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

This paper deals with the stability problem of a delay differential system of the form x′(t)=-ax(t-τ)-by(t)x′(t)=-ax(t-τ)-by(t), y′(t)=-cx(t)-ay(t-τ)y′(t)=-cx(t)-ay(t-τ), where a, b, and c   are real numbers and ττ 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 ττ 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.