Global asymptotic stability for a nonlinear density-dependent mortality Nicholson’s blowflies system involving multiple pairs of time-varying delays

In our article, a nonlinear density-dependent mortality Nicholson’s blowflies system with patch structure has been investigated, in which the delays are time-varying and multiple pairs. Based upon the fluctuation lemma and differential inequality techniques, some sufficient conditions are found to ensure the global asymptotic stability of the addressed model. Moreover, a numerical example is provided to illustrate the feasibility and effectiveness of the obtained findings, and our consequences are new even when the considered model degenerates to the scalar Nicholson’s blowflies equation.


Introduction
Recently, Berezansky and Braverman [1] pointed out that several important classes of infinite dimensional dynamical systems arising from biological and medical sciences are special cases of the following general scalar delay differential equation: where m and l are positive integers. Here G is considered to be instantaneous mortality, F j (j ∈ I := {1, 2, . . . , m}) describes the feedback controls depending on the values of the stable variable with respective delays τ 1 (t), τ 2 (t), . . . , τ l (t). Clearly, (1.1) includes the following nonlinear density-dependent mortality Nicholson's blowflies model: x(t-g j (t)) , t ≥ t 0 , (1.2) which in the case h j ≡ g j coincide with the classical models [2][3][4][5][6]. In particular, the nonlinear density-dependent mortality term, a(t)x(t) b(t)+x(t) is referred to as population mortality, β j (t)x(th j (t))e -γ j (t)x(t-g j (t)) designates the time-dependent birth function with maturation delay h j (t) and incubation delay g j (t), and gets the maximum reproduces rate 1 γ j (t) , and j ∈ I. For the past decade or so, for the special case of (1.2) with h j ≡ g j (j ∈ I), not only the dynamic behaviors of time-delay Nicholson's blowflies models, such as existence, persistence, oscillation, periodicity and stability, but also the variants of the models have aroused current research interest, and some useful results have been obtained in the existing papers; for example, see [7][8][9][10][11][12][13][14][15]. In addition, it is proved that more than one delay involved in the identical nonlinear function F j can cause chaotic oscillations in [1], and an example is given to represent that two delays, rather than one delay, can produce a continuous oscillation. As a matter of fact, when more than one delay occurs, the delay feedback function F j should be regarded as a multi-variable function. This will make it more difficult to study the dynamic behaviors of (1.1) and (1.2).
On the other hand, it is of great practical significance to investigate the dynamic behaviors of a Nicholson's blowflies model with patch structure. Consequently, the scalar equation (1.1) can be naturally generalized as the following nonlinear density-dependent mortality Nicholson's blowflies model with patch structure: which in the classical case τ ij ≡ σ ij (i ∈ Q, j ∈ I) has been widely studied in the literature of the past [16][17][18][19][20]. In the ith patch, a ii (t)x i (t) b ii (t)+x i (t) labels the death rate of the the current population level x i (t); β ij (t)x i (tτ ij (t))e -γ ij (t)x i (t-σ ij (t)) designates the time-dependent birth function which requires maturation delays τ ij (t) and incubation delays σ ij (t), and gets the maximum reproduction rate 1 γ ij (t) ; for i, j ∈ Q and j = i, the weight function b ij (t)+x j (t) designates the population cooperative connection between jth patch and ith patch.
It should be mentioned that, up to now, the models (1.1), (1.2) and (1.3) relate to the global stability analysis of two or more delays are very few [1,[21][22][23][24]. For the special case of (1.2) with h j ≡ g j (j ∈ I), some delay-independent criteria ensuring the global asymptotic stability have been established in [25]. More precisely, the author in [25] obtained the global asymptotical stability of (1.2) on C([-τ , 0], (0, +∞)) and under the following assumptions: where τ := max{max 1≤j≤m g + j , max 1≤j≤m h + j } > 0, and g + and gbe defined as The deficiency is that we can find some errors in the process of proving the main consequence in [25]. In fact, as pointed in [26], in lines 3-4 of page 856 in [25], letting t → η(ϕ) cannot result in lim sup t→+∞ m j=1 β j (t) γ j (t)a(t) 1 e ≥ 1 because of the fact that η(ϕ) = +∞ has not been proved. This suggests that the above-described literature leaves space for improvement.
Based on the above considerations, we study a nonlinear density-dependent mortality Nicholson's blowflies system involving multiple pairs of time-varying delays described in (1.3). We shall establish a delay-independent criterion to ensure the global asymptotic stability of (1.3) without τ ij ≡ σ ij (i ∈ Q, j ∈ I), which has not been investigated till now. Moreover, our consequences generalize and improve all known consequences in [25,26], and the error mentioned above has been corrected in Lemma 2.1.
For convenience, we suppose that a ii , b ii , γ ij : R → (0, +∞), a ij (i = j), b ij (i = j), β ij , τ ij , σ ij : R → [0, +∞) for all i ∈ Q, j ∈ I are bounded and continuous functions, and we denote Furthermore, it will be considered the following admissible initial conditions: We denote x(t; t 0 , ϕ) as a solution of (1.3) with the initial value problem (1.5), and let [t 0 , η(ϕ)) be the maximal right-interval of existence of x(t; t 0 , ϕ). Moreover, by employing the local Lipschitz property of the right side function with regard to the nonnegative function space, we find that x(t; t 0 , ϕ) exists and is unique.

Preliminary results
We first present the global existence of solutions for (1.3) with the admissible initial value problem (1.5).
Proof We first assert that Suppose to the contrary that Eq. (2.3) does not hold, then there exist ω ∈ Q andt ω ∈ (t 0 , η(ϕ)) such that Based on the fact that which is a contradiction and results in the above assertion.
which suggests that Hence, by the Gronwall-Bellman inequality, we obtain It follows from Theorem 2.3.1 in [27] that η(ϕ) = +∞, and then Note that, for t ≥ t 0 , it follows that where M i 0 (t) > 2τ + t 0 .
Letting t → +∞, from the facts which is a contradiction and proves that x(t) is bounded for all t ∈ [t 0 , +∞). The proof is complete.

Global asymptotic stability for (1.3)
Theorem 3.1 For all i ∈ Q, j ∈ I, let (2.2) and be satisfied. Then the zero equilibrium point of (1.3) is globally asymptotically stable on C 0 + .
We first claim that the zero equilibrium point is stable. Without loss of generality, let 0 < < 1 satisfy Choose 0 < δ < , we claim that, for ϕ < δ, In the contrary case, there exist t * ∈ (t 0 , +∞) and i * ∈ Q such that which is absurd and proves (3.3). Therefore, the zero equilibrium point is stable. Next, we just need to prove that u = max i∈Q lim sup t→+∞ x i (t) = 0. From the fluctuation lemma [28,Lemma A.1], one can pick a sequence {t k } k≥1 and i * ∈ Q such that Moreover, from the boundedness of the coefficient and delay functions in (1.3), we can suppose that, for j ∈ I, and Furthermore, from (1.3), (2.2), (2.4), we get where t k > 2τ + t 0 . If u ≥ 1, from (2.2), (3.1), (3.6), (3.8) and the facts that u b * ii +u ≥ 1 b * ii +1 and sup u≥0 ue -u = 1 e , letting k → +∞ leads to which is a contradiction and we have the result that 0 ≤ u < 1.

and the fact that
xe -x is monotonously increasing on [0, 1], we have and then which is absurd and proves that u = 0. The proof is complete.
Remark 3.1 Obviously, for the scalar equation (1.2), all the results of [25,26] are special cases in Theorem 3.1 because the adopted assumptions are weaker.

A numerical example
This section presents a numerical example to illustrate the applicability of the analytical results derived in this article.

Conclusions
In the present manuscript, we studied nonlinear density-dependent mortality Nicholson's blowflies systems with patch structure, in which the delays are time-varying and come in multiple pairs. Here, we develop a method based on differential inequality techniques combining the application of the fluctuation lemma to obtain some sufficient conditions for the global asymptotic stability of the given system. The derived results of this manuscript complement some earlier publications to some extent. To the best of our knowledge, it is the first time one deals with this aspect. In addition, the method used in this paper provides a possible method for studying the global asymptotic stability of other patch structure population dynamic models with multiple pairs of different time-varying delays.