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Stability switches in a system of linear differential equations with diagonal delay. (English) Zbl 1171.34346

Summary: 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.

MSC:
34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
92D25Population dynamics (general)
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