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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 $$x'(t)=-ax(t-\tau )-by(t), \qquad y'(t)=-cx(t)-ay(t-\tau ),$$ 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.

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