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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1063–1079 | Cite as

Behaviors of Solutions for a Singular Prey–Predator Model and its Shadow System

  • Arnaud Ducrot
  • Jong-Shenq Guo
  • Masahiko Shimojo
Article

Abstract

We study the asymptotic behaviors and quenching of the solutions for a two-component system of reaction–diffusion equations modeling prey–predator interactions in an insular environment. First, we give a global existence result for the solutions to the corresponding shadow system. Then, by constructing some suitable Lyapunov functionals, we characterize the asymptotic behaviors of global solutions to the shadow system. Also, we give a finite time quenching result for the shadow system. Finally, some global existence results for the original reaction–diffusion system are given.

Keywords

Singular prey–predator model Shadow system Quenching Blow-up 

Mathematics Subject Classification

Primary 35K51 35K55 Secondary 35K57 35K67 

References

  1. 1.
    Courchamp, F., Langlais, M., Sugihara, G.: Controls of rabbits to protect birds from cat predation. Biol. Conserv. 89, 219–225 (1999)CrossRefGoogle Scholar
  2. 2.
    Courchamp, F., Sugihara, G.: Modelling the biological control of an alien predator to protect island species from extinction. Ecol. Appl. 9, 112–123 (1999)CrossRefGoogle Scholar
  3. 3.
    Ducrot, A., Guo, J.-S.: Quenching behavior for a singular predator-prey model. Nonlinearity 25, 2059–2073 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ducrot, A., Langlais, M.: A singular reaction-diffusion system modelling prey-predator interactions: invasion and co-extinction waves. J. Differ. Equ. 253, 502–532 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ducrot, A., Langlais, M.: Global weak solution for a singular two component reaction-diffusion system. Bull. Lond. Math. Soc. 46, 1–13 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gaucel, S.: Analyse mathématique et simulation d’un système prédateur-proies en milieu insulaire hétérogène, Thèse, Université Bordeaux 1, (2005)Google Scholar
  7. 7.
    Gaucel, S., Langlais, M.: Some remarks on a singular reaction-diffusion arising in predator-prey modelling. Discret. Contin. Dyn. Syst. Ser. B 8, 61–72 (2007)CrossRefMATHGoogle Scholar
  8. 8.
    Hale, J.K., Sakamoto, K.: Shadow systems and attractors in reaction-diffusion equations. Appl. Anal. 32, 287–303 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kaplan, S.: On the growth of solutions of quasi-linear parabolic equations. Comm. Pure Appl. Math. 16, 305–330 (1963)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ni, W.M., Suzuki, K., Takagi, I.: The dynamics of a kinetic activator-inhibitor system. J. Differ. Equ. 229, 426–465 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rothe, F.: Uniform bounds from bounded \(L^p\)-functionals in reaction-diffusion equations. J. Differ. Equ. 45, 207–233 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Arnaud Ducrot
    • 1
    • 2
  • Jong-Shenq Guo
    • 3
  • Masahiko Shimojo
    • 4
  1. 1.IMB, UMR 5251University of BordeauxBordeauxFrance
  2. 2.IMB, UMR 5251CNRSTalenceFrance
  3. 3.Department of MathematicsTamkang UniversityNew Taipei CityTaiwan
  4. 4.Department of Applied MathematicsOkayama University of ScienceOkayamaJapan

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