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 GuoEmail author
  • Masahiko Shimojo


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.


Singular prey–predator model Shadow system Quenching Blow-up 

Mathematics Subject Classification

Primary 35K51 35K55 Secondary 35K57 35K67 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Arnaud Ducrot
    • 1
    • 2
  • Jong-Shenq Guo
    • 3
    Email author
  • 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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