Almost Periodic Solution of a Competition System with Dominated Infinite Delays

Xue-zhong He^{∗} School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, AUSTRALIA

Abstract. In this paper, we consider an n-species almost periodic Lotka-Volterra competition system with dominated infinite delays. By constructing suitable Lyapunov functionals, we are able to show that, under a set of algebraic conditions, the system has a unique positive almost periodic solution which is globally attractive.

1991 Mathematics Subject Classification: 34K15, 34K20, 92A15

Key Words: Lotka-Volterra competition system, almost periodic solution, infinite delays, Lyapunov functionals, global attractivity.

Typeset by AMS-TEX

∗E-mail address: he t@maths.su.oz.au

1. Introduction

In this paper, we consider an almost periodic Lotka-Volterra system

ẋ_{i}(t) = x_{i}(t)[b_{i}(t) − ∑_{j=1}^{n} a_{ij}(t) ∫_{−∞}^{t} K_{ij}(t − s)x_{j}(s) ds],  i = 1, ⋯, n, (1.1)

which decrisbes a model of the dynamics of an n-species competition in mathematical ecology. When system (1.1) has delay-independent dominated terms, it takes the following form

ẋ_{i}(t) = x_{i}(t)[b_{i}(t) − a_{ii}(t)x_{i}(t) − ∑_{j=1,j≠i}^{n} a_{ij}(t) ∫_{−∞}^{t} K_{ij}(t − s)x_{j}(s) ds],
i = 1, ⋯, n.  (1.2)

Recently, Gopalsamy [3] discussed the system (1.2) with ω-periodic coefficients b_{i}, a_{ij} (i, j = 1, ⋯, n) and proved that, under a set of delay-independent algebraic conditions, the system (1.2) has a unique globally attractive ω-periodic solution. Murakami [10] generalized the discussion to the system (1.2) with almost periodic parameters b_{i}, a_{ij}(i, j = 1, ⋯, n). By investigating the stability properties of the solutions of system (1.2), Murakami [10] was able to show that (1.2) has an almost periodic solution. We also refer to Hamaya [7] and Hamaya and Yoshizawa [8] for further discussion on the periodic and almost periodic system (1.2) respectively. As one can see easily, when such delay-independent dominated terms are not present, the argument used in Gopalsamy [3], Hamaya [7], Hamaya and Yoshizawa [8] and Murakami [10] can not been used for (1.1). For (1.1), when n = 1, the related problem has been done recently, we refer to Gopalsamy et al [5] for periodic case and Gopalsamy and He [4] and Seifert [11] for almost periodic case. We also refer to He and Gopalsamy [9] for the discussion on the periodic system (1.1) with n = 2. However, when n ≥ 2, whether system (1.1) has a unique globally attractive positive almost periodic solution is still an open problem (see also Gopalsamy and He [6] and He and Gopalsamy [9]). It is the purpose of this paper to solve this problem. Motivated by our recent work (see Gopalsamy and He [4,6]), we firstly give an estimates on the uniform upper and lower bounds of positive solutions of (1.1). Then, by constructing some Lyapunov functionals, we obtain a set of algebraic conditions, under which the systems (1.1) has a unique positive almost periodic which is globally attractivie. As in the case n = 1 (see Gopalsamy and He [4]), the sufficient conditions are delay-dependent, which characterises the competition systems with delay-dominated terms, while the conditions for (1.2) are often delay-independent (see Gopalsamy [3], Hamaya [7], Hamaya and Yoshizawa [8] and Murakami [10]).

In what follows, we denote by R^{n} the n-dimensional real Euclidean space and by |x| the norm of x ∈ R^{n}. Given x = (x_{1}, ⋯, x_{n}) ∈ R^{n} and y = (y_{1}, ⋯, y_{n}) ∈ R^{n}, we put x > y if x_{i} > y_{i} and x ≥ y if x_{i} ≥ y_{i} for all i ∈ I = {1, 2, ⋯, n}. R_{+}^{n} will denote the nonnegative cone of R^{n}. Throughout this paper, we assume that the functions b_{i}, a_{ij} and K_{ij} in (1.1) are real-valued functions on R and that the following conditions are satisfied: (H_{1}) a_{ij} and b_{i} are continuous, almost periodic functions, and inf_{t∈R} a_{ij}(t) ≥ 0

for i ≠ j, inf_{t∈R} a_{ii}(t) > 0 and inf_{t∈R} b_{i}(t) > 0 for i, j ∈ I; (H_{2}) K_{ij} is nonnegative piecewise continuous, ∫_{0}^{∞} K_{ij}(s) ds = 1, ∫_{0}^{∞} sK_{ij}(s) ds <

∞ and ∫_{0}^{∞} s^{2}K_{ij}(s) ds < ∞ for i, j ∈ I.

Consequently, define constants b_{i}^{l}, b_{i}^{u}, a_{ij}^{l}, a_{ij}^{u} (i, j ∈ I) by the following:

b_{i}^{l} = inf_{t∈R} b_{i}(t), b_{i}^{u} = sup_{t∈R} b_{i}(t), a_{ij}^{l} = inf_{t∈R} a_{ij}(t), a_{ij}^{u} = sup_{t∈R} a_{ij}(t); i, j ∈ I.

Let BC^{+} be the set of all bounded nonnegative continuous functions from R_{−} = (−∞, 0] into R_{+}^{n} satisfying ϕ(0) > 0. Set ∥ϕ∥ = sup_{s∈R_{−}} ϕ(s) for ϕ(s) ∈ BC^{+}. We assume the system (1.1) is supplemented with initial condition of the form

x(s) = ϕ(s) ∈ BC^{+}  for  s ∈ R_{−}.  (1.3)

2. Uniform upper and lower bounds

In this section, following the idea in Gopalsamy and He [4], we obtain a priori upper and lower bounds of the positive solutions of (1.1) and (1.3). One can see that, under the conditions (H_{1}) and (H_{2}), the solutions of (1.1) and (1.3) exist for all t ∈ [0, r) (r ≤ +∞) and remain positive. From the following Lemma 2.1, we know that the solutions of (1.1) and (1.3) are continuable to t = ∞. It follows from the positivity of solution x(t) of (1.1) and (1.3) that

x′(t) ≤ x(t)[b_{i}(t) − a_{ii}(t) ∫_{0}^{∞} K_{ii}(s)x_{i}(t − s) ds],  i ∈ I,  (2.1)

from which, using the same argument as in Theorem 2.1 in Gopalsamy and He [4], we can derive the following estimate on the uniform upper bound of the solutions of (1.1) and (1.3).

Lemma 2.1. Under assumptions (H_{1}) and (H_{2}), the solutions x(t) = (x_{1}(t), ⋯, x_{n}(t)) of (1.1) and (1.3) satisfy x(t) > 0 for all t ≥ 0, and furthermore

lim sup_{t→∞} x_{i}(t) ≤ M_{i} =^{def} \frac{b_{i}^{u}}{a_{ii}^{l} ∫_{0}^{∞} K_{ii}(s)e^{−b_{i}^{u}s} ds}  for  i ∈ I.  (2.2)

Using Lemma 2.1 and the idea of Theorem 2.2 in Gopalsamy and He [4], we now have the following estimate on the uniform lower bound of the solutions of (1.1) and (1.3).

Lemma 2.2. Assume the system (1.1) satisfies (H_{1}), (H_{2}) and (H_{3})

b_{i}^{l} > ∑_{j=1,j≠i}^{n} a_{ij}^{u}M_{j}  (2.3)
with M_{i} defined by (2.2) and i, j ∈ I;

(H_{4}) there exists δ > 0 such that

∫_{0}^{∞} K_{ii}(s) exp [ − (b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}M_{j})s + δs] ds < ∞.  (2.4)

Then the solution x(t) of (1.1) and (1.3) satisfies

lim inf_{t→∞} x_{i}(t) ≥ m_{i} =^{def} \frac{b_{i}^{l} − ∑_{j=1,j≠i}^{n} a_{ij}^{u}M_{j}}{a_{ii}^{u} ∫_{0}^{∞} K_{ii}(s) exp[−(b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}M_{j})s] ds}
for  i ∈ I.  (2.5)

Proof. The proof is similar to that of Theorem 2.2 in Gopalsamy and He [4] and we indicate it briefly.

Let x(t) = (x_{1}(t), ⋯, x_{n}(t)) be any solution of (1.1) and (1.3). It follows from Lemma 2.1 and the condition (H_{4}) that, for ϵ < δ, there exists a t_{1} > 0 such that

x_{i}(t) ≤ M_{i} + ϵ  for  t ≥ t_{1} and  i ∈ I  (2.6)

with M_{i} defined by (2.2). Then, following the positivity of x(t) and (1.1) and (2.6), one can see that, for t ≥ t_{1},

x′_{i}(t) ≥ x_{i}(t)[b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}( ∫_{0}^{t−t_{1}} K_{ij}(s)x_{j}(t − s) ds

+ ∫_{t−t_{1}}^{∞} K_{ij}(s)x_{j}(t − s) ds)]  (2.7)

≥ x_{i}(t)[b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}([M_{j} + ϵ] ∫_{0}^{t−t_{1}} K_{ij}(s) ds

+ ∫_{t−t_{1}}^{∞} K_{ij}(s)x_{j}(t − s) ds)]
= x_{i}(t)c_{i}(t)  (2.8)

with

\begin{aligned}c_{i}(t) = b_{i}^{l}− & ∑_{j=1}^{n} a_{ij}^{u}([M_{j} + ϵ] ∫_{0}^{t−t_{1}} K_{ij}(s) ds + ∫_{t−t_{1}}^{∞} K_{ij}(s)x_{j}(t − s) ds), \\ & t ≥ t_{1}, i ∈ I. \end{aligned}

By the boundedness of x(t) and assumption (H_{2}),

lim_{t→∞} c_{i}(t) = C_{i} =^{def} b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}(M_{j} + ϵ),  i ∈ I.  (2.9)

It follows from (2.8) that

x_{i}(t − s) ≤ x_{i}(t) exp[− ∫_{t−s}^{t} c_{i}(s) ds]  for  t − s > t_{1},

which, together with (2.7), implies that

x′_{i}(t) ≥ x_{i}(t)[b_{i}^{l} − a_{ii}^{u}( ∫_{0}^{t−t_{1}} K_{ii}(s) exp [ − ∫_{t−s}^{t} c_{i}(r) dr] ds)x_{i}(t)

− ∑_{j=1,j≠i}^{n} a_{ij}^{u}(M_{j} + ϵ) ∫_{0}^{t−t_{1}} K_{ij}(s) ds

− ∑_{j=1}^{n} a_{ij}^{u} ∫_{t−t_{1}}^{∞} K_{ij}(s)x_{j}(t − s) ds].  (2.10)

Note that

lim_{t→∞} ∑_{j=1,j≠i}^{n} a_{ij}^{u}(M_{j} + ϵ) ∫_{0}^{t−t_{1}} K_{ij}(s) ds = ∑_{j=1}^{n} a_{ij}^{u}(M_{j} + ϵ), i ∈ I,

lim_{t→∞} ∑_{j=1}^{n} a_{ij}^{u} ∫_{t−t_{1}}^{∞} K_{ij}(s)x_{j}(t − s) ds = 0

and

a_{ii}^{u}( ∫_{0}^{t−t_{1}} K_{ii}(s) exp [ − ∫_{t−s}^{t} c_{i}(r) dr] ds) ≤ a_{ii}^{u} ∫_{0}^{∞} K_{ii}(s)e^{−C_{i}s}e^{ϵ_{1}s} ds

for t ≥ t_{2} ≥ t_{1} and some ϵ_{1} with 0 < ϵ_{1} < δ. Then we can see from (2.10) that

x′_{i}(t) ≥ x_{i}(t)[(b_{i}^{l} − ϵ_{2}) − ∑_{j=1,j≠i}^{n} a_{ij}^{u}(M_{j} + ϵ) − a_{ii}^{u}( ∫_{0}^{∞} K_{ii}(s)e^{−C_{i}s}e^{ϵ_{1}s} ds)x_{i}(t)]

(2.11)

for some ϵ_{2} > 0 sufficiently small and all large t. Then, similar to the proof of Theorem 2.2 in Gopalsamy and He [4], one can show from (2.11) that x_{i}(t) staisfies (2.5). This completes the proof.

It is noticed that, the upper and lower bounds obtained in Lemmas 2.1 and 2.2 depend on the diagonal delay terms only. As a special case of (1.1), it can take the following form with finite discrete delays:

ẋ_{i}(t) = x_{i}(t)[b_{i}(t) − a_{ii}(t)x_{i}(t − τ_{i}) − ∑_{j=1,j≠i}^{n} a_{ij}(t) ∫_{−∞}^{t} K_{ij}(t − s)x_{j}(s) ds],
i = 1, ⋯, n,  (2.12)

where 0 ≤ τ_{i} (i ∈ I) are finite constants. Consequently, from Lemmas 2.1 and 2.2, we have the following bound estimate on the solutions of (2.12), which can also be found in Gopalsamy and He [6]. This result will be used in our later discussion.

Corollary 2.3. Under assumptions (H_{1}) − (H_{2}), if

b_{i}^{l} > ∑_{j=1,j≠i}^{n} a_{ij}^{u} \frac{b_{j}^{u}}{a_{jj}^{l}} e^{b_{j}^{u}τ_{i}}  for  i ∈ I,  (2.13)

then the positive solutions x(t) of (2.12) satisfy

0 < m_{i} ≤ lim inf_{t→∞} x(t) ≤ lim sup_{t→∞} x_{i}(t) ≤ M_{i}

with

\begin{matrix}M_{i} & = \frac{b_{i}^{u}}{a_{ii}^{l}} e^{b_{i}^{u}τ_{i}} \\ m_{i} & = \frac{b_{i}^{l} − ∑_{j=1,j≠i}^{n} a_{ij}^{u}M_{j}}{a_{ii}^{u}} exp[(b_{i}^{l} − ∑_{j=1}^{n} a_{ij}^{u}M_{j})τ_{i}] \end{matrix}  (i ∈ I).  (2.14)

3. Existence and attractivity of almost periodic solution

In this section, we will show that, under a set of algebraic conditions, the system (1.1) has a positive almost periodic solution which is globally attractive.

Lemma 3.1. Suppose system (1.1) satisfies (H_{1}) − (H_{4}). Then there exists a solution x(t) = (x_{1}(t), ⋯, x_{n}(t)) of (1.1) and (1.3) on R such that 0 < m_{i} − ϵ ≤ x_{i}(t) ≤ M_{i} + ϵ (i ∈ I) for t ∈ R and sufficiently small ϵ > 0, here M_{i} and m_{i} (i ∈ I) are defined by (2.2) and (2.5) respectively.

Lemma 3.1 can be proved by repeating almost same argument as in Lemma 2 in Murakami [10] and Lemm 4 in Seifert [11], so we omit the details.

Theorem 3.2. Assume (H_{1}) − (H_{4}) are satisfied. Suppose that

(M_{i}a_{ii}^{u})^{2}σ_{i} < a_{ii}^{l} ∫_{0}^{∞} K_{ii}(s)e^{−b_{i}^{u}s} ds (i = 1, ⋯, n)

with σ_{i} = ∫_{0}^{∞} sK_{ii}(s) ds and E = (e_{ij})_{n×n} is an M-matrix, where

e_{ij} = \{ \begin{matrix}a_{ii}^{l} ∫_{0}^{∞} K_{ii}(s)e^{−b_{i}^{u}s} ds − (M_{i}a_{ii}^{u})^{2}σ_{i}, & for i = j \\ −[1 + M_{i}^{2}a_{ii}^{u}σ_{i}]a_{ij}^{u}, & for i ≠ j. \end{matrix}

Then system (1.1) is extreme stable (see Yoshizawa [12, 13]); that is, for any two positive solutions x(t) and y(t) of (1.1) and (1.3), we have

lim_{t→∞}[x(t) − y(t)] = 0.

Proof. Since the matrix E = (e_{ij})_{n×n} is an M-matrix, we know that (see [2,6]) there exist α = (α_{1}, ⋯, α_{n}) > 0 and ϵ_{0} > 0 such that

α_{i}(e_{ii} − ϵ_{o}) > ∑_{j=1,j≠i}^{n} α_{j}(|e_{ji}| + ϵ_{o}),  i ∈ I.  (3.1)

Clearly there exists an ϵ ∈ (0, ϵ_{o}) such that

b_{i}^{l} − ϵ ≤ b_{i}(t) ≤ b_{i}^{u} + ϵ, 0 < a_{ii}^{l} − ϵ ≤ a_{ii}(t), a_{ij}(t) ≤ a_{ij}^{u} + ϵ

for all large t and

e_{ii} − ϵ_{o} < e_{ii}(ϵ),  e_{ij} − ϵ_{o} < e_{ij}(ϵ)  (3.2)

with

\begin{matrix}e_{ii}(ϵ) & = (a_{ii}^{l} − ϵ) ∫_{0}^{∞} K_{ii}(s)e^{−(b_{i}^{u}+ϵ)s} ds − [(a_{ii}^{u} + ϵ)(M_{i} + ϵ)]^{2}σ_{i} \\ e_{ij}(ϵ) & = −(a_{ij}^{u} + ϵ)[1 + (M_{i} + ϵ)^{2}(a_{ii}^{u} + ϵ)σ_{i}]  i ≠ j \end{matrix} \} .  (3.3)

We first know from Lemma 3.1 that there exists a solution, say y(t) = (y_{1}(t), ⋯, y_{n}(t)) of (1.1) and (1.3) satisfying

0 < m_{i} − ϵ ≤ y_{i}(t) ≤ M_{i} + ϵ  for  t ∈ R  and  i ∈ I.  (3.4)

To prove the extreme stability of (1.1), it is enough to show that for any positive solution x(t) = (x_{1}(t), ⋯, x_{n}(t)) of (1.1) and (1.3) and the solution y(t) satisfying (3.4),

lim_{t→∞}[x_{i}(t) − y_{i}(t)] = 0,  i ∈ I.  (3.5)

Define u(t) = (u_{1}(t), ⋯, u_{n}(t)), v(t) = (v_{1}(t), ⋯, v_{n}(t)) and w(t) = (w_{1}(t), ⋯, w_{n}(t)) as follows:

u_{i}(t) = ℓn[x_{i}(t)],  v_{i}(t) = ℓn[y_{i}(t)],  w_{i}(t) = u_{i}(t) − v_{i}(t),  i ∈ I. (3.6)

Then, from (1.1) and (3.6),

\frac{d}{dt}[u_{i}(t) − v_{i}(t)] = − ∑_{j=1}^{n} a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)[e^{u_{j}(t−s)−v_{j}(t−s)} − 1] ds. (3.7)

For i ∈ I, equation (3.7) can be written as

\frac{dw_{i}(t)}{dt} = − ∑_{j=1}^{n} a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)[e^{w_{j}(t−s)} − 1] ds

= −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds][e^{w_{i}(t)} − 1]

+ a_{ii}(t) ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s)( ∫_{t−s}^{t} e^{w_{i}(s_{1})}ẇ_{i}(s_{1}) ds_{1} ds)

− ∑_{j=1,j≠i}^{n} a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)[e^{w_{j}(t−s)} − 1] ds

= −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds][e^{w_{i}(t)} − 1]

− ∑_{j=1,j≠i}^{n} a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)[e^{w_{j}(t−s)} − 1] ds

− a_{ii}(t) ∑_{j=1}^{n} ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s)( ∫_{t−s}^{t} e^{w_{i}(s_{1})}a_{ij}(s_{1})

× ( ∫_{0}^{∞} K_{ij}(s_{2})y_{j}(s_{1} − s_{2})[e^{w_{j}(s_{1}−s_{2})} − 1] ds_{2}) ds_{1} ds). (3.8)

Let

V_{i1}(w(t)) = |w_{i}(t)|  (i ∈ I).  (3.9)

Then, it follows from (3.9) and (3.8) that the upper right derivative \frac{D^{+}}{Dt} V_{i1}(w) of V_{i1}(w) along the solutions of (3.8) is given by

\frac{D^{+}}{Dt} V_{i1}(w(t)) ≤ −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds]|e^{w_{i}(t)} − 1|

+ ∑_{j=1,j≠i}^{n} a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)|e^{w_{j}(t−s)} − 1| ds

+ a_{ii}(t) ∑_{j=1}^{n} ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s)[ ∫_{t−s}^{t} e^{w_{i}(s_{1})}a_{ij}(s_{1})

× ( ∫_{0}^{∞} K_{ij}(s_{2})y_{j}(s_{1} − s_{2})|e^{w_{j}(s_{1}−s_{2})} − 1| ds_{2}) ds_{1} ds].

(3.10)

Let

V_{i2}(w)(t) = ∑_{j=1,j≠i}^{n} ∫_{0}^{∞} K_{ij}(s) ∫_{t−s}^{t} a_{ij}(s_{1} + s)y_{j}(s_{1})|e^{w_{j}(s_{1})} − 1| ds_{1} ds

+ ∑_{j=1}^{n} ∫_{0}^{∞} K_{ii}(s) ∫_{t−s}^{t} a_{ii}(s_{3} + s)y_{i}(s_{3}) ∫_{s_{3}}^{t} e^{w_{i}(s_{1})}a_{ij}(s_{1})

× ∫_{0}^{∞} K_{ij}(s_{2})y_{j}(s_{1} − s_{2})|e^{w_{j}(s_{1}−s_{2})} − 1| ds_{2} ds_{1} ds_{3} ds.

(3.11)

Then, from (3.10) and (3.11),

\frac{D^{+}}{Dt} [V_{i1} + V_{i2}](w(t))

≤ −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds]|e^{w_{i}(t)} − 1|

+ ∑_{j=1,j≠i}^{n} ( ∫_{0}^{∞} K_{ij}(s)a_{ij}(t + s) )y_{j}(t)|e^{w_{j}(t)} − 1| ds

+ ∑_{j=1}^{n} ( ∫_{0}^{∞} K_{ii}(s) ∫_{t−s}^{t} a_{ii}(s_{1} + s)y_{i}(s_{1}) ds_{1} ds)

× e^{w_{i}(t)}a_{ij}(t) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)|e^{w_{j}(t−s)} − 1| ds. (3.12)

Denote

d_{i}(t) = ∫_{0}^{∞} K_{ii}(s) ∫_{t−s}^{t} a_{ii}(s_{1} + s)y_{i}(s_{1}) ds_{1} ds.

Note that, for all large t, e^{w_{i}(t)} ≤ M_{i} + ϵ (i ∈ I) and also, from (3.4),

d_{i}(t) ≤ (M_{i} + ϵ)(a_{ii}^{u} + ϵ) ∫_{0}^{∞} sK_{ii}(s) ds = (M_{i} + ϵ)(a_{ii}^{u} + ϵ)σ_{i}.

It follows from (3.12) and (3.13) that, for all large t,

\frac{D^{+}}{Dt} [V_{i1} + V_{i2}](w(t))

≤ −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds]|e^{w_{i}(t)} − 1|

+ ∑_{j=1,j≠i}^{n} ( ∫_{0}^{∞} K_{ij}(s)a_{ij}(t + s) )y_{j}(t)|e^{w_{j}(t)} − 1| ds

+ (a_{ii}^{u} + ϵ)(M_{i} + ϵ)^{2}σ_{i}

× ∑_{j=1}^{n}(a_{ij}^{u} + ϵ) ∫_{0}^{∞} K_{ij}(s)y_{j}(t − s)|e^{w_{j}(t−s)} − 1| ds. (3.13)

Now, let

V_{i}(w)(t) = [V_{i1} + V_{i2} + V_{i3}](w(t)),  i ∈ I  (3.14)

with

\begin{aligned}V_{i3}(w(t)) & = (a_{ii}^{u} + ϵ)(M_{i} + ϵ)^{2}σ_{i} \\ & × ∑_{j=1}^{n}(a_{ij}^{u} + ϵ) ∫_{0}^{∞} K_{ij}(s) ∫_{t−s}^{t} y_{j}(s_{1})|e^{w_{j}(s_{1})} − 1| ds_{1} ds. \end{aligned}

Then, one can derive from (3.14) and (3.13) that, for all large t,

\frac{D^{+}}{Dt} V_{i}(w)(t) ≤ − a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds]|e^{w_{i}(t)} − 1|

+ ∑_{j=1,j≠i}^{n} [ ∫_{0}^{∞} K_{ij}(s)a_{ij}(t + s) ds]y_{j}(t)|e^{w_{j}(t)} − 1|

+ (a_{ii}^{u} + ϵ)(M_{i} + ϵ)^{2}σ_{i} ∑_{j=1}^{n}(a_{ij}^{u} + ϵ)y_{j}(t)|e^{w_{j}(t)} − 1|.

(3.15)

On the other hand, one have from (1.1) that

y′_{i}(t) ≤ b_{i}(t)y_{i}(t),

which implies

y_{i}(t) ≤ y_{i}(t − s)e^{∫_{t−s}^{t} b_{i}(s_{1}) ds_{1}}  (i ∈ I).  (3.16)

From (3.16), for all large t,

−a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)y_{i}(t − s) ds] ≤ −a_{ii}(t)[ ∫_{0}^{∞} K_{ii}(s)e^{− ∫_{t−s}^{t} b_{i}(s_{1}) ds_{1}} ds]y_{i}(t)
≤ −(a_{ii}^{l} − ϵ) ∫_{0}^{∞} K_{ii}(s)e^{−(b_{i}^{u}+ϵ)s} dsy_{i}(t). (3.17)

Note that y_{i}(t)|e^{w_{i}(t)} − 1| = |x_{i}(t) − y_{i}(t)|. Therefore, from (3.15) and (3.17), for all large t,

\frac{D^{+}}{Dt} V_{i}(w)(t) ≤ − (a_{ii}^{l} − ϵ)[ ∫_{0}^{∞} K_{ii}(s)e^{−(b_{i}^{u}+ϵ)s} ds]|x_{i}(t) − y_{i}(t)|

+ ∑_{j=1,j≠i}^{n}(a_{ij}^{u} + ϵ)|x_{j}(t) − y_{j}(t)|

+ (a_{ii}^{u} + ϵ)(M_{i} + ϵ)^{2}σ_{i} ∑_{j=1}^{n}(a_{ij}^{u} + ϵ)|x_{j}(t) − y_{j}(t)|

= − ∑_{j=1}^{n} e_{ij}(ϵ)|x_{j}(t) − y_{j}(t)|,  (3.18)

where e_{ij}(ϵ) is defined by (3.3). Now, let

V (w(t)) = ∑_{j=1}^{n} α_{i}V_{i}(w(t)).

Then, from (3.18), (3.2) and (3.1), for all large t,

\frac{D^{+}}{Dt} V (w(t)) ≤ − ∑_{i=1}^{n} α_{i} ∑_{j=1}^{n} e_{ij}(ϵ)|x_{j}(t) − y_{j}(t)|

= − ∑_{i=1}^{n} [ ∑_{j=1}^{n} α_{j}e_{ji}(ϵ)]|x_{i}(t) − y_{i}(t)|

≤ − ∑_{i=1}^{n} [α_{i}(e_{ii} − ϵ_{o}) − ∑_{j=1,j≠i}^{n} α_{j}(|e_{ji}| + ϵ_{o})]|x_{i}(t) − y_{i}(t)|

= − ∑_{i=1}^{n} β_{i}|x_{i}(t) − y_{i}(t)|,  (3.19)

where β_{i} = α_{i}[e_{ii} −ϵ_{o}]−∑_{j=1,j≠i}^{n} α_{j}(|e_{ji}|+ϵ_{o}) > 0 (by (3.1)). Note that V (w(t)) ≥ 0. Integrating (3.19) from t_{1} (large enough) to t ≥ t_{1}, we have

∑_{i=1}^{n} β_{i} ∫_{t_{1}}^{t} |x_{i}(t) − y_{i}(t)| ds < ∞,

from which, one can conclude that (see, also, Gopalsamy [3] and Gopalsamy and He [4]) (3.5) holds. This completes the proof.

Now consider the system

x′_{i}(t) = x_{i}(t)[b_{i}^{∗}(t) − ∑_{j=1}^{n} a_{ij}^{∗}(t) ∫_{0}^{∞} K_{ij}(s)x_{j}(t − s) ds],  i ∈ I,  (3.20)

where K_{ij} as in (1.1) and for some sequence {t_{n}},

b_{i}(t + t_{n}) → b_{i}^{∗}(t),  a_{ij}(t + t_{n}) → a_{ij}(t)  as  n → ∞,

uniformly for t ∈ R. Then, using our Theorem 3.2 and the arguments of Lemma 6 and Theorem in Seifert [11], one can conclude the following result on the existence and global attractivity of positive almost periodic solution of (1.1). For more details, we refer to Seifert [11] (see, also, Fink [1] and Yoshizawa [13, 14]) and we omit the proof.

Theorem 3.3. Under the assumptions of Theorem 3.2, system (1.1) has a positive almost periodic solution, which is globally attractive.

We conclude this paper with the following remark. The conditions of Theorem 3.2 depend only on the size of diagonal delays, which are measured by σ_{i} (i ∈ I). For (2.12), which is a special case of (1.1), σ_{i} = τ_{i} and the conditions of Theorem 3.2 become (2.13), (M_{i}a_{ii}^{u})^{2}τ_{i} < a_{ii}^{l}e^{−b_{i}^{u}τ_{i}} and that E^{∗} = (e_{ij}^{∗})_{n×n} is an M-matrix with e_{ii}^{∗} = a_{ii}^{l}e^{−b_{i}^{u}τ_{i}} − (M_{i}a_{ii}^{u})^{2}τ_{i} and e_{ij}^{∗} = −[1 + M_{i}^{2}a_{ii}^{u}τ_{i}]a_{ij}^{u} for i ≠ j i, j ∈ I, where M_{i} (i ∈ I) are defined by (2.14). In particularly, when τ_{i} = 0 (i ∈ I), system (2.12) becomes (1.2) and the corresponding conditions become

b_{i}^{l} > ∑_{j=1,j≠i}^{n} a_{ij}^{u} \frac{b_{j}^{u}}{a_{jj}^{l}}  i ∈ I  (3.21)

and that E = (e_{ij})_{n×n} with e_{ii} = a_{ii}^{l} and e_{ij} = −a_{ij}^{u} (i ≠ j) is an M-matrix. Using the properties of M-matrix (see Gopalsamy and He [6]), one can verify that condition (3.21) implies that E is an M-matrix. In fact, in addition to (3.21), under the condition

a_{ii}^{l} > ∑_{j=1,j≠i}^{n} a_{ji}^{u}  i ∈ I,  (3.22)

the existence of a strictly positive almost periodic solution has been shown by Gopalsamy [3] for the periodic case in (1.2) and by Murakami [10] for the almost periodic case in (1.2). It is shown by Hamaya and Yoshozawa [8] that, for the almost periodic system (1.2), the condition (3.22) is not necessary. Therefore, when (1.1) takes the form (1.2), our conditions are reduced to the one for (1.2). It is in this sense that our result is a significant generalization of the known results.

Acknowledgement. The author thinks Dr. S. Ruan for a careful reading of the original draft and suggesting a number of corrections.

References

1. A.M. Fink, Almost periodic differential equations, vol. 377, Lecture Notes in Math., Springer-Verlag, 1974.

2. K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dordrecht, 1992.

3. K. Gopalsamy, Global asymptotic stability in a periodic integrodifferential system, Tohoku Math. J. 37 (1985), 323-332.

4. K. Gopalsamy and X. Z. He, Dynamics of an almost periodic logistic integrodifferential equation, Meth. Appl. Anal. 2 (1995), 38-66.

5. K. Gopalsamy, X. Z. He and L. Wen, Global attractivity and “level-crossing” in a periodic logistic integrodifferential equation, Math. Nachr. 156 (1992), 25-44.

6. K. Gopalsamy and X. Z. He, Global stability in n-species competition modelled by “puredelay-type” systems II: nonautonomous case, Resech Report 96-15, School of Math. & Stat., University of Sydney (1996).

7. Y. Hamaya, Periodic solutions of nonlinear integrodifferential equations, Tohoku Math. J. 41 (1989), 105-116.

8. Y. Hamaya and T. Yoshizawa, Almost periodic solutions in an integrodifferential equation, Proc. Roy. Soc. Edinburgh 114 A (1990), 151-159.

9. X. Z. He and K. Gopalsamy, Persistence, stability and level-crossing in an integrodifferential system, J. Math. Biol. 32 (1994), 395-426.

10. S. Murakami, Almost periodic solutions of a system of integrodifferential equations, Tohoku Math. J. 39 (1987), 71-79.

11. G. Seifert, Almost periodic solutions for delay logistic equations with almost periodic time dependence, Diff. Integ. Equs. 9 (1996), 335-342.

12. T. Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rat. Mech. Anal. 17 (1964), 148-170.

13. T. Yoshizawa, Stability theory by Lyapunov’s second method, The Math. Soc. Japan, Tokyo, 1966.

14. T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, Berlin, 1975.