Novel stability criteria on a patch structure Nicholson’s blowflies model with multiple pairs of time-varying delays

Abstract: This paper investigates a patch structure Nicholson’s blowflies model involving multiple pairs of different time-varying delays. Without assuming the uniform positiveness of the death rate and the boundedness of coefficients, we establish three novel criteria to check the global convergence, generalized exponential convergence and asymptotical stability on the zero equilibrium point of the addressed model, respectively. Our proofs make substantial use of differential inequality techniques and dynamical system approaches, and the obtained results improve and supplement some existing ones. Last but not least, a numerical example with its simulations is given to show the feasibility of the theoretical results.

Unfortunately, there exist two mistakes in the proof of the main results in [39]. The first mistake is on lines 1-2 of page 3. More precisely, let can not lead to |ϕ i (t)|}. The same mistake also appeared in [40] [see, p. 4], [41] [see, p. 5] and [43] [see, p. 4]. The logical error for the above mistake is that the left side of (1.6) is the whole maximum value for all i, but the right side of (1.6) is only for a fixed i. Fortunately, this error will be corrected in the Lemma 2.1 of this present paper. The second mistake in literature [39] is on line 26 of page 3 that the inequality which is absurd. Actually, by taking ϕ = 100 and letting t → +∞, one can find that is a contradiction.
Furthermore, we consider the following concrete example and its numerical simulations in Figure 1 to illustrate the above mistakes. Obviously, the assumptions (W 1 ) and (W 2 ) are satisfied in (1.7). Therefore, from Theorem 1.1, we have that the zero equilibrium point of system (1.7) is globally generalized exponentially stable. However, the numerical solutions with three different initial values shown in Figure 1 reveal that three numerical trajectories of (1.7) are convergent to 0 as t → +∞, but they are not generalized exponential convergence to 0. This makes us doubt whether the conclusions of Theorem 1.1 are correct. x 2 (t) Figure 1. Numerical solutions of (1.7) for differential initial values.
Based on the above theoretical analysis and numerical simulations, two problems naturally arise. One is whether the assumptions (W 1 ) and (W 2 ) can guarantee that every solution of (1.3) and (1.4) is convergent to 0 as t → +∞. The other is what kind of conditions can ensure the global generalized exponential convergence of the zero equilibrium point of (1.3).
Regarding the above discussions, in this manuscript, we first establish the global convergence of model (1.3) under the original conditions (W 1 ) and (W 2 ). It is worth noting that, two or more delays appearing in the same time-dependent birth function will cause complex dynamic behaviors, even chaotic oscillations. It is impossible to establish the global exponential convergence of (1.1) without appropriate restrictions on the distinctive delays and some corresponding examples are also given in [18]. Therefore, we add a new delay-dependent assumption to obtain some new criteria for the generalized exponential convergence and global exponential asymptotic stability on the zero equilibrium point of (1.3). In a nutshell, our results not only correct the errors in the existing literature [39][40][41]43], but also improve and complement the existing conclusions in the recent publications [23,24,26,32,39], and the effectiveness is demonstrated by a numerical example.
Proof. First, we prove that every solution x(t) = x(t; t 0 , ϕ) is bounded on [t 0 , +∞). By (W2), we have and hence, 1−σ 2 > 0. Therefore, there exists T * > t 0 − r i such that n j=1, j i Clearly, Furthermore, we state that Otherwise, there exist i * ∈ Q and T * * > T * such that which is a contradiction and proves that x(t) is bounded on [t 0 , +∞). Now, it is sufficient to show that u = max i∈Q lim sup t→+∞ x i (t) = 0.
For any ε > 0, one can choose Λ > T * such that Thus, By taking the upper limits, (W 1 ) leads to which, together with the arbitrariness of ε, implies that u = 0. This finishes the proof of Theorem 3.1. We now make the following assumptions: for all i ∈ Q and j ∈ I. and there exists λ > 0 such that where t ∈ [T * , +∞) and i ∈ Q. Let us consider the following function: where t ≥ t 0 − r i , κ = e λ T * t 0 α(s)ds , and M(ϕ) is defined in (3.2). In view of (3.4) and the fact that where t ∈ [T * , +∞) and i ∈ Q.
Hereafter, we show that (3.6) It is easy to see that Next, we prove that Suppose, for the sake of contradiction, there exist t > T * and i 0 ∈ Q satisfying that Then, which together with (1.3) and (3.3), we gain which yields a contradiction and shows (3.7) holds. Consequently, (3.6) holds, and the proof of Theorem 3.2 is now completed. For ε > 0, we choose δ ∈ (0, ε). Next, we claim that, for ϕ < δ, x i (t) = x i (t; t 0 , ϕ) < ε for all t ∈ [t 0 , +∞) and i ∈ Q. (3.9) In the contrary case, there exist t * ∈ (t 0 , +∞) and i * ∈ Q such that x i * (t * ) = ε, x j (t) < ε for all t ∈ [t 0 − r j , t * ) and j ∈ Q. (3.10) Consequently, (1.3), (3.8) and (3.10) involve that which is absurd and proves (3.9). Therefore, the zero equilibrium point is globally asymptotically stable. Remark 3.1. One can easily see that all convergence results in references [23,24,26,32] are special ones of Theorems 3.1, 3.2 and 3.3 with n = 1 and g i j (t) ≡ h i j (t). In addition, the errors in the existing literatures [39][40][41]43] have been corrected in the proof of Lemma 2.1 and Theorem 3.2. Specially, we add a new delay-dependent condition W 3 to ensure the correctness of the main conclusions Theorems 3.1 and 3.2 in [39]. Therefore, the obtained results of this present paper improve and complement the above mentioned references [23,24,26,[39][40][41]43].

A numerical example
In this section, we present an example to check the validity of the main results obtained in Section 3. Example 4.1. Consider the following patch structure Nicholson's blowflies model with multiple pairs of time-varying delays:   In addition, the global stability of the patch structure Nicholson's blowflies model with multiple pairs of time-varying delays has not been touched in . This implies that all the results in [23][24][25][26][27][28][29][30][31][32][33] and  cannot be used to show the global convergence and stability on system (3.1) where τ i j (t) h i j (t)(i ∈ Q, j ∈ I). Moreover, the proposed techniques could be taken into consideration in the dynamics research on other patch structure population models involving two or more delays in the same time-dependent birth function.

Conclusions
In the present manuscript, we investigate the asymptotic behavior for a patch structure Nicholson's blowflies model involving multiple pairs of different time-varying delays. Here, we obtain some novel results about the asymptotic behavior on the zero equilibrium point of the addressed model without assuming the uniform positiveness of the death rate and the boundedness of coefficients, which complement some earlier publications to some extent. 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.