Existence and stability of a spike in the central component for a consumer chain model in a two-dimensional domain. (August 2022)

• Record Type: Journal Article
• Authors: Ao, Weiwei; Peng, Yunjie; Winter, Matthias
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: Consumer chain model -- Reaction–diffusion systems -- Spiky solutions -- Stability; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112880
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study a three-component consumer chain model which is based on Schnakenberg type kinetics in a two-dimensional domain. In the model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. Through a rigorous analysis, we show that there exist two different single spike solutions if the feed rates are small. Further, we also establish the stability results: If the time-relaxation constants for both producer and the second consumer vanish, the large amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are positive. We show a new effect that if the time-relaxation constant of the second consumer is bounded, the large-amplitude spike solution is still stable while it is unstable in the one-dimensional case.