Persistence and Stability for a Class of Forced Positive Nonlinear Delay-Differential Systems

Persistence and stability properties are considered for a class of forced positive nonlinear delay-differential systems which arise in mathematical ecology and other applied contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes in an ecological setting), disturbances induced by seasonal or environmental variation, or migration. We provide necessary and sufficient conditions under which the states of these models are semi-globally persistent, uniformly with respect to the initial conditions and forcing terms. Under mild assumptions, the model under consideration naturally admits two steady states (equilibria) when unforced: the origin and a unique non-zero steady state. We present sufficient conditions for the non-zero steady state to be stable in a sense which is reminiscent of input-to-state stability, a stability notion for forced systems developed in control theory. In the absence of forcing, our input-to-sate stability concept is identical to semi-global exponential stability.


Introduction
The present paper considers boundedness, convergence, persistence and stability properties for a class of forced positive nonlinear delay-differential systems. The forcing arises from exogenous terms which, depending on the context, are interpreted as a control action or disturbance. Specifically, we consider the systeṁ (1.1) state (equilibrium) of system (1.1) when unforced (u constant, v = 0). Generally speaking, persistency, which is now a well-established concept [13-15, 38, 39], captures the extent to which non-zero solutions are bounded away from 0 which corresponds to the avoidance of extinction in population models. Under fairly natural assumptions, the unforced version of (1.1) admits two steady states -0 and a unique non-zero non-negative steady state x s . If the system is persistent, then, obviously, 0 is not stable and the immediate stability question is: do there exist natural and easily checkable conditions under which the non-zero steady state x s is stable, and attracts all (non-zero) solutions in the absence of forcing? Furthermore, what are the effects of the potentially persistent forcing terms? The motivation for including these terms is to study a framework where disturbances (unintended effects), such as temporal parameter variation, or control actions (intended effects) can be accommodated.
Our main results relate to persistence and stability of (1.1). For the former, our main result is Theorem 4.3 which provides sufficient conditions for (1.1) to be semi-globally persistent, uniformly with respect to time. Here the term "semi-global" refers to initial conditions as well as forcing functions. Furthermore, under mild additional assumptions, we show that the sufficient persistency conditions of Theorem 4.3 are necessary, see Proposition 4.4. In relation to stability, our main result is Theorem 5.2, which provides estimates of x(t) − x s in terms of the initial condition and the forcing functions u and v. The stability conditions are based on the interplay of the constant −c T A −1 b and an associated sector condition for f . The concept of a sector condition is ubiquitious in the theory of Lur'e systems [22,23,29,47] and is related to that of an enveloping condition [6,34] in the difference equations literature. In the absence of any forcing, our stability results guarantee that x s is semi-globally exponentially stable. The theoretical results are illustrated by detailed discussions of three classes of examples from population dynamics and chemical reaction networks, namely: delayed recruitment models, dispersal of a population with unique breeding region (modelled by a so-called Nicholson system) and self-regulated biochemical reactions.
The feature which distinguishes our work from much of the literature on population dynamics is the inclusion of the forcing terms u and v (modelling disturbances, control actions or certain parameter variations) in the stability analysis to estimate the effects these terms may have on the unforced dynamics. To do so, we make use of concepts and techniques from the field of nonlinear control theory, namely the input-to-state stability paradigm [8,42,43], initiated in [40], and one of the major developments in nonlinear control theory over the last 30 years. To make the paper understandable to readers without any background in control theory, we explain and state the relevant control theoretic concepts and results in some detail.
The paper is organised as follows. Sections 2 and 3 gather mathematical preliminaries and introduce the model we consider, respectively. Sections 4 and 5 comprise the technical heart of the paper and contain persistence and stability results, respectively. Three classes of examples are considered in depth in Sect. 6. Some technical aspects are relegated to the Appendices.
For M = (m ij ) ∈ R n×q , we write M ≥ 0 if M ∈ R n×q + , M > 0 if M ≥ 0 and M = 0 and M 0 if m ij > 0 for all i and j . If M 0, then we also say that M is strictly positive. The matrix M is said to be Metzler if it is square and m ij ≥ 0 for i = j . We recall that a square matrix is called Hurwitz if every eigenvalue has negative real part.
The i-th canonical basis vector of R n is denoted by e i , that is, e i is the vector in R n the i-th component of which is equal to 1 and with all other components being equal to 0. Obviously, e i > 0, but e i is not strictly positive. For the purposes of this paper, it is convenient to endow R n with the 1-norm · 1 , that is, for z = (z 1 , . . . , z n ) T ∈ R n , the norm z 1 is given by z 1 = n i=1 |z i |. Since the 1-norm on R n is used throughout, we will simply write z := z 1 . Occasionally, the maximum norm z ∞ = max 1≤i≤n |z i | will also be used.
For r = 1, ∞ and h > 0, we define Consider the following linear system with output delaẏ where A ∈ R n×n , b, c ∈ R n , h ≥ 0, v f and v are forcing (input, control, disturbance) functions and y is the so-called output (measurement, observation). In the rest of this paper, the input v f will be generated by nonlinear output feedback (see below). The impulse response associated with the delay-free linear systeṁ will be denoted by G, that is, Note that if x(0) = 0, then the output (or response) of system (2.2) corresponding to a Dirac delta input v f is given by G, hence the terminology. Denoting the corresponding transfer function (the Laplace transform of G) by G, we have that where s is a complex variable. It is clear that G is a rational function which vanishes at ∞.
Application of nonlinear output feedback of the form v f (t) = N(t, y(t)) to (2.1) leads tȯ where v ∈ L 1 loc (R + , R n ). It is assumed that N : R + × R → R is locally integrable in its first variable, that is, for each z ∈ R, the function t → N(t, z) is locally integrable, and N is locally Lipschitz in its second variable, in the sense that, for every z ∈ R, there exist a locally integrable function λ : R + → R + and an open interval J ⊂ R containing z such that If h = 0 (in which case we may identify M 1 with R n and the initial condition in (2.3) reduces to x(0) = ξ 0 ∈ R n ), then it is well known that (2.3) has a unique maximally defined solution x : [0, τ ) → R n , and, if τ < ∞, then x(t) → ∞ as t → τ , see, for example [41,Appendix C.3] or [48,§10,Supplement II]. Furthermore, if N satisfies an affine-linear bound in the sense that there exist nonnegative locally integrable functions α and β such that |N(t, z)| ≤ α(t) + β(t)|z| for all t ≥ 0 and z ∈ R, then the maximal interval of existence is equal to [−h, ∞).
In the next result, we consider the case wherein h > 0. In particular, it turns out that if the function t → N(t, ξ 1 (t − h)) is integrable on [0, h], then blow up in finite time is not possible. Proposition 2.1 is well known, but it is hard to find a precise reference, and therefore we provide a proof in Appendix B.

Proposition 2.1 Assume that h > 0 and let
It follows from the assumptions imposed on N that , and thus, in this case, the initial-value problem (2.3) has a unique solution on [−h, ∞) for every ξ ∈ M 1 .
The following so-called input-to-state stability result will play an important role in the paper.

Proposition 2.2
Assume that there exists l ≥ 0 such that For each initial condition ξ ∈ M 1 and each v ∈ L ∞ loc (R + , R n ) there exists a unique solution of (2.3) on [−h, ∞), and the following statements hold.
(1) Let τ > 0. There exists a constant ≥ 1 (depending on (A, b, c), l, h and τ ) such that, for each ξ ∈ M 1 and each v ∈ L ∞ loc (R + , R n ), the unique solution x : [−h, ∞) → R n of (2.3) satisfies (2) If A is Hurwitz and l G L 1 < 1, then there exist constants ≥ 1 (depending on (A, b, c), h and l) and γ > 0 (depending on (A, b, c) and l) such that, for each ξ ∈ M 1 and

Corollary 2.3
Assume that A is Hurwitz and there exist constants a ≥ 0 and l > 0 with l G L 1 < 1 and such that (2.5) Then there exist constants ≥ 1 (depending on (A, b, c), h and l), γ > 0 (depending on (A, b, c) and l) and θ ≥ 0 (depending on (A, b, c), a and l) such that, for each initial condition ξ ∈ M 1 and each v ∈ L ∞ loc (R + , R n ), the unique solution x : [−h, ∞) → R n of (2.3) satisfies The proofs of the above proposition and the corollary can be found in Appendix B.

A Class of Forced Positive Delay-Differential Systems
In the rest of the paper, we will be interested in non-negative systems of the form (1.1), that is, A is Metzler and b, c, f and ξ are non-negative. More specifically, we will make the following assumptions. (L1) A ∈ R n×n is Metzler and Hurwitz.
(L2) b, c ∈ R n + , b = 0 and c = 0. In the context of (L1), it is interesting to note that a Metzler matrix A is Hurwitz if, and only if, A is invertible and −A −1 ≥ 0. Furthermore, if A is Metzler and the spectral abscissa a of A is positive (implying in particular that A is not Hurwitz), then (1.1) (see also (3.2) below) has exponentially growing solutions. Indeed, as is well-known, there exists non-zero η ∈ R n + such that Aη = aη, and so, with ξ = (η, 0) and non-negative u, v, and f , it is immediate that the solution x of (3.2) satisfies We mention that if (L1) and (L2) hold, then G(t) = c T e At b ≥ 0 for all t ≥ 0 and thus An application of the feedback law w = f (u, y) to the linear system (2.1) leads to the following initial-value probleṁ where h ≥ 0 and f : denotes the space of Lebesgue measurable functions R + → U , should be considered as forcing terms (which, depending on the context, are interpreted as a control, input or disturbance). The nonlinearity is continuous and locally Lipschitz in its second argument uniformly with respect to its first argument, that is: for all z ∈ R + , there exists a relatively open set Z ⊂ R + and a constant λ > 0 such that z ∈ Z and For the following, it is convenient to define
(2) Let β > 0. Assume that (L1) and (L2) hold and σ G(0) < 1. Then there exists ρ > 0 (depending on β, (A, b, c), f and h) such that, for all v ∈ L ∞ + and all ξ = (ξ 0 , ξ 1 ) ∈ M 1 The hypotheses on f imply that N u satisfies the assumptions imposed on N in Sect. 2. Furthermore, for l > σ , there exists a ≥ 0 such that Consequently, the function t → N u (t, c T ξ 1 (t − h)) is integrable on [0, h] and statement (1) of Proposition 2.1 guarantees that the initial-value probleṁ has a unique solution x on [−h, ∞). It is sufficient to show that x(t) ≥ 0 for all t ≥ 0, because in this case x is also a solution of (3.2). By the variation-of-parameters formula, Consequently, using the Metzler property of A and the non-negativity of b, c, N u , ξ and v, an application of the above identity for k = 0 shows that x(t) ≥ 0 for all t ∈ [0, h]. This argument can now be repeated with k = 1 and (x(h), x h ) taking the role of ξ , to obtain that Continuing in this way, we obtain that x(t) ≥ 0 for all t ≥ 0.
Typical scenarios for f are given by: • f (w, z) = g(wz)z, where g : (0, ∞) → R + is continuous and such that lim z→0 g(z)z exists and is finite; In each of the above cases, U is a compact subset of R + .
We impose a further positivity assumption on the linear system underlying (3.2). (L3) For every i ∈ {1, . . . , n}, there exists τ i > 0 such that c T e Aτ i e i > 0.
For a good understanding of hypothesis (L3), it is useful to recall some basic facts about observability, see, for example, also [29,41]. The observed systeṁ is said to be observable if, for all x 0 = 0, the function t → y(t) = c T e At x 0 is not identically equal to 0. In the following, the above observed system will be denoted by (c T , A is the so-called observability matrix (see, for example, [29]). Furthermore, we recall that Hypothesis (L3) simply means that, for each i ∈ {1, . . . , n}, the observation y of (3.5) corresponding to the initial condition x(0) = e i does not vanish identically on R + , or, equivalently, e i / ∈ ker O(c T , A). Conversely, assuming that A is Metzler and c is non-negative, the condition that e i / ∈ ker O(c T , A) for all i ∈ {1, . . . , n}, implies that (L3) holds. As will be shown further below (see Corollary 4.5), under natural assumptions, (L3) is equivalent to c *persistency of (3.2). The following result provides a number of important characterizations of (L3).

Proposition 3.2 Assume that A is Metzler and
Under these conditions, assumption (L3) is equivalent to each of the following properties.
It is not difficult to find examples which show that (L3) does not enforce observability. Indeed, consider and note that A is Metzler and Hurwitz (the eigenvalues of A are −3, −2 and −1) and Since ker O(c T , A) = {ρ(0, 1, −1) T : ρ ∈ R}, we see that (c T , A) is not observable, but (L3) holds since ker O(c T , A) ∩ R 3 + = {0}. As for characterization (7), we point out that (L3) does not imply that A + dc T is irreducible for all non-zero d ∈ R n + . A counterexample is given by for which it is easily shown that (L3) holds and A + dc T is reducible.

Proof of Proposition 3.2
To prove the characterizations of condition (L3), we proceed in several steps, the roles of which are outlined in the diagram below:
To this end, set g(t) := c T e At for all t ≥ 0. Obviously, the i-th component g i of g is given by g i (t) = c T e At e i and g i (t) ≥ 0 for all t ≥ 0 and all i ∈ {1, . . . , n}.

g(t) = c T e At = c T e Aθ e A(k−1)θ e A(t−kθ) .
As c T e Aθ 0 and e A(k−1)θ e A(t−kθ) is a non-negative invertible matrix, we conclude that g(t) = c T e At 0 for all t > θ.  it is clear that (3) implies (4). Conversely, if (4) holds, then, by the above identity, for every i ∈ {1, . . . , n}, g i is not the zero function. Hence (L3) holds, and so, invoking Steps 1 and 2, we see that (3) is satisfied.
Step 7: (6) ⇔ (7). Assume that (7) holds, that is, there exists d ∈ R n + such that A + dc T is irreducible. Then, as A + dc T is Metzler, we have that e (A+dc T )t 0 for all t > 0, see, for example, [46,Theorem 8.2]. Consequently, c T e (A+dc T )t 0 for all t > 0. It now follows from Step 5 that (6) holds. Conversely, assume that (6) is satisfied. Seeking a contradiction, suppose that there does not exist a vector d ∈ R n + such that A + dc T is irreducible. Then, in particular, A + 1c T is reducible, where 1 = (1, 1, . . . , 1) T ∈ R n + . Hence there exist nonempty disjoint subsets I and J of {1, . . . , n} such that I ∪ J = {1, . . . , n} and where the a ij are the entries of A and the c j are the components of c. For a pair (i, j ) ∈ I × J we have that i = j , and so a ij ≥ 0. As c j ≥ 0, we conclude from (3.7) that a ij = 0 for all Hence, By the same argument, with o 1j replaced by o 2j , we see that o 3j = 0 for all j ∈ J . Continuing this line of reasoning, we conclude that, for each j ∈ J , the j - (7). Conversely, suppose that (7) holds, that as follows from (3.6) and arguments used in the above proof. If A is not only Metzler, but also Hurwitz, then the identity is valid for ν = 0, and thus, For the analysis of the behaviour of the solutions of (3.2), it is useful to consider the following linear system of homogeneous delay-differential equationṡ where q ≥ 0. It is well-known that (3.9), a special case of (1.1), induces a strongly continuous solution semigroup on M 1 which we shall denote by T q (t) t≥0 (see, for example, for all t ≥ 0. We record some consequences of the assumptions (L1)-(L3) in the proposition below.
and define a functional F ∈ L(M 1 , R) by The following statements hold.
We note that statement (6) says that the functional F is constant along orbits of the semigroup (T p (t)) t≥0 , or, equivalently, F is a first integral of the linear delay-differential

Proof of Proposition 3.3 (1)
As A is Hurwitz, (3.8) holds and the claim follows immediately from the non-negativity of b.
For the rest of this paper, it will always be assumed that (L1) and (L2) hold, in which case, by Lemma 3.3, the constant p defined in (3.10) satisfies 0 < p ≤ ∞ (and p < ∞ if (L3) holds).

Persistence Results
We consider persistence properties of the initial value problem (3.2). Assume that (L1) and (L2) hold and that f (w, z) ≥ 0 for all w ∈ U and z ≥ 0.
In the following, let : M 1 → R by a bounded linear functional and let D ⊂ M 1 + × L(R + , U) × L ∞ + . We say that the system (3.2) is uniformly -persistent with respect to D if there exist τ ≥ 0 and δ > 0 such that, for all (ξ, u, v) ∈ D, the solution x of (3.2) has the property that It is clear that if (3.2) is uniformly -persistent with respect to D (and thus (4.1) holds), then for all (ξ, u, v) ∈ D, the solution x of (3.2) satisfies In the following, we shall associate with a vector d ∈ R n + a corresponding bounded linear functional d * which is given by Of particular interest will be d * -persistence when d = c.
We now introduce two assumptions on the nonlinearity f .
Obviously, the interpretation of (N1) and (N2) depends on the particular context, but we refer the reader to [14,Sect. 3] for more discussion of these types of conditions for a class of ecological population models. Briefly, the quantity p acts as a stability threshold for the linear delay system (3.9). Indeed, if q < p, then solutions of (3.9) converge to zero exponentially by statement (2) of Proposition 2.2 and (3.1). If p < ∞ (that is, G(0) > 0) and q ≥ p, then the trivial solution of (3.9) is not exponentially stable. Indeed, in this case, there exists s * ≥ 0 (with s * = 0 if, and only if, q = p) such that qG(s * )e −s * h = 1 from which it follows via a routine calculation that A well-known result (see, for example, [7, Theorem 5.1.7]) now yields that the trivial solution of (3.9) is not exponentially stable. We conclude that the "smallest" parameter value for which an additive perturbation of the form qbc T x(t − h), where q ≥ 0, destabilizesẋ = Ax is given by q = p.
We remark that the case p = ∞ (or, equivalently G(0) = 0) is of little interest in the current context, because, in this case, under reasonable assumptions, persistency cannot be expected. For example, assuming that (L1) and (L2) hold and f (w, , the solution of (3.2) approaches 0 as time goes to infinity. For β > 0 and r = 1, ∞, we set The next result shows that c * -persistence implies e * i -persistence whenever −e T i A −1 b > 0.
Proof Assume that (3.2) is uniformly c * -persistent with respect to D. Together with statement (2) of Lemma 3.1 this implies that there exist δ 2 > δ 1 > 0 and τ ≥ 0 such that, for all (ξ, u, v) ∈ D, the solution x of (3.2) satisfies Setting follows from the variation-of-parameters formula that, for all t ≥ τ + h, and thus We note that the positivity of −e T i A −1 b does not imply that of e T i b: indeed, for the simple example The following lemma will be a key tool for the persistency analysis of (3.2).
Furthermore, the following statements hold.
For the rest of the proof, we set Setting y † := ρ c ∞ , it follows that Furthermore, by the properties of f , and consequently, and thus and let x be the corresponding solution of (3.2). It follows from (4.4) and the differential equation in (3.2) that there exists λ > 0 such that, for all Note that, on the interval [h, 2h], the above bound holds because the L ∞ -norm of ξ 1 is uniformly bounded for all (ξ, u, v) ∈ D ∞ (β). By (4.5), there exists ε ∈ (0, y # ) such that We set I 1 := [0, y # + ε] and I 2 := [y # − ε, y † ]. By (4.6), the family of all y generated by data (ξ, u, v) ∈ D ∞ (β) is equi-continuous on [h, ∞), and therefore, there exists τ > 0 such that, for every t ≥ h and every such y, and Let t ≥ h. We distinguish two cases.
, we obtain (4.10) Let us first assume that h > 0. Then, without loss of generality, we may further assume that 2τ ≤ h. Let z : [−h, ∞) denote the unique solution of the linear initial-value probleṁ where, in the last step, we have used that 2τ ≤ h. Now e As z(0) + p s 0 e A(s−θ) bc T z(θ − h)dθ = z(s) for all s ∈ [0, 2τ ] and thus, (4.11)) and Together with (4.11) this yieldš Consequently, an application of statement (6) of Proposition 3.3 yields, If h = 0, thenx(t) = x(t), T p (t) = e (A+pbc T )t and z(s) = e (A+pbc T )s x(t), and (4.11) follows easily from the variation-of-parameters formula and (4.10). We conclude that (4.12) continues to hold in the delay-free case. CASE 2: y(t) ∈ (y # , y † ]. By (4.9), y(t + s) ∈ I 2 for all s ∈ [0, 2τ ], and so Hence, If h > 0, then, without loss of generality, we may assume that τ ≤ h, and so Combining the last inequality with (4.13) and setting η := (y # − ε)τ > 0, we arrive at The positivity of κ 2 is a consequence of (L3) and Proposition 3.2. We see that (4.14) continues to hold in the delay-free case (now with η = κ 1 κ 2 ).
Combining (4.12) and (4.14) from Cases 1 and 2, respectively, we obtain Consequently, with t = t 0 + kτ , where t 0 ≥ h and k ∈ Z + , and s = τ , it follows that and thus, For t ≥ t 0 , let k ∈ Z + be such that t = t 0 + kτ + s, where 0 ≤ s < τ. The last inequality together with (4.15) leads to
An inspection of the above proof shows that (L3) was used only in the delay-free case: if h > 0, then Lemma 4.2 remains true without assuming (L3).
For β > α > 0 and r = 1, ∞, we set and note that the following inclusions hold: We are now in the position to state and prove our main results on persistence.  (2) If (L3) holds and f is bounded, then there exists δ > 0 such that, for all (ξ, u, v) ∈ E 1,0 (α, β), the solution x of (3.2) satisfies (4.16). ( The following result can be considered as a "converse" of Theorem 4.3.  Statements (1) and (2) of Corollary 4.5 provide a considerable improvement of the results in [3] where, for the undelayed case and under the assumption that A + bc T is irreducible, persistence-like properties were proved.
(2) The proof is identical to that of statement (1), we only need to invoke statement (2) (instead of statement (1)) of Lemma 4.2.
(3) Assume that c 0. Then, trivially, (L3) holds, and moreover, min 0≤s≤h c T e As b > 0 . (4.21) As the L ∞ -norm of ξ 1 is uniformly bounded for all (ξ, u, v) ∈ E ∞ (α, β), it follows from the properties of f that there exists q > 0 such that Since, by the variation-of-parameters formula, we conclude that Consequently, for all (ξ, u, v) ∈ E ∞ (α, β), the solution x of (3.2) satisfies with κ given by (4.18) and where the positivity of λ follows from (4.21). By the strict positivity of c, there exists 0) . Using (4.22), we may conclude that there exists λ 2 > 0 such that, for (ξ, u, v) ∈ E ∞ (α, β), the solution x of (3.2) satisfies, Therefore, by statement (5) of Proposition 3.3, Appealing to statement (1) of Lemma 4.2, we see that there exists η > 0 such that, for all showing that (3.2) is uniformly F -persistent with respect to E ∞ (α, β). The proof can now be completed by invoking a contradiction argument identical to that used in the proof of statement (1).

Stability Results
We consider stability properties of the initial value problem (3.2). Particularly, we formulate conditions under which (3.2) (in the absence of forcing) admits a unique, constant nonzero solution. We then provide conditions under which this solution is stable in a sense we describe.
In the following, let p be the constant given by (3.10) and let u s ∈ U , where U ⊂ R is compact. The number u s plays the role of a target or nominal value for the variable u. Further below, we will be interested in the steady states (equilibria) of (1.1) when u = u s and v = 0 and that is the motivation for the superscript "s".
The nonlinearity f appearing in (1.1) is assumed to satisfy the following conditions. The existence of y s > 0 such that f (u s , y s ) = py s follows from the continuity of f and (N2), whilst uniqueness of y s is a consequence of (5.1). Note that (5.1) is a sector condition and means that the graph F of y → f (u s , z) is "sandwiched" between the lines L + = {(z, pz) : z ≥ 0} and L − = {(z, −pz + 2py s ) : z ≥ 0}, see Fig. 1 for an illustration.
Obviously, the only points the graph F has in common with L + or L − are (y s , py s ) and, the origin (0, 0) if f (u s , 0) = 0. Condition (5.2) implies that the intersections of F with L + and L − at the point (y s , py s ) are non-tangential, whilst (N2) ensures that if f (u s , 0) = 0, then F is non-tangential to L + at (0, 0). Finally, it follows from the continuity of f , assumption (N2), (5.1) and (5.2) that, for every δ > 0, there exists q ∈ (0, p) such that In the following lemma we will consider the delay-differential equatioṅ where κ > 0 is such that A + κI ≥ 0. It follows immediately from the definitions of x s and p that c T x s = y s . Consequently, since y s > 0, we conclude that x s = 0, and so x s > 0. To show that x s is a steady state, we invoke (N3) to note that Ax s + bf (u s , c T x s ) = Ax s + bf (u s , y s ) = Ax s + bpy s = 0 .
As for uniqueness, let x † be an another non-zero vector in R n + satisfying , and thus, f (u s , c T x † ) = pc T x † . Now c T x † = 0 (otherwise, due to (5.6), 0 would be an eigenvalue of A which is not possible by (L1)), and so, since y s is the unique positive solution of the equation f (u s , y) = py, we see that c T x † = y s = c T x s , whence Let us now additionally assume that A + bc T is irreducible. As we see that Since A + pbc T is Metzler and irreducible, e (A+pbc T )t 0 for all t > 0 (see, for example, [46, Theorem 8.2]), and as x s > 0, it now follows from (5.7) that x s 0.
It is convenient to define, for given u s ∈ U and ρ > 0, the following function ϕ ρ : U → R + , We are now in the position to state a stability theorem relating to the non-zero steady state associated with the system (3.2).
Note that whilst the above criterion guarantees stability independent of the delay parameter h, the "quality" of the stability will in general depend on h because of the h-dependence of the constants M, μ and ρ > 0.
Proof By invoking (5.18), we obtain the following estimate for the function d defined in (5.11) which replaces (5.12). Otherwise the proof is very similar to that of Theorem 5.2 and we omit the details. Statement (2) considers data triple (ξ, u, v) ∈ E 1,0 (α, β) and so allows for unbounded ξ 1 . Note that there is no counterpart to statement (2) in Theorem 5.2: the reason is that in general there does not exist a finite ρ such that (5.9) holds if ξ 1 is not essentially bounded (see proof of Theorem 5.2).
Nonlinearities which satisfy (N3) and (5.18) are quite common in mathematical ecology as the following example shows.  Table 5.1] that if au − > p, then (N3) holds for f 1 and f 2 with y s = (au s − p)/(pku s ) and y s = (au s − p)/(pk), respectively.
(2) This example focuses on the nonlinearities f 1 and f 2 induced by the Ricker-type function g(z) = aze −z via f 1 (w, z) = g(wz) and f 2 (w, z) = wg(z) for w ∈ [u − , u + ] and z ≥ 0, where u + > u − > 0. Under the assumption au + e 2 < p < au − , (5.20) it is well-known that, for every u s ∈ [u − , u + ], (N3) is satisfied for f 1 and f 2 with y s = ln(au s /p)/u s and y s = ln(au s /p), respectively, see [13,  If the nonlinearity f does not satisfy the sector condition (N3), then it may still satisfy some sector condition and we will now explore this in some more detail. For which purpose, assume, for simplicity, that f (w, z) = f (z) does not depend on w. For q > 0, we denote by S q the set of all locally Lipschitz functions f : R + → R + for which there exist affine-linear functions l + , l − : R → R with slopes q and −q, respectively, and y > 0 (all depending on f ) such that l + (z) < f (z) < l − (z) ∀ z ∈ [0, y ) and l − (z) < f (z) < l + (z) ∀ z > y , (5.21) and, furthermore, Note that l + (y ) = l − (y ) = f (y ), l + (0) = f (y ) − qy , l − (0) = f (y ) + qy , and, by (5.21), Obviously, this looks similar to (5.1), but here z = 0 is included, and, in general, f (y ) = qy . If f ∈ S q , then we say that f satisfies a sector condition with abscissa y and slope q. The set S q constitutes a rich class of functions. For example: • any bounded differentiable function f : R + → R + such that lim sup z→∞ |f (z)| < q is an element in S q ; • any locally Lipschitz function f : R + → R + such that f (z) converges to a finite limit as z → ∞ and for which there existq ∈ (0, q) and a sequence of intervals [y k , is in S q .

Example 5.6
Consider the functions f 1 , f 2 : R + → R + given by which are plotted in Fig. 2. Since the function f 1 does not even satisfy (N2), for any p > 0, and so (N3) cannot hold. The function f 2 does not satisfy (N3) when p = 1, see Fig. 2(b). However, the functions f 1 and f 2 are clearly bounded and differentiable, with and so belong to S q for all q > 0. Figure 2 illustrates graphically that f 1 , f 2 ∈ S 1 . ♦ The next result shows that if f ∈ S p , then there exists a constant forcing function v such that the resulting forced system has nice stability and convergence properties  on (A, b, c), f , h and y ) and γ > 0 (depending on (A, b, c), f and y ) such that, for all (ξ, v) ∈ M 1 + × L ∞ + , the unique solution x : [−h, ∞) → R n of (2.3) satisfies We emphasize here that x is not a steady state of the unforced In the context of scalar (n = 1) instances of (3.2), in the absence of the forcing term u, it has been shown in the chaos control literature (see, for example, [24]) that constant additive control may enforce convergence in systems which show chaotic behaviour when unforced. Proposition 5.7 identifies a general scenario in which the application of such control action results in dynamics which are stable in the sense of (5.23).

Proof of Proposition 5.7
As f ∈ S p , the function N : R → R defined by Let (ξ, v) ∈ M 1 + × L ∞ + and let x be the unique solution x of (3.2). Setting e(t) := x(t) − x , it follows thaṫ y for all t ≥ 0, the above equation can re-written asė The claim now follows from (5.24), the fact that p = 1/G(0) = 1/ G L 1 and statements (2) and (3) of Proposition 2.2.
We conclude the section by mentioning that Proposition 5.7 may be extended to the case wherein f = f (u, z) does depend on two arguments, provided that f (u , ·) ∈ S p for some u ∈ U . For the sake of brevity, we do not give a formal statement.

Examples
The results presented in the previous sections allow the analysis of mathematical models arising in a great variety of contexts. To demonstrate this, we consider three different models in this section. The first two are related to population dynamics and the last one to selfregulated biochemical reactions.

Delayed Recruitment Models
Recruitment models typically assume that the dynamics of sexually mature individuals of a population are driven by the difference between the rate at which new members are recruited and the mortality rate. If the maturation process takes a constant time h > 0, competition occurs only in a specific age cohort and the mortality rate is constant, then the following population model is obtaineḋ Here x(t) denotes the number of mature individuals at time t , μ > 0 is the mortality rate, the production function f depends on the competition between individuals, u(t) ∈ U := [u − , u + ] ⊆ (0, ∞) and v(t) ≥ 0 are forcing terms which model the effect of environmental fluctuations affecting recruitment. A derivation of (6.1) without forcing may be found in, for example, [4] and we refer the reader to [20,Sect. 1] for an interesting discussion on delays in population models.
where k > 0, see part (1) of Example 5.5. This corresponds to a so-called contest competition setting, in which resources available are monopolized by some individuals. In this situation the production function f tends to the maximum a/k as z → ∞, uniformly in w. Model (6.1) with f given by (6.2) is a special case of (1.1) with n = 1, A = −μ, b = 1 and c = 1. It is straightforward to verify that assumptions (L1), (L2) and (L3) are satisfied and, trivially, p = 1/G(0) = μ. We have shown in Example 5.5 that, for every u e ∈ [u − , u + ], condition (N3) holds, provided that u − > μ. Thus, we can use Theorem 4.3, Corollary 5.3 and Theorem 5.4 to obtain persistence, convergence and stability results, respectively, for the forced equation (6.1) with f given by (6.2). For instance, Theorem 5.4 implies that the deviation of x(t) from x s is bounded in the uniform manner (5.19), whereas Corollary 5.3 shows that the equation satisfies a converging-input converging-state propertynamely, that if lim t→∞ u − u s L ∞ (t,∞) = 0 and lim t→∞ v L ∞ (t,∞) = 0, then x(t) → x s as t → ∞. ♦ The above example shows that, under contest competition, the mature population tends to a constant value, provided that the forcing functions u(t) and v(t) converge as t → ∞. However, it is known that this is not the case if resources are equally allocated among individuals, that is, under so-called scramble competition (characterized by a unimodal production function which tends to zero at high population sizes, see [4,5]). In this case, even in the absence of fluctuating external forcing, the solutions of model (6.1) might show persistent fluctuations which, in many practical situations, are undesirable. Interestingly, a constant control, which adds a constant amount of mature individuals to the population per unit time, can have a stabilizing effect. This was shown in the context of (6.1) with a specific f (Mackey-Glass equation) in [24]. In the example below, we obtain the stabilizing effect of constant control action as a consequence of Proposition 5.7. Example 6.2 Consider model (6.1) with μ = 1, so that p = 1, and where the production function f , assumed to be independent of its first variable, is given by for fixed parameters a, k > 0 and s > 0. Whilst f trivially satisfies (N2) whenever a > k, Fig. 3(a) illustrates that condition (N3) does not hold for f with k = 2, s = 3 and a = 7 , (6.4) and, consequently, Corollary 5.3 does not apply for these values. Nevertheless, for any fixed a > 0 and k > 0, the bounded and differentiable function f belongs to S q for any q > 0, since |f (z)| → 0 as z → ∞.
In particular, f ∈ S 1 and y = 2.65 is a sector abscissa, see  In this simulation, only one initial condition is used from (6.5), namely ξ 1 . As ensured by Proposition 5.7, bounded oscillations around x are observed which, as expected, increase with increasing k. Figure 4(c) shows simulations of (6.1) with constant forcing term v(t) ≡ ψ . As ensured by Proposition 5.7, we observe that the addition of sufficiently large constant forcing has the effect of making solutions converge to a positive limit. ♦

Dispersal of a Population with a Unique Breeding Region
Consider the following model of a population spatially structured over n discrete patcheṡ Here The model (6.7) is a so-called Nicholson system, as the case n = 1 reduces to the wellknown Nicholson's blowfly equation [18]. A related Nicholson system is studied in [11]. However, the results in [11] focus on the case wherein the birth of new individuals occurs in every patch. Here, we consider a different situation commonly seen in nature, in which there is a single breeding patch. System (6.7) is a special case of (1.1) with The matrix A is Metzler by construction and, since, without newborns, the population should become extinct, it is natural to assume that A is Hurwitz. Under this assumption, (L1) and (L2) hold. Verifying condition (L3) in general without further assumptions on A and c seems analytically cumbersome. Proposition 3.2 provides characterisations of (L3) which may be used in specific situations. The Ricker nonlinearity f satisfies (N3) provided (5.20) holds, see Example 5.5.
We note that A is Metzler and Hurwitz and since c = (c 1 , c 2 , c 3 ) T > 0 by assumption. From the block-structure of A it is immediate that property (7) .7) is uniformly e * i -persistent with respect to E r,0 (α, β) (r = 1, ∞ and β > α > 0) as follows from Proposition 4.1.
We now explore the case wherein c 3 = 0, that is, (L3) does not hold. Then, by Corollary 4.5, the population is not uniformly c * -persistent with respect to E ∞,0 (α, β), E 1,0 (α, β) or E ∞ (α, β) for any β > α > 0. Indeed, by inspection of the model structure, for all initial conditions ξ with ξ 0 i = 0 and ξ 1 i = 0 for i ∈ {1, 2}, all functions u : R + → [u − , u + ] and for v = 0, the first two components of the solution x are equal to 0, showing that the system fails to be c * -persistent. Interestingly, if we consider the dynamics of just the first two states, which decouple from the third, then we obtain the subsystem 9) which is another special case of (6.7) and (6.8) with Since c 1 + c 2 > 0 as c > 0, it is straightforward to show that (L3) always holds for the subsystem (6.9). It is routine to calculate that Consequently, if (5.20) holds, then subsystem (6.9) is ultimately c * s -persistent and Theorems 5.2 and 5.4 are applicable to (6.9). Finally, Proposition 4.1 guarantees that (6.9) is also ultimately e * 1 -and e * 2 -persistent. ♦

Self-Regulated Biochemical Reactions
Consider a chain of chemical reactions which converts a (first) substance with concentration s 1 into an end-product with concentration s n via several intermediate substances with concentrations s 2 , . . . , s n−1 . Let τ i ≥ 0 be the time needed for a substance s i to affect the production of the next substance in the chain s i+1 . The existence of delays is common in biochemical reactions of gene expression and are related to transcriptional and translational processes [31]. We assume that the reaction is self-regulated in the sense that an increase of concentration of substance n in the chain induces or represses the production of the first substance. More specifically, we consider the following model where d i > 0, i = 1, . . . , n is the decay rate of substance i and a i−1 > 0, i = 2, . . . , n, is the production rate of substance i from substance i − 1. The nonlinearity f : U × R + → R + is the self-regulation function for the reaction, where U = [u − , u + ] ⊂ (0, ∞) and the functions u and ω i model external disturbances. Without delays and external disturbances, model (6.10) was proposed in [16] and further studied in [45]. The authors of [3] also studied a version of (6.10) with additive forcing, but without delays and for f not depending on u. With delays, model (6.10) was first considered in [27], although without considering forcing terms. System (6.10) is not in the form of (1.1) but, by setting h := n j =1 τ j , it is routine to check thaṫ which is a special case of system (1.1), with We see immediately that assumptions (L1) and (L2) are satisfied. Assumption (L3) also holds, as follows from Proposition 3.2 by noting that A + bc T is irreducible. Hence, Proposition 3.3 implies that G(0) > 0. Owing to the particular structure of A, b and c, the latter can also be seen directly by noting that Depending on whether an increase of the concentration x n of substance n inhibits or activates the production of the first substance, the function f is assumed to be decreasing or increasing in its second variable, respectively, see [1]. We illustrate the stability theory developed in Sect. 5 in the case of an increasing f , namely, Before we do this, we state a simple lemma which provides a sufficient condition for (N3) to hold. The proof is straightforward and is left to the reader. Writing f given by (6.12) as f (w, z) = wg(z) with and thus, by Lemma 6.4, f satisfies (N3) if and so, by Lemma 6.4, (N3) holds. Consequently, the stability theory developed in Sect. 5 is applicable to this example. If we reduce d 1 to d 1 = 1, but keep the other parameter values as in (6.13), then p = 1, and statement (2) of Lemma 6.4 does not apply. In fact, condition (N3) does not hold in this case as Fig. 2(b) shows. Furthermore, in this case, the system with u(t) ≡ u s and v = 0 seems to be bi-stable, see the simulations in Fig. 5(a) with initial conditions given by All numerical solutions in this example were computed using the dde23 command in MAT-LAB.
Nevertheless, the function z → f (u s , z) was considered in Example 5.6, and shown to belong to S 1 . Furthermore, y = 2.453 is a sector abscissa of f (u s , ·) with ψ = y − f (u s , y ) ≈ 0.5 > 0, see Fig. 2 is "stable" in the sense of (5.23). To illustrate this result numerically, Fig. 5(b) plots a numerical simulation of (6.11) with parameter values given by (6.13) and d 1 = 1, u(t) ≡ u s = 0.25, and with oscillatory forcing term v given by The initial conditions are given by (6.14a), (6.14b). Bounded oscillations around x are observed.
Moreover, Proposition 5.7 ensures that, for every (ξ, v) ∈ M 1 + × L ∞ + such that lim t→∞ v − ψb L ∞ (t,∞) = 0, the unique solution x of system (6.11) satisfies x(t) → x as t → ∞. Thus, Fig. 5(c) plots a numerical simulation of the same model, only now with a convergent forcing term v given by v(t) = bψ(1 − 0.3e −0.5t cos(0.5t)) ∀ t ∈ R + . (6.16) Convergence x(t) → x as t → ∞ is observed. Hence, the inclusion of such a convergent additive control removes the bi-stability and forces the solutions to converge to a limit x independently of the initial condition in M 1 + and delay h ≥ 0. ♦

Appendix A: List of Function Spaces Used
For ease of reference, we provide a list of the function spaces introduced in the paper. (2) If x is a solution of (2.3) on [−h, τ ) for some τ > h, then bw =ẋ − Ax − v ∈ L 1 ([0, h], R n ), and, as b = 0, it follows that w ∈ L 1 ([0, h], R), establishing the claim.
We note that in the above proof the Lipschitz property has not been used in the argument establishing uniqueness. In fact, an inspection of the proof shows that Proposition 2.1 holds if N is measurable in its first argument and continuous in its second and, for every z ∈ R, there exist a locally integrable function ν : R + → R + and an open interval J ⊂ R containing z such that |N(t, z)| ≤ ν(t)|z| for all z ∈ J and t ≥ 0. These conditions are weaker than the assumptions imposed on N in Sect. 2. However, the latter are required in the delay-free case.
The claim can now follows by a routine argument which is based on the repeated application of the above estimate, starting with k = 0 and stopping with the smallest k such that kh ≥ τ .
(2) By the Hurwitz property of A and the assumption that l G L 1 < 1, there exists μ > 0 such that A + μI is Hurwitz and The claim now follows with γ = μ from a straightforward argument based on (B.5)-(B.7) and the fact that ṽ L 1 (0,t) ≤ μ −1 e μt v L ∞ (0,t) .
Let ξ ∈ M 1 and v ∈ L ∞ loc (R + , R n ) and let x be the corresponding solution of (2.3). Setting d(t) := N(t, c T x(t − h)) −Ñ(t, c T x(t − h)) for all t ≥ 0, then we have that d L ∞ ≤ θ anḋ An application of statement (2) of Proposition 2.2 to the above system yields the claim.
The following simple lemma was used in the proof of Theorem 4.3. it follows that and the claim follows with γ := (νε)/2.