Global Exponential Stability of Almost Periodic Solutions for Nicholson’s Blowflies System With Nonlinear Density-Dependent Mortality Terms and Patch Structure

Abstract

This paper considers a generalized Nicholson’s blowflies system with nonlinear density-dependent mortality terms and patch structure. Under appropriate conditions, we establish some criteria to ensure that the solutions of this system exist and converge globally exponentially to a positive almost periodic solution. The results complement another case of nonlinear density-dependent mortality terms in Chen and Wang [5].

Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term

“…Remark 1. Note that x * (t) ≥ 3 does not belong to [κ, κ] ≈ [0.7215355, 1.342276], and γ 1 (t) = γ 2 (t) = 1 2 does not satisfy the following condition: γ j (t) ≥ 1, for all t ∈ R, j ∈ S, which has been considered as fundamental for the considered periodicity and almost periodicity of delayed non-autonomous Nicholson's blowflies models in [10,11,12,14,15,16,17,18,19,20]. Hence, all the results in these above mentioned references cannot be applicable to show the global exponential stability on the positive almost periodic solution of (4.1).…”

“…For example, some sufficient conditions ensuring the global exponential stability of positive periodic solutions and almost periodic solutions of classical Nicholson's blowflies equation with time-varying delays have been established in [10] and [11,12] respectively. Furthermore, the global exponential stability of positive almost periodic solutions for Nicholson's blowflies models involving nonlinear density-dependent mortality terms has been obtained in [14,15,16,17,18,19,20]. The results included in each paper of [10,11,12,14,15,16,17,18,19,20] gave an answer to the open problem: 3338 CHUANGXIA HUANG, HUA ZHANG AND LIHONG HUANG Find global stability conditions for the positive periodic solution of delayed non-autonomous Nicholson's blowflies equation, which was proposed by Berezansky et al [21].…”

“…It should be pointed out that, to a large extent, all results involving the global exponential stability of periodic solutions and almost periodic solutions for delayed non-autonomous Nicholson-type equations established in [10,11,12,14,15,16,17,18,19,20] are based on that the solutions exist in a smaller interval [κ, κ] ≈ [0.7215355, 1.342276], where…”

“…The definition on almost periodic function can be found in [1,3]. For more details on biological explanations of the coefficients of (1.1), we refer the reader to [12,14,15,16,17,18,19,20,21]. The significance of this paper is as follows.…”

New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson’s blowflies model with multiple pairs of time-varying delays

“…For the past decade or so, for the special case of (1.1) with h j ≡ g j (j ∈ I), the existence of positive solutions, permanence, oscillation, periodicity, and stability of such equations and similar models have been studied extensively [7][8][9][10][11][12][13][14]. In particular, the authors in [1] illustrated that two or more delays involved in the same nonlinear function F j can lead to chaotic oscillations, and they also gave some examples to show that having two delays instead of one can produce sustainable oscillations.…”