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
This is a preview of subscription content, log in to check access.
Courchamp, F., Langlais, M., Sugihara, G.: Controls of rabbits to protect birds from cat predation. Biol. Conserv. 89, 219–225 (1999)CrossRefGoogle Scholar
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
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