Taiwanese Journal of Mathematics

Invasion Entire Solutions for a Three Species Competition-diffusion System

Guang-Sheng Chen and Shi-Liang Wu

Full-text: Open access


The purpose of this paper is to study a three species competition model with diffusion. It is well known that there exists a family of traveling wave solutions connecting two equilibria $(0,1,1)$ and $(1,0,0)$. In this paper, we first establish the exact asymptotic behavior of the traveling wave profiles at $\pm \infty$. Then, by constructing a pair of explicit upper and lower solutions via the combination of traveling wave solutions, we derive the existence of some new entire solutions which behave as two traveling fronts moving towards each other from both sides of $x$-axis. Such entire solution provides another invasion way of the stronger species to the weak ones.

Article information

Taiwanese J. Math., Volume 22, Number 4 (2018), 859-880.

Received: 29 July 2017
Revised: 8 October 2017
Accepted: 16 October 2017
First available in Project Euclid: 26 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations 92D30: Epidemiology 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]

traveling wave competition system invasion entire solution existence


Chen, Guang-Sheng; Wu, Shi-Liang. Invasion Entire Solutions for a Three Species Competition-diffusion System. Taiwanese J. Math. 22 (2018), no. 4, 859--880. doi:10.11650/tjm/171001. https://projecteuclid.org/euclid.twjm/1508983230

Export citation


  • X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212 (2005), no. 1, 62–84.
  • J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 193–212.
  • J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math. 19 (2015), no. 6, 1805–1829.
  • J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J. (2) 62 (2010), no. 1, 17–28.
  • F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999), no. 10, 1255–1276.
  • ––––, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$, Arch. Ration. Mech. Anal. 157 (2001), no. 2, 91–163.
  • X. Hou and Y. Li, Traveling waves in a three species competition-cooperation system, Commun. Pure Appl. Anal. 16 (2017), no. 4, 1103–1119.
  • L.-C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3691–3700.
  • M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol. 45 (2002), no. 3, 219–233.
  • W.-T. Li, N.-W. Liu and Z.-C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl. (9) 90 (2008), no. 5, 492–504.
  • G. Lv, Entire solutions of delayed reaction-diffusion equations, Z. Angew. Math. Mech. 92 (2012), no. 3, 204–216.
  • Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations 18 (2006), no. 4, 841–861.
  • Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal. 40 (2009), no. 6, 2217–2240.
  • Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations 251 (2011), no. 3, 551–581.
  • Y. Wang and X. Li, Some entire solutions to the competitive reaction diffusion system, J. Math. Anal. Appl. 430 (2015), no. 2, 993–1008.
  • Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2047–2084.
  • M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity 23 (2010), no. 7, 1609–1630.
  • X. Wang and G. Lv, Entire solutions for Lotka-Volterra competition-diffusion model, Int. J. Biomath. 6 (2013), no. 4, 1350020, 13 pp.
  • C.-H. Wu. A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations 28 (2016), no. 2, 317–338.
  • S.-L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations 25 (2013), no. 2, 505–533.
  • H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci. 39 (2003), no. 1, 117–164.