Advertisement

Journal of Dynamical and Control Systems

, Volume 20, Issue 2, pp 229–240 | Cite as

Coexistence of Three Competing Species with Non-negative Cross-diffusion rate

  • Wonlyul Ko
  • Kimun Ryu
  • Inkyung AhnEmail author
Article

Abstract

In this paper, we report on our investigation of the existence and non-existence of positive solutions to a 3 × 3 competition interaction system with non-negative cross-diffusion under Dirichlet boundary conditions. First, it is shown that the system with constant diffusions can have a positive solution under suitable assumptions even though each of three 2 × 2 sub-systems coupled from the three equations of the system does not have a positive solution. Second, we show that the emergence of cross-diffusion in one equation of the system may generate a positive solution in case that the corresponding competition interaction system without cross-diffusion does not have a positive solution.

Keywords

Positive solutions Competition model Non-negative cross-diffusion Fixed point index 

Mathematics Subject Classifications (2010)

35J60 35Q80 

Notes

Acknowledgments

The authors thank the referee for the careful reading and valuable comments which have helped to improve the presentation of this paper. This work was supported by a Korea University Grant.

References

  1. 1.
    Alvesem C, De Figueiredo D. Nonvariational elliptic systems. Discrete Contin Dyn Syst. 2002;8:289–302.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Chi C, Hsu S, Wu L. On the asymmetric May-Leonard model of three competing species. SIAM J Appl Math. 1998;58(1):211–26.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dancer EN. On the indices of fixed points of mappings in cones and applications. J Math Anal Appl. 1983;91(1):131–51.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dancer EN, Du Y. Positive solutions for a three-species competition system with diffusion–I. General existence results. Nonlinear Anal. 1995;24(3):337–57.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Driessche P, Zeeman M. Three-dimensional competitive Lotka-Volterra systems with no periodic orbits. SIAM J Appl Math. 1998;58(1):227–34.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Feng W, Ruan WH. Coexistence, permanence, and stability in a three species competition model. Acta Math Appl Sinica (English Ser). 1996;12(4):443–46.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ghoreishi A, Logan R. Positive solutions of a class of biological models in a heterogeneous environment. Bull Austral Math Soc. 1991;44:79–94.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kuto K, Yamada Y. Multiple coexistence states for a prey-predator system with cross-diffusion. J Differ Equat. 2004;197(2):315–48.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Li L. Coexistence theorems of steady states for predator-prey interacting systems. Trans Amer Math Soc. 1988;305(1):143–66.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Li L, Ghoreishi A. On positive solutions of general nonlinear elliptic symbiotic interacting systems. Appl Anal 1991;40:281–95.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Li L, Logan R. Positive solutions to general elliptic competition models. Differ Integr Equat. 1991;4:817–34.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Lou Y, Martínez S, Ni WM. On 3 × 3 Lotka-Volterra competition systems with cross-diffusion. Discret Contin Dynam Syst. 2000;6(1):175–90.zbMATHGoogle Scholar
  13. 13.
    Lou Y, Ni WM. Diffusion, self-diffusion and cross-diffusion. J Differ Equat. 1996; 131(1):79–131.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    May R, Leonard W. Nonlinear aspects of competition between three species. SIAM J Appl Math. 1975;29:243–53.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Pao CV. Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion. Nonlinear Anal. 2005;60(7):1197–1217.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Phillipson P, Schuster P, Johnston R. An analytic study of the May-Leonard equations. SIAM J Appl Math. 1985;45(4):541–54.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ruan WH, Feng W. On the fixed point index and multiple steady-state solutions of reaction-diffusion systems. Differ Integr Equat. 1995;8(2):371–91.zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ryu K, Ahn I. Positive steady-states for two interacting species models with linear self-cross diffusions. Discret Contin Dyn Syst. 2003;9(4):1049–61.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ryu K, Ahn I. Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics. J Math Anal Appl. 2003;283(1):46–65.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Schuster P, Sigmund K, Wolff R. On ω-limit for competing between three species. SIAM J Appl Math. 1979;37(1):49–54.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Shigesada N, Kawasaki K, Teramoto E. Spatial segregation of interacting species. J Theoret Biol. 1979;79(1):83–99.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Wang M, Li ZY, Ye QX. Existence of positive solutions for semilinear elliptic system. In: School on qualitative aspects and applications of nonlinear evolution equations (Trieste, 1990). River Edge: World Sci. Publishing; 1991, pp. 256–259.Google Scholar
  23. 23.
    Wolkowicz G. Interpretation of the generalized asymmetric May-Leonard model of three species competition as food web in a chemostat. Fields Inst Comm. 2006;48:279–89.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsKorea UniversitySeoulKorea
  2. 2.Department of Mathematical EducationCheongju UniversityJochiwonKorea
  3. 3.Department of Information and MathematicsKorea UniversityJochiwonKorea

Personalised recommendations