Permanence for nonautonomous differential systems with delays in the linear and nonlinear terms

In this paper, we obtain sufficient conditions for the permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed delays in both the linear and nonlinear terms, and where typically the nonlinear terms are nonmonotone. Applications to generalized Nicholson or Mackey-Glass systems are given.


Introduction
In this paper, we investigate the persistence and permanence for a class of multidimensional nonautonomous delay differential equations (DDEs), which includes a wide range of structured models used in population dynamics, neural networks, physiological mechanisms, engineering and many other fields.
We start by setting the abstract framework for the DDEs which we deal with in the next sections. For τ ≥ 0, consider the Banach space C := C([−τ, 0]; R n ) with the norm φ = max θ∈[−τ,0] |φ(θ)|, where | · | is a fixed norm in R n . We shall consider DDEs written in the abstract form where x t ∈ C denotes the segment of a solution x(t) given by x t (θ) = x(t + θ), −τ ≤ θ ≤ 0, L(t) : C → R n is linear bounded and the nonlinearities are given by continuous functions f : For simplicity, we set t 0 = 0. As in many mathematical biology models, we shall assume the existence and dominance of diagonal linear instantaneous negative feedback terms in (1.1) and that each component f i of f = (f 1 , . . . , f n ) depends only on t and on the component i of the solution: f (t, φ) = (f 1 (t, φ 1 ), . . . , f n (t, φ n )) for t ≥ 0, φ = (φ 1 , . . . , φ n ) ∈ C. (1.2) Recently, there has been a renewed interest in questions of persistence and permanence for DDEs. A number of methods has been proposed to tackled different situations, depending on whether the equations are autonomous or not, scalar or multi-dimensional, monotone or nonmonotone. See [1,2,4,6,8,9,[11][12][13][14][15]20] and references therein, also for explanation of the models and motivation from real world applications.
Here, the investigation concerning permanence in [9,11] is pursued. In [9] only cooperative systems were considered, whereas in [11] sufficient conditions for the permanence of systems x i (t − τ ik (t))), i = 1, . . . , n, t ≥ 0, (1.3) were established. Clearly, nonautonomous differential equations with multiple time-varying discrete delays are a particular case of (1.3). In this paper, the more general framework of systems (1.1) with (possibly distributed) delays in both L and f is considered, although sharper results will be obtained for models of the form a ij (t)x j (t) + f i (t, x i,t ), i = 1, . . . , n, t ≥ 0.
The criteria for permanence in [4,8,11] and many other works demand that all the coefficients are bounded. More recently, some authors have relaxed this restriction [3,[13][14][15], though still under some boundedness requirements. Here, the boundedness of all the coefficients in (1.1) will not be a priori assumed. We also emphasize that typically the nonlinearites f i (t, φ i ) in (1.3) are not monotone in the second variable -which is the case of Nicholson-type systems, for example. Nevertheless, some techniques for cooperative systems will be used. Our results extend and improve some recent achievements in the literature [4,6,14,15,19,22], which mostly deal with scalar DDEs and/or cooperative n-dimensional models. We now introduce some standard notation. In what follows, R + = [0, ∞), the matrix I n , or simply I, denotes the n × n identity matrix and 1 = (1, . . . , 1) ∈ R n . For τ > 0, the set C + = C([−τ, 0]; (R + ) n ) is the cone of nonnegative functions in C and ≤ the usual partial order generated by C + : φ ≤ ψ if and only if ψ − φ ∈ C + . A vector v ∈ R n is identified in C with the constant function ψ(θ) = v for −τ ≤ θ ≤ 0. For τ = 0, we take C = R n , C + = [0, ∞) n ; a vector v ∈ R n is positive if all its components are positive, and we write v > 0. We write φ < ψ if ψ(θ) < φ(θ) for θ ∈ [−τ, 0]; the relations ≥ and > are also defined in the usual way.
For nonlinear DDEs (1.1) under conditions of existence and uniqueness of solutions, x(t, σ, φ) denotes the solution of (1.1) with initial condition x σ = φ, for (σ, φ) ∈ R + × C. For models inspired by mathematical biology applications, we shall consider as the set of admissible initial conditions. Without loss of generality, we shall restrict the analysis to solutions x(t, 0, φ) with φ ∈ C + 0 , and assume that f is sufficiently regular so that such solutions are defined on R + . If the set C + 0 is (positively) invariant for (1.1), the notions of (uniform) persistence, permanence and stability always refer to solutions with initial conditions in C + 0 . In this way, we say that the system is uniformly persistent (in C + 0 ) if there exists a positive uniform lower bound for all solutions with initial conditions in C + 0 ; i.e., there is m > 0 such that all solutions x(t) = x(t, 0, φ) with φ ∈ C + 0 are defined on R + and satisfy x i (t, 0, φ) ≥ m for t ≫ 1 and i = 1, . . . , n. The system (1.1) is said to be permanent if there exist positive constants m, M such that all solutions x(t) = x(t, 0, φ) with φ ∈ C + 0 are defined on R + and satisfy m ≤ x i (t) ≤ M for t ≫ 1 and i = 1, . . . , n. As usual, the expression t ≫ 1 means "for t > 0 sufficiently large". For short, here we say that a DDE x ′ (t) = F (t, x t ) is cooperative if F = (F 1 , . . . , F n ) satisfies the quasi-monotone condition (Q) in [21]: if φ, ψ ∈ C + and φ ≥ ψ, then F i (t, φ) ≥ F i (t, ψ) for t ≥ 0, whenever φ i (0) = ψ i (0) for some i.
The remainder of this paper is divided into three sections. In Section 2, we establish sufficient conditions for the uniform persistence and permanence for a large family of nonlinear system (1.1). To illustrate the results, generalized Nicholson and Mackey-Glass systems are considered in Section 3, together with examples, as well as counter-examples showing the necessity of some hypotheses. The paper ends with a short section of conclusions and open problems.

Persistence and permanence for a class of nonautomous DDEs
In this section, we establish explicit and easily verifiable criteria for both the persistence and the permanence of systems (1.1) with nonlinearities f expressed by (1.3).
Let C := C([−τ, 0]; R n ) with the supremum norm be the phase space. We start with a general nonutonomous linear differential equation in C, where L : R → L(C, R n ), L(C, R n ) is the usual space of bounded linear operators from C to R n equipped with the operator norm, and t → L(t)φ is Borel measurable for each φ, with L(t) bounded on R + by a function m(t) in L 1 loc (R + ; R). Assuming the exponential asymptotic stability of (2.1), next theorem provides conditions for the dissipativeness and extinction of perturbed nonlinear systems. Its proof is easily deduced from the variation of constant formula [17] and arguments similar to the ones for ODEs, thus it is omitted.
Theorem 2.1. Assume that the system (2.1) is exponentially asymptotically stable, and consider the perturbed equation where f : [0, ∞) × S → R n is continuous and S is a (positively) invariant set for (2.2).
For (2.1), we now suppose that L = (L 1 , . . . , L n ) is given by with d i (t) > 0 and L ij (t) bounded linear functionals. Although it is not relevant for our results, we may assume that L ii (t) is non-atomic at zero (see [17] for a definition). For (2.1), define the n × n matrix-valued functions where a ij (t) = L ij (t) , t ≥ 0, i, j ∈ {1, . . . , n}.
For (2.1), the general hypotheses below will be considered: (H2) there exist a vector v > 0 and a constant δ > 0 such that Instead of (H2), one may assume: (H2*) there exist a vector v > 0 and a constant α > 1 such that D(t)v ≥ αA(t)v for t ≫ 1 .
Henceforth, we consider delay differential systems written as with the linear functionals L ij (t) nonnegative (i.e., L ij (t)φ j ≥ 0 for φ j ≥ 0) and f i (t, φ i ) continuous and satisfying same requirements formulated below. Recall that, by the Riesz representation theorem, the nonnegative bounded functionals L ij (t) have a representation where a ij (t) = L ij (t) , the functions ν ij (t, s) are defined for (t, s) ∈ R + × [−τ, ∞), are continuous in t, left-continuous and nondecreasing in s, and normalized so that In the case of no delays in (2.6), then L ij (t)x j,t = a ij (t)x j (t) with a ij (t) ≥ 0. Clearly, this framework includes the particular case of DDEs with multiple time discrete delays: where the coefficients and delays are all continuous and nonnegative.
Systems (2.5) are sufficiently general to encompass many relevant models from mathematical biology and other fields. In some contexts, they are interpreted as structured models for populations distributed over n different classes or patches, with migration among the patches, where x i (t) is the density of the species on class i, a ij (t) (j = i) is the migration coefficient from class j to class i, d i (t) the coefficient of instantaneous loss for class i, and f i (t, φ i ) is the birth function for class i. Although most models do not include delays in the migration terms, structured models where delays intervene in the linear terms have deserved the attention of a number of researchers, see e.g. Takeuchi et al. [22]. We also observe that in biological models most situations require a single delay for each population, however multiple or distributed delays naturally appear in neural networks models, or generalizations of the classic Mackey-Glass equation used as hematopoiesis models. We refer the reader to [1-4, 6, 14, 21, 22], for real interpretation of the DDEs under consideration and some applications.
In what follows, for φ = (φ 1 , . . . , φ n ) ∈ C + we use the notation To establish the permanence of (2.5), we further impose that the nonlinearities satisfy the following conditions: (H3) the functions f i : R + × C + → R + are completely continuous and locally Lipschitzian in the second variable, i ∈ {1, . . . , n}; (H4) there exist continuous functions β i : with D(t), A(t) as above and β i (t) as in (H4).
Instead of (H5), we shall often assume: Some comments about these assumptions are given in the remarks below.
In the study of stability for nonautonous DDEs, a condition as (H2*) with v = 1 has been often presented (see e.g. [9,14]) in the equivalent form lim inf t→∞ 0 is a vector as in (H5) or (H5*), we obtain a new system In this way, and after dropping the hats for simplicity, we may consider an original system (2.5) and take v = 1 in (H5) or (H5*).
The main criterion for the permanence of (2.5) is now established. Theorem 2.3. For (2.5), assume (H1)-(H4). Furthermore, let the following conditions hold: . . , n and t ≥ 0 (in other words, there are no delays in (2.1)), or L ij (t) are nonnegative and a ij (t) = L ij (t) are bounded on R + , i, j = 1, . . . , n; (ii) either (H5) is satisfied and lim sup t→∞ β i (t) < ∞ for all i, or (H5*) is satisfied and lim inf Proof. The proof follows along the main ideas in [11,Theorem 3.3], however new arguments are used to take into account the more general form of (2.5), that delays are allowed in the linear part and that the coefficients d i (t) are not required to be bounded -as well as a ij (t), if there are no delays in L ij (t).
, it is clear that F is continuous, locally Lipschitzian in the second variable and bounded on bounded sets of R + × C + . From Theorem 2.2, (2.1) is exponentially asymptotically stable (for the case of no delays in the linear functionals L ij (t), recall that the boundedness of a ij (t) is not required). Theorem 2.1 implies that (2.5) is dissipative. Observe that the solutions of (2.5) satisfy the ordinary differential inequalities 0 are positive for t ≥ 0. From (ii) and Remark 2.1, both (H5) and (H5*) are satisfied, with a common vector v > 0. By the scaling described in Remark 2.3, without loss of generality we may take v = 1 in (H5), (H5*).
We now derive the uniform persistence of (2.13) by showing that, for any solution x(t) = x(t, t 0 , φ, G) of (2.13), there exists T ≥ t 0 such that (2.14) This is proven in several steps.
Step 2. We first prove that the ordered interval [m, Note that both the functions H i (x) and the operators L ij (t) are nondecreasing and If φ ∈ [m, ∞) n and φ i (0) = m for some i, from (2.11) we therefore obtain, for t ≥ T 0 , From [21,Remark 5.2.1], it follows that the set [m, ∞) n ⊂ C is positively invariant for (2.13).
If s 1 ≤ s 0 , then Assuming that s 1 ≤ s 0 and which is a contradiction. This shows that s 1 > s 0 whenever s 1 < m.
Step 4. Define the sequence For the sake of contradiction, assume that s k < m for all k ∈ N 0 . Thus, reasoning as in Step 3, (s k ) is strictly increasing. Let t k ∈ I k := [T 0 + kτ, T 0 + (k + 1)τ ] be such that s k = x i k (t k ), for some i k ∈ {1, . . . , n}. By jumping some of the intervals I k and considering a subsequence of (t k ), still denoted by (t k ), we may consider a unique i ∈ {1, . . . , n} such that s k = x i (t k ). Denote ℓ = lim k s k > 0.
Let α > 1 and δ > 0 as in (2.12). We now claim that Otherwise, suppose that there is k such that s k+1 < α min j H j (s k ). We distinguish two situations: either there are no delays in (2.1), or a ij (t) are all bounded in R + -in which case we suppose that α is chosen so that it also satisfies 1 First, we treat the case of no delays in the linear part L of (2.1). In this situation, L ij (t)x j,t = a ij x j (t) ≥ a ij (t)s k for t ∈ I k (k ∈ N 0 ). Estimate (2.11) leads to which is not possible. Thus, (2.15) holds.
When delays are allowed in the linear part, we can write L ij x j,t ≥ a ij (t)s k for t ∈ I k+1 , and which is not possible. Thus, claim (2.15) is proven. From (2.15), we obtain m ≥ ℓ ≥ α min j H j (ℓ) = α min j h − j (ℓ) > ℓ, which is not possible. Therefore, s k ≥ m for some k, and the result follows by Step 2.

Remark 2.4.
For v = (v 1 , . . . , n) and m > 0 as in the above proof, one concludes that any solution It is clear that assumption (H2) was used in the above proof only to derive that (2.5) is dissipative. In the case of bounded nonlinearities, Theorem 2.1 shows that Theorem 2.3 is still valid if one replaces (H2) by the requirement of having (2.1) exponentially asymptotically stable, as stated below.
The previous arguments also allows us to derive sufficient conditions for the uniform persistence of (2.5) without requiring that the system is dissipative, nor that the coefficients β i (t) are bounded.
Theorem 2.6. For (2.5), assume (H1), (H3), (H4) and (H5*), and the following conditions: . . , n and t ≥ 0, or L ij (t) are nonnegative and a ij (t) = L ij (t) are bounded on R + , i, j = 1, . . . , n; ( When the linearities do not have delays, the above proof only requires the use of assumption (H5*) to show the uniform persistence, but not of (H5). This observation and Theorem 2.2(iii) allow us to conclude the following:  (H4), for all i, then (2.16) is uniformly persistent. We end this section with two remarks, leading to more precise and general results.
Remark 2.5. More explicitly, we could have written the linear DDE (2.1) as and a ij (t), ν ij (t, s) as above, with s → ν ij (t, s) non atomic at zero, and apply more precise criteria for its exponential asymptotic stability, see [10]. Namely, the criteria in Theorem 2. Remark 2.6. Consider nonlinearites which also incorporate a strictly sublinear negative feedback term of the form −K i (t, x i (t)), so that (2.5) reads as where K i (t, x) ≥ 0 are continuous and K i (t, x) ≤ κ i (t)g i (x) for some continuous functions κ i , g i : By comparing below and above the solutions of (2.17) with solutions of cooperative systems and from Theorem 2.1, it follows (2.17) is dissipative and that C + 0 is forward invariant for (2.17). On the other hand, for any fixed ε > 0 small, there is m 0 > 0 such that 0 ≤ K i (t, x) ≤ εx for x ∈ [0, m 0 ]. A careful analysis shows that the arguments in the proof of Theorem 2.3 carry over to (2.17) if one chooses ε ∈ (0, δ), for δ > 0 as in (H5), so that (2.12) is satisfied with d i (t) replaced by d i (t) + ε. In this way, one may conclude that the permanence results stated in Theorems 2.3, 2.4 and Corollary 2.1 are still valid for (2.17). This more general framework allows in particular to consider structured models with harvesting.

Applications and examples
We now apply our results to generalized Nicholson and Mackey-Glass systems. The literature on generalized Nicholson and Mackey-Glass models is very extensive, here we only mention a few selected references dealing with the persistence and permanence for either scalar or multidimensional Nicholson equations [6,11,12,19,20] and Mackey-Glass equations [3,4,11], and references therein.
Consider systems given by λ ik (s)g ik (s, x i (s)) ds, i = 1, . . . , n, The functions d i (t), a ij (t), b ik (t), σ i (t), τ ik (t), λ ik (t), c ik (t) are assumed to be continuous and nonnegative, with σ i (t), τ ik (t) ∈ [0, τ ] (for some τ > 0), and d i (t) > 0, c ik (t) > 0, for all i, j, k and t ≥ 0. For g ik as in (3.2) a modified Nicholson-type system is obtained, whereas the choice (3.3) provides a Mackey-Glass-type system. We suppose that the linear operators L ij (t) are nonnegative, thus a ij (t) = L ij (t) as before, and define From Theorem 2.3, we derive sufficient conditions for the permanence of (3.1). Proof. Clearly, system (3.1) has the form (2.5), with Since g ik and β i (t) are bounded, from Theorem 2.1 the system is dissipative. Choose M > 0 such that 0 < x i (t) ≤ M, t ∈ R + , i = 1, . . . , n, for any solution x(t) with initial condition in  For the situation without delays in the linear part, from Corollary 2.1 we obtain: We emphasize that this corollary gives a sharper criterion for permanence than the one in [11], and moreover applies to a much larger family of delayed structured models. For instance, in the case of Nicholson systems, the result in Corollary 3.1 was established in [11,Theorem 3.5] only for the case of Nicholson systems (3.4) with discrete delays and all coefficients bounded.
Some illustrative examples, as well as counter-examples showing the necessity of our assumptions, are now presented.
Example 3.1. This counter-example is based on a counter-example due to Győri and Horváth [16], and shows that if (H2) holds but the coefficients a ij (t) are not bounded, then even the asymptotic stability of (2.1) may fail.
permanence implies the existence of a positive periodic solution -in this context, a stability result will show that such a periodic solution is a global attractor of all positive solutions. It is worthwhile mentioning that, in the last few years, the stability of nonautonomous linear DDEs has received a great deal of attention, and several methods have been used to obtain explicit sufficient conditions for the asymptotic and exponential asymptotic stability of a general linear system (2.1), see e.g. [5,10,16] and references therein. Actually, both delay independent and delay-dependent criteria for the stability of linear DDEs with possible infinite delays were given in [10]. Since the exponential stability of (2.1) is a key ingredient to show the permanence of (2.5), this leads us to two natural lines of future research, explained below.
The first one is to replace assumption (H2) or (H2*) -which forces (2.1) to possess diagonal terms without delay which dominate the effect of the delayed terms -by a condition depending on the size of delays in such a way that (2.1) maintains the exponential asymptotic stability, and further analyse how such a condition interplays with the assumption (H5).
Another open problem is to study the persistence and permanence of systems of the form (2.5) with unbounded delays. DDEs with infinite delay are surely more challenging: not only an admissible phase space satisfying some fundamental set of axioms should be chosen [18], but most techniques for finite delays do not apply for such equations. There has been some work on permanence for scalar nonautonomous DDEs with infinite delay, see e.g. [16]. In the case of multidimensional DDEs with infinite delay, the work in [9] only contemplates situations of cooperative systems, namely of the form x) sublinear in x ∈ R + . For the case of nonmonotone nonlinearities in (2.5), it is clear that the technique developed in the proof of Theorem 2.3 does not apply to systems with infinite delay, since it relies on a step-wise iterative argument on intervals of lenght τ , where τ is the supremum of all delays -thus, new tools and arguments to tackle the difficulty must be proposed. This open problem is a strong motivation for a next future investigation.
The treatment of mixed monotonicity models, in what concerns questions of permanence, is another topic deserving attention, since they appear naturally in real-world applications. In fact, there has been an increasing interest in DDEs with mixed monotonicity, where the nonlinear terms involve one or more functions with different delays e.g. of the form f (t, x(t − τ (t)), x(t − σ(t))), with f (t, x, y) monotone increasing in the variable x and monotone decreasing in y. As as illustrated by Berezansky and Braveman [3], though small delays are in general harmless, the presence of two or more delays in the same nonlinear function may change drastically the global properties of the solutions. The permanence and stability of DDEs with nonlinearities of mixed monotonicity have been analyzed in [2,3,7,13,15]. As far as the author knows, only the case of discrete delays has been dealt with. As seen, systems (2.5) encompass models with noncooperative nonlinearities, nevertheless cooperative techniques were used in our arguments. Therefore, new tools are required to handle the case of mixed monotonicity in the nonlinear terms.