Stability switches in a system of linear differential equations with diagonal delay

摘要

This paper deals with the stability problem of a delay differential system of the form x′(t)=-ax(t-τ)-by(t), y′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if 0<4a<-bc; and from instability to stability to instability if --bc<2a<0. As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka–Volterra systems.