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.
\[ x'(t)=-ax(t-\tau )-by(t), \qquad y'(t)=-cx(t)-ay(t-\tau ), \]
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.
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34K06 | Linear functional-differential equations |
| 92D25 | Population dynamics (general) |
Keywords:
delay differential equations; asymptotic stability; stability criteria; diagonal delay; characteristic equationReferences:
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