Stability and attractivity for Nicholson systems with time-dependent delays

We analyse the stability and attractivity of a class of n-dimensional Nicholson systems with constant coefficients and multiple time-varying delays. Delayindependent sufficient conditions on the coefficients are given, for the existence and absolute global exponential stability of a unique positive equilibrium N∗, generalizing and improving known results for autonomous systems. We further establish delaydependent criteria for N∗ to be a global attractor of all positive solutions. In the latter case, upper bounds on the size of the delays which do not require an a priori explicit knowledge of the equilibrium N∗ are also derived.

Multi-dimensional Nicholson systems are a natural extension of the famous Nicholson's blowflies equation N (t) = −dN(t) + βN(t − τ)e −N(t−τ) (d, β, τ > 0), for which it is well known that the positive equilibrium N * = log(β/d) exists and is globally attractive if 1 < β/d ≤ e 2 .Moreover, for this scalar Nicholson's equation with β/d > e 2 , several criteria on the size of the delay τ have been established for the global attractivity of N * [8,14,16].For a nice survey on the subject and further references, see [1].
Only recently have Nicholson systems with patch structure and multiple delays deserved some attention from researchers.Most studies are centred on autonomous systems (or at least with constant coefficients), the main focus under investigation being the existence and global attractivity of a positive equilibrium [5,7,10,12,13].
Here, we investigate the stability and global attractivity of a positive equilibrium N * for (1.1).Most of the results will be proven for systems (1.1) with c i = 1 for all i, i.e., β ik N i (t − τ ik (t))e −N i (t−τ ik (t)) , i = 1, . . ., n, t ≥ 0, (1.2) since, as we shall see, a simple scaling shows that the conclusions obtained for (1.2) hold with small adjustments for the more general family (1.1).We start with a brief overview of some recent and selected results regarding the global asymptotic behaviour of solutions to Nicholson systems.Among them, emphasis is given to the papers of Faria and Röst [7] and Jia et al. [10], which strongly motivated the present work.
In [12], Liu considered an autonomous Nicholson system of the form with d, β, r i > 0, a ij ≥ 0 for i = j, 1 ≤ i, j ≤ n, and [a ij ] an irreducible matrix.Moreover, it was assumed in [12] that n ∑ j=1 a ij = 0, i = 1, . . ., n, so that, when β > d, is the positive equilibrium of (1.3).Under these conditions, the global attractivity of N * was proven in [12] for the case β/d ∈ [e, e 2 ].This result was later extended by the same author [13] to more general systems with multiple time-dependent delays of the form (1.2), but again with the same requirements on the coefficients, except that β was replaced by the constant β := ∑ k β ik ; note that d, β were still assumed to be independent of i and ∑ j a ij = 0.These constraints were relaxed in [5,7], however only the autonomous version, β ik N i (t − τ ik )e −N i (t−τ ik ) , i = 1, . . ., n, t ≥ 0, (1.4) was treated.Here, as well as in [5,7], without loss of generality we assume that a ii = 0 for i = 1, . . ., n, since each of these coefficients may be incorporated in d i .
With the terms d i − ∑ j =i a ij and ∑ k β ik depending naturally on i, a prime concern is to ensure the existence of a positive equilibrium N * , since it cannot be explicitly computed.This was established in [7] by imposing that the ODE system x i = −d i x i + ∑ n j=1,j =i a ij x j is asymptotically stable (a natural requirement from a biological point of view) and the following condition on the community matrix, defined here as where β i = ∑ k β ik (see Section 2 for further details): there exists a positive vector v such that Mv > 0. In the case of [a ij ] an irreducible matrix (a constraint not imposed in [5,7]), this condition turns out to be equivalent to saying that the community matrix M has an eigenvalue with positive real part, and is indeed a necessary and sufficient condition for the existence of a positive equilibrium N * ; otherwise, the equilibrium 0 is a global attractor of all positive solutions of (1.4).Furthermore, under the stronger condition it was also shown in [7] that N * is globally asymptotically stable.
The first purpose of this paper is to recover and generalize the results on the existence and global attractivity of the positive equilibrium N * given in [7], so that they apply to the nonautonomous version (1.2), and more generally to (1.1).Note that, although this system has autonomous coefficients, the delays τ ik (t) are time-dependent, and consequently known results and techniques for autonomous Nicholson systems do not apply directly to (1.1); thus, new arguments should be used.
On the other hand, it is well known that the introduction of large delays may induce instability, oscillations, unbounded solutions; contrarily, small delays are expected to be negligible.Delay-dependent criteria for the global attractivity of equilibria are in general more difficult to obtain, even for scalar delay differential equations (DDEs), for which several conjectures on local stability implying global asymptotic stability remain open.For multi-dimensional DDEs, clearly this topic is even harder to address, and only a few results have been produced.See e.g.[17], for 3/2-criteria for the global attractivity of delayed Lotka-Volterra systems.
To the best of our knowledge, for Nicholson systems, criteria for the global attractivity of the positive equilibrium N * depending on the size of the delays were established for the first time in 2017, in two very recent papers [4,10].In Jia et al. [10], the quite restrictive assumption was still imposed, and consequently the positive equilibrium N * = (N * 1 , . . ., N * n ) exists and has all its components equal to the same constant, N * = (log c, . . ., log c).On the other hand, El-Morshedy and Ruiz-Herrera [4] gave a result for the global attractivity of the positive equilibrium N * of the autonomous system (1.4) which does not depend on knowing N * explicitly; nevertheless the authors had to assume that such an equilibrium exists.This brings us to the second main task of this paper: to generalize the result in [10], by establishing a more general criterion for the global attractivity of N * depending on the size of the delays τ ik (t), which in particular not only does not require an a priori explicit knowledge of the equilibrium N * , much less that the components of N * are all equal.
The main lines of the work in this paper are as described above, and its organization is as follows.In Section 2, we introduce some notation and recall some preliminary results on persistence and existence of a positive equilibrium N * for (1.2).In Section 3, sufficient conditions for the absolute global exponential stability of N * are given.In Section 4, we establish a delay-dependent criterion for the global attractivity of N * , without assuming condition (1.6), which generalizes the result in [10].A comparison with the criterion in [4] will also be given.
Two illustrative examples will be given at the end.

Preliminaries
Consider a Nicholson system with patch structure of the form under the following general assumption on the coefficients and delays: These systems are in general used in population dynamics or disease modelling, as they serve as models for the growth of biological populations distributed over n classes or patches, with migration among them: x i (t) denotes the density of the ith-population, a ij is the rate of the population moving from class j to class i, d i is the coefficient of instantaneous loss for class i (which integrates both the death rate and the dispersal rates of the population in class i moving to the other classes), and β ik N i (t − τ ik (t))e −N i (t−τ ik (t)) are birth functions for class i; as usual, delays are included in the "birth terms", and our model prescribes time-dependent delays.Due to this biological interpretation, it is natural to assume that a ii = 0 for all i; however, for different settings, one may still suppose that a ii = 0, since, for each i, the term a ii N i (t) may be incorporated in the term −d i N i (t).In biological terms, it is also natural to take d i = m i + ∑ j =i a ji , where m i > 0 is the death rate for class i, although here a weaker version of this condition will be required.
Set τ = max i,k sup t≥0 τ ik (t) > 0. As the phase space for (1.1), take the Banach space A vector v ∈ R n will be identified with the constant function ϕ(t) ≡ v in C. System (1.1) can be written as an abstract DDE in C, N (t) = f (t, N t ), where N t denotes the function in C given by N t (θ) = N(t + θ) for −τ ≤ θ ≤ 0. By C + we denote the cone in C of nonnegative functions, and write ϕ ≥ 0 for ϕ ∈ C + .By a positive vector v ∈ R n , we mean a vector whose components are all positive, and write v > 0. In a similar way, we denote ϕ > 0 for a function in C whose components are positive for all t ∈ [−τ, 0].
Bearing in mind the biological interpretation of the family (1.1), the set is taken as the set of admissible initial conditions.For simplicity, here initial conditions are given at time t = 0, its solutions are defined for all t ≥ 0 and are eventually bounded in norm by a common positive constant; in other words, there exists M > 0 such that lim sup t→∞ |x(t, 0, ϕ)| ≤ M for all ϕ ∈ C + 0 .The DDE x (t) = f (t, x t ), t ≥ 0, is said to be persistent (in C + 0 ) if all its solutions are defined and bounded below away from zero on [0, ∞), i.e., lim inf t→∞ x i (t, 0, ϕ) > 0 for all 1 ≤ i ≤ n, ϕ ∈ C + 0 ; and (2.1) is uniformly persistent if all positive solutions are defined on [0, ∞) and there is a uniform lower bound m > 0, i.e., lim inf t→∞ x i (t, 0, ϕ) ≥ m for all 1 ≤ i ≤ n, ϕ ∈ C + 0 .To simplify the notation, define the n × n matrices where a ii := 0 (1 ≤ i ≤ n), and the so-called community matrix The properties of the matrices D − A and M play an important role in the global asymptotic behaviour of solutions of (2.1).The following algebraic concept is timely.Definition 2.1.A square matrix N = [n ij ] with nonpositive off-diagonal entries (i.e., n ij ≤ 0 for i = j) is said to be a non-singular M-matrix if all its eigenvalues have positive real parts.
Clearly, this is equivalent to saying that the ODE x = −(D − A)x is asymptotically stable.
Throughout this paper, we shall assume a stronger hypothesis: (H1) there exists a vector v = (v 1 , . . ., v n ) > 0 such that Since β i > 0 for all i, note that (H1) is equivalent to saying that there is a positive vector v that satisfies both (D − A)v > 0 and Mv > 0.
Remark 2.2.Since M is a cooperative matrix (also called a Metzler matrix), i.e., all its off-diagonal entries are nonnegative, by using the theory of Perron-Frobenius one can show that, if Mc > 0 for some vector c > 0, then the spectral bound s(M) = max{Re λ : λ ∈ σ(M)} of M is positive; in fact, the converse is also true when M is irreducible.Moreover, when D − A is a nonsingular M-matrix, (H1) is satisfied if and only if there exists a positive vector c such that Mc > 0 (see [7]).This means that, if there are positive vectors u, w such that (D − A)u > 0 and Mw > 0, then there is a positive vector v for which both conditions (D − A)v > 0 and Mv > 0 are satisfied.Remark 2.3.From a biological viewpoint, it is quite natural to assume that D − A is a nonsingular M-matrix.In fact, as mentioned above, for models from population dynamics we take , where m i is the death rate for the population in patch i.Thus D − A T is diagonally dominant, i.e., [D − A T ]1 > 0 where 1 := (1, . . ., 1).In particular, the matrix D − A T is a non-singular M-matrix, which implies that D − A is a non-singular M-matrix as well.
The main purpose of this paper is to investigate the stability and global attractivity of a positive equilibrium N * , when it exists.Some standard definitions are given below.Definition 2.4.A positive equilibrium N * of (2.1) is said to be globally attractive (in C + 0 ) if N(t, 0, ϕ) → N * as t → ∞, for all solutions of (2.1) with initial conditions N 0 = ϕ ∈ C + 0 ; N * is globally asymptotically stable if it is stable and globally attractive.If there are K > 0, α > 0 such that |N(t, 0, ϕ) − N * | ≤ Ke −αt ϕ − N * , for all t ≥ 0, ϕ ∈ C + 0 , then N * is said to be globally exponentially stable.
In what concerns the existence and uniqueness of an equilibrium N * > 0, observe that the equilibria of (2.1) coincide with the equilibria of the autonomous ODE (2.6) The nonlinearity h(x) = xe −x is bounded on [0, ∞), hence a simple use of the variation of constants formula shows that, if the linear ODE x = −[D − A]x is exponentially stable, then (2.6) is dissipative.Since R n + is positively invariant for (2.6) and the system is dissipative, by [9] there is at least a saturated equilibrium of (2.6) in R n + .Under the assumption (H1), by exploiting the properties of the cooperative matrix M, it was shown in [7] that such an equilibrium is forcefully positive and unique.
Some preliminary results on the global asymptotic behaviour of (2.1) are collected in the theorem below.In spite of the situation with time-dependent delays, the statements are easily deduced by repeating the arguments in [7], so the proofs are omitted (see also [5,10]).Theorem 2.5.For system (2.1), assume (H0) and that D − A is a non-singular M-matrix.Then: (ii) if s(M) ≤ 0, the equilibrium 0 is globally asymptotically stable; (iii) if (H1) holds, (2.1) is uniformly persistent and there is a unique positive equilibrium N * ; (iv) if (1.5) holds, the positive equilibrium N * is globally asymptotically stable.Remark 2.6.One can check that the statements in Theorem 2.5 (i)-(iii) are valid with h(x) = xe −x replaced in each equation by smooth functions h i (x) with h i (x) > 0 for x > 0, h i (0) = 0, h i (0) = 1, h i (∞) = 0 and h i (x)/x decreasing on (0, ∞).Nevertheless, good criteria for the attractivity of the equilibrium N * > 0 would depend heavily on the shape of the nonlinearities h i (x).

Absolute exponential stability of the positive equilibrium
If the coefficients γ i (v) defined in (2.5) satisfy suitable upper bounds, the positive equilibrium N * has its components in the interval (0, 2), where the nonlinearity h(x) = xe −x has very specific properties.This allows us to derive the absolute global exponential stability of the positive equilibrium N * of (2.1), where as usual the term 'absolute' refers to the fact that such a stability holds regardless of the size of the delay functions τ ik (t), provided that they remain bounded.
Before the main theorem of this section, we state two auxiliary results.The first lemma is a simplified version of [6, Lemma 3.2], while the second refers to properties of the nonlinearity xe −x .Lemma 3.1 ([6]).Let S ⊂ C be the set of initial conditions for a DDE x Then the solutions x(t) of x (t) = f (t, x t ) with initial conditions x t 0 = ϕ ∈ S are defined and bounded for t ≥ t 0 and, if x t ∈ S for all t ≥ t 0 , the solution satisfies |x(t)| ≤ x t 1 for all t ≥ t 1 ≥ t 0 .
where γ i (v) are defined as in (2.5).
Then, the positive equilibrium N * of (2.1) is uniformly stable.Moreover, if the positive equilibrium N * of (2.1) is globally exponentially stable.In particular, if N * is globally exponentially stable.
Proof.Assume (H2).The existence and uniqueness of a positive equilibrium N * is guaranteed by Theorem 2.5.The equilibrium N * = (N * 1 , . . ., N * n ) is determined by the system Hence from (3.1) we have On the other hand, as all coordinates of N * lie in (0, 2], from Lemma 3.2 we have for all x > −1, x = 0, where as before h(x) = xe −x .
Returning to (2.1), through the change of variables system (2.1) becomes where âij = is the set of admissible initial conditions for the transformed system (3.7).
Let ϕ be as above and fix i such that |ϕ(0)| = |ϕ i (0)|.We only consider the case ϕ i (0) > 0, since the case ϕ i (0) < 0 is treated in a similar way.For t ≥ 0, the estimates in (3.5) yield and consequently (3.8) From Lemma 3.1, it follows that, for any solution x(t) of (3.7) with initial condition in S, the function t → x t is non-increasing.This shows that the equilibrium N * of (2.1) is uniformly stable.
Next, we assume the strict inequalities in (3.2).Proceeding as in (3.4), one obtains that all the components N * i of N * are in the interval (0, 2).From the boundedness and persistence of solutions to (2.1), one may fix m, L > 0 such that the components of the solution x(t) of (3.7) satisfy −1 + m ≤ x i (t) ≤ L for t sufficiently large.On the other hand, since |h where g i (x) is the continuous function given by g where r i = max 1≤k≤m sup t≥0 τ ik (t).
Analysis of the above proof shows that, under the existence of the positive equilibrium N * , hypotheses (H2) and (H2*) were used only to derive that all its components N * i are in the interval (0, 2], respectively (0, 2).Therefore, a weaker version of Theorem 3.3 is obtained as follows.
(i) If there exists v > 0 such that the positive equilibrium N * of (1.1) is uniformly stable.
(ii) If the positive equilibrium N * of (1.1) is globally exponentially stable.
Proof.Effect the scalings Ñi (t) = c i N i (t), i = 1, . . ., n. System (1.1) is transformed into a system of the form (2.1), with the coefficients a ij replaced by ãij := c i c j a ij , i, j = 1, . . ., n, j = i.
Remark 3.6.For the case of (2.1) with autonomous discrete delays τ ik (t) ≡ τ ik , the global asymptotic stability (but not the exponential stability) of the positive equilibrium was proven in [7] under (H2) with v = 1 := (1, . . ., 1): However, the arguments in [7] do not carry out for the present situation, since properties of ω-limit sets for autonomous DDEs were employed to derive the result.Although Theorem 3.3 and Theorem 3.5 address the more general situation of Nicholson systems with time-varying discrete delays, with γ i (1) replaced by γ i (v) for some positive vector v, the global attractivity of N * cannot be derived when γ i (1) = e 2 for some i.On the other hand, combining the techniques above with the ones in [7], for the autonomous case of (1.1) with τ ik (t) ≡ τ ik it follows that (3.10) is a sufficient condition for the global asymptotic stability of N * .

Global attractivity under small delays
In this section, the goal is to prove that a condition on the size of the delays, together with (H1), implies that the positive equilibrium is a global attractor.Here, some ideas in So and Yu [16] (for the scalar case) and Jia et al. [10] (for the n-dimensional case) are followed.However, significant adjustments to the arguments in [10] have to be performed, in order to eliminate the quite restrictive assumption (1.6).We emphasize that, without imposing (1.6), not only are the components of N * in general different from each other, but also N * cannot be computed explicitly.
To simplify some arguments, we write (2.1) as where m 1 , . . ., m n ∈ N, all the coefficients and delays are as in (2.1) and moreover we now demand that β ik > 0 for all i = 1, . . ., n, k = 1, . . ., m i .In what follows, as before we always assume a ii = 0, and denote Assume (H0), (H1), and effect again the change of variables (3.6), which transforms system (2.1) into (3.7), also written as In this way, the global attractivity of the equilibrium N * for (2.1) is equivalent to the global attractivity of the trivial solution for (4.2).
We start with a useful lemma, whose elementary proof can be checked by the reader or found in [11, p. 122] or [16].
By the fluctuation lemma [15], there exists an increasing sequence (t q ) such that t q → ∞, x i 1 (t q ) > 0 with x i 1 (t q ) → µ, x i 1 (t q ) → 0. We divide the rest of the proof into several steps.
Step 1.We first prove that there exists q 0 ∈ N such that, whenever q ≥ q 0 , there is l q ∈ [t q − r i 1 , t q ) such that x i 1 (l q ) = 0 and x i 1 (t) > 0, for t ∈ (l q , t q ).Suppose the assertion is false.Then there is a subsequence of t q , which we also denote by t q , such that x i 1 (t) > 0, for all t ∈ [t q − r i 1 , t q ).We have as by assumption x i 1 (t q − τ i 1 k (t q )) > 0. Now we claim that, for all k, lim x i 1 (t q − τ i 1 k (t q )) = µ.On the one hand, lim sup x i 1 (t q − τ i 1 k (t q )) ≤ µ.On the other hand, taking lim inf on (4.3), observing again that x i 1 (t q − τ i 1 k (t q )) > 0 and with α k := lim inf x i 1 (t q − τ i 1 k (t q )), we get However, since α k ≤ µ, one also has 1 This is only possible with α k = µ, for all k.Therefore, for every k = 1, ..., m i 1 , and consequently lim x i 1 (t q − τ i 1 k (t q )) exists and is equal to µ, for all k.Taking limits in (4.3) yields 0 ≤ (µ + 1) which is not possible.This finishes Step 1.
Step 2. Next, we show that λ, µ satisfy By the definition of λ and µ, there exists for all i, k and t sufficiently large.For l q as in Step 1, multiplying the i 1 -equation of (4.2) by e d i 1 t and integrating over the interval [l q , t q ] gives (1 + x i 1 (t q ))e d i 1 t q − e d i 1 l q = A q + B q , ( where and From (4.4), we obtain Inserting these upper bounds for A q and B q in (4.5), we derive Letting q → ∞ and ε → 0 In particular, this implies that λ < 0.
Reasoning as above, we take an increasing sequence (s q ) such that s q → ∞ and x i 1 (s q ) → λ, x i 1 (s q ) → 0. Next, as in Step 1, we can find q 2 > q 1 such that, if q ≥ q 2 , there is p q ∈ [s q − r i 1 , s q ) such that x i 1 (p q ) = 0 and x i 1 (t) < 0, for t ∈ (p q , s q ).As above in this step, similar arguments now show that and consequently, taking limits as q → ∞ and ε → 0 + , λ ≥ (e d i 1 r i 1 − 1) We finally apply hypothesis (H3).With (H3), the estimates (4.6) and (4.7) lead to which proves the claim.
As in Section 3, the above theorem is easily extended to more general Nicholson systems (1.1) by effecting the scalings N i (t) → c i N i (t), i = 1, . . ., n.For such systems, Theorem 4.2 reads as follows.
Condition (H3) is a condition on the size of the delays, as lim r i →∞ (e d i r i − 1) = ∞, lim r i →0 + (e d i r i − 1) = 0, thus (H3) fails to be true if the delays are too large.On the other hand, (H3) involves the a priori knowledge of N * , therefore it is relevant to obtain criteria that do not depend on it.Then, the positive equilibrium N * of (1.1) is a global attractor (in C + 0 ).
Corollary 4.5.Assume that (H1) holds, for some v > 0 for which where γ i (v) are as in (2.5).Then there is a positive equilibrium N * of (2.1), which is a global attractor Proceeding as in the proof of Theorem 3.3, we obtain e Moreover, , where âij = a ij N * j /N * i for j = i.Finally note that, for any given j, and thus for any i = 1, . . ., n we have (e d i r i − 1) The statement is now a consequence of Theorem 4.2.
We now adapt e.g.where γ i (c −1 ) := Then there is a positive equilibrium N * of (1.1), which is a global attractor (in C + 0 ).
Proof.As before, effect the change of variables Ñi (t) = c i N i (t), i = 1, . . ., n, and recall that (1.1) is transformed into a system of the form (2.1), with the coefficients a ij replaced by ãij := c i c j a ij .With the notation in (3.12), we have γi Hence, the result follows from the above corollary.Remark 4.8.In [10], the authors considered (2.1) but, instead of assuming a ii = 0 and incorporating this coefficient in d i , for each i, they assumed that ∑ n j=1 a ij = 0. Changing accordingly to the notation followed here, it was assumed in [10] that γ i = γ i (1) is the same constant c > 1 for all i, i.e., condition (1.6) is satisfied, so that N * = (log c, . . ., log c) is the positive equilbrium.With this notation, Jia et al. [10] proved the global attractivity of N * under the additional condition (e d i τ − 1) where τ = max 1≤i≤n r i .Note however that, in this situation, (4.11) is equivalent to and c −1 log c = e −N * i N * i for i = 1, . . ., n.This shows that the criterion in [10] is just a particular case of our Theorem 4.2.Remark 4.9.In the recent paper [4], El-Morshedy and Ruiz-Herrera considered an abstract setting, in which they developed a geometric method to prove the global attractivity of nontrivial equilibria for systems of autonomous DDEs.They also gave an application to an autonomous Nicholson system of the form (1.4), where τ ik (t) ≡ τ ik ≥ 0 for all i, k.However, two major constraints are imposed in [4]: first, the authors assume the existence of a positive equilibrium N * of (1.4); secondly, the matrix D − A is assumed to be not only a non-singular M-matrix, but diagonally dominant, i.e., d i − ∑ j =i a ij > 0 for i = 1, . . ., n.For γ i as above, and denoting ρ = max 1≤i,k≤m (τ ik d i ), it was shown in [4] that N * is a global attractor for (1.4) if This criterion is not always comparable with the ones presented in Theorem 4.2 and its corollaries, in the sense that for different concrete values of the coefficients and delays our results might provide better criteria than the one in [4], and vice-versa.On the other hand, condition (4.12) provides both a delay-independent and a delay-dependent criterion for global attractivity.In fact, observe that the function g(ρ) = 1 + e ρ e ρ −1 is decreasing on (0, ∞) with g(0 + ) = ∞, g(∞) = 2, thus (4.12) is always satisfied if 1 < γ i ≤ e 2 for i = 1, . . ., n.This means that the nice result in [4] in particular recovers the requirement in [7] for the absolute global attractivity of N * (see also Remark 3.6); at the same time, (4.12) is clearly satisfied if τ = max 1≤i,k≤m τ ik is small.