Hiroshima Mathematical Journal

Semi--exact equilibrium solutions for three-species competition--diffusion systems

Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura, Makoto Tohma, and Daishin Ueyama

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Abstract

We consider a three-species competition-diffusion system, in order to discuss the problem of competitor-mediated coexistence in situations where one exotic competing species invades a system that already contains two strongly competing species. It is numerically shown that, under some conditions, there exist stable non-constant equilibrium solutions that indicate the coexistence of two strongly competing species. This result motivates us to develop a semi-exact representation for finding these equilibrium solutions from an analytical viewpoint.

Article information

Source
Hiroshima Math. J., Volume 43, Number 2 (2013), 179-206.

Dates
First available in Project Euclid: 25 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1372180511

Digital Object Identifier
doi:10.32917/hmj/1372180511

Mathematical Reviews number (MathSciNet)
MR3072951

Zentralblatt MATH identifier
06214248

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 92D40: Ecology

Keywords
Reaction-diffusion equations competitor--mediated coexistence semi--exact equilibrium solutions

Citation

Chen, Chiun-Chuan; Hung, Li-Chang; Mimura, Masayasu; Tohma, Makoto; Ueyama, Daishin. Semi--exact equilibrium solutions for three-species competition--diffusion systems. Hiroshima Math. J. 43 (2013), no. 2, 179--206. doi:10.32917/hmj/1372180511. https://projecteuclid.org/euclid.hmj/1372180511


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References

  • M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), 835–840.
  • C.-C. Chen, L.-C. Hung, M. Mimura, and D. Ueyama, Exact traveling wave solutions of three species competition-diffusion systems, to appear in Discrete and Continuous Dynamical Systems B.
  • E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X. J. Wang, and C. H. Zhang, AUTO-07p: Continuation and bifurcation software for ordinary differential equations.Version 0.7, 2010, Concordia Univ., http://sourceforge.net/projects/auto-07p/files/auto07p
  • S.-I. Ei, R. Ikota, and M. Mimura, Segregating partition problem in competition-diffusion systems, J. Interfaces and Free Boundaries, 1 (1999), 57–80.
  • M. Gyllenberg, and P. Yan, On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle, Differential Equations and Applications, 1(4) (2009), 473–490.
  • M. Gyllenberg, P. Yau, and Y. Wang, A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of conjecture by Hofbauer and So, Applied Mathematics Letters, 19 (2006), 1–7.
  • T. G. Hallam, L. J. Svoboda and T. C. Gard, Persistence and extinction in three species Lotka-Volterra competitive system, Math. Biosci., 46(1-2) (1979), 117–124.
  • J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7(6) (1994), 65–70.
  • Y. Kan-on, Existence of standing waves for competition–diffusion equations, Japan J. Ind. Appl. Math. 13 (1996), 117–133.
  • Y. Kan-on, Bifurcation structure of stationary solutions of a Lotka–Volterra competition model with diffusion, SIAM J. Math. Anal. 29(2) (1998), 424–436.
  • Y. Kan-on, A note on the propagation speed of traveling waves for a Lotka–Volterra competition model with diffusion, J. Math. Anal. Appl. 217 (1998), 693–700.
  • J. Kastendiek, Competitor-mediated coexistence: interactions among three species of benthic macroalgae, J. Exp. Mar. Biol. Ecol., 62(3) (1982), 201–210.
  • K. Kishimoto, The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution., J. Math. Biol., 16(1) (1982), 103–122.
  • M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J. 30(2) (2000), 257–270.
  • M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math. 18(3) (2001), 657–696.
  • A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society London. Series B, 237(604) (1952), 37–72.
  • Q. Wang, W. Huang, and H. Wu, Bifurcation of limit cycles for 3D Lotka-Volterra competitive systems, Acta Appl. Math. 114 (2011), 207–218.
  • Wolfram Research, Inc., Mathematica, Version 5.0, Champaign, IL (2003).
  • M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems Dynam. Stability Systems 8(3) (1993), 189–217.