ASYMPTOTIC BEHAVIOUR FOR A CLASS OF DELAYED COOPERATIVE MODELS WITH PATCH STRUCTURE

. For a class of cooperative population models with patch structure and multiple discrete delays, we give conditions for the absolute global as- ymptotic stability of both the trivial solution and – when it exists – a positive equilibrium. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. As a by-product, we obtain a criterion for the existence of positive traveling wave solutions for an associated reaction-diﬀusion model with patch structure. Our results improve and generalize criteria in the recent literature.


1.
Introduction. Many systems of one or multiple biological species are composed of a network of connected different patches, with migration of the populations among them. Due to several features of an heterogeneous environment, the growth of the populations often depends on the resources of each particular patch, and on the dispersal and interactions of the populations distributed over the different patches of the entire system. Patch-structured models are also used to capture the dynamics of species which go through several different life stages according to age or size, or in disease models with transitions between stages of normal and infected cells. Frequently, the past history of the species is important for their dynamics, to account for the time of the spatial dispersion of the populations from one patch to another, or to represent maturation periods, hunting delays, etc., and therefore these models should incorporate time-delays. For these reasons, in recent years population dynamics models with patch structure and delays have attracted the attention of an increasing number of mathematicians and biologists.
In this paper, we study some aspects of the long time behaviour of solutions for a class of n-dimensional cooperative delay differential equations (DDEs) of the form with a i ∈ R, b i > 0, c i ≥ 0, d ij ≥ 0 and discrete delays σ i ≥ 0, τ ij ≥ 0, i, j = 1, . . . , n. Actually, we can consider more general models given by ij ), i = 1, . . . , n, ij ≥ 0, for i, j = 1, . . . , n, p = 1, . . . , m. For simplicity, we shall consider (1.1) instead of (1.2), but all our results apply to (1.2)  The case c i = 0 for all i, constitutes an important subclass of (1.1): since it satisfies the sublinearity assumption (see Section 2 for the definition), sharper results are obtained for (1.3). An interesting application was studied by Takeuchi et al [8,9], who considered the following delayed population model with patch structure: (1.4) there are no diagonal delays. System (1.4) models the growth of a single species population distributed over n different foodrich patches, where x i (t) denotes the density of the species in patch i, τ ij is the time the species takes to move from patch j to patch i, γ ij is the death rate during the dispersion, and a i , b i are the intrinsic rate and the regulation capacity in each patch i, respectively. The natural growth in each patch is of logistic type, however without requiring a i > 0. Another significant model, which has the form (1.2) with c i = m p=1 c (p) i > 0, was studied by Liu [5] and will be later investigated in this paper.
Our purpose is to study two aspects of the global dynamics of (1.1): the global stability of equilibria, and the existence of a positive heteroclinic solution, connecting the trivial equilibrium to a positive equilibrium, when it exists.
If the matrix D = [d ij ] is irreducible, the theory of cooperative systems in [6] and the results by Zhao and Jing [11] can be used to obtain a sharp criterion of exchange of global attractivity between the zero solution and a positive equilibrium for the sublinear system (1.3), as the spectral bound of a certain matrix, named here as the (linear) community matrix, changes sign (cf. [9]). As regards the stability analysis, the novelty in this paper is not only to consider the more general system (1.1) (or (1.2)), but also to address the case of a reducible community matrix which is generally not treated in the literature. For b i > c i for all i, we further show that the positive equilibrium of (1.1) is always a global attractor of all positive solutions whenever it exists. For stability results for non-cooperative patch-structured models, see [1,5], also for further references.
If D is irreducible, we also give conditions for the linearized equation at zero to have a positive dominant eigenvalue, with a positive associated eigenfunction. We can therefore use some recent results in [3] on the existence of a positive heteroclinic solution for (1.1), connecting the trivial equilibrium to a positive equilibrium. In fact, in the vicinity of −∞ this heteroclinic solution was constructed in [3] as a perturbation of a positive eigenfunction. Adding a spatial variable to system (1.1) and diffusion terms, we obtain an associated reaction-diffusion model, for which the existence of positive traveling wave solutions for large wave speeds also follows from [3] . Such traveling fronts were obtained in [3] via a contraction principle argument as perturbations of the heteroclinic solution for the system without diffusion.
The paper is organized as follows. In Section 2, we set some notation and recall preliminary results for sublinear systems from the literature. The absolute (i.e., independent of the size of the delays) local and global asymptotic stabilities of the equilibria of (1.1) are studied in Section 3. Section 4 is devoted to the existence of a positive heteroclinic solution for (1.1), as well as to positive traveling fronts for a diffusive version of (1.1). Several examples of application are given at the end of Sections 3 and 4; these include patch-structured models studied in [5,9].
2. Notation and preliminary results. We set some standard notation. Let τ > 0 and define C = C([−τ, 0]; R n ) as the space of continuous functions from [−τ, 0] to R n , equipped with the supremum norm for some fixed norm in R n , ϕ = max θ∈[−τ,0] |ϕ(θ)| for ϕ ∈ C. For an abstract autonomous DDE in C for which uniqueness of solutions is assumed, x(t; ϕ) designates the solution of (2.1) with initial condition x 0 = ϕ (ϕ ∈ C). As usual, x t is the function in C given by If v ∈ R n is a positive (respectively non-negative) vector, i.e., all its components are positive (respectively non-negative) and f (v) = 0, then v is a positive (respectively nonnegative) equilibrium of (2.1).
For (1.1), we take the phase space C = C([−τ, 0]; R n ) where τ is the maximum of the delays, τ = max{σ i , τ ij : 1 ≤ i, j ≤ n}, and assume τ > 0. However, the case τ = 0 is included in our analysis. Due to its biological interpretation, only non-negative solutions are meaningful. For (1.1), we shall restrict the set of admissible initial conditions to either the positive cone C + = {ϕ ∈ C : ϕ i (θ) ≥ 0 for all θ ∈ [−τ, 0], i = 1, . . . , n}, or the subset of C + of functions which are strictly positive at zero, C + 0 = {ϕ ∈ C + : ϕ i (0) > 0, i = 1, . . . , n}. A non-negative equilibrium v of system (1.1) is said to be stable (in a set S ⊂ C + ) if for any ε > 0 there is δ = δ(ε) > 0 such that x t (ϕ) − v < ε for all ϕ ∈ S with ϕ − v < δ and t ≥ 0; and v is said to be globally attractive (relative to S) if x(t) → v as t → ∞, for all admissible solutions x(t) of (1.1) (i.e., solutions with initial conditions x 0 = ϕ ∈ S); we say that v is globally asymptotically stable if it is stable and globally attractive.
. . , f n ) and This implies that f satisfies the quasi-monotonocity condition on p. 78 of [6]. Thus, solutions x(t; ϕ) with initial conditions ϕ ∈ C + remain nonnegative for t ≥ 0 whenever they are defined; moreover Typically, in population dynamics the stability of equilibria is closely related to the algebraic properties of some kind of competition matrix of the community. Denote A = diag (a 1 , . . . , a n ), D = [d ij ]. For convenience, here we shall refer to M = A + D as the (linear) community matrix: If D is irreducible, then M is also irreducible; in this case, (1.1) is called an irreducible system [6] p. 88, and the semiflow ϕ → x t (ϕ) is eventually strongly monotone.
Let σ(M ) be the spectrum of M . As usual, we define the spectral bound of M and the spectral radius of M given respectively by s(M ) = max{Re λ : λ ∈ σ(M )} and r(M ) = max{|λ| : λ ∈ σ(M )}.
For (1.3), it is known that all positive solutions are defined for t ≥ 0 and are uniformly bounded (see e.g. [10,8]). Wang et al. [10] and Takeuchi et al. [9] studied the stability of sublinear systems in the form (1.3), as well as of their periodic versions, however with more restrictions on the sign of the coefficients in (1.3). Regarding the stability of equilibria for autonomous systems, the following result is deduced from [10].

5)
then there is a unique positive equilibrium of (1.3), which is globally asymptotically stable (with respect to C + 0 ). Since f (ϕ) defined in (2.4) is cooperative and sublinear, and f 0 (x) is strictly sublinear with f 0 (0) = 0, if M = Df 0 (0) is irreducible we can apply the theory in [11] to (1.3). From Theorem 2.1 in [9] we easily obtain the following sharp criterion:   Proof. Assume first that M is irreducible. Since M has nonnegative off-diagonal entries, then µ 0 ∈ σ(M ) and there is a positive vector Consider now the case of a reducible matrix M . After a simultaneous permutation of rows and columns, M can be reduced to the form M = M 11 0 M 21 M 22 with M kk n k × n k square matrices, k = 1, 2, and M 11 either zero or irreducible. Clearly, after reordering the conditions in (2.5) according to the above mentioned permutation, from β i > 0 for i = 1, . . . , n 1 we obtain that M 11 is irreducible and For (1.1), the linearized equation at zero reads as (3.1) To establish the local asymptotic stability about the equilibrium 0, we apply a result in [2] which uses the properties of M-matrices.
Recall that a square matrix A = [a ij ] with non-positive off-diagonal entries is said to be an M-matrix if all the eigenvalues of A have a non-negative real part, or, equivalently, if all its principal minors are non-negative; and A is said to be a nonsingular M-matrix if all the eigenvalues of A have positive real part, or, equivalently, if all its principal minors are positive (cf. [4]).

2)
is absolutely (i.e., for all choices of delay functions) asymptotically stable if and only if det N = 0 andN is an M-matrix. For solutions x(t) of the sublinear DDE (1.3), we obtain the inequalities  Proof. For f 0 (x) = f (x), x ∈ R n , where f is as in (2.2) and 1 ≤ i ≤ n, observe that easily conclude that all admissible solutions are bounded [6] It suffices to prove that L i := lim sup t→∞ y i (t) = 0 for 1 ≤ i ≤ n. Let L i = max 1≤j≤n L j , and suppose that L i > 0. From (3.3), we can choose ε > 0 such that Let T > 0 be such that y j (t) ≤ L i + ε for all t ≥ T − τ and 1 ≤ j ≤ n, and separate the cases of y i (t) eventually monotone and not eventually monotone. If y i (t) is eventually monotone, then y i (t) → L i as t → ∞, and for t ≥ T we obtain as t → ∞. Since γ i < 0, this implies that lim t→∞ y i (t) = −∞, which is not possible. If y i (t) is not eventually monotone, there is a sequence t n → ∞ such that y(t n ) → L i , y i (t n ) = 0. For t n ≥ T , we obtain (3.4) with t replaced by t n , again a contradiction. This proves that L i = 0.
When the trivial solution attracts all non-negative solutions, the population is driven to extinction in all the patches. An interesting open question is whether s(M ) = 0 still implies the global attractivity of 0 if M is reducible. The situation when a positive equilibrium x * = (x * 1 , . . . , x * n ) (x * i > 0) of (1.3) exists is however biologically more significant. Next, we show that, it b i > c i for all i, then the positive equilibrium is globally asymptotically stable whenever it exists, and give conditions for its existence. . , x * n ) of (1.1), and that b i > c i for i = 1, . . . , n. Then x * is hyperbolic and locally asymptotically stable.
Proof. For x * an equilibrium of (1.1), we have After the change y(t) = x(t) − x * , equation (1.1) becomes with linearization at zero given by With the notation in Theorem 3.2, the matrices N,N are given by . . , n. In particular, x * > 0 and N x * > 0, which implies that N is a non-singular M-matrix (cf. Theorem 5.1] of [4]). Now applying Theorem 3.2, we conclude that all characteristic roots of (3.7) have negative real parts, hence x * is locally asymptotically stable as a solution of (1.3). Lemma 3.3. Suppose there is a positive equilibrium x * = (x * 1 , . . . , x * n ) of (1.1), and that b i > c i for i = 1, . . . , n. Then all solutions x(t; ϕ) of (1.1) with ϕ ∈ C + 0 satisfy lim inf t→∞ x i (t; ϕ) ≥ x * i for 1 ≤ i ≤ n.
Next, we prove that i ≥ 1 for all i = 1, . . . , n. Choose i such that i = min 1≤j≤n j , and suppose that i < 1. Let T > 0 and ε > 0 be chosen so that x j (t) ≥ i − ε for all t ≥ T − τ and 1 ≤ j ≤ n, and If x i (t) is eventually monotone, then x i (t) → i , and for t ≥ T we have leading to x i (t) → ∞ as t → ∞, which is a contradiction. If x i (t) is not eventually monotone, there is a sequence t n → ∞ with x i (t n ) → i and x i (t n ) = 0. For t n ≥ T , we obtain the inequality above for t n instead of t, which yields 0 = x i (t n ) ≥ m i , again a contradiction. This proves that i ≥ 1.
We are ready to state the main results of this section: If there is a positive equilibrium x * of (1.1), then x * is hyperbolic and globally asymptotically stable (with respect to C + 0 ).
For the sake of contradiction, suppose that L i = max j L j > 1. Choose ε > 0 and t > τ , such that x j (t) ≤ L i + ε for all t ≥ T − τ and 1 ≤ j ≤ n, and Separating the cases of x i (t) eventually monotone and not eventually monotone, and reasoning as in the proofs of Theorem 3.3 and Lemma 3.3, we obtain a contradiction. Details are omitted.
We now give sufficient conditions for the existence of a positive equilibrium of (1.1).
Theorem 3.5. Consider (1.1) and suppose that Then, there is a positive equilibrium of (1.1), which is globally asymptotically stable (in C + 0 ). Proof. Arguing as in the proof of Lemma 3.3 with β i = a i + j d ij instead of β * i , if β i > 0 for all i = 1, . . . , n, we easily conclude that lim inf t→∞ x(t; ϕ) > 0 for any ϕ ∈ C + 0 . Next, denote 1 = (1, . . . , 1) ∈ C + as before. Let l > 0 be large enough so that f i (l1) = l a i + j d ij − (b i − c i )l ≤ 0, for f i as in (2.2). Since (1.1) is cooperative, then the components of the solution x(t) = x(t; l1) are non-increasing (see Corollary 5.2.2 of [6]), hence there is lim t→∞ x i (t) := x * i > 0. Clearly, both x i and x i are bounded and uniformly continuous on [0, ∞). From Barbalat's Lemma, lim t→∞ x i (t) = 0, and this yields that x * = (x * 1 , . . . , x * n ) is an equilibrium of (1.1). The global asymptotic stability of x * follows now from the previous theorem.
For the sublinear case (1.3), from Theorems 3.4 and 3.5, we immediately conclude the result below, which generalizes Theorem 2.1: If there is a positive equilibrium x * of (1.3), then x * is hyperbolic and globally asymptotically stable (with respect to C + 0 ). Moreover, such a positive equilibrium always exists if (2.5) holds.
Observe however that condition µ 0 := s(M ) > 0 is strictly weaker than (2.5), and does not imply the existence of a positive equilibrium. For instance, for M = −1 0 1 1 clearly µ 0 > 0, but the system  [8,9], the matrix M reads as (3.10) If s(M ) < 0, from Theorem 3.3 we conclude that 0 is a global attractor for (1.4) in C + . (Note that s(M ) < 0 implies that a i − d ji < 0 for 1 ≤ i ≤ n.) If a i + j [ε ij d ij − d ji ] > 0 for 1 ≤ i ≤ n, from Theorem 3.5 there is a positive equilibrium x * of (1.4) which attracts all positive solutions. We recall that only the special case of M irreducible was studied in [9].
Example 3.2. Consider the system with a i > 0, b i > 0, d ij ≥ 0 and delays τ ij ≥ 0. For β i as in (2.5), we have β i = a i , and thus system (3.11) has a positive equilibrium which is globally asymptotically stable.
Example 3.3. Consider the patch-structured DDE of logistic type estudied by Liu [5], a ij x j (t), i = 1, . . . , n, (3.12) for a i0 ∈ R, b i0 > 0, b ik ≤ 0, a ij ≥ 0 for i = j, and σ ik > 0, i, j = 1, . . . , n, k = 1, . . . , m. System (3.12) has the form (1.2) with a i = a i0 + a ii , so we adapt conditions (3.9) and Theorem 3.5 (stated for (1.1)) to this situation by replacing c i with − m k=1 b ik , and derive the following criterion: Remark 3.1. Theorem 3.6 strongly improves Theorem 3.2 of Liu [5], who considered (3.12) with severe additional restrictions: in [5], it is assumed that conditions (3.13) are fulfilled, that the matrix [a ij ] (and hence also M ) is irreducible and that α1 b1 = α2 b2 = · · · = αn bn =: K. Under these conditions, Liu proved that the equilibrium x * = (K, K, . . . , K) is globally attractive. 4. Positive heteroclinic solutions. In this section, we always assume that M is irreducible. Under conditions that guarantee the existence of a globally attractive equilibrium x * > 0, the goal is to prove that there is a positive heteroclinic solution for (1.1) connecting the two equilibria 0 and x * . We first need to show that if µ 0 > 0 then there is a positive eigenfunction for the linearization of (1.1) at zero. Lemma 4.1. Assume that M is an irreducible matrix, with s(M ) > 0. Then there is a dominant characteristic eigenvalue of (3.1) (i.e., λ 0 is a characteristic eigenvalue, and Re λ < 0 for all the other characteristic eigenvalues λ = λ 0 of (3.1)) and a positive eigenvector associated with λ 0 .
Next, we claim that λ 0 is the positive dominant root of (4.1). The proof follows along the lines of the proof of Lemma 4.3 in [1]. Let λ = a + ib be a solution of (4.1), with λ = λ 0 and a > 0. Since λ ∈ σ(M (λ)), we get a ≤ µ λ .
We now state the existence of positive heteroclinic solutions.
Proof. From the results in Section 3, the positive equilibrium x * is hyperbolic, locally asymptotically stable, and a global attractor relative to the set of solutions with initial conditions in C + 0 . The result is now a consequence of Lemma 4.1 and Theorem 2.1 of [3] (cf. Remark 4.1 below).
The work in [3] also motivates the consideration of a diffusive version of system (1.1), given by the reaction-diffusion equation  for t ∈ R, x ∈ Ω ⊂ R p , with diffusion coefficients δ i > 0 and all other coefficients as in (1.1).
Theorem 4.2. Assume that M is an irreducible matrix with s(M ) > 0, and that det ∆ (λ 0 ) = 0, for λ 0 the dominant eigenvalue of (3.1). If c i > 0 for some i, assume in addition that (3.9) is satisfied. Then, for c > 0 sufficiently large, system (4.5) has a positive traveling wave solution of the form u(x, t) = ψ(ct + w · x) for each unit vector w ∈ R p , with ψ(−∞) = 0 and ψ(∞) = x * , where x * is the positive equilibrium of (1.1).