Liu, Xinzhi Stability results for impulsive differential systems with applications to population growth models. (English) Zbl 0808.34056 Dyn. Stab. Syst. 9, No. 2, 163-174 (1994). The paper deals with impulsive differential systems, i.e. ordinary differential equations systems \(dx/dt = f(t,x)\), \(x(t_ 0) = x_ 0\) \((t \in \mathbb{R}\), \(x \in \mathbb{R}^ n)\) subjected to (additive) impulses at fixed times \((t_ 0<) t_ 1 < t_ 2< \cdots\). The impulse at time \(t_ k\) instantaneously changes \(x(t_ k)\) by an amount that may depend on \(x(t_ k)\) (but not on \(t_ k)\), after which the state variable is again governed by the system \(dx/dt = f(t,x)\). The paper uses a general concept of stability (and of asymptotic stability and also of instability) with respect to maps that includes several stability concepts found in the literature. Several theorems give sufficient conditions for stability, asymptotic stability, and instability of the impulsive system. Examples show that impulses can stabilize an otherwise unstable system and destablilize an otherwise stable system. A population growth model that allows for “impulsive” fishing is studied. Reviewer: C.A.Braumann (Evora) Cited in 1 ReviewCited in 75 Documents MSC: 34D20 Stability of solutions to ordinary differential equations 92D25 Population dynamics (general) 34A37 Ordinary differential equations with impulses Keywords:impulsive differential systems; stability; with respect to maps; population growth model; fishing PDFBibTeX XMLCite \textit{X. Liu}, Dyn. Stab. Syst. 9, No. 2, 163--174 (1994; Zbl 0808.34056) Full Text: DOI References: [1] DOI: 10.1017/S033427000000850X · Zbl 0881.35006 [2] Freedman H. I., Deterministic Mathematical Models in Population Ecology (1987) · Zbl 0448.92023 [3] DOI: 10.1142/0906 [4] DOI: 10.1016/0022-247X(89)90265-5 · Zbl 0688.34031 [5] Liu X. Z, Differential Equations, Stability and Control pp 61– (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.