Abstract
This paper is concerned with a class of Nicholson’s blowflies model involving nonlinear density-dependent mortality terms and multiple pairs of time-varying delays. By using differential inequality techniques and the fluctuation lemma, we establish a delay-independent criterion on the global asymptotic stability of the addressed model, which improves and complements some existing ones. The effectiveness of the obtained result is illustrated by some numerical simulations.
Highlights
1 Introduction Just as pointed out by Berezansky and Braverman [1], in the study of mathematical biology, many models of population dynamics can be characterized by the following delayed differential equation: m x (t) = Fj t, x t – τ1(t), . . . , x t – τl(t) – G t, x(t), t ≥ t0, (1.1)
It should be mentioned that some delay-independent criteria ensuring the global asymptotic stability of for the Nicholson’s blowflies model (1.2) with hj ≡ gj(j ∈ I) have been established in [17]
Remark 4.1 It should be mentioned that the global asymptotic stability on the Nicholson’s blowflies model involving nonlinear density-dependent mortality terms and multiple pairs of time-varying delays has not been touched in the previous literature
Summary
Just as pointed out by Berezansky and Braverman [1], in the study of mathematical biology, many models of population dynamics can be characterized by the following delayed differential equation:. If two or more delays occur, the time delay feedback function Fj should be considered as a function of several variables This will add difficulty when studying the dynamics of (1.1) and (1.2). It should be mentioned that some delay-independent criteria ensuring the global asymptotic stability of for the Nicholson’s blowflies model (1.2) with hj ≡ gj(j ∈ I) have been established in [17]. 2. Inspired by the above discussions, in this paper, we consider the nonlinear densitydependent mortality Nicholson’s blowflies model with multiple pairs of time-varying delays described in (1.2). We develop an approach based on differential inequality techniques coupled with an application of the Fluctuation Lemma to establish a delayindependent criterion to ensure the global asymptotic stability of (1.2) in the important, yet difficult case where the two delays are asymptotically apart, i.e., hj ≡ gj (j ∈ I). Our analysis can be applied to the nonautonomous Mackey–Glass equation, and our work partially solves an open problem posed for the Mackey–Glass equation in [1]
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