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Zbl 1069.34109
Liz, Eduardo; Tkachenko, Victor; Trofimchuk, Sergei
A global stability criterion for scalar functional differential equations. (English)
[J] SIAM J. Math. Anal. 35, No. 3, 596-622 (2003). ISSN 0036-1410; ISSN 1095-7154/e

The authors consider scalar delay differential equations $$ x'(t) = -\delta x(t)+f(t,x_t)\tag1 $$ with nonlinear $f$ satisfying a sort of negative feedback condition combined with a boundedness condition. The well-known Mackey-Glass-type equations, equations satisfying the Yorke condition, and equations with maxima are special cases of (1). Here, a criterion is established for the global asymptotical stability of a unique steady state to (1). As an example, Nicholson's blowflies equation is studied where the computations support the Smith conjecture about the equivalence between global and local asymptotical stabilities in this population model.
[Anatoly Martynyuk (Ky{\"\i}v)]

MSC 2000:: *34K20 Stability theory of functional-differential equations
92D25 Population dynamics

Keywords: delay differential equations; global stability; Yorke condition; Schwarz derivative; Nicholson's blowflies equation

Cited in: Zbl 1093.34038 Zbl 1068.34076

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