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

Abstract

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 < 4 a < - bc ; and from instability to stability to instability if - - bc < 2 a < 0 . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka–Volterra systems.