Global Stability in Nonautonomous Lotka-Volterra Systems of “Pure-Delay-Type”

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

Abstract. In this paper, nonautonomous Lotka-Volterra systems of “pure-delay-type” are considered and some sufficient conditions on the global asymptotical stability are obtained. As a corollary, we show that, under the conditions of Theorem 2.1 in Kuang (Diff. Integ. Equs., 9 (1996)), the system remains globally asymptotically stable provided the delays are sufficient small. Both finite and infinite delays are allowed in the systems. Our results give an affirmative answer to the two open problems due to Kuang. The results are established by constructing suitable Lyapunov functionals.

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

Key Words: Lotka-Volterra nonautonomous system, puredelay-type, Lyapunov functionals, global asymptotical stability.

Typeset by AMS-TEX

∗E-mail address: he t@maths.su.oz.au

1. Introduction.

We will be concerned with the solutions of the following nonautonomous “puredelay-type” Lotka-Volterra system with both distributed and discrete, finite and infinite delays

x′_{i}(t) = b_{i}(x_{i}(t))[r_{i}(t) − a_{i}(t)x_{i}(t − τ_{i}) − c_{i}(t) ∫_{−∞}^{0} x_{i}(t + s)h_{i}(s) ds  (1.1)

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(t)x_{j}(t − τ_{ijl}) + ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} x_{j}(t + s)k_{ij}(t, s) ds

+ ∑_{j=1}^{n} d_{ij}(t) ∫_{−∞}^{0} x_{j}(t + s)h_{ij}(s) ds], i = 1, ⋯, n

associated with initial condition

x_{i}(s) = ϕ(s),  s < 0,  ϕ_{i}(0) > 0,  i = 1, ⋯, n,  (1.2)

where ϕ_{i} ∈ C((−∞, 0], R_{+}) are bounded and R_{+} = [0, ∞). System (1.1) is called “pure-delay-type” since it is not necessary for (1.1) to have delay-independent intraspecific and interspecific effects (see Kolmanovskii et al [9] and Gopalsamy and He[3, 8]). We assume that for i, j = 1, ⋯, n, l = 1, ⋯, l_{ij}: (H_{1}) b_{i}(·) is continuousely differentiable such that b_{i}(0) = 0, b′_{i}(x) > 0 for x ≥ 0; (H_{2}) r_{i}(t), a_{i}(t), c_{i}(t) and b_{ijl}(t) are bounded continuous functions for t > 0;

k_{ij}(t, s) are integrable with respect to s ∈ [−τ_{ij}, 0] and continuous and bounded with respect to t; h_{i}(s)(≥ 0), |h_{ij}(s)|, |s|h_{i}(s), |sh_{ij}(s)| are integrable with respect to s ∈ (−∞, 0]; τ_{i}, τ_{ij}, τ_{ijl} are nonnegative constants.

For i, j = 1, ⋯, n, l = 1, ⋯, l_{ij}, we set

a_{i} = lim inf_{t→∞} a_{i}(t), A_{i} = lim sup_{t→∞} a_{i}(t),

c_{i} = lim inf_{t→∞} c_{i}(t), C_{i} = lim sup_{t→∞} c_{i}(t), D_{ij} = lim sup_{t→∞} |d_{ij}(t)|,

B_{ijl} = lim sup_{t→∞} |b_{ijl}(t)|, R_{i} = lim sup_{t→∞} |r_{i}(t)|,

K_{ij}(s) = lim sup_{t→∞} |k_{ij}(t, s)|, K_{ij} = ∫_{−τ_{ij}}^{0} K_{ij}(s) ds,

τ = max{τ_{i}, τ_{ij}, τ_{ijl}; i, j = 1, ⋯, n, l = 1, ⋯, l_{ij}}.

It is well known that (1.1) and (1.2) have a unique positive soution x(t) = (x_{1}(t), ⋯, x_{n}(t)) for t > 0. As in Kuang [11], we say solutions of (1.1) and (1.2) are eventually uniformly bounded above and below by positive constant vectors Δ = (Δ_{1}, ⋯, Δ_{n}) and δ = (δ_{1}, ⋯, δ_{n}), respectively if, for any solution x(t) = (x_{1}(t), ⋯, x_{n}(t)) of (1.1) and (1.2), we have, for all large t, δ_{i} ≤ x_{i}(t) ≤ Δ_{i} and i = 1, ⋯, n. We say system (1.1) is globally asymptotically stable if for any two solutions x(t), y(t) of (1.1) and (1.2), we have

lim_{t→∞}(x(t) − y(t)) = 0.

The global asymptotical behaviours of the solutions of system (1.1) have been studied extensively, particularly for certain special cases of (1.1). The model systems considered are basically of two classes; some models contain an instantaneous (without delay) negative stabilising density dependent feedback which dominates other intraspecific and interspecific interaction effects with and without delays (we call this class as no-pure-delay-type); in the other category such an instantaneous stabilising negative feedback mechanism is not present (we refer this class as puredelay-type). By using either Lyapunov functional method or Razumikhin type function argument, various results on the global asymptotical stability of both types systems have been obtained. For autonomous systems, we refer to Gopalsamy [2] and He and Gopalsamy [8] for the competition systems and He [5, 6] for predatorprey systems. When the systems are nonautonomous, we refer to Gopalsamy and He [3] and He [4, 7] for pure-delay-type competition systems and Kuang [10, 11] and Kuang and Smith [12] for more general systems. It is noticed that the systems studied in Kuang and Smith [12] are required to have a saturated equilibrium. Recently, Kuang [11] droped this unnatural requirement and studied the no-pure-delay-type system (1.1) with τ_{i} = 0, c_{i}(t) = 0 and d_{ij}(t) = 0 for i, j = 1, ⋯, n and t ∈ R. In this case, the corresponding initial condition has the form

x_{i}(s) = ϕ(s),  s ∈ [−τ, 0],  ϕ_{i}(0) > 0,  i = 1, ⋯, n,  (1.3)

with ϕ_{i} ∈ C([−τ, 0], R_{+}) and R_{+} = [0, ∞). Kuang [11] derived the following result (Theorem 2.1, Kuang [11]) on the global asymptotical stability of the solutions.

Theorem A. Assume system (1.1) with τ_{i} = 0, c_{i}(t) = 0 and d_{ij}(t) = 0 (i, j = 1, ⋯, n) satisfies (H_{1}), (H_{2}) and (H_{3}) solutions of (1.1) and (1.3) are eventually uniformly bounded above and

below by positive constant vectors Δ and δ, respectively; (H_{4}) the matrix M = (m_{ij})_{n×n} with

m_{ii} = a_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii}, m_{ij} = − ∑_{l=1}^{l_{ij}} B_{ijl} − K_{ij}, i ≠ j

is an M-matrix. Then system (1.1) is globally asymptotically stable.

The purpose of this paper is to generalise the above Theorem A to the “puredelay-type” system (1.1) with both finite and infinite delays. Some sufficient conditions on the global asymptotical stability of (1.1) are obtained. As a corollary, we show that, for the pure-delay-type system (1.1) with finite delays (τ > 0, c_{i}(t) = 0 and d_{ij} = 0 (i, j = 1, ⋯, n)), if the conditions of Theorem A are satisfied and τ_{i} are sufficiently small, then system (1.1) with (1.3) remains globally asymptotically stable. We also show that similar results can be obtained for systems (1.1) with infinite delays. Consequently, an affirmative answer is given to the two open problems due to Kuang [11]. All the results are established by constructing suitable Lyapunov functionals. It is shown that the Lyapunov functionals method is a very effective tool to the study of the global asymptotical stability of (1.1).

2. Finite delay case.

In this section, we consider system (1.1) with finite delays; that is system (1.1) with c_{i}(t) = 0 and d_{ij}(t) = 0 for i, j = 1, ⋯, n and t ∈ R.

Theorem 2.1. Assume system (1.1) with c_{i}(t) = 0 and d_{ij}(t) = 0 ( i, j = 1, ⋯, n, t ∈ R) satisfies (H_{1}), (H_{2}), (H_{3}) and (H_{5}) the matrix M = (m_{ij})_{n×n} with

\{ \begin{matrix}m_{ii} = & a_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii} \\ − A_{i}τ_{i}\{R_{i}B_{i} + b_{i}(Δ_{i})[A_{i} + ∑_{l=1}^{l_{ii}} B_{iil} + K_{ii}] \\ + B_{i}[A_{i}Δ_{i} + ∑_{j=1}^{n} Δ_{j}( ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij})]\} \\ m_{ij} = & − [1 + A_{i}τ_{i}b_{i}(Δ_{i})][ ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij}],  i ≠ j \end{matrix}  (2.1)

is an M-matrix, where B_{i} = max_{δ_{i}≤s≤Δ_{i}} |b′_{i}(s)|.

Then system (1.1) is globally asymptotically stable.

Remark 2.2. It is noticed that condition (H_{5}) depends on τ_{i} and the ultimate upper bound Δ of the solutions of (1.1), from which, one can derive an estimate on the allowed size of delays for the global asymptotical stability of (1.1). If τ_{i} = 0 for i = 1, ⋯, n, then condition (H_{5}) is reduced to (H_{4}). Thus Theorem 2.1 is a generalisation of Theorem A. When τ_{i} are sufficiently small, we have the following corollary, which gives an affirmative answer to the open problem 1 due to Kuang [11].

Corollary 2.3. For system (1.1) with c_{i}(t) = 0 and d_{ij}(t) = 0 (for i, j = 1, ⋯, n and t ∈ R), if the conditions of Theorem A are satisfied and τ_{i} are sufficiently small, then system (1.1) remains globally asymptotically stable.

Proof of Theorem 2.1. Since M is an M-matrix, we know that (see He and Gopalsamy [8] and Kuang [10]) there exist α_{i} > 0 and ϵ > 0 such that

α_{i}(m_{ii} − ϵ) > ∑_{j=1,j≠i}^{n} α_{j}(|m_{ji}| + ϵ),  i = 1, ⋯, n.  (2.2)

Clearly there is an ϵ_{1} ∈ (0, ϵ) such that

m_{ii} − ϵ < m_{ii}(ϵ_{1}), m_{ij} − ϵ < m_{ij}(ϵ_{1}), i, j = 1, ⋯, n,  (2.3)

where

\begin{matrix}m_{ii}(ϵ_{1}) = & a_{i} − ϵ_{1} − ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) − (K_{ii} + ϵ_{1}) − τ_{i}(A_{i} + ϵ_{1})[(R_{i} + ϵ_{1})B_{i} \\ + b_{i}(Δ_{i})((A_{i} + ϵ_{1}) + ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) + (K_{ii} + ϵ_{1})) \\ + B_{i}(Δ_{i}(A_{i} + ϵ_{1}) + ∑_{j=1}^{n} Δ_{j}( ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1})))] \\ m_{ij}(ϵ_{1}) = & − [1 + τ_{i}b_{i}(Δ_{i})(A_{i} + ϵ_{1})]( ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1})), i ≠ j \end{matrix} \} .

(2.4)

Denote

v_{i}(p, q) = ∫_{q}^{p} \frac{ds}{b_{i}(s)}, i = 1, ⋯, n.

For i = 1, 2, let x^{(i)}(t) = (x_{1}^{(i)}(t), ⋯, x_{n}^{(i)}(t)) be any two solutions of (1.1) and (1.3). Then

\frac{d}{dt}v_{i}(x_{i}^{(1)}(t), x_{i}^{(2)}(t)) = \frac{1}{b_{i}(x_{i}^{(1)}(t))} \frac{d}{dt}x_{i}^{(1)}(t) − \frac{1}{b_{i}(x_{i}^{(2)}(t))} \frac{d}{dt}x_{i}^{(2)}(t)

= −a_{i}(t)[x_{i}^{(1)}(t − τ_{i}) − x_{i}^{(2)}(t − τ_{i})] + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(t)[x_{i}^{(1)}(t − τ_{ijl}) − x_{i}^{(2)}(t − τ_{ijl})]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[x_{j}^{(1)}(t + s) − x_{j}^{(2)}(t + s)]k_{ij}(t, s) ds

= −a_{i}(t)[x_{i}^{(1)}(t) − x_{i}^{(2)}(t)] + a_{i}(t)w_{i}(x^{(1)}(t), x^{(2)}(t))

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(t)[x_{j}^{(1)}(t − τ_{ijl}) − x_{j}^{(2)}(t − τ_{ijl})]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[x_{j}^{(1)}(t + s) − x_{j}^{(2)}(t + s)]k_{ij}(t, s) ds,  (2.5)

where

w_{i}(x^{(1)}(t), x^{(2)}(t)) = [x_{i}^{(1)}(t) − x_{i}^{(2)}(t)] − [x_{i}^{(1)}(t − τ_{i}) − x_{i}^{(2)}(t − τ_{i})],  (2.6)

which can be written as follows

w_{i}(x^{(1)}(t), x^{(2)}(t)) = ∫_{t−τ_{i}}^{t} \frac{d}{du}[x_{i}^{(1)}(u) − x_{i}^{(2)}(u)] du
= ∫_{t−τ_{i}}^{t} {r_{i}(u)[b_{i}(x_{i}^{(1)}(u) − b_{i}(x_{i}^{(2)}(u)]

− a_{i}(t)[b_{i}(x_{i}^{(1)}(u))[x_{i}^{(1)}(u − τ_{i}) − x_{i}^{(2)}(u − τ_{i})]

+ x_{i}^{(2)}(u − τ_{i})[b_{i}(x_{i}^{(1)}(u)) − b_{i}(x_{i}^{(2)}(u))]]

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(u)[b_{i}(x_{i}^{(1)}(u))[x_{j}^{(1)}(u − τ_{ijl}) − x_{j}^{(2)}(u − τ_{ijl})]

+ x_{j}^{(2)}(u − τ_{ijl})[b_{i}(x_{i}^{(1)}(u)) − b_{i}(x_{i}^{(2)}(u))]]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} [b_{i}(x_{i}^{(1)}(u))[x_{j}^{(1)}(u + s) − x_{j}^{(2)}(u + s)]

+ x_{j}^{(2)}(u + s)[b_{i}(x_{i}^{(1)}(u)) − b_{i}(x_{i}^{(2)}(u))]]k_{ij}(t, s) ds} du. (2.7)

From (H_{1}) − (H_{3}), we know that there exists a large T_{1} > 0 such that, for t ≥ T_{1}, δ_{i} ≤ x_{i}^{(1)}(t), x_{i}^{(2)}(t) ≤ Δ_{i}, b_{i}(s) ≤ b_{i}(Δ_{i}) and |b′_{i}(s)| ≤ B_{i} for s ∈ [δ_{i}, Δ_{i}], and a_{i} − ϵ_{1} ≤ a_{i}(t) ≤ A_{i} + ϵ_{1}, |r_{i}(t)| ≤ R_{i} + ϵ_{1}, |b_{ijl}(t)| ≤ B_{ijl} + ϵ_{1} and |k_{ij}(t, s)| ≤ K_{ij}(s) + ϵ_{1}/(1 + τ_{ij}). Let

V_{i1}(x_{i}^{(1)}, x_{i}^{(2)}) = |v_{i}(x_{i}^{(1)}, x_{i}^{(2)})|.  (2.8)

Then it follows from (2.5)-(2.8) that, for t ≥ T_{2} ≥ T_{1}+2τ, the upper right derivative of V_{i1} along the solutions of (1.1) is governed by

\frac{d}{dt}V_{i1}(x_{i}^{(1)}(t), x_{i}^{(2)}(t))
≤ −(a_{i} − ϵ_{1})|x_{i}^{(1)}(t) − x_{i}^{(2)}(t)| + (A_{i} + ϵ_{1})|w_{i}(x_{i}^{(1)}(t), x_{i}^{(2)}(t)|

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|x_{j}^{(1)}(t − τ_{ijl}) − x_{j}^{(2)}(t − τ_{ijl})|

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} |x_{j}^{(1)}(t + s) − x_{j}^{(2)}(t + s)|(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds, (2.9)

where

|w_{i}(x^{(1)}(t), x^{(2)}(t))|
≤ ∫_{t−τ_{i}}^{t} {(R_{i} + ϵ_{1})B_{i}|x_{i}^{(1)}(u) − x_{i}^{(2)}(u)|

+ (A_{i} + ϵ_{1})[b_{i}(Δ_{i})|x_{i}^{(1)}(u − τ_{i}) − x_{i}^{(2)}(u − τ_{i})|

+ Δ_{i}B_{i}|x_{i}^{(1)}(u)) − x_{i}^{(2)}(u)|]

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})[b_{i}(Δ_{i})|x_{j}^{(1)}(u − τ_{ijl}) − x_{j}^{(2)}(u − τ_{ijl})|

+ Δ_{j}B_{i}|x_{i}^{(1)}(u)) − x_{i}^{(2)}(u)|]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} [b_{i}(Δ_{i})|x_{j}^{(1)}(u + s) − x_{j}^{(2)}(u + s)|

+ Δ_{j}B_{i}|x_{i}^{(1)}(u) − x_{i}^{(2)}(u)|](K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds\} du

=^{def} ∫_{t−τ_{i}}^{t} W_{i1}(u) du  (2.10)

in which, the mean-valued theorem is used. Let y_{i}(t) = x_{i}^{(1)}(t) − x_{i}^{(2)}(t), i = 1, ⋯, n,

V_{i2}(x^{(1)}(t), x^{(2)}(t)) = (A_{i} + ϵ_{1}) ∫_{t−τ_{i}}^{t} ∫_{v}^{t} W_{i1}(u) du dv

+ (A_{i} + ϵ_{1})^{2}b_{i}(Δ_{i})τ_{i} ∫_{t−τ_{i}}^{t} |y_{i}(u)| du

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})[b_{i}(Δ_{i})τ_{i}(A_{i} + ϵ_{1}) + 1] ∫_{t−τ_{ijl}}^{t} |y_{j}(u)| du

+ ∑_{j=1}^{n}[1 + (A_{i} + ϵ_{1})b_{i}(Δ_{i})τ_{i}] ∫_{−τ_{ij}}^{0} ∫_{t+s}^{t} |y_{j}(u)| du(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds

(2.11)

and

V_{i}(x^{(1)}(t), x^{(2)}(t)) = (V_{i1} + V_{i2})(x^{(1)}(t), x^{(2)}(t)).  (2.12)

Then V_{i} ≥ 0 and V_{i} is differentiable. For t ≥ T_{3} ≥ T_{2} + 2τ, one have from (2.9)- (2.12) and (2.3), (2.4) that

\frac{d}{dt}V_{i}(x^{(1)}(t), x^{(2)}(t))

≤ − (a_{i} − ϵ_{1})|y_{i}(t)| + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|y_{j}(t)| + ∑_{j=1}^{n}(K_{ij} + ϵ_{1})|y_{j}(t)|

+ τ_{i}(A_{i} + ϵ_{1}){(R_{i} + ϵ_{1})B_{i}|y_{i}(t)| + (A_{i} + ϵ_{1})[b_{i}(Δ_{i}) + Δ_{i}B_{i}]|y_{i}(t)|

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})[b_{i}(Δ_{i})|y_{j}(t)| + Δ_{j}B_{i}|y_{i}(t)|]

+ ∑_{j=1}^{n}[b_{i}(Δ_{i})|y_{j}(t)| + Δ_{j}B_{i}|y_{i}(t)|][K_{ij} + ϵ_{1}]}

={ − (a_{i} − ϵ_{1}) + ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) + (K_{ii} + ϵ_{1})

+ τ_{i}(A_{i} + ϵ_{1})[(R_{i} + ϵ_{1})B_{i} + (A_{i} + ϵ_{1})[b_{i}(Δ_{i}) + Δ_{i}B_{i}]

+ b_{i}(Δ_{i}) ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) + B_{i} ∑_{j=1}^{n} Δ_{j} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})

+ b_{i}(Δ_{i})(K_{ii} + ϵ_{1}) + B_{i} ∑_{j=1}^{n} Δ_{j}(K_{ij} + ϵ_{1})]}|y_{i}(t)|

+ ∑_{j=1,j≠i}^{n} { ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1})

+ τ_{i}(A_{i} + ϵ_{1})b_{i}(Δ_{i})[ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1})]}|y_{j}(t)|

= − ∑_{j=1}^{n} m_{ij}(ϵ_{1})|y_{j}(t)|

≤ − (m_{ii} − ϵ)|y_{i}(t)| + ∑_{j=1,j≠i}^{n}(|m_{ij}| + ϵ)|y_{j}(t)|.  (2.13)

Now we define Lyapunov functional

V (x^{(1)}, x^{(2)}) = ∑_{i=1}^{n} α_{i}V_{i}(x^{(1)}, x^{(2)}).  (2.14)

Then V is well defined and, for t ≥ T_{3}, one can see from (2.13) and (2.14) that

\frac{d}{dt}V (x^{(1)}, x^{(2)})(t) ≤ − ∑_{i=1}^{n} α_{i}[ − (m_{ii} − ϵ)|y_{i}(t)| + ∑_{j=1,j≠i}^{n}(|m_{ij}| + ϵ)|y_{j}(t)|]

= − ∑_{i=1}^{n} (α_{i}(m_{ii} − ϵ) − ∑_{j=1,j≠i}^{n} α_{j}(|m_{ji}| + ϵ))|y_{i}(t)|. (2.15)

Let

β_{i} = α_{i}(m_{ii} − ϵ) − ∑_{j=1}^{n} α_{j}(|m_{ji}| + ϵ).

Then it follows from (2.2) that β_{i} > 0. Integrating (2.15) from T_{3} to t > T_{3}, we have

∑_{i=1}^{n} β_{i} ∫_{T_{3}}^{t} |y_{i}(s)|ds = ∑_{i=1}^{n} β_{i} ∫_{T_{3}}^{t} |x_{i}^{(1)}(s) − x_{i}^{(2)}(s)|ds ≤ V (x^{(1)}, x^{(2)})(T_{3}) < ∞,

which implies that |x_{i}^{(1)}(t) − x_{i}^{(2)}(t)| ∈ L_{1}[T_{3}, ∞). Also, from (H_{3}), |\frac{d}{dt}[x_{i}^{(1)}(t) −x_{i}^{(2)}(t)]| exists and is uniformly bounded for all t ≥ T_{3}. Thus, we have

lim_{t→∞} |x_{i}^{(1)}(t) − x_{i}^{(2)}(t)| = 0, i = 1, ⋯, n.

This completes the proof. ■

Remark 2.4. Following the proof of our Theorem 2.1 and the argument used for Theorem 3.1 in Kuang [11], one can extend our Theorem 2.1 to a more general nonautonomous Kolmogrov type delay systems, we refer to Kuang [11] for details.

3. Infinite delay case.

In this section, we generalise our discussion in section 2 and consider the system (1.1) with both finite and infinite delays. Consequently, we solve the open problem 2 in Kuang [11]. We will first consider a special case of system (1.1) with c_{i}(t) = 0 for i = 1, ⋯, n.

Theorem 3.1. Let c_{i}(t) = 0 (i = 1, ⋯, n) in (1.1). Suppose that

∫_{−∞}^{0} |h_{ij}(s)| ds = 1, ∫_{−∞}^{0} |sh_{ij}(s)| ds < ∞, (i, j = 1, ⋯, n).

If system (1.1) satisfies (H_{1}), (H_{2}), (H_{3}) and (H_{6}) the matrix P = (p_{ij})_{n×n} with

\{ \begin{matrix}p_{ii} = & a_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii} − D_{ii} \\ − A_{i}τ_{i}\{R_{i}B_{i} + b_{i}(Δ_{i})[A_{i} + ∑_{l=1}^{l_{ii}} B_{iil} + K_{ii} + D_{ii}] \\ + B_{i}[Δ_{i}A_{i} + ∑_{j=1}^{n} Δ_{j}( ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij} + D_{ij})]\} \\ p_{ij} = & − [1 + A_{i}τ_{i}b_{i}(Δ_{i})][ ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij} + D_{ij}],  i ≠ j \end{matrix}  (3.1)

is an M-matrix, where B_{i} = max_{δ_{i}≤s≤Δ_{i}} |b′_{i}(s)|;

(H_{7}) There exists a solution x̃(t) = (x̃_{1}(t), ⋯, x̃_{n}(t)) of (1.1) and (1.2) with

δ_{i} ≤ x̃_{i}(t) ≤ Δ_{i} for all t ∈ R and i = 1, ⋯, n, then system (1.1) is globally asymptotically stable.

Remark 3.2. Condition (H_{7}) is needed since infinite delays are allowed in system (1.1). In the case the system (1.1) has a positive saturated steady state, (H_{7}) is obviousely satisfied. In the case when the systems are periodic or almost periodic Lotka-Volterra type systems, condition (H_{7}) can also be removed. In fact, for the almost periodic system (1.1), following the argument used for Lemma 2 in Murakami [14], one can prove that, under condition (H_{3}), system (1.1) has a solution x̃(t) staisfying (H_{7}). Then, employing the argument used in Fink [1], Gopalsamy and He [3] and Seifert [14], one can derive the existence of positive almost periodic solution of (1.1). We refer to Gopalsamy and He [3] and He [4, 7] for the related discussion on pure-delay-type competition systems, where we constructed different Lyapunov functionals and obtained some different results.

Condition (H_{6}) depends only on the finite delays τ_{i}. When τ_{i} (i = 1, ⋯, n) are sufficiently small, we have following corollary.

Corollary 3.3. If condition (H_{6}) in Theorem 3.1 is replaced by that the matrix P̃ = (p̃)_{n×n} with

p̃_{ii} = a_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii} − D_{ii},  p̃_{ij} = −[ ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij} + D_{ij}]  (i ≠ j)

is an M-matrix, then the conclusion of Theorem 3.1 remains true provided delays τ_{i} (i = 1, ⋯, n) are sufficiently small.

Proof of Theorem 3.1. The proof of this theorem is similar to Theorem 2.1 and we indicate it briefly.

Let x^{(1)}(t) be any solution of (1.1) and (1.2) and x^{(2)}(t) be the solution which satisfies condition (H_{7}); that is δ_{i} ≤ x_{i}^{(2)}(t) ≤ Δ_{i} for all t ∈ R and i = 1, ⋯, n. Then it is sufficient to prove that

lim_{t→∞}(x_{i}^{(1)}(t) − x_{i}^{(2)}(t)) = 0.

Let v_{i}(p, q) be defined as in section 2 and denote

v_{i}(x_{i}(t)) = v_{i}(x_{i}^{(1)}(t), x_{i}^{(2)}(t)),  y_{i}(t) = x_{i}^{(1)}(t) − x_{i}^{(2)}(t),
B_{i}(y_{i}(t)) = b_{i}(x_{i}^{(1)}(t)) − b_{i}(x_{i}^{(2)}(t)),  i = 1, ⋯, n.  (3.2)

Then, from (1.1) and (3.2),

\frac{d}{dt}v_{i}(x_{i}(t)) = − a_{i}(t)y_{i}(t) + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(t)y_{i}(t − τ_{ijl}) + ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} y_{j}(t + s)k_{ij}(t, s) ds
+ ∑_{j=1}^{n} d_{ij}(t) ∫_{−∞}^{0} y_{j}(t + s)h_{ij}(s) ds + a_{i}(t)w_{i}(y_{i}(t))  (3.3)

with

w_{i}(y_{i}(t)) = ∫_{t−τ_{i}}^{t} y′_{i}(u) du =^{def} ∫_{t−τ_{i}}^{t} W_{i2}(u) du  (3.4)

and

W_{i2}(u) =r_{i}(u)B_{i}(y_{i}(u)) − a_{i}(t)[b_{i}(x_{i}^{(1)}(u))y_{i}(u − τ_{i}) + x_{i}^{(2)}(u − τ_{i})B_{i}(y_{i}(u))]

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(u)[b_{i}(x_{i}^{(1)}(u))y_{j}(u − τ_{ijl}) + x_{j}^{(2)}(u − τ_{ijl})B_{i}(y_{i}(u))]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[b_{i}(x_{i}^{(1)}(u))y_{j}(u + s) + x_{j}^{(2)}(u + s)B_{i}(y_{i}(u))]k_{ij}(t, s) ds

+ ∑_{j=1}^{n} d_{ij}(u) ∫_{−∞}^{0}[b_{i}(x_{i}^{(1)}(u))y_{j}(u + s) + x_{j}^{(2)}(u + s)B_{i}(y_{i}(u))]h_{ij}(s) ds.

(3.5)

Let V_{i1} be defined by (2.8), then it follows from (3.3)-(3.5) that, for all large t,

\frac{d}{dt}V_{i1}(t) ≤ − (a_{i} − ϵ_{1})|y_{i}^{(1)}(t)| + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|y_{j}(t − τ_{ijl})|

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} |y_{j}(t + s)|(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0} |y_{j}(t + s)||h_{ij}(s)|ds

+ (A_{i} + ϵ_{1}) ∫_{t−τ_{i}}^{t} |W_{i2}(u)| du.  (3.6)

Define

V_{i3}(y_{i}(t)) = ∫_{t−τ_{i}}^{t} ∫_{v}^{t} |W_{i2}(u)|dudv.  (3.7)

Noting that (from (H_{3})), for all large t, x_{i}^{(1)}(t), x_{i}^{(2)}(t) ≤ Δ_{i}, b_{i}(x_{i}^{(1)}(t)) ≤ b_{i}(Δ_{i}) and |B_{i}(y_{i}(t))| ≤ B_{i}|y_{i}(t)| (i = 1, ⋯, n). From (3.5)-(3.7) and (H_{3}), one can see that, for all large t,

\frac{d}{dt}(V_{i1} + V_{i3})(y_{i}(t))
=\frac{d}{dt}V_{i1}(y_{i}(t)) + (A_{i} + ϵ_{1})[ − ∫_{t−τ_{i}}^{t} |W_{i2}(u)| du + τ_{i}|W_{i2}(t)|]

≤ − (a_{i} − ϵ_{1})|y_{i}^{(1)}(t)| + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|y_{j}(t − τ_{ijl})|

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} |y_{j}(t + s)|(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0} |y_{j}(t + s)||h_{ij}(s)|ds

+ (A_{i} + ϵ_{1})τ_{i}{(R_{i} + ϵ_{1})B_{i}|y_{i}(t)| + (A_{i} + ϵ_{1})[b_{i}(Δ_{i})|y_{i}(t − τ_{i})| + Δ_{i}B_{i}|y_{i}(t)|]

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl}(t) + ϵ_{1})[b_{i}(Δ_{i})|y_{j}(t − τ_{ijl})| + Δ_{j}B_{i}|y_{i}(t)|]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[b_{i}(Δ_{i})|y_{j}(t + s)| + Δ_{j}B_{i}|y_{i}(t)|](K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0}[b_{i}(Δ_{i})|y_{j}(u + s)| + x_{j}^{(2)}(t + s)B_{i}|y_{i}(t)|]|h_{ij}(s)| ds}.

(3.8)

For the last term in (3.8), we use (H_{7}) and have x_{j}^{(2)}(t + s) ≤ Δ_{j} for t + s ∈ R.

Note that ∫_{−∞}^{0} |h_{ij}(s)|ds = 1(i, j = 1, ⋯, n). Thus, for all large t,

\frac{d}{dt}(V_{i1} + V_{i3})(y_{i}(t))
≤ − (a_{i} − ϵ_{1})|y_{i}^{(1)}(t)|

+ (A_{i} + ϵ_{1})τ_{i}{(R_{i} + ϵ_{1})B_{i}|y_{i}(t)| + (A_{i} + ϵ_{1})[b_{i}(Δ_{i})|y_{i}(t − τ_{i})| + Δ_{i}B_{i}|y_{i}(t)|]

+ B_{i} ∑_{j=1}^{n} Δ_{j}[ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1}) + (D_{ij} + ϵ)]|y_{i}(t)|}

+ [1 + (A_{i} + ϵ_{1})τ_{i}b_{i}(Δ_{i})] ∑_{j=1}^{n} [ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|y_{j}(t − τ_{ijl})|

+ ∫_{−τ_{ij}}^{0} |y_{j}(t + s)|(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}})ds

+ (D_{ij} + ϵ_{1}) ∫_{−∞}^{0} |y_{j}(t + s)||h_{ij}(s)|ds].  (3.9)

Let

V_{i}(y_{i}(t)) = (V_{i1} + V_{i3} + V_{i4})(y(t))  (3.10)

with

V_{i4}(y(t)) =(A_{i} + ϵ_{1})^{2}τ_{i}b_{i}(Δ_{i}) ∫_{t−τ_{i}}^{t} |y_{i}(u)| du

+ [1 + (A_{i} + ϵ_{1})τ_{i}b_{i}(Δ_{i})] ∑_{j=1}^{n} [ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) ∫_{t−τ_{ijl}}^{t} |y_{j}(u)| du

+ ∫_{−τ_{ij}}^{0} ( ∫_{t+s}^{t} |y_{j}(u)|du)(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}})ds

+ (D_{ij} + ϵ_{1}) ∫_{−∞}^{0} ( ∫_{t+s}^{t} |y_{j}(u)| du)|h_{ij}(s)|ds].  (3.11)

Then, for all large t,

\frac{d}{dt}V_{i}(y(t)) ≤ −(a_{i} − ϵ_{1})|y_{i}^{(1)}(t)|
+ (A_{i} + ϵ_{1})τ_{i}{[(R_{i} + ϵ_{1})B_{i} + (A_{i} + ϵ_{1})[b_{i}(Δ)_{i} + Δ_{i}B_{i}]]

+ B_{i} ∑_{j=1}^{n} Δ_{j}[ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1}) + (D_{ij} + ϵ)]}|y_{i}(t)|

+ [1 + (A_{i} + ϵ_{1})τ_{i}b_{i}(Δ_{i})] ∑_{j=1}^{n} [ ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) + (K_{ij} + ϵ_{1})

+ (D_{ij} + ϵ_{1})]|y_{j}(t)|.  (3.12)

Now, let

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

Then, similar to the last part of the proof of Theorem 2.1, we can derive the conclusion. This completes the proof. ■

Note that condition (H_{6}) depends only on the finite delays τ_{i}, not the infinite delays. Therefore, the terms with infinite delays are dominated by the terms with finite delays τ_{i}. One can see from the following Theorem 3.4 that this is no longer the case when there is no such dominated terms with finite delays.

Theorem 3.4. Let a_{i}(t) = 0 (i = 1, 2, ⋯, n) in system (1.1). Assume that h_{i}(s) ≥ 0 for s ∈ (−∞, 0] and, for i = 1, ⋯, n,

∫_{−∞}^{0} h_{i}(s) ds = 1,  σ_{i} = ∫_{−∞}^{0} |s|h_{i}(s) ds < ∞,  ∫_{−∞}^{0} s^{2}h_{i}(s) ds < ∞.

If system (1.1) satisfies (H_{1}) − (H_{3}), (H_{7}) and (H_{8}) the matrix Q = (q_{ij})_{n×n} with

\{ \begin{matrix}q_{ii} & = c_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii} − D_{ii} \\ & − C_{i}σ_{i}[R_{i}B_{i} + b_{i}(Δ_{i})(C_{i} + ∑_{l=1}^{l_{ii}} B_{iil} + K_{ii} + D_{ii}) \\ & + B_{i}(C_{i}Δ_{i} + ∑_{j=1}^{n} Δ_{j}[ ∑_{l=1}^{l_{ij}} B_{ijl} + (K_{ij} + D_{ij})])]; \\ q_{ij} & = −[1 + C_{i}σ_{i}b_{i}(Δ_{i})][ ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij} + D_{ij}],  i ≠ j \end{matrix}

is an M-matrix, where B_{i} = max_{δ_{i}≤s≤Δ_{i}} |b′_{i}(s)|,

then system (1.1) is globally asymptotically stable.

Proof. Similar to the proof of Theorem 3.1, one can derive from (1.1) and (3.2) that

\frac{d}{dt}v_{i}(x_{i}(t)) = −c_{i}(t)y_{i}(t) + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(t)y_{j}(t − τ_{ijl}) + ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} y_{j}(t + s)k_{ij}(t, s)ds

+ ∑_{j=1}^{n} d_{ij}(t) ∫_{−∞}^{0} y_{j}(t + s)h_{ij}(s) ds + c_{i}(t) ∫_{−∞}^{0} ( ∫_{t+v}^{t} y′_{i}(u)du)h_{i}(v)dv

(3.13)

and

∫_{−∞}^{0} ( ∫_{t+v}^{t} y′_{i}(u)du)h_{i}(v)dv = ∫_{−∞}^{0} ( ∫_{t+v}^{t} W_{i3}(u)du)h_{i}(v)dv

with

W_{i3}(u) =r_{i}(u)B_{i}(y_{i}(u))

− c_{i}(u) ∫_{−∞}^{0}[b_{i}(x_{i}^{(1)}(u)y_{i}(u + s) + x_{i}^{(2)}(u + s)B_{i}(y_{i}(u))]h_{i}(s) ds

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}} b_{ijl}(u)[b_{i}(x_{i}^{(1)}(u)y_{j}(u − τ_{ijl}) + x_{j}^{(2)}(u − τ_{ijl})B_{i}(y_{i}(u))]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[b_{i}(x_{i}^{(1)}(u)y_{j}(u + s) + x_{j}^{(2)}(u + s)B_{i}(y_{i}(u))]k_{ij}(u, s)ds

+ ∑_{j=1}^{n} d_{ij}(u) ∫_{−∞}^{0}[b_{i}(x_{i}^{(1)}(u)y_{j}(u + s) + x_{j}^{(2)}(u + s)B_{i}(y_{i}(u))]h_{ij}(s)ds.

(3.14)

Define V_{i1} as in (2.8), we have from (3.13) and (3.14) that, for all large t,

\frac{d}{dt}V_{i1}(y_{i}(t)) ≤ −(c_{i} − ϵ_{1})|y_{i}(t)| + ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})|y_{j}(t − τ_{ijl})|

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} |y_{j}(t + s)|(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}})ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0} |y_{j}(t + s)||h_{ij}(s)| ds

+ (C_{i} + ϵ_{1}) ∫_{−∞}^{0} ( ∫_{t+v}^{t} |W_{i3}(u)du)h_{i}(v)dv  (3.15)

From (H_{3}), we know that, for all large t, x_{i}^{(1)}(t), x_{i}^{(2)}(t) ≤ Δ_{i}, b_{i}(x_{i}^{(1)}(t)) ≤ b_{i}(Δ_{i}) and |B_{i}(y_{i}(t))| ≤ B_{i}|y_{i}(t)| (i = 1, ⋯, n). Then, using (H_{7}), one can verify that, for all large t,

|W_{i3}(t)| ≤(R_{i} + ϵ_{1})B_{i}|y_{i}(t)|

+ (C_{i} + ϵ_{1}) ∫_{−∞}^{0}[b_{i}(Δ_{i})|y_{i}(t + s)| + Δ_{i}B_{i}|y_{i}(t)|]h_{i}(s) ds

+ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1})[b_{i}(Δ_{i})|y_{j}(t − τ_{ijl})| + Δ_{j}B_{i}|y_{i}(t)|]

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0}[b_{i}(Δ_{i})|y_{j}(t + s)| + Δ_{j}B_{i}|y_{i}(t)|](K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}} ) ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0}[b_{i}(Δ_{i})|y_{j}(t + s)| + Δ_{j}B_{i}|y_{i}(t)|]|h_{ij}(s)| ds.

Let

V_{i}(y)(t) = (V_{i1} + V_{i5})(y)(t)

with

V_{i5}(y(t)) = [1 + (C_{i} + ϵ_{1})σ_{i}b_{i}(Δ_{i})]{ ∑_{j=1}^{n} ∑_{l=1}^{l_{ij}}(B_{ijl} + ϵ_{1}) ∫_{t−τ_{ijl}}^{t} |y_{j}(u)|du

+ ∑_{j=1}^{n} ∫_{−τ_{ij}}^{0} ( ∫_{t+s}^{t} |y_{j}(u)|du)(K_{ij}(s) + \frac{ϵ_{1}}{1 + τ_{ij}}) ds

+ ∑_{j=1}^{n}(D_{ij} + ϵ_{1}) ∫_{−∞}^{0} ( ∫_{t+s}^{t} |y_{j}(u)|du)|h_{ij}(s)|ds}

+ (C_{i} + ϵ_{1})^{2}σ_{i}b_{i}(Δ) ∫_{−∞}^{0} ( ∫_{t+s}^{t} |y_{i}(u)|)h_{i}(s) ds

+ (C_{i} + ϵ_{1}) ∫_{−∞}^{0} ( ∫_{t+v}^{t} ∫_{w}^{t} |W_{i3}(u)| du dw)h_{i}(v) d.,  (3.16)

Then we have from (3.16) that, for all large t,

\frac{d}{dt}V_{i}(y)(t) ≤\{ − (c_{i} − ϵ_{1}) + ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) + (K_{ii} + ϵ_{1}) + (D_{ii} + ϵ_{1})

+ (C_{i} + ϵ_{1})σ_{i}[(R_{i} + ϵ_{1})B_{i} + b_{i}(Δ_{i})((C_{i} + ϵ_{1})

+ ∑_{l=1}^{l_{ii}}(B_{iil} + ϵ_{1}) + (K_{ii} + ϵ_{1}) + (D_{ii} + ϵ_{1}))

+ B_{i}(Δ_{i}(C_{i} + ϵ_{1}) + ∑_{j=1}^{n} Δ_{j} ∑_{l=1}^{l_{ij}}[(B_{ijl} + ϵ_{1})

+ (K_{ij} + ϵ_{1}) + (D_{ij} + ϵ_{1})])]}|y_{i}(t)|

+ ∑_{j=1,j≠i}^{n}[1 + (C_{i} + ϵ_{1})σ_{i}b_{i}(Δ_{i})][ ∑_{l=1}^{l_{ij}}[(B_{iil} + ϵ_{1}) + (K_{ij} + ϵ_{1})

+ (D_{ij} + ϵ_{1})]]|y_{j}(t)|.  (3.17)

From (3.17), one can use (H_{8}) and the same argument as in Theorems 2.1 and 3.1 to show that the system (1.1) is globally asymptotically stable. ■

As indicated in Remark 3.2, condition (H_{7}) in Theorem 3.4 can be removed when (1.1) is an almost periodic system. One can see that condition (H_{8}) depends only on σ_{i} = ∫_{−∞}^{0} |s|h_{i}(s) ds (i = 1, ⋯, n), which measure the size of the infinite delays. When they are sufficiently small, we have the following corollary.

Corollary 3.5. If condition (H_{8}) in Theorem 3.4 is replaced by that the matrix Q̃ = (q̃)_{n×n} with

q̃_{ii} = c_{i} − ∑_{l=1}^{l_{ii}} B_{iil} − K_{ii} − D_{ii},  q̃_{ij} = −[ ∑_{l=1}^{l_{ij}} B_{ijl} + K_{ij} + D_{ij}]  (i ≠ j)

is an M-matrix, then the conclusion of Theorem 3.4 remains true provided delays σ_{i} (i = 1, ⋯, n) are sufficiently small.

4. Discussion.

By constructing suitable Lyapunov functionals in this paper, we are able to generalise the results in Kuang [11] to the pure-delay-type system (1.1) and establish the global asymptotical stability of (1.1). Consequently, we give affirmative answer to the two open problems in Kuang [11]. As in Kuang [11], we can also apply our results to some nonautonomous pure-delay-type Lotka-Volterra competition and predator-prey systems. Our conditions are different from that obtained in Kuang [11] on the no-pure-delay-type systems. In Kuang [11], the conditions are generally delay independent, which characterises the systems of no-pure-delay-type; while for the pure-delay-type systems, the conditions are delay dependent. The size of the allowed delays to maintain the global asymptotical stability of the systems is relied on the estimates of the ultimate upper bounds of the solutions. For Lotka-Volterra competition and prey-predator systems of pure-delay-type, some estimates on the ultimate upper bound of the solution can be done explicitely; see Gopalsamy [2], Gopalsamy and He [3], He [4, 5] and Kuang [10]. However, for Lotka-Volterra cooperation systems of pure-delay-type, to prove the uniform persistence and to find the ultimate upper bounds are still unsolved problems. With the estimation on the upper and lower bounds of the solutions of the systems, we could locate the attractive region of the solutions and, when the systems are periodic or almost periodic, indicate the region for the attractive periodic and almost periodic solutions. We leave this as our further study.

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