Permanence and exponential stability

Permanence and exponential stability

for generalised nonautonomous Nicholson systems

Teresa Faria

Departamento de Matemática and CMAFCIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

Received 10 November 2020, appeared 7 February 2021 Communicated by Leonid Berezansky

Abstract. The paper is concerned with nonautonomous generalised Nicholson sys- tems under conditions which imply their permanence: by refining the assumptions for permanence, explicit lower and upper uniform bounds for all positive solutions are provided, as well as criteria for the global exponential stability of these systems. In particular, for periodic systems, conditions for the existence of a globally exponentially attractive positive periodic solution are derived.

Keywords: delay differential equations, Nicholson systems, exponential stability, per- manence.

2020 Mathematics Subject Classification: 34K12, 34K25, 34K20, 92D25.

1 Introduction

In a recent paper [9], the permanence for a family of multidimensional nonautonomous and noncooperative delay differential equations (DDEs), which includes a large spectrum of struc- tured models used in population dynamics and other fields, was investigated. Once the permanence is established, several question about the global behaviour of solutions arise. To further analyse the stability and other features of such models, it is, however, clear that the conditions to be imposed depend heavily on the shape and properties of the nonlinear terms.

Nicholson-type systems constitute a specific case included in such family. Here, we consider a nonautonomous generalised Nicholson system with bounded distributed delays given by

x_i⁰(t) =−d_i(t)x_i(t) +

∑

a_ij(t)x_j(t) +

mi

k

∑

b_ik(t)

Z _t

t−τ_ik(t)λ_ik(s)x_i(s)e^{−cik(s)xi(s)}ds, t≥ t₀, i=_{1, . . . ,}n,

(1.1)

where all the coefficients and delays are continuous, nonnegative and satisfy some additional conditions described in the next section.

BEmail: teresa.faria@fc.ul.pt

Since the introduction of the classic Nicholson’s blowflies equation

x⁰(t) =−dx(t) +px(t−τ)e^{−ax(t−τ)} (a,d,p,τ>0), (1.2) by Gurney et al. [12], as a model based on the experimental data of Nicholson [18] and con- structed to study the Australian sheep blowfly pest, the original equation (1.2) as well as a large number of modified and generalised scalar models have been extensively used in population dynamics and other mathematical biology contexts – yet, many open problems concerning the asymptotic behaviour of solutions to scalar Nicholson equations remain un- solved [1]. In recent years, Nicholson-type systems have received much attention in view of their applications as models for populations structured in several patches or classes (see e.g. [2] for some concrete applications). Significant progress has been made, addressing top- ics such as the extinction, permanence, existence of positive equilibria or periodic solutions, stability of solutions, global attractivity of equilibria or periodic solutions. Systems with au- tonomous coefficients (and either autonomous or time-dependent delays) were investigated in [2,3,6,7,11,14,25], whereas the works [4,8–10,15,16,21,22,24] were concerned with nonau- tonomous versions of such systems.

The purpose of this paper is to complement the studies in [8,9], with more results on the large time behaviour of solutions to (1.1), by providing criteria for their global exponen- tial stability, as well as explicit uniform lower and upper bounds for all positive solutions.

The results on stability are obtained by refining the assumptions for permanence established previously in [9]. In [8], the existence of a positive periodic solution for periodic Nicholson’s blowflies systems was analysed, and, in the case of systems with all discrete delays multiples of the period, criteria for the global attractivity of such a positive periodic solution established.

Here, we provide sufficient conditions for the exponential stabilityof any positive solution of (1.1), without any constraint on the type of delays.

We emphasize that, in spite of the recent interest in nonautonomous Nicholson systems, only a few authors have exhibited criteria for their stability, usually for periodic or almost periodic Nicholson equations or systems withdiscretetime-delays; see [5,8,13,15–17,21,23,24]

and references therein. Typically, conditions have been imposed in such a way that convenient lower and upper bounds for all solutions hold. Here, as we shall see, the permanence is still a key ingredient to prove the stability, however, only an explicitupper boundfor solutions of such systems will be required. The criteria enhance and extend some recent achievements in the literature in several ways: not only are the imposed assumptions less restrictive than the ones found in recent papers, but (1.1) is much more general: namely, it incorporatesdistributed delays, not all coefficients are required to be bounded and the global exponential stability is studied for a model that is not necessarily periodic or almost periodic.

This paper is organized as follows: Section 2 is devoted to the study of uniform lower and upper bounds for the positive solutions of (1.1). Section 3 addresses the global stability of (1.1). Examples and a comparison with recent results in the literature [13,16,21,23] are also given, in particular for periodic systems. A brief section of conclusions ends the paper.

2 Permanence: uniform bounds for the solutions

For simplicity of exposition, and without loss of generality, take t₀ = 0 in (1.1) and let τ = sup{τ_ik(t) : t ≥ 0,i = 1, . . . ,n,k = 1, . . . ,m_i} > 0. Take C := C([−τ, 0];Rⁿ) with the supremum normkφk=_{maxθ∈[−τ,0]}|φ(θ)|as the phase space. In abstract form, system (1.1) is

written as the DDE

x_i⁰(t) =−d_i(t)x_i(t) +

∑

a_ij(t)x_j(t) + f_i(t,x_i,t), t ≥0, i=1, . . . ,n, (2.1) where the nonlinearities take the form

f_i(t,x_i,t) =

m_i k

∑

b_ik(t)

Z _t

t−τik(t)λ_ik(s)x_i(s)e^{−cik(s)xi(s)}ds, i=1, . . . ,n. (2.2) For (1.1), define then×nmatrices

D(t) =diag(d₁(t), . . . ,d_n(t)), A(t) =^ha_ij(t)ⁱ B(t) =_diag(β₁(t)_{, . . . ,}β_n(t))_, t≥0,

(2.3) where we may suppose that a_ii(t) ≡ 0 (since a_ii(t) may be incorporated in d_i(t)) and β_i(t) denotes

β_i(t):=

mi

k

∑

b_ik(t)

Z _t

t−τ_ik(t)λ_ik(s)ds, t≥0, i=1, . . . ,n;

The following assumptions will be considered:

(h1) d_i(t),a_ij(t),b_ik(t),τ_ik(t), λ_ik(t),c_ik(t) are continuous and nonnegative with d_i(t) > 0, c_ik(t)≥ c_i >0,β_i(t)> 0,τ_ik(t)∈ [0,τ], c_ik(t)are bounded, fori,j=1, . . . ,n,k= 1, . . . ,m_i andt≥0;

(h2) there is a positive vectorusuch that lim inft→_∞

D(t)−A(t)u>_0;

(h3) there are a positive vector v and T > 0,α > 1 such that B(t)v ≥ α[D(t)−A(t)]v for t ≥T.

The particular case of (1.1) withc_ik(t)≡1 for 1≤i≤n, 1≤k≤ m_i, is expressed by x⁰_i(_t) =−d_i(_t)_xi(_t) +

∑

a_ij(_t)_xj(_t) +

m_i k

∑

b_ik(_t)

Z _t

t−τ_ik(t)

λ_ik(_s)_h(_xi(_s))_{ds, 1}≤i≤n, (2.4) for h(x) = xe^−x, x ≥0. Note that the nonlinearityh is unimodal, e⁻¹ = h(1) =max_x≥0h(x), h(_∞) =0 andx=2 is its unique inflexion point.

We now set the usual orders in Rⁿ and C. Rⁿ may be seen as the subset of constant functions inC. We suppose thatRⁿis equipped with the maximum norm| · |. LetR⁺= [0,∞). A vector v ∈ _Rⁿ is nonnegative, with notation v ≥ 0 (respectively, positive, denoted by v > 0), ifv ∈ (_R⁺)ⁿ (respectivelyv ∈ (0,∞)ⁿ). We denote~₁ = (1, . . . , 1). Consider the cone C⁺ = C([−τ, 0];(_R⁺)ⁿ)of nonnegative functions in C and the partial order inC yielded by C⁺: φ ≤ ψif and only if ψ−φ ∈ C⁺. Thus, φ ≥ 0 if and only if φ ∈ C⁺. We write φ > _{0 if} φ(θ)>0 for−τ≤θ ≤0. The relations≤and<are defined in the obvious way. Foru,v ∈_Rⁿ with u≤v,[u,v]⊂Cdenotes the ordered interval [u,v] ={φ∈C:u≤ φ≤v}.

Due to the real-world interpretation of our models, we take C₀⁺={φ∈C⁺ :φ(₀)>₀}

as the set of admissible initial conditions, and only consider solutions x(t) =x(t,t₀,φ)_{of (1.1)} with initial conditionsxt₀ =φ, φ∈C₀⁺. It is clear that such solutions are defined and positive on R⁺.

The definitions of permanence and global stability are recalled below.

Definition 2.1. Consider a DDE x⁰(t) = f(t,x_t) in C for which all solutions x(t) = x(t, 0,φ) with φ ∈ C₀⁺ are defined on R⁺. The DDE is said to be permanent if there exist positive constantsm,M such that all solutionsx(t) =x(t, 0,φ)withφ∈C₀⁺ satisfy

m≤lim inf

t→_∞ x_i(t), lim sup

t→_∞

x_i(t)≤ M fori=1, . . . ,n.

For short, we say thatx⁰(t) = f(t,xt)isglobally attractive(inC⁺₀) if all positive solutions are globally attractive: for anyφ,ψ∈C₀⁺,

x(t, 0,φ)−x(t, 0,ψ)→_{0 as}t →_∞;

and the DDEx⁰(t) = f(t,x_t)is said to be (eventually) globally exponentially stableif there existδ>0,M >0 such that, for anyφ∈ C₀⁺, there isT≥0 such that

|x(t,t₀,φ)−x(t,t₀,ψ)| ≤ Me^{−δ(t−t0)}kφ−ψk, fort≥t₀≥ T, ψ∈C₀⁺. Note thatδ,Mdo not depend ont₀,φ, though a prioriT depends onφ.

Although the nonlinear terms in (1.1) are nonmonotone, results for cooperative systems from [19] will be used.

Definition 2.2. A DDE x⁰(t) = f(t,x_t) _is cooperative if f = (f₁, . . . ,f_n) satisfies the quasi- monotone condition(Q) in [19], as follows:

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 somei.

In [9], the permanence of generalised Nicholson systems was established.

Theorem 2.3([9, Corollary 3]). Assume (h1)–(h3) and that β_i(t)are bounded onR⁺. Then(1.1)is permanent.

Remark 2.4. When lim inf_t→_∞d_i(t) > 0, for all i, Theorem 2.3 is still valid if one replaces (h2) by the assumptions D(t)u ≥ αA(t)u,t 1, for some vector u > 0 and constant α > _1.

Similarly, (h3) can be replaced by the condition lim inft→_∞

B(t)−D(t) + A(t)v > 0, for some vectorv>0, whenβ_i(t)are all bounded. In fact, if β_i(t)are bounded below and above by positive constants, for alli, conditions lim inf_t→_∞

B(t)−D(t) +A(t)v > 0 and (h3) are equivalent. See [9] for details.

Remark 2.5. In fact, instead of (2.1), more general Nicholson systems with possible delays in the linear terms were considered in [9]:

x⁰_i(t) =−d_i(t)x_i(t) +

∑

L_ij(t)x_j,t+ f_i(t,x_i,t), t ≥0, i=1, . . . ,n, (2.5)

where f_iare as in (2.2) and L_ij(t)are linear bounded functionals,nonnegative(i.e.L_ij(t)(ψ)≥₀ forψ ∈ C([−τ, 0];R⁺)) and continuous in t. With kL_ij(t)k = a_ij(t), the permanence of such systems was also established in [9], if in addition to (h1)–(h3) a_ij(t) are bounded and β_i(t) bounded below and above by positive constants.

When (h2) and (h3) are satisfied simultaneously by a same vector v = (v₁, . . . ,v_n) > 0, there areδ,αsuch that

lim inf

t→_∞

d_i(t)v_i−

∑

j

a_ij(t)v_j

≥δ >0, lim inf

t→_∞

β_i(t)v_i

d_i(t)v_i−_∑ja_ij(t)v_j ≥ α>1, i=1, . . . ,n.

This motivates the following definition: for t ≥ 0 and v = (v₁, . . . ,v_n) > 0 such that [D(t)−A(t)]v6=0, set

γ_i(t,v) = ^βi(t)v_i

d_i(t)v_i−_∑ja_ij(t)v_j, i=1, . . . ,n. (2.6) For the particular casev=~_{1 :}= (1, . . . , 1), we obtain

γ_i(t):=γ_i(t,~₁) = ^βi(t)

d_i(t)−_∑ja_ij(t)^{, i}=1, . . . ,n. (2.7) Next result gives sufficient conditions, expressed in terms of γ_i(t,v), for the positive in- variance of some specific intervals under (1.1), and also provides explicit uniform lower and upper bounds for all solutions.

Theorem 2.6. For(1.1), assume (h1), and that c_ik(t)are bounded below and above onR⁺by positive constants, and denote c_i,c_i such that

0<c_i ≤c_ik(t)≤c_i fort ∈_R⁺, 1≤i≤n, 1≤k≤m_i.

Suppose that there are constants a,b with0 < a≤ b, t₀ ≥0and a vector v = (v₁, . . . ,v_n)>0such that

e^a ≤γ_i(t,v)≤e^b, 1≤i≤ n, t ≥t₀, (2.8) and define

C=C(v):= min

1≤i≤n(c_iv_i), C=C(v):= max

1≤i≤n(c_iv_i). (2.9) Then:

(a) The ordered interval[mC⁻¹v,C⁻¹e^b−1v] ={φ= (φ₁, . . . ,φ_n)∈C :mC⁻¹v_i ≤ φ_i ≤C⁻¹e^b−1v_i, i=1, . . . ,n} ⊂C, where mC⁻¹C∈(0, 1)is such that

m≤a and h(mc_iv_iC⁻¹)≤ h(c_iv_iC⁻¹e^b−1), i=1, . . . ,n, (2.10) is positively invariant for(1.1)and t≥t₀.

(b) If β_i(t) are also bounded below and above by positive constants, any positive solution x(t) = (x₁(t), . . . ,xn(t))of (1.1)satisfies

mC⁻¹v_i ≤lim inf

t→_∞ x_i(t)≤lim sup

t→_∞

x_i(t)≤e^b−1C⁻¹v_i, i=1, . . . ,n. (2.11)

Proof. (a) Write (1.1) asx⁰(t) = F(t,xt), with the componentsF_i of Fgiven by F_i(t,φ) =−d_i(t)φ_i(0) +

∑

j

a_ij(t)φ_j(0) +

mi

k

∑

b_ik(t)

Z ₀

−τ_ik(t)λ_ik(t+s)h_ik(t+s,φ_i(s))ds, i=1, . . . ,n,

where h_ik(s,x) := xe^−cik(s)x. Let c_i,c_i be such 0 < c_i ≤ c_ik(s) ≤ c_i for s ∈ _R⁺ and all i,k, and take the functions h⁻_i (x) = xe^−cix,h⁺_i (x) = xe^−cix. For h(x) = xe^−x as before, we have h⁻_i (x) = (c_i)⁻¹h(c_ix),h⁺_i (x) = (c_i)⁻¹h(c_ix). Clearly, h⁻_i (x)≤ h_ik(s,x) ≤ h⁺_i (x)≤ (c_ie)⁻¹, for s,x≥0.

We know already that the set(0,∞)ⁿis forward invariant. We now compare the solutions of (1.1) from above with the solutions of the cooperative system x⁰(t) = F^u(t,x_t), where the components ofF^u are given by F_i^u(t,φ) = −d_i(t)φ_i(0) +_∑ja_ij(t)φ_j(0) +β_i(t)(c_ie)⁻¹. Clearly F_i(t,φ) ≤ F_i^u(t,φ) for all φ ∈ C⁺. From [19], this implies that x(t,t0,φ,F) ≤ x(t,t0,φ,F^u), where x(t,t₀,φ,F)and x(t,t₀,φ,F^u) are the solutions of x⁰(t) = F(t,x_t) andx⁰(t) = F^u(t,x_t) with initial conditionx_t0 = φ ∈ C⁺₀, respectively. If φ∈ [0,C⁻¹e^b−1v] andφ_i(0) = C⁻¹e^b−1v_i for somei, the use of (2.8) implies

F_i^u(t,φ)≤ C⁻¹e^b−1h

−d_i(t)v_i+

∑

j

a_ij(t)v_ji

+β_i(t)(c_ie)⁻¹

≤ ^hd_i(t)v_i−

∑

j

a_ij(t)v_jih

−C⁻¹e^b−1+γ_i(t,v)(c_iv_ie)⁻¹ⁱ

≤ e^b−1h

d_i(t)v_i−

∑

j

a_ij(t)v_ji

(−C⁻¹+ (c_iv_i)⁻¹)≤0.

From [19, p. 82], the set(_0,C⁻¹e^b−1v]⊂ Cis positively invariant for (1.1).

Next, we start by observing that, for any a,b> 0 witha ≤b, we havea < e^a−1 ≤ e^b−1 for alla6=1. By considering the casesa< e^b−1≤ 1,a<1≤e^b−1or 1 ≤a<e^b−1, it is possible to choosem∈(0,CC⁻¹)such that conditions (2.10) are fulfilled. We get

h⁻_i (C⁻¹e^b−1v_i) = (c_i)⁻¹h(C⁻¹e^b−1c_iv_i)≥(c_i)⁻¹h(mC⁻¹c_iv_i) =h⁻_i (mC⁻¹v_i)_.

As 1>mc_iv_iC⁻¹andhis increasing on(0, 1), forφ_i such thatmC⁻¹v_i ≤φ_i(s)≤C⁻¹e^b−1v_i we therefore obtain

h⁻_i (φ_i(s))≥ h⁻_i (mC⁻¹v_i)

andF_i(t,φ)≥ F_i^l(t,φ):=−d_i(t)φ_i(0) +_∑ja_ij(t)φ_j(0) +β_i(t)h⁻_i (mC⁻¹v_i)fori=1, . . . ,n.

Consider the interval ˆI = [mC⁻¹v,C⁻¹e^b−1v] ⊂ C. For φ ∈ I^ˆ with φ_i(0) = mC⁻¹v_i for somei, the lower bound in (2.8) leads to

F_i^l(t,φ)≥^hd_i(t)v_i−

∑

j

a_ij(t)v_ji h

−mC⁻¹+γ_i(t,v)v⁻_i ¹h⁻_i (mC⁻¹v_i)ⁱ

≥d_i(t)v_i−

∑

j

a_ij(t)v_jh

−mC⁻¹+γ_i(t,v)(v_ic_i)⁻¹h(mC⁻¹c_iv_i)ⁱ

=mC⁻¹

d_i(t)v_i−

∑

j

a_ij(t)v_jh

−1+γ_i(t,v)e^−mC

−1

c_iv_ii

≥mC⁻¹

d_i(t)v_i−

∑

j

a_ij(t)v_j −1+e^ae^−m

≥0.

Hence, from [19] it follows that ˆI is positively invariant for (1.1).

(b) Next, assume also that 0< β≤β_i(t)≤ βfort ≥0. From (2.8), d_i(t)v_i−

∑

j

a_ij(t)v_j ≥e^−bβv_i, β_i(t)v_i ≥e^a[d_i(t)v_i−

∑

j

a_ij(t)v_j],

hence (h2)–(h3) are satisfied. From Theorem2.3, (2.4) is permanent.

Fix any positive solutionx(t)of (2.4), define

¯

x_i =lim sup

t→_∞

x_i(t), i=1, . . . ,n,

and let max_j(v⁻_j ¹x¯_j) = v⁻_i ¹x¯_i for some i. By the fluctuation lemma, there exists a sequence t_k →_∞such that x_i(t_k)→x¯_i andx⁰_i(t_k)→0. Without loss of generality, we can also suppose that v⁻_i ¹x_i(t_k) = max_1≤j≤nv⁻_j ¹x_j(t_k)fork large – otherwise, we choose t_k → _∞ such that, for some subsequence,v⁻_i ¹x_i(t_k) =max₁≤j≤nmax_{t∈[kτ,(k+1)τ]}v⁻_j ¹x_j(t). Thus, reasoning as in (a),

x_i⁰(t_k)≤ −d_i(t_k)x_i(t_k) +

∑

a_ij(t_k)v⁻_i ¹v_jx_i(t_k) +β_i(t_k)(c_ie)⁻¹

≤v⁻_i ¹

d_i(t_k)v_i−

∑

a_ij(t_k)v_jh

−x_i(t_k) +γ_i(t,v)(c_ie)⁻¹ⁱ

≤v⁻_i ¹

d_i(t_k)v_i−

∑

a_ij(t_k)v_jh

−x_i(t_k) +e^b−1(c_iv_i)⁻¹v_ii

≤v⁻_i ¹

d_i(t_k)v_i−

∑

a_ij(t_k)v_jh

−x_i(t_k) +e^b−1C⁻¹v_ii .

(2.12)

Consider a subsequence of(t_k), still denoted by(t_k), for which d_i(t_k)−_∑ⁿ_j=1a_ij(t_k) → ` >0.

By letting k → _∞, we obtain 0 ≤ −x¯_i+e^b−1C⁻¹v_i, thus ¯x_i ≤ e^b−1C⁻¹v_i. For j 6= i, it follows that ¯x_j ≤v_jv⁻_i ¹x¯_i ≤e^b−1C⁻¹v_j.

Proceeding as in (a), in a similar way one can now show that lim inft→_∞x(t)≥mC⁻¹vfor all positive solutions. This proves (b).

Remark 2.7. For the simpler case (2.4), where the nonlinearities are all given in terms of h(x) =xe^−x, under (h1) and

e^a ≤γ_i(t)≤ e^b, 1≤i≤n, t≥t0 (2.13) (i.e., v =~_{1 in γi}(t,v)), we haveC = C = 1; thus, the interval [m,e^b−1]ⁿ is forward invariant, wherem>0 is chosen so thatm<1,m≤a andh(m)≤h(e^b−1).

We also derive the following auxiliary result.

Lemma 2.8. For(1.1), assume (h1) and that0<c_i ≤c_ik(t)≤c_i for t∈_R⁺, 1≤i≤n, 1≤ k≤m_i. Suppose also that there are a vector v = (v₁, . . . ,v_n)>0,t ≥t₀and a constantγsuch that

0< γ_i(t,v)≤γ, 1≤i≤n, t ≥t₀. (2.14) For C,C as in (2.9), the interval (0,γ(Ce)⁻¹v] ⊂ C is positively invariant for (1.1) (t ≥ t0). In particular, if (2.14)holds with

γ<2eC C⁻¹,

there exist solutions of (1.1)such that0< x_i(t)<₂(c_i)⁻¹_, t ≥t₀, i=_{1, . . . ,}n.

Proof. The invariance of the intervalI := (_0,γ(Ce)⁻¹v]for (1.1) was shown in the above proof.

If in addition γ < 2e CC⁻¹, then I ⊂ (0, 2C⁻¹v), and in particular the solutions with initial conditionsφ∈ I satisfy 0< c_ix_i(t)<_{2 for}t≥_{0, 1}≤i≤n.

Remark 2.9. Consider e.g. the Nicholson system (2.4). If 0 <γ_i(t,v)≤e^bv_i for alli, for some b> 0 and a vector v= (v₁, . . . ,vn) >0, from the proof of Theorem 2.6 the interval(0,e^b−1v] is positively invariant. With v =~_{1 and 0} < γ_i(t) ≤ γ < e and the boundedness conditions in Theorem 2.6(b), lim sup_t→∞x_i(t) ≤ γe⁻¹ < 1 for all positive solution; this means that (1.1) has a cooperative behaviour, because the nonlinearityh(x)is monotone on [0, 1]. Note, however, that (2.8) with e.g. v =~_{1 and e} < γ = e^b does not imply that the interval [1,b]ⁿ is positively invariant. In fact, for simplicity take n = 1 and consider the Nicholson equation x⁰(t) = −d(t)x(t) +e^bd(t)x(t−τ)e^−x(t−τ), for someb> 1. For an initial condition 1≤φ≤ b such thatφ(0) =bandφ(−τ) = 1, thenx⁰(0) =d(0)[−b+e^b−1] =d(0)e^b[−h(b) +h(1)]>0, thus x(t) > bfor t > 0 sufficiently small. Nevertheless, we conjecture that if (2.13) is satis- fied with γ < e² and all coefficients are bounded, then all positive solutions of (2.4) satisfy lim sup_t→∞x_i(t)<2 for alli. See also Remark3.9.

3 Stability

In this section, sufficient conditions for the global exponential stability of Nicholson systems (1.1) are established.

In the sequel, the following auxiliary lemma will play an important role.

Lemma 3.1([8]). Fix m∈(0, 1)and define G_m :(0, 2)×[0,∞)→_Rby Gm(x,y) =

(_h(y)−h(x)

y−x , y6=x

(1−x)e^−x, y= x

where h(x) = xe^−x,x≥ 0. Then, G_m(x,y)is continuous and, for any x∈ (0, 2), there isM_m(x):= max_y≥m|G_m(x,y)|<e^−x.

As a consequence, for a functionh_c(x)_:= xe^−cx =c⁻¹h(cx)_{for some}c>0, it follows that for any fixedx ∈(0, 2c⁻¹)andm∈(0,c⁻¹), we have

|hc(y)−hc(x)| ≤M_m(cx)|y−x| for all y≥m, (3.1) whereM_m(x)is the function defined in the lemma above. Moreover,M_m :(0, 2)→(0,e⁻²)is continuous.

We first establish a criterion for the global attractivity of (1.1).

Theorem 3.2. Consider (1.1) under (h1)–(h3) and suppose that the coefficients β_i(t)_,c_ik(t) are all bounded below and above by positive constants onR⁺, for all i,k. Assume in addition that there exists a positive solution x^∗(t)such that

lim sup

t→_∞

c_ik(t)x^∗_i(t)<2, i=1, . . . ,n, k =1, . . . ,m_i. (3.2) Then, any two positive solutions x(t)_,y(t)of (1.1)satisfy

tlim→_∞(x(t)−y(t)) =0.

Proof. From Theorem 2.3, system (1.1) is permanent. Leth_ik(t,x) = xe^−cik(t)x fort,x ≥ 0 and alli,k.

Write 0 < β ≤ β_i(t) ≤ β, 0 < c ≤ c_i ≤ c_ik(t) ≤ c_i ≤ c for t ∈ _R⁺ and all i,k. From the permanence of (1.1), there are m,M with 0 < m < 1 ≤ M, such that any solution x(t) = x(t, 0,φ) with φ ∈ C₀⁺ satisfies m ≤ x_i(t) ≤ M for i = 1, . . . ,n and t ≥ T, for some T = T(φ) > 0. Fix a positive solution x^∗(t) as in (3.2), let m₀ := cm and ε > 0 small, so that m0 ≤ c_ik(t)x^∗_i(t)≤ 2−ε for all i= 1, . . . ,n,k = 1, . . . ,m_i andt 1. In Lemma3.1, take the functionM:=M_m0.

Effecting the changes of variablesz_i(t) = ^xi(t)

x^∗_i(t)−₁ (₁≤i≤n), system (1.1) becomes z⁰_i(t) = ¹

x^∗_i(t) n

x_i⁰(t)−(1+z_i(t))(x^∗_i)⁰(t)^o

= ¹

x^∗_i(t)

n−d^∗_i(t)z_i(t) +

∑

j

a_ij(t)x^∗_j(t)z_j(t) +

m_i k

∑

b_ik(t)

Z _t

t−τik(t)λ_ik(s)^hh_ik s,x^∗_i(s)(1+z_i(s))−h_ik s,x_i^∗(s)ⁱdso ,

(3.3)

fori=1, . . . ,n,t≥0, where d^∗_i(t) =

∑

j

a_ij(t)x^∗_j(t) +

mi

k

∑

b_ik(t)

Z _t

t−τ_ik(t)λ_ik(s)h_ik s,x^∗_i(s)ds.

Letz(t) = (z₁(t), . . . ,zn(t))be any solution of (3.3) with initial conditionz₀ ≥ −1,z(0)>

−1. Define −v_i = lim inf_t→_∞z(t),u_i = lim sup_t→∞z(t). From the permanence of (1.1), in particular −₁ < −v_i ≤ u_i < _∞ and, as observed, x_i^∗(t) ≥ m and x^∗_i(t)(₁+z_i(t)) ≥ m for t> 0 large. Consideru =max_iu_i,v= max_iv_i. A priori,−v,u can be both nonnegative, both nonpositive, or have different signs, nevertheless it is sufficient to show that max(u,v) =0.

Let max(u,v) = u. In this case, u ≥ 0. Assume for the sake of contradiction that u > _0.

Chooseisuch thatu= u_i and take a sequencet_k →_∞withz_i(t_k)→u,z⁰_i(t_k)→0.

From (3.1), we have

h_ip s,x^∗_i(s)(1+z_i(s))−h_ip s,x^∗_i(s)

≤M(c_ip(s)x^∗_i(s)x^∗_i(s)|z_i(s)|,

for 1 ≤ i ≤ n, 1 ≤ p ≤ m_i and s ≥ 0 sufficiently large. As previously, fork large we may suppose that z_j(t_k)≤z_i(t_k)for all j, and from (3.3) we get

z⁰_i(t_k)≤ ¹ x^∗_i(t_k)

h−d^∗_i(t_k) +

∑

j

a_ij(t_k)x^∗_j(t_k)ⁱz_i(t_k) +

m_i p

∑

b_ip(t_k)

Z _tk

tk−τip(tk)λ_ip(s)M c_ip(s)x^∗_i(s)x^∗_i(s)|z_i(s)|ds

= ¹

x^∗_i(t_k)

−z_i(t_k)

m_i p

∑

b_ip(t_k)

Z _tk

tk−τip(tk)λ_ip(s)h_ip s,x^∗_i(s)ds (3.4) +

mi

p

∑

b_ip(t_k)

Z _tk

t_k−τ_ip(t_k)λ_ip(s)M c_ip(s)x^∗_i(s)x^∗_i(s)|z_i(s)|ds

= ¹

x^∗_i(t_k)

mi

p

∑

b_ip(t_k)

Z _tk

t_k−τ_ip(t_k)λ_ip(s)x^∗_i(s)

−z_i(t_k)e^{−cip(s)x∗i(s)}+M c_ip(s)x_i^∗(s)|z_i(s)|

ds.

By the mean value theorem for integrals, we obtain z⁰_i(t_k)≤ ¹

x^∗_i(t_k)

mi

p

∑

x^∗_i(s_k,p)B_kpb_ip(t_k)

Z _tk

t_k−τ_ip(t_k)λ_ip(s)ds, (3.5) where

B_kp =−z_i(t_k)e^{−cip(sk,p)x∗i(sk,p)}+M c_ip(s_k,p)x^∗_i(s_k,p)|z_i(s_k,p)|, for somes_k,p ∈[t_k−τ_ip(t_k),t_k].

For some subsequence of (s_k,p)_k∈N(1 ≤ p ≤ m_i), still denoted by (s_k,p), there exist the limits lim_kc_ip(s_k,p)x^∗_i(s_k,p) = ξ_p ∈ [m₀, 2−ε] and lim_kz_i(s_k,p) = w_p ∈ [−v,u]. SinceM(x)is continuous, this leads to

limk B_kp =−ue^−ξp +M ξ_p

|wp| ≤ −e^−ξp +M ξ_p u<0,

since Lemma3.1asserts thatM(ξ)< e^−ξ for anyξ ∈ (0, 2). In particular,B_kp < 0 fork large, p=1, . . . ,m_i, and from (3.5) we derive that

z⁰_i(t_k)≤ ^m

Mβ_i(t_k) max

1≤p≤m_iB_kp ≤ ^m

Mβ_i max

1≤p≤m_iB_kp. By lettingk→∞, this estimate yields

0≤ max

1≤p≤m_i −e^−ξp +M ξp u<0, which is not possible. Thus,u=0.

Similarly, consider the situation when max(u,v) = v (which impliesv ≥ 0), and suppose thatv>0. By choosingisuch thatv= v_iand a sequencet_k →_∞withz_i(t_k)→ −v,z⁰_i(t_k)→0, for anyε>0 small andksufficiently large, reasoning as above we obtain

z⁰_i(t_k)≥ − ¹ x_i^∗(t_k)

mi

p

∑

b_ip(t_k)

Z _tk

t_k−τ_ip(t_k)λ_ip(s)x_i^∗(s)

z_i(t_k)e^{−cip(s)xi∗(s)}+M c_ip(s)x^∗_i(s)|z_i(s)|

ds

≥ −^m

Mβ_i(t_k) max

1≤p≤mi

C_kp, where now

C_kp =z_i(t_k)e^{−cip(sk,p)x∗i(sk,p)}+M c_ip(s_k,p)x^∗_i(s_k,p)|z_i(s_k,p)|

for some subsequences s_k,p ∈ [t_k−τ_ip(t_k),t_k]. In an analogous way, by taking convergent subsequences of the sequences c_ip(s_k,p)x_i^∗(s_k,p) and z_i(s_k,p), we obtain a contradiction from Lemma3.1. Consequently,v =0. This completes the proof.

Note that hypotheses (h2), (h3) in the statement of Theorem 3.2 were imposed only to derive the permanence of (1.1). In fact, the above proof applies if, instead of the permanence, all solutions are bounded and persistent; in other words, if for anyφ∈C₀⁺there are constants m(φ),M(φ), such that 0<m(φ)≤lim inft→_∞x(t, 0,φ)≤lim sup_t→∞x(t, 0,φ)≤ M(φ).

We are ready to state our main result, on the global exponential stability of (1.1).

Theorem 3.3. Suppose that the hypotheses of Theorem 3.2 are satisfied. Then, (1.1) is (eventually) globally exponentially stable: there existδ>0,L>0such that, for anyφ^∗∈ C₀⁺, there is T =T(φ^∗) such that

|x(t,t₀,φ)−x(t,t₀,φ^∗)| ≤Le^{−δ(t−t0)}kx_t0(_0,φ)−x_t0(_0,φ^∗)k_, t≥ t₀ ≥T, φ∈C₀⁺. (3.6)