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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Courchamp, F., Langlais, M., Sugihara, G.: Controls of rabbits to protect birds from cat predation. Biol. Conserv. 89, 219–225 (1999)
Courchamp, F., Sugihara, G.: Modelling the biological control of an alien predator to protect island species from extinction. Ecol. Appl. 9, 112–123 (1999)
Ducrot, A., Guo, J.-S.: Quenching behavior for a singular predator-prey model. Nonlinearity 25, 2059–2073 (2012)
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)
Ducrot, A., Langlais, M.: Global weak solution for a singular two component reaction-diffusion system. Bull. Lond. Math. Soc. 46, 1–13 (2014)
Gaucel, S.: Analyse mathématique et simulation d’un système prédateur-proies en milieu insulaire hétérogène, Thèse, Université Bordeaux 1, (2005)
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)
Hale, J.K., Sakamoto, K.: Shadow systems and attractors in reaction-diffusion equations. Appl. Anal. 32, 287–303 (1989)
Kaplan, S.: On the growth of solutions of quasi-linear parabolic equations. Comm. Pure Appl. Math. 16, 305–330 (1963)
Ni, W.M., Suzuki, K., Takagi, I.: The dynamics of a kinetic activator-inhibitor system. J. Differ. Equ. 229, 426–465 (2006)
Rothe, F.: Uniform bounds from bounded \(L^p\)-functionals in reaction-diffusion equations. J. Differ. Equ. 45, 207–233 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Jong-Shenq Guo is partially supported by the Ministry of Science and Technology of Taiwan under the Grant 102-2115-M-032-003-MY3. Masahiko Shimojo is supported in part by JSPS KAKENHI Grant Number 16K17634. Part of this work was done during a visit of the Masahiko Shimojo to Taiwan. We thank the support of the National Center for Theoretic Sciences for his visit.
Rights and permissions
About this article
Cite this article
Ducrot, A., Guo, JS. & Shimojo, M. Behaviors of Solutions for a Singular Prey–Predator Model and its Shadow System. J Dyn Diff Equat 30, 1063–1079 (2018). https://doi.org/10.1007/s10884-017-9587-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-017-9587-1