Non-autonomous Differential Systems with Delays: A Global Attraction Analysis

In this paper, we derive criteria of global attractivity of a (possibly constant) positive periodic solution in non-autonomous systems of delay differential equations. Our approach can be viewed as the extension for non-autonomous systems of the folkloric connection between discrete dynamics and scalar delay differential equations. It is worth mentioning that we provide delay-dependent criteria of global attraction that cover the best delay independent conditions. We apply our results to non-autonomous variants of several classical models such that Nicholson’s blowfly equation, Goodwin’s model oscillator, the Mackey–Glass equation and systems with patch structure.


Introduction
The seasonal fluctuations of the environmental conditions play a central role in the regulation of populations, the structuring of ecological communities and the functioning of ecosystems (Lou and Zhao 2017;Lou et al. 2019). In epidemiology, the transmission of most infectious diseases also depends on several temporal variables (Barrientos et al. 2017;Li et al. 2020). For example, seasonal influenza generally recurs with a large epidemic in winter and a negligible presence in summer. Given the importance of the seasonal fluctuations, non-autonomous equations are certainly useful in any life-system. On the other hand, time delays are rather common in applied Communicated by Mary Pugh.
B Alfonso Ruiz-Herrera ruizalfonso@uniovi.es 1 Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain sciences to model, for instance, age-structure, maturation periods or hatching times (Lou et al. 2019;Ruiz-Herrera 2019). Taking together both frameworks, we arrive at non-autonomous delay differential systems. These models have attracted much attention in the last decades, see Berezansky et al. (2010), Faria (2021Faria ( , 2017, Faria et al. (2018), Li et al. (2020), Lou and Zhao (2017) and Lou et al. (2019) and the references therein. The main reason for this interest is that the interplay between time-delays and seasonality brings great challenges to the mathematical analysis.
When one faces with a particular model, the natural question is to study the existence of a globally attracting solution. There are mainly two approaches for this problem: the construction of Lyapunov functions (McCluskey 2015) and the theory of monotone systems (Smith 2011). In the context of autonomous scalar delay differential equations, Ivanov and Sharkovsky (1992) and Mallet-Paret and Nussbaum (1986) proposed an alternative methodology. Specifically, they proved thatx is a globally attracting solution in providedx is a globally attracting fixed point for the difference equation is the function that determines the equilibria in (1). The connection between discrete and continuous equations was a considerable step in the understanding of delay differential equations. The obvious advantage is that we can handle an equation where the initial conditions belong to an infinite dimensional space via an equation with initial conditions in an one-dimensional space. Moreover, the connection typically leads to the best delay independent condition of global attraction. It is worth mentioning that the direct extension for systems is not possible. In general, it is necessary to add extra conditions, see Example 3 in Ruiz-Herrera (2020).
In this paper, we propose a connection between non-autonomous delay differential systems and discrete equations similar to that in Ivanov and Sharkovsky (1992), Mallet-Paret and Nussbaum (1986). Our methodology is new and follows the next steps: First, we prove the existence of a (possibly constant) positive periodic solution x * (t). Then, we employ a change of variable and identify a class of "amenable" nonlinearities. Finally, we construct an adequate function depending on the upper and lower bounds of the possible positive periodic solutions of the system. A strength of our results is that we recover the best delay independent conditions of global attraction and some classical delay dependent criteria (see Gyori and Trofimchuk 1999) when we study autonomous equations. We stress that the function that "codes" the dynamical behavior in is not clear. This first difficulty could explain why the analysis in Ivanov and Sharkovsky (1992) or Mallet-Paret and Nussbaum (1986) has not been extended to non-autonomous systems yet.
Another motivation of the paper is to examine critical issues such as the influence of the seasonal fluctuations of the environment and time-delays on the creation/suppression of oscillations or the density of population, i.e., whether or not a population is adversely affected by a periodic environment. Apart from these issues, we derive sufficient conditions for the existence of a globally attracting periodic solution in the Mackey-Glass equation and Nicholson's blowfly model with periodic coefficients. Despite the variety of methods and tools that have been proposed in the literature for these models, we obtain sharper criteria than the existing ones, see Faria (2017) and the references therein. We also analyze the Goodwin oscillator model (Ruoff and Rensing 1996;El-Morshedy and Ruiz-Herrera 2020) and some classical metapopulation models subject to seasonal fluctuations of the environment (El-Morshedy and , 2020Faria 2014).
The structure of the paper is as follows. In Sect. 2, we give some useful lemmas on discrete dynamics. In Sect. 3, we derive criteria of global attractivity of a (possibly constant) positive periodic solution in scalar delay differential equations. In Sect. 4, we extend the results to systems. We finish the paper with a discussion on our findings. A critical tool in this paper will be the fluctuation lemma. We recall its statement for the reader's convenience. lim n→+∞ ϕ(t n ) = lim sup x→+∞ ϕ(x) and lim n→+∞ ϕ (t n ) = 0. lim n→+∞ ϕ(s n ) = lim inf x→+∞ ϕ(x) and lim n→+∞ ϕ (s n ) = 0.
To conclude this section, we introduce some notation. Given a subset A ⊂ R N and two positive constants τ, T > 0 with τ ≥ T , we define for all t, t + T ∈ [−τ, 0], we write φ for the T -periodic function defined in R which coincides with φ on [−τ, 0]. We denote by C T (A) the set of T -periodic continuous functions φ : R −→ A, which can be identified as a subset of C([−τ, 0], A) respectively, with the same topology.

Proposition 2.1 (Lemma 2.5 in El-Morshedy and Lopez 2008)
Proposition 2.2 Assume that ϕ is a decreasing or unimodal function of class C 3 with negative Schwarzian derivative, that is, The previous result can be found in Corollary 2.10 of El-Morshedy and Lopez (2008) for unimodal functions. For decreasing maps, we can deduce the result by a simple adaptation of the arguments in Singer (1978). It is worth mentioning that ϕ(x) = xe ρ(1−x) with ρ > 0 and ϕ(x) = 1+ρ γ 1+(ρx) γ x with ρ > 0 and γ > 1 are unimodal functions with negative Schwarzian derivative.

Scalar Equations with Periodic Coefficients
In this section, we derive criteria of global attractivity of a positive T -periodic solution in where d, β : R −→ (0, +∞) are continuous and T -periodic; τ > 0 and h : We stress that the term T -periodic function in this paper encompasses the constant functions. Additionally, we impose that h satisfies conditions (A), (B) and (C). Our approach has two important ingredients: • The connection of (4) with a suitable discrete equation.
• An a-priori estimate of upper and lower bounds for the possible positive T -periodic solutions of (4).
In Sect. 3.3, we will apply our results to the classical Nicholson's blowfly equation with periodic coefficients. The reader can consult (Liz and Ruiz-Herrera 2013;Zou 2008, 2010) for different approaches relating discrete dynamics and continuous equations. We stress that the positive periodic solutions in (4) are generally nonconstant. In Sect. 3.4, we illustrate how to apply our tools when the positive periodic solution in (4) is a constant function, i.e., d(t) = kβ(t) for some constant k > 0. 0]. Based on the variation of the constant formula, Eq. (4) can be written as:
Our next goal is to prove that the positive solutions of (4) are uniformly bounded.

Proposition 3.1 Assume (A). Then,
for any positive solution x(t) of (4) with 2 the constant given in (5).
Proof Take x(t) a positive solution of (4). As mentioned above, if x(t 0 ) ≥ 2 for some t 0 > 0, then x(t) is strictly decreasing in a neighborhood of t 0 . We can also deduce that if x(t 1 ) ≤ 2 for some t 1 > 0, then x(t) ≤ 2 for all t ≥ t 1 . Next we prove that there is a time t 2 > 0 so that x(t 2 ) ≤ 2 . Assume, by contradiction, that x(t) > 2 for all t > 0. In such a case, x (t) < 0 for all t > 0 by (6). Hence, there exists ξ ≥ 2 so that lim t−→+∞ x(t) = ξ . Moreover, we can take a sequence t n −→ +∞ so that lim n−→+∞ x (t n ) = 0. Evaluating (4) at t n , we obtain that Making n −→ +∞ and using that d(t) is T -periodic and positive, we conclude that This implies that ξ < 2 , a contradiction.
To guarantee the uniform boundedness away from zero for the positive solutions, we impose the following condition: (P) There are two constants c > 0 and η > 1 so that We stress that if h(0) > 0, then (P) automatically holds. On the other hand, if The next result shows that any constant 1 > 0 satisfying min with 2 and c the upper bound given in (5) and the constant in property (P), respectively, is an uniform lower bound for the positive solutions of Eq. (4).

Proof
We fix x(t) a positive solution of (4). We split the proof into two steps: Step 1 lim inf t−→+∞ x(t) > 0. Assume, by contradiction, that lim inf t−→+∞ x(t) = 0. In this case, we can take a sequence {s n } −→ +∞ with the following properties: (S1) x (s n ) ≤ 0 for all n ∈ N. for all n ∈ N. By (S1) and the expression of Eq. (4), we have that for all n ∈ N, or equivalently, for all n ∈ N. If x(s n −τ ) 0 as n → +∞, then there are ξ 1 > 0 and a subsequence x(s σ (n) −τ ) so that x(s σ (n) − τ ) → ξ 1 as n → +∞, (recall that x(t) is bounded). In light of (8), x(s σ (n) ) cannot tend to 0 as n → +∞. This is a contradiction with (S3). If x(s n − τ ) → 0 as n → +∞, then x(s n − τ ) ∈ (0, c) for n large enough where (0, c) is the interval given in (P). Now by (P), (S2) and (8), we obtain that with η > 1. This contradiction completes the proof of the first step.
By the previous step, lim inf t−→+∞ x(t) = L > 0 for a suitable constant L. By Lemma 1.1, there is a sequence {s n } tending to +∞ so that x(s n ) −→ L and x (s n ) −→ 0. By the expression of Eq. (4), we have that It is not restrictive to assume that Making n −→ +∞ in (9) and using that d(t) is strictly positive and T -periodic, we conclude that or equivalently, If L ∈ (0, c), we have that L > η L with η > 1, (see condition (P)). This is a contradiction with L ≤ L. Thus, L ∈ [c, 2 ]. As a consequence of (7), we conclude that L > 1 .
Next we recall a result on the existence of positive T -periodic solutions for Eq. (4). As a direct consequence of Propositions 3.1 and 3.2, the positive T -periodic solutions of (4) are bounded and bounded away from zero in an uniform manner. In the rest of this subsection, we take θ min > 0 and θ max > 0 so that for any positive T -periodic solution x * (t) of (4). We also define Next, we introduce an extra condition regarding the function g: As we will see, this last condition is satisfied in most classical models. Fix x * (t) a positive T -solution of (4). The critical step in our arguments is to employ the change of variable y(t) = x(t) x * (t) . After some straightforward computations, we arrive at Our aim now is to prove that Eq. (11) admits an unique positive equilibrium.
Proof Let y = a > 0 be an equilibrium of (11). Then, for all t ∈ R, or equivalently, for all t > 0. By Lemma 2.1 with θ 0 = θ min , we conclude that a = 1.

Proposition 3.3 Assume conditions (A), (B), (C), (P) and (Q)
. Fix x * (t) > 0 a Tperiodic solution of (4). Suppose that there exists a positive solution x(t) of (4) so that x(t) − x * (t) does not converge to 0 as t → +∞. Then, there are four positive constants L, S, L and S with the following properties: (iv) f ( L) ≤ L and f ( S) ≥ S (the function given in (10)).

Proof
Define are bounded and positive. On the other hand, as a direct consequence of Propositions 3.1 and 3.2, we have that y(t) is bounded and lim inf t−→+∞ y(t) > 0. Hence, using that 1 is the unique positive equilibrium of (11) by Lemma 3.1 and y(t) does not converge to 1 as t → +∞, we conclude that Set L = lim inf t−→+∞ y(t) and S = lim sup t−→+∞ y(t). By Lemma 1.1, we can take a sequence {t n } −→ +∞ satisfying: It is not restrictive to assume, after taking sub-sequences if necessary, that Now we evaluate Eq. (11) at t n , that is, Making n −→ +∞ and using that or equivalently Arguing in an analogous manner with 0 < L = lim inf t−→+∞ y(t), we can find two constants L ∈ [L, S] and L 1 ∈ [θ min , θ max ] so that The constants L, S, L and S satisfy (i) and (iii). Finally, we apply Lemma 2.2 with θ 0 = θ min and θ 1 = θ max to deduce (ii) and (iv).
Now we are ready to give the main delay independent criterion of global attraction for Eq. (4).

Theorem 3.2 Assume conditions (A), (B), (C), (P) and (Q).
If 1 is a global attractor in (0, +∞) for the difference equations then there exists a positive T -periodic solution x * (t) of (4) which is globally attracting, that is, for all x(t) positive solution of (4), Proof By Theorem 3.1, we can take x * (t) a positive T -periodic solution of (4). Assume, by contradiction, that there exists a positive solution x(t) of (4) so that x * (t) − x(t) does not converge to 0 as t −→ +∞. Then, by Proposition 3.3, there are four positive constants S, L, L and S with the following properties: . The existence of this interval contradicts Proposition 2.1.
Let us refine Proposition 3.3 in order to obtain a delay-dependent criterion of global attraction. First, we fix a positive constant ω so that with x * (t) the positive T -periodic solution of (4) fixed previously and M an upper bound of h (see (A)). Then, we define where τ > 0 is the delay of Eq. (4) and f is given in (10).

Proposition 3.4 Assume (A), (B), (C), (P) and (Q)
. Fix x * (t) > 0 a T -periodic solution of (4). Suppose that there exists a positive solution x(t) of (4) so that x(t) − x * (t) does not converge to 0 as t −→ +∞. Then, there are four positive constants L, S, ρ 1 and ρ 2 with the following properties: Proof Arguing in the same manner as in the proof of Proposition 3.3, we have that does not converge to 1 as t → +∞. As mentioned there, Let x * (t) .
With this notation, Eq. (13) now writes as: Using the variation of the constants formula, we know that On the other hand, since 1 is the unique equilibrium of (13), (see Lemma 3.1), we observe that 0 < L < S with L = lim inf t→+∞ y(t) and S = lim sup t→+∞ y(t). Again, as mentioned in the proof of Proposition 3.3, there exists a sequence {t n } → +∞ satisfying the following conditions: We evaluate Eq. (14) at t n , that is, for all n ∈ N. Dividing and multiplying by h(x * (s − τ )) in the last integral term, we arrive at for all n ∈ N. After simple computations, we obtain that Let ξ n ∈ [t n − τ, t n ] be a sequence of points so that for all n ∈ N. We can assume, after passing to sub-sequences if necessary, the following properties: ) is periodic, positive and continuous. It is worth noting that (see definition of ω in (12)). Making n −→ +∞ in Eq. (15), we deduce that Using that S < 1 < S, we obtain that It is also clear that Hence, using (16), (17) and S < 1, we conclude that On the other hand, by (B) together with θ 1 ∈ [θ min , θ max ] and ρ 1 < 1, we have that Inserting this inequality in (19), we arrive at Arguing in a similar manner with lim inf t−→+∞ y(t), we can find ρ 2 > 1 with ρ 2 ∈ [L, S] so that

Theorem 3.3 Assume conditions (A), (B), (C), (P) and (Q).
If 1 is a global attractor in (0, +∞) for the difference equations then there exists a positive T -periodic solution x * (t) of (4) which is globally attracting, that is, for all x(t) positive solution of (4), Proof The proof of this result is exactly the same as that in Theorem 3.2 using Proposition 3.4 instead of Proposition 3.3.

Estimating Upper and Lower Bounds for the Positive T-Periodic Solutions of Eq. (4)
The main results of the previous subsection are expressed in terms of the global attraction of a suitable discrete equation. In turn, this equation depends on θ max and θ min , upper and lower bounds (non-necessarily optimal) of the positive T -periodic solutions of (4). In this subsection, we provide an estimate of these bounds when h is strictly decreasing or of the form strictly decreasing. Recall that we always assume that h is of class C 1 . We focus on the second class of functions. The first class can be treated analogously and we omit the details. We introduce some notation to simplify the statement of the results:

Proposition 3.5 Assume (A) and (P). Moreover, suppose that h
By the expression of (4), To prove (ii), we observe that by (20), Using that Hence, Using (20) and the previous inequality, it is clear that Arguing as above, we can deduce that for all t ∈ [0, T ].

Example: Nicholson's Blowfly Equation with Periodic Coefficients
In this subsection, we apply the previous theoretical results to where d, β : [0, +∞) −→ (0, +∞) are continuous and T -periodic and τ > 0. We assume that for all t ∈ [0, T ]. In this framework, it is straightforward to check conditions (A), (B), (C) and (P). Notice that h(x) = xe −x and As a consequence of Theorem 3.1, Eq. (21) is an unimodal function with negative Schwarzian derivative. As a direct consequence of Proposition 2.2, 1 is a global attractor in (0, +∞) for the difference equation if | f (1)| ≤ 1. This last condition is equivalent to By this discussion and using Theorem 3.3 (see (23) for the definition of ω) and Proposition 3.5, we have the following result: Theorem 3.4 Assume conditions (22), τ = kT with k ∈ N and − ln min Then, there exists a T -periodic solution x * (t) of (21) with x * (t) > 0 for all t ∈ [0, T ] which is globally attracting, that is, for any positive solution x(t) of (21), Notice that 1 is a global attractor in (0, +∞) for the difference Eq. (24) if θ max ≤ 2. Using this fact, we can obtain the following delay independent criterion of global attraction. (22), τ = kT with k ∈ N and

Corollary 3.1 Assume conditions
Then, there exists a T -periodic solution x * (t) of (21) with x * (t) > 0 for all t ∈ [0, T ] which is globally attracting, that is, for any positive solution x(t) of (21), To assess the potential of our approach, we recall that the best delay independent conditions for global attractivity of the positive equilibrium in the autonomous Nicholson's blowfly equation Informally speaking, Theorem 3.4 can be viewed as the extension of the results developed in Gyori and Trofimchuk (1999) for (21). To the best of our knowledge, there are no results in the literature regarding delay-dependent criteria of global attraction that cover the best delay independent conditions, see the different comparisons in Faria (2017).
Next we derive criteria of global attraction when the delay is not a multiple of T . (22) and

Theorem 3.5 Assume conditions
Then, there exists a T -periodic solution x * (t) of (21) with x * (t) > 0 for all t ∈ [0, T ] which is globally attracting, that is, for any positive solution x(t) of (21), Proof Observe that the constants of Proposition 3.5 (ii) satisfy As above, we can obtain the following delay independent criterion of global attraction. (22) and

Corollary 3.2 Assume condition
Then, there exists a T -periodic solution x * (t) of (21) with x * (t) > 0 for all t ∈ [0, T ] which is globally attracting, that is, for any positive solution x(t) of (21),

Nicholson's Blowfly Equation with Periodic Coefficients and a Positive Constant Solution
In the previous subsections, we always stress that the positive T -periodic solution of (4) can be in fact a constant function. This happens when the equation is autonomous, or more generally, when β(t) = rd(t) for some positive constant r . In this subsection, we show that Theorem 3.4 can be strengthened for this particular case because we can work with better estimates of the upper and lower bounds of the positive Tperiodic solutions. Generally speaking, better (a-priori) bounds of the positive Tperiodic solutions lead to sharper results. Consider where d : [0, +∞) → (0, +∞) is continuous and T -periodic, r > 1 and τ = kT with k ∈ N. We first observe that x * (t) = ln r is the unique positive T -periodic solution of (25). Indeed, fix x * (t) a T -periodic solution of (25) and take t 0 , t 1 ∈ [0, T ] so that By Eq. (25) and using that x * (t − τ ) = x * (t) for all t, we conclude that After this remark, we can take θ max = θ min = ln r and Repeating the argument made in Theorem 3.4, we obtain the following result: Theorem 3.6 Assume r > 1, τ = kT with k ∈ N and Then, for any positive solution x(t) of (25), lim t−→+∞ x(t) = ln r .

Systems of Delay Differential Equations with Periodic Coefficients
Many ideas developed in the previous section also work in systems of delay differential equations. We illustrate this fact with two classical examples: Goodwin's model oscillator (Ruoff and Rensing 1996) and systems with patch structure (El-Morshedy and Ruiz-Herrera 2017; Faria 2017). We analyze models with nonlinearities different from h(x) = xe −x to show the versatility of our results. Throughout this section, we say that a vector v = (v 1 , . . . , v N ) is positive is v i > 0 for all i = 1, . . . , N . As mentioned above, the constant functions are trivially T -periodic.
Our next goal is to prove that the positive solutions of (26) are uniformly bounded. for any positive solution (x(t), y(t)) of (26).

Proposition 4.1 Assume (A) and (G1). Take a constant ϒ 2 so that
Proof Take (x(t), y(t)) a positive solution of (26). As mentioned above, if y(t 0 ) > ϒ 2 for some t 0 > 0, then y(t) is strictly decreasing in a neighborhood of t 0 . This implies that if y(t 1 ) ≤ ϒ 2 , then y(t) ≤ ϒ 2 for all t ≥ t 1 . Arguing as in the proof of Proposition 3.1, we deduce that there is t > 0 so that y(t) < ϒ 2 for all t ≥ t. Now, by the first equation of (26), we have that for all t ≥ t + σ 1 . Repeating the argument of the proof of Proposition 3.1, we can find t * > t so that Next we prove that these solutions are bounded away from 0 in an uniform manner.
Proof We divide the proof into two steps.
Step 1 min{lim inf t−→+∞ x(t), lim inf t−→+∞ y(t)} > 0 for all (x(t), y(t)) positive solution of (26). Assume, by contradiction, that there exists a positive solution (x(t), y(t)) so that Then, we can take {t n } −→ +∞ so that one of the following sets of conditions is satisfied: We refer to the proof of Proposition 3.1 for the construction of the previous sequence {t n }.
Assume that the first block of conditions (i.e., (X1)-(X3)) holds. By the first equation of (26) and (G1), we deduce that for all n ∈ N. Thus, it is not restrictive to assume that the second block (i.e., (Y1)-(Y3)) holds. Notice that by Proposition 4.1, we can take t 0 > 0 large enough so that for all t ≥ t 0 . This implies that there exists n 0 ∈ N so that x(t n − σ 2 ) ≤ ϒ 2 and y(t n ) ≤ ϒ 2 for all n ≥ n 0 . By the second equation of (26) and (Y1), we have that for all n ∈ N. If x(t n − σ 2 ) ≥ c for all n ≥ n 0 , we deduce by (30) that y(t n ) ≥ ϒ 1 for all n ≥ n 0 . This is a contradiction with (Y3). If there exists n with n ≥ n 0 so that x(t n − σ 2 ) < c, we also have a contradiction. Indeed, notice that using (G2) in (30), we have that with η > 1. Thus, y(t n ) > x(t n − σ 2 ). On the other hand, we know by (Y2) that x(t n − σ 2 ) ≥ y(t n ).
Take (x(t), y(t)) a positive solution of (26). Let Assume that L = lim inf t→+∞ y(t). By the fluctuation lemma (see Lemma 1.1), we can take a sequence {t n } → +∞ so that lim n→+∞ y (t n ) = 0 and lim n→+∞ y(t n ) = lim inf t→+∞ y(t). Evaluating the second equation of (26) at t n , we obtain that It is not restrictive, after passing to subsequences if necessary, to assume that with ϒ 2 the constant given in (28). Making n −→ +∞ and using that d(t) is positive and T -periodic, we conclude that If L ∈ (0, c), we deduce by (G2) that L ≥ η L with η > 1, a contradiction with the fact L ∈ [L, ϒ 2 ]. Therefore, L ∈ [c, ϒ 2 ]. In this case, L ≥ ϒ 1 as a direct consequence of (31) and (29). If L = lim inf t→+∞ x(t), then, by the fluctuation lemma (see Lemma 1.1), we can take a sequence {t n } → +∞ so that lim n→+∞ x (t n ) = 0 and lim n→+∞ x(t n ) = lim inf t→+∞ x(t). Then, evaluating the first equation of (26) at t n , we obtain that Using (G1), we have that In this case, the limit of a subsequence of y(t n − σ 1 ) is less or equal than L. Thus, it is not restrictive to assume that L = lim inf t→+∞ y(t).
The next result guarantees the existence of a positive T -periodic solution for (26). The method of proof is basically the adaptation of the ideas developed in Theorem 3.1 in Faria (2017).
Step 1 P(x, y)(t) is a T -periodic function provided (x(t), y(t)) is a T -periodic function.
Notice that In the last equality, we have employed the change of variable s = s + T . Using that a, y, and b are T -periodic, we conclude that We can reason in an analogous manner with P 2 (x, y)(t). Thus, Step 2 Define with M an upper bound of h, (see (A)). Note that by (G1), Q 1 ≥ Q 2 . We prove that Analogously we can prove that P 2 (x, y)(t) ≤ Q 2 . The proof of this step is completed. Using (G2), we can take γ > 0 so that We define Step 3 P(B γ ) ⊂ B γ .
Take (x(t), y(t)) ∈ B γ . Then, Step 4 P is equicontinuous in B γ . Take t 1 , t 2 ∈ [−σ, 0] and (x, y) ∈ B γ . We analyze the first component of P (the analysis of the second component is analogous).
Notice that the last integral term is smaller than In light of this type of estimates, we can deduce that P is equicontinuous. The conclusion follows from the classical Schauder's theorem. We stress that B γ is a convex set.
The positive T -periodic solutions of (26) are bounded and bounded away from zero in an uniform manner (see Propositions 4.1 and 4.2). As in the scalar case, we take positive constants θ min and θ max so that for all positive T -periodic solution (x * (t), y * (t)) of (26). We define We assume that g : [0, +∞) −→ [0, +∞) satisfies condition (Q) introduced in Sect. 3. Next we fix a positive T -periodic solution (x * (t), y * (t)) of (26) and employ the change of variable and z 2 (t) = y(t) y * (t) .
Notice that if, for example, L = lim inf t→+∞ z 1 (t), then by the fluctuation lemma (see Lemma 1.1), there would exist a sequence {t n } tending to +∞ satisfying that lim n−→+∞ z 1 (t n ) = 0 and lim n−→+∞ z 1 (t n ) = L. Evaluating the first equation of (34) at t n , we obtain that Since a(t), y * (t) and x * (t) are positive and T -periodic functions, we have that lim n−→+∞ z 2 (t n − σ 1 ) = lim n−→+∞ z 1 (t n ) = L. On the other hand, if (x(t), y(t)) − (x * (t), y * (t)) does not converge to (0, 0) as t −→ +∞, then (z 1 (t), z 2 (t)) does not converge to (1, 1) as t −→ +∞. Note that Observe that if L = S, then lim t→+∞ z 2 (t) = L = S = lim t→+∞ z 1 (t). This is a contradiction because (1, 1) is the unique positive equilibrium of (34). Hence, L < S. Moreover, the solutions of (34) are bounded and bounded away from zero by Propositions 4.1 and 4.2. Hence, L > 0 and S < +∞. The rest of the proof is exactly the same as that in Proposition 3.3. Specifically, we take a sequence {t n } tending to +∞ so that z 2 (t n ) −→ 0 and z 2 (t n ) −→ S. Then, we arrive at with S 1 ∈ [θ min , θ max ] and S ∈ [L, S]. Repeating the argument with L, we obtain that with L 1 ∈ [θ min , θ max ] and L ∈ [L, S]. The proof follows from Lemma 2.2.
Repeating the argument of the proof of Theorem 3.2, we obtain the following result.
To complete this section, we apply the previous theorem when h is decreasing.

Models for Populations with Patch Structure
Consider the system

We define by A(t), B(t), D(t), M(t)
the T -periodic square matrices of order N given by where a ii (t) = 0 for 1 ≤ i ≤ N . We also need the following condition: The previous assumptions guarantee the existence and uniqueness of solutions for ( (x 1 (t), . . . , x N (t)) which initial condition φ satisfies that for all t ≥ t 0 and i = 1, . . . , N . Our aim is to prove the existence of a globally attracting positive T -periodic solution for system (36). As in previous sections, we assume that h also satisfies (B) and (C).
Fix x * (t) = ( p 1 (t), . . . , p N (t)) a positive T -periodic solution of (36). The change of variable . . . , N transforms system (36) into for i = 1, . . . , N . Since the positive T -periodic solutions of (36) are bounded and bounded away from zero in an uniform manner, we can take two positive constants θ min and θ max so that positive T -periodic solution of system (36). We also define f (x) = H (θ max , x) and g(x) = H (θ min , x).
Our first aim is to prove that (37) has a unique positive equilibrium. Proof First we prove that if ζ = (a, . . . , a) is an equilibrium of (37) with a > 0, then for all i = 1, . . . , N and for all t ∈ [0, T ]. Since β ik (t) p i (t) > 0 for all t ∈ [0, T ], we can find two times t 0 , t 1 and two indices i, k so that By Theorem 4.4, 0 < L and S ∈ R. We know that Using that x(t) − x * (t) does not converge to 0 as t −→ +∞, we deduce that (y 1 (t), . . . , y N (t)) does not converge to (1, . . . , 1). Notice that L < S. To see this claim, we observe that if L = S, then lim t→+∞ y i (t) = L = S for all i = 1, . . . , N . However, this is not possible because (1, . . . , 1) is the unique nontrivial equilibrium of (37) by Lemma 4.2 and we know that (y 1 (t), . . . , y N (t)) does not converge to (1, .., 1).
By Lemma 1.1, we can take {t n } a sequence tending to +∞ so that lim n→+∞ y i 0 (t n ) = 0 and lim n→+∞ y i 0 (t n ) = S. It is not restrictive to assume that there exists n 0 ∈ N large enough so that for all n ≥ n 0 and j ∈ {1, . . . , N }. We can also suppose that for all k ∈ {1, . . . , m}. Evaluating the i 0 -th equation of (37) at t n , we obtain that Using that y j (t n ) − y i 0 (t n ) ≤ 0 and a i 0 j (t n ) > 0 for all n ≥ n 0 and j ∈ {1, . . . , N }, we conclude that m k=1 Making n → +∞ in the previous expressions and using (P3), we conclude that there exists k 0 ∈ {1, . . . , m} so that That is, S ≤ H (S k 0 , S k 0 ). Repeating the analogous argument with L, we deduce the existence of an index j 0 ∈ {1, . . . , m}, and two constants L j 0 ∈ [L, S] and L j 0 ∈ [θ min , θ max ] so that L ≥ H (L j 0 , L j 0 ). The conclusion follows from Lemma 2.2.
then there exists a positive T -periodic solution x * (t) = ( p 1 (t), . . . , p N (t)) of (36) which is globally attracting, that is, for any

Estimating an Upper Bound for the Positive T-Periodic Solutions of (36) and Applications
In this subsection, we translate the abstract criterion developed in Theorem 4.6 into a more applied one. We suppose that h(x) is a bounded function of class C 1 . We also impose that d i , a i j , β ik : R −→ [0, +∞) are continuous and T -periodic and τ ik ≥ 0. For simplicity, in this subsection we work with functions of the form h(x) = xq(x) where q : [0, +∞) −→ (0, +∞) is strictly decreasing and q(0) = 1. Since h is bounded, it is clear that lim x−→+∞ q(x) = 0. In this framework, (P4) holds and the rest of conditions, when u = v = (1, . . . , 1), can be re-written in the following manner: , j =i a i j (t) > 0 for all t ∈ R and i ∈ {1, . . . , N }. As discussed in Faria (2017), it is easy to manage the general case from the choice u = v = (1, . . . , 1).
Next, we take and M an upper bound of h(x). First, we estimate an uniform upper bound for the positive T -periodic solutions of (36). and is T -periodic and τ i 0 k = n i 0 k T for all k = 1, . . . , m. Using the expression of the i 0 -th equation of (36) and (38), we have that Thus, . Now, the conclusion of (i) is clear. To prove (ii), we take t 0 and i 0 as above. We observe that or equivalently, Remark 4.1 Informally speaking, Proposition 4.5 says that we can take θ max = q −1 ( 1 ) in (i) and θ max = M 1 in (ii). To conclude this section, we apply Theorem 4.6 in system (36) with h(x) = x 1+x 2 . In this case, It is clear that (A), (B) and (C) hold. Moreover, condition (Q) is satisfied for any θ > 0. On the other hand, for each t > 0, we have that Thus, for each θ max > 0, f (x) = H (θ max , x) is a function with negative Schwarzian derivative and | f (1)| < 1. By Proposition 2.2, we conclude that 1 is a global attractor in (0, +∞) for the difference equation x n+1 = f (x n ).
In conclusion, if we consider system (36) with h(x) = x 1+x 2 and (A1)-(A4) are satisfied, then there is a positive T -periodic solution that is globally attracting for all positive solutions of (36).
In general, for h(x) = x 1+x γ with γ ≥ 2, the global attraction of a T -periodic solution in system (36) is guaranteed if 2 ≥ (γ − 2)θ γ max with θ max the upper bound given in Remark 4.1.
In Section 4 in Faria (2017), Faria analyzed the existence of a global attracting positive T -periodic solution in (36) when h(x) = xe −x and all delays are multiple of the period. Following our strategy with this nonlinearity, we recover exactly the same results as those in Faria (2017). Our contribution in comparison with Faria (2017) is that our approach is not restricted to h(x) = xe −x . We stress that the main result in Faria (2017) on the existence of a positive T -periodic solution, the key contribution of that paper, is not restricted to Nicholson's system.

Discussion
A major challenge in theoretical biology is to understand the influence of the seasonal fluctuations of the environment on the evolution of the species (Lou and Zhao 2017; To approach this problem, we offer new criteria of global attractivity of a positive periodic solution in non-autonomous systems of delay differential equations. Generally speaking, our approach can be viewed as the extension for non-autonomous systems of the popular connection between scalar delay differential equations and discrete equations developed in Ivanov and Sharkovsky (1992) and Mallet-Paret and Nussbaum (1986). As particular examples, we have studied non-autonomous variants of some classical models that include Nicholson's blowfly equation or Mackey-Glass model (Berezansky et al. 2010), the Goodwin oscillator for chemical reactions (Ruoff and Rensing 1996) and models with patch structure (Faria 2017). The main advantages of our results in comparison with those in Faria (2021) (1) We cover the common nonlinearities employed in mathematical biology.
(2) We provide delay-dependent criteria of global attraction that include the best delay independent results. (3) We can apply our results in non-monotone models.
To apply our results, it is critical to determine an (uniform) upper bound for the positive T -periodic solutions of the model. Moreover, a better estimate leads to a sharper result. In light of some numerical simulations (see Fig. 1), it seems that the estimates given in Proposition 3.5 when the delays are not multiple of the period can be improved. We will study this issue in future works.
A controversial result in ecology deduced from equation is that seasonality has a deleterious influence on the overall population size (Henson and Cushing 1997). That is, the average of the population size is always less than the average of the carrying capacity. When we analyze a similar question with Nicholson's blowfly equation, we have observed that the delay can promote or reverse that deleterious influence, see Fig. 2. Particularly, time delays in (21) can stimulate a more efficient use of the resources.
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