneity I. Commun. Pure Appl. Math. 69, 981–1014 (2016)
He, X., Ni, W.M.: Global dynamics of the Lotka–Volterra competition-diffusion system with equal amount of total resources, II. Calc. Var. Partial Differ. Equ. 55, 25 (2016)
He, X., Ni, W.M.: Global dynamics of the Lotka–Volterra competition-diffusion system with equal amount of total resources, III. Calc. Var. Partial Differ. Equ. 56, 132 (2017)
Huang, W.: Uniqueness of the bistable traveling wave for mutualist species. J. Dyn. Differ. Equ. 13, 147–183 (2001)
Huang, W., Han, M.: Non-linear determinacy of minimum wave speed for a Lotka–Volterra competition model. J. Differ. Equ. 251, 1549–1561 (2011)
Lam, K.Y., Ni, W.M.: Uniqueness and complete dynamics in heterogenous competition diffusion systems. SIAM J. Appl. Math. 72, 1695–1712 (2012)
Lou, Y.: On the effects of migration and spatial heterogeneity on single and multiple species. J. Differ. Equ. 223, 400–426 (2006)
Murray, J.D.: Mathematical Biology. Springer, Heidelberg (1989)
Xin, X.: Existence and uniqueness of travelling waves in a reaction–diffusion equation with combustion nonlinearity. Indiana Univ. Math. J. 40, 985–1008 (1991)
Xin, X.: Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Differ. Equ. 3, 541–573 (1991)
Xin, X.: Existence of planar flame fronts in convective-diffusive periodic media. Arch. Ration. Mech. Anal. 121, 205–233 (1992)
Xin, X.: Existence and nonexistence of traveling waves and reaction–diffusion front propagation in periodic media. J. Stat. Phys. 73, 893–926 (1993)
Yu, X., Zhao, X.Q.: Propagation phenomena for a reaction–advection–diffusion competition model in a periodic habitat. J. Dyn. Differ. Equ. 29, 41–66 (2017)
Zhang, Y., Zhao, X.Q.: Bistable travelling waves for a reaction and diffusion model with seasonal succession. Nonlinearity 26, 691–709 (2013)
Zhao, X.Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
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. Pures Appl. 96, 627–671 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Philipp M Altrock.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the Canada NSERC discovery Grant (RGPIN04709-2016).
Rights and permissions
About this article
Cite this article
Wang, H., Ou, C. Propagation Speed of the Bistable Traveling Wave to the Lotka–Volterra Competition System in a Periodic Habitat. J Nonlinear Sci 30, 3129–3159 (2020). https://doi.org/10.1007/s00332-020-09646-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-020-09646-5
Keywords
- Reaction diffusion equations
- Bistable Lotka–Volterra competition system
- Traveling wave
- Propagation direction