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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

${x}^{\text{'}}\left(t\right)=-ax\left(t-\tau \right)-by\left(t\right),\phantom{\rule{2.em}{0ex}}{y}^{\text{'}}\left(t\right)=-cx\left(t\right)-ay\left(t-\tau \right),$

where $a,b$, and $c$ are real numbers and $\tau$ 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 $\tau$ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if $0<4a<\sqrt{-bc}$; and from instability to stability to instability if $-\sqrt{-bc}<2a<0$. As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K06 Linear functional-differential equations 92D25 Population dynamics (general)
##### References:
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