Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1063–1079

# 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

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## 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