On permanence of all subsystems of competitive Lotka–Volterra systems with delays

Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301

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On permanence of all subsystems of competitive Lotka–Volterra systems with delays Zhanyuan Hou ∗ Faculty of Computing, London Metropolitan University, North Campus, 166-220 Holloway Road, London N7 8DB, UK

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Article history: Received 21 January 2010 Accepted 1 May 2010

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Permanence Lotka–Volterra Competitive systems Distributed delays Perturbed systems

1. Introduction In the context of differential equations, a class of systems of Lotka–Volterra type differential equations have the form x0i (t ) = xi (t )fi (t , x),

i ∈ IN ,

(1)

where, for any positive integer m, Im = {1, 2, . . . , m} and for a fixed positive integer N and each i ∈ IN , fi is vaguely described as a continuous function of (t , x) (without delays) or functional (with finite or infinite delays). Since a particular case of (1) was established and analyzed in the 1920s as a mathematical model of population growth of a community of N species, where xi (t ) denotes the population size of the ith species at time t, the study of such systems is mainly restricted to the nonnegative cone

RN+ = {x ∈ RN : ∀i ∈ IN , xi ≥ 0}. In particular, solutions of (1) are mostly considered in the interior of RN+ , int RN+ = {x ∈ RN+ : ∀i ∈ IN , xi > 0}. Ecologically, one concern about the community is whether all species can coexist uniformly, i.e. each species has an ultimate positive lower bound and an upper bound regardless of its starting population. This is mathematically gauged by the concept of permanence. A system is called permanent if there are r > 0 and M > r such that every solution of the system in int RN+ satisfies r ≤ xi (t ) ≤ M for all i ∈ IN and all sufficiently large t.

∗

Tel.: +44 20 71334582. E-mail address: [email protected].

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.05.015

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Ahmad and Lazer [1] (or [2]) showed by an example that a permanent system may have non-permanent subsystems (see Section 2 for another example). However, in some situation, a condition for permanence of a system also ensures permanence of all subsystems. Earlier observations on this point can be found in [3]. The main focus of this paper is on a condition for all subsystems to be permanent. The simplest instance of (1) is the autonomous Lotka–Volterra system of ordinary differential equations having the form x0i = xi (bi − Ai x),

i ∈ IN ,

(2)

where Ai is the ith row of a matrix A = (aij ) and the bi and aij are constant real numbers. For (2) to be permanent, sufficient conditions as well as necessary conditions can be found in [4,5]. For autonomous systems of the form x0i = xi fi (x), Schreiber [6] investigated C r robust permanence (i.e., the permanence of the system and its sufficiently small C r perturbations). Based on this theory, Mierczyński and Schreiber [7] obtained necessary and sufficient conditions for all subsystems of a Kolmogorov system (including the system itself) to be (C 1 ) robustly permanent. The following theorem is the result of a simple application of one of the conditions to (2). Theorem 1 ([7, Corollary 3.1]). Assume (2) is dissipative. Then all subsystems of (2) are robustly permanent if and only if for every I ⊂ IN there is a unique equilibrium x∗I in CI0 = {x ∈ RN+ : xj > 0 if and only if j ∈ IN \ I }

(3)

such that Ai x∗I < bi for all i ∈ I. Parallel to system (2), the nonautonomous instances of (1) without delays are x0i (t ) = xi (t )[ri (t ) − Ai x(t )],

i ∈ IN

(4)

and x0i (t ) = xi (t )[˜ri (t ) − Ai (t )x(t )],

i ∈ IN ,

(5)

where the ri (t ) and r˜i (t ) are continuous functions from R0 = (−∞, ∞), (c , ∞) or [c , ∞) for some c ∈ R to R, the Ai are the same as in (2) and the Ai (t ) = (ai1 (t ), . . . , aiN (t )) are continuous from R0 to RN+ . Although there are many investigations dealing with permanence related problems of (4), (5) and their variations, we only quote [8,9,1,10–15] and the references therein as examples, with most of them providing sufficient or necessary conditions for permanence. A commonly used technique is the employment of the average m(g , t1 , t2 ) =

Z

1 t2 − t1

t2

g (t )dt

(6)

t1

of a function g. Ahmad and Lazer [11] investigated the permanence of competitive systems (4) and (5) under the assumptions that the ri (t ), r˜i (t ) and aii (t ) have positive lower and upper bounds, that the aij (t ) are bounded and nonnegative, and that r : R0 → RN+ satisfies

∀ i ∈ IN ,

lim m(ri , t0 , t ) = bi uniformly for t0 ∈ R0 .

t →∞

(7)

It is noted from [11] that a large class of functions, including periodic and almost periodic functions, satisfy (7). System (5) can be viewed as an ε -perturbation of (4) if

∀i, j ∈ IN , ∀t ∈ R0 ,

|˜ri (t ) − ri (t )| ≤ ε,

|aij (t ) − aij | ≤ ε.

(8)

Ahmad and Lazer [11] introduced the concept of total permanence of (4) as the existence of ε > 0 such that all subsystems of (5) satisfying (8) are permanent. Obviously, the example given in [1] or [2] shows that permanence does not imply total permanence. As an analogue of Theorem 1 for (2), we have the following nice result for (4) and (5). Theorem 2 ([11]). System (4) with (7), b = (b1 , . . . , bN )T ∈ int RN+ , A ∈ RN+×N , aii > 0 for i ∈ IN is totally permanent if and only if b and A satisfy the (I − J )-conditions (to be defined in Section 3). The aim of this paper is to investigate permanence of competitive Lotka–Volterra systems with delays and extend Theorems 1 and 2 to such systems. As another instance of (1) we consider the following autonomous system with distributed delays,

" x0i (t ) = xi (t ) bi −

N X j=1

Z

0

aij

# xj (t + θ )dξij (θ ) ,

i ∈ IN ,

(9)

−τ

where the bi and aij are real valued constants, τ > 0, and the ξij are nondecreasing functions from any open interval containing [−τ , 0] to R+ satisfying ξij (0+ ) − ξij (−τ − ) = 1 for all i, j ∈ IN . For some particular cases of (9) when N = 2, some necessary, sufficient or necessary and sufficient conditions for the permanence of (9) can be found in [16–22] and the references therein. Kuang observed the relationship between (2) and (9) and made the conjecture that system (9) is permanent if and only if (2) is permanent. Chen, Lu and Wang [21] showed by examples that this conjecture is not true for

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cooperative systems. Is it true for competitive systems and predator–prey systems? We cannot answer this question directly but, indirectly, we shall see that the property of (2) secured by Theorem 1 is not affected by the delays in (9). Finally, another large class of systems as instances of (1) are nonautonomous systems with delays. Consider the systems

" xi (t ) = xi (t ) ri (t ) − 0

N X

Z

#

0

xj (t + θ )dξij (θ ) ,

aij

i ∈ IN

(10)

−τ

j =1

and

" xi (t ) = xi (t ) r˜i (t ) − 0

N Z X j =1

#

0

xj (t + θ )dθ ηij (t , θ ) ,

i ∈ IN ,

(11)

−τ

where the ri (t ) and r˜i (t ) are the same as in (4) and (5), the aij and ξij are the same as in (9), and each ηij (t , θ ) is of bounded

R0

variation in θ ∈ [−τ , 0] for each fixed t ∈ R0 and −τ ϕ(θ )dθ ηij (t , θ ) is continuous in t for each fixed ϕ ∈ C ([−τ , 0], R). Various conditions for permanence of (10), (11) or systems of other forms with delays can be found in [23–25,22,26–29] and the related references quoted there. In this paper, we shall extend Theorem 2 for (4) to systems (10) and (11). This will effectively show that the property of (4) described by Theorem 2 is independent of the delays in (10) and (11). 2. Another example of a permanent system with a non-permanent subsystem It is known from [1] or [2] that a permanent system may have non-permanent subsystems. For convenience, we present another example in this section. Consider system (2) with N = 3, A=

1 0.9 0.8

0.5 1 0.9

2 det(A + A ) = 1.4 1.7

1.4 2 1.2

1.7 1.2 = 0.52 > 0 1.7

1 1 , 1

!

b=

0.9 0.3 . 0.85

!

(12)

Since T



and the principal minors of A + AT are all positive, A + AT is positive definite. As x∗ = ( 430 , 230 , 120 )T is an equilibrium of 653 653 653 ∗ 3 (2) with (12), by [17, Theorem 3.2.1] x is globally asymptotically stable in int R+ so the system is permanent. Clearly, (2) with (12) has the following subsystem x01 = x1 (1 − x1 − 0.9x3 ),

x03 = x3 (1 − 0.8x1 − 0.85x3 ).

From a phase portrait of (13) we see that (x1 , x3 ) = (0,

20 17

(13)

) is a global attractor of (13) in int R+ so (13) is not permanent. 2

3. Main results We shall rewrite the equations appeared in Section 1 for convenience. First we consider autonomous competitive Lotka–Volterra systems with delays having the form

" xi (t ) = xi (t ) bi − 0

N X

Z

#

0

xj (t + θ )dξij (θ ) ,

aij

i ∈ IN ,

(14)

−τ

j =1

where τ ≥ 0, the ξij are nondecreasing satisfying

∀i, j ∈ IN ,

ξij (0+ ) − ξij (−τ − ) = 1,

(15)

and

∀i, j ∈ IN ,

bi > 0,

aii > 0,

aij ≥ 0.

(16)

For any nonempty subset J ⊂ IN , the system

" xi (t ) = xi (t ) bi − 0

X

Z

xj (t + θ )dξij (θ ) ,

aij

j∈J

#

0

i∈J

(17)

−τ

is a |J |-dimensional subsystem of (14). Let

πi = {x ∈ RN+ : xi = 0}, γi = {x ∈ R+ : Ai x = bi }, N

i ∈ IN , i ∈ IN .

(18) (19)

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Then each solution of (17) is formed by the components xj (t ), j ∈ J, of a solution x(t ) of (14) in

T

i∈IN \J

πi . For the same

b ∈ int RN+ and A = (aij ) ∈ RN+×N as in (14), the autonomous competitive Lotka–Volterra system without delays can be written as x0i = xi (bi − Ai x),

i ∈ IN ,

(20)

where Ai = (ai1 , . . . , aiN ) is the ith row of A. Definition 1. A vector b ∈ int RN+ and a matrix A ∈ RN+×N are said to satisfy the (I − J )-conditions if for each proper subset I ⊂ IN with J = IN \ I, if there exist x∗i > 0 for all i ∈ I satisfying

∀i ∈ I ,

X

aik x∗k = bi

(21)

k∈I

then they also satisfy

∀j ∈ J ,

X

ajk x∗k < bj .

(22)

k∈I

Remark 1. Under the assumption (16), we claim that b and A satisfy the (I − J )-conditions if and only if (20) satisfies Condition (C): For each proper subset J ⊂ IN , system (20) has a unique equilibrium x∗J in CJ0 (defined by (3)) such that

∀j ∈ J ,

Aj x∗J < bj .

(23)

Indeed, if (20) satisfies Condition (C) then each x∗J in CJ0 satisfies both (21) and (22) so b and A satisfy the (I − J )-conditions. On the other hand, for each i ∈ IN , x∗i = a i > 0 and satisfies aii x∗i = bi . By the (I − J )-conditions, x∗i also satisfies aji x∗i < bj ii for all j ∈ IN \ {i}. Then, for I = {1, 2}, by applying [11, Lemma 2.7] to the system a11 x1 + a12 x2 = b1 , a21 x1 + a22 x2 = b2 ∗ ∗ T we know that it has a unique solution (x1 , x2 ) ∈ int R2+ , i.e. (21) holds. By the (I − J )-conditions, (x∗1 , x∗2 ) also satisfies (22). Repeatedly using [11, Lemma 2.7], we see that for each proper I ⊂ IN with J = IN \ I, there is a unique x∗J ∈ CJ0 satisfying both (21) and (22) so x∗J is an equilibrium of (20) satisfying (23). Therefore, (20) satisfies Condition (C). b

To interpret the (I − J )-conditions or Condition (C) geometrically, we note that γi defined by (19) is the ith nullcline plane of (20) in RN+ and πi defined by (18) is the ith coordinate plane. For any (N − 1)-dimensional plane γ , if the origin 0 6∈ γ , then γ divides RN+ into three mutually exclusive connected sets γ − , γ and γ + with 0 ∈ γ − . A point x ∈ RN+ is said to be below (on or above) γ if x ∈ γ − (x ∈ γ or x ∈ γ + ). A set S is said to be below (on or above) γ if every point in S is so. Remark 2. Condition (C) has a clear geometric explanation: for each i ∈ IN every equilibrium of (20) on πi is below γi . Indeed, under this geometric condition, the author [30, Lemma 5] showed the truth of Condition (C) and the existence of the unique equilibrium x∗ of (20) in int RN+ , i.e. x∗ = A−1 b ∈ int RN+ (see also [11, Lemma 2.7] for the last bit). This shows that, under assumption (16), the (I − J )-conditions, Condition (C) and the condition of Theorem 1 are all equivalent. In [30] the author obtained a sufficient condition (Condition (C) plus an extra) for every subsystem of (20) to have a globally attractive equilibrium and further conjectured that Condition (C) alone might be enough to guarantee the global attractivity of all subsystems. This is true when N = 2 and N = 3. But since Condition (C) will be a necessary and sufficient condition for the permanence of all subsystems of (20), this indirectly suggests that the above conjecture is not true for general N-dimensional systems. This is confirmed by examples in [31]. Theorem 3. All subsystems of (14) are permanent if and only if b and A satisfy the (I − J )-conditions. Remark 3. When N = 2, the (I − J )-conditions become b1 a22 > b2 a12 ,

b2 a11 > b1 a21 .

(24)

By Lemmas 1 and 2 given in the next section, every one-dimensional subsystem of (14) is always permanent regardless of the (I − J )-conditions under the assumption (16). Thus, when N = 2, all subsystems of (14) are permanent if and only if (14) is permanent. Further, Theorem 3 is simplified as ‘‘(14) is permanent if and only if (24) hold’’. This is consistent with a result given in [20]. Since (20) can be viewed as a special case of (14) with τ = 0, by Theorem 1 and Remarks 1 and 2 we immediately obtain the following. Corollary 1. The following statements are equivalent. (i) (ii) (iii) (iv)

The b and A satisfy the (I − J )-conditions. All subsystems of (14) are permanent. All subsystems of (20) are permanent. All subsystems of (20) are robustly permanent.

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From Corollary 1 we see that the sizes and the ways of distribution of delays do not affect the permanence property of all subsystems of (20). When all subsystems of (14) are permanent, a small perturbation on b and A does not affect this property of (14). Next we consider nonautonomous competitive Lotka–Volterra systems

" xi (t ) = xi (t ) ri (t ) − 0

N X

Z

#

0

xj (t + θ )dξij (θ ) ,

aij

i ∈ IN

(25)

−τ

j =1

and

" xi (t ) = xi (t ) r˜i (t ) − 0

N Z X j =1

#

0

xj (t + θ )dθ ηij (t , θ ) ,

i ∈ IN ,

(26)

−τ

where the aij and ξij are the same as in (14), the ri (t ) and r˜i (t ) are continuous satisfying

∀ t ∈ R 0 , ∀ i ∈ IN ,

0 < riL ≤ ri (t ) ≤ riM < ∞,

∀t ∈ R0 , ∀i ∈ IN , 0 < r˜iL ≤ r˜i (t ) ≤ r˜iM < ∞, ∀i ∈ IN , lim m(ri , t0 , t ) = bi > 0 uniformly for t0 ∈ R0 . t →∞

(27) (28) (29)

We assume that the ηij (t , θ ) are nondecreasing in θ ∈ [−τ , 0] for each fixed t ∈ R0 satisfying

∀ i ∈ IN , ∀i, j ∈ IN ,

0 < αii ≤ aii (t ) = ηii (t , 0+ ) − ηii (t , −τ − ) ≤ βii < ∞, 0 ≤ aij (t ) = ηij (t , 0 ) − ηij (t , −τ ) ≤ βij < ∞, +

−

(30) (31)

R0

and that the aij (t ) and −τ ϕ(θ )dθ ηij (t , θ ) for each fixed ϕ ∈ C ([−τ , 0], R) are continuous in t. The corresponding systems without delays are x0i (t ) = xi (t )[ri (t ) − Ai x(t )],

i ∈ IN

(32)

and, with Ai (t ) = (ai1 (t ), . . . , aiN (t )), x0i (t ) = xi (t )[˜ri (t ) − Ai (t )x(t )],

i ∈ IN .

(33)

Theorem 4. Assume that system (25) satisfies (27), (29) and (15) . Then all subsystems of (25) and its small perturbations on A are permanent if and only if b and A satisfy the (I − J )-conditions. Note that (33) can be viewed as an ε -perturbation of (32) if (8) holds and that (32) is totally permanent if there exists an ε > 0 such that all subsystems of an ε -perturbation of (32) are permanent. Since (32) can be viewed as a special case of (25) with τ = 0, the corollary below follows from Theorem 2. Corollary 2. The following statements are equivalent. (i) The b and A satisfy the (I − J )-conditions. (ii) All subsystems of (25) and its small perturbations on A are permanent. (iii) System (32) is totally permanent. From Corollary 2 it is clear that the sizes and the ways of distribution of delays in (25) do not affect the permanence property of all subsystems of (32). Moreover, when this property holds for (25), a small perturbation on r (t ) and A in the sense of (8) does not affect this property. However, in general, it is not appropriate to define (26) as an ε -perturbation of (25) by (8). Definition 2. System (26) is called an ε -perturbation of (25) if every solution x(t ) of (26) in RN+ satisfies, for every i ∈ IN ,

N Z 0 X
lim sup r˜i (t ) − ri (t ) + xj (t + θ )dθ aij ξij (θ ) − ηij (t , θ ) < ε. t →∞ j=1 −τ Note that this definition not only includes perturbations on r (t ) and A in the sense of (8) but also incorporate perturbations on delays such as xi (t − τij (t )). This will be demonstrated by an example in Section 7. Theorem 5. Assume that b and A satisfy the (I − J )-conditions. Then there is an ε > 0 such that all subsystems of (26) as an ε -perturbation of (25) are permanent. The proof of Theorem 5 is left to Section 6. Since (14) is a special case of (25) and (25) can be regarded as an ε -perturbation of itself for any ε > 0, the ‘‘if’’ part of Theorem 3 and that of Theorem 4 are covered by Theorem 5. The proofs of the ‘‘only if’’ part of Theorem 3 and that of Theorem 4 are left to Section 5.

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4. Preliminaries In this section, as a preparation for the proofs of the theorems we present a few simple results for system (26) satisfying (28), (30) and (31). Since systems (14) and (25) can be regarded as special cases of (26), certainly these preliminary results can be applied to (14) and (25). For ϕ ∈ C ([−τ , 0], RN+ ) and t0 ∈ R0 , the solution of (26) with x t0 = ϕ

(34)

is denoted by x(t , t0 , ϕ) with xt (t0 , ϕ) ∈ C ([−τ , 0], RN+ ) or by x(t ) for simplicity. For autonomous system (14), we always replace t0 by 0 in (34) and write x(t , ϕ) with xt (ϕ) ∈ C ([−τ , 0], RN+ ) for the solution. Let d0 = min{˜riL : i ∈ IN },

(35)

0

d = max{˜riM : i ∈ IN },

(36)

α = min{αii : i ∈ IN }, β = max{βij : i ∈ IN , j ∈ IN },

(37)

0 d0 τ

ρ0 = d e

(38)

/α.

(39)

Lemma 1. For each t0 ∈ R0 and every ϕ ∈ C ([−τ , 0], RN+ ), the solution of (26) with (34) satisfies 0

∀t ∈ [0, τ ], ∀i ∈ IN , ∀ t ≥ τ , ∀ i ∈ IN ,

xi (t + t0 ) ≤ ϕi (0)ed t ,

xi ( t + t 0 ) ≤

(40) 0 d0 τ

ϕi (0)d e 0 d0 ed (τ −t )

+ ϕi (0)α(1 − ed0 (τ −t ) )

.

(41)

Thus, lim supt →∞ xi (t ) ≤ ρ0 for all i ∈ IN . Moreover, for any ρ > ρ0 , there is a T = T (ϕ) ≥ 0 such that

∀i ∈ IN , ∀t ≥ t0 + T ,

0 ≤ xi (t , t0 , ϕ) ≤ ρ.

(42)

Proof. For each i ∈ IN , since (40) and (41) are obvious if ϕi (0) = 0, we suppose ϕi (0) > 0. Then, for t ≥ t 0 ≥ t0 , the solution of (26) with (34) satisfies xi (t ) = xi (t 0 ) exp

"Z

t t0

r˜i (s)ds −

N Z t Z X j=1

t0

0

 # xj (s + θ )dθ ηij (s, θ ) ds .

(43)

−τ

From this we have 0 0 xi (t ) ≤ xi (t 0 )ed (t −t )

∀i ∈ IN , ∀t ≥ t 0 ≥ t0 ,

(44)

and then (40) follows from (44) with the replacements of t 0 by t0 and t by t + t0 . Rearrangement of (44) gives

∀i ∈ IN , ∀t ≥ t0 + τ , ∀θ ∈ [−τ , 0],

0 xi (t + θ ) ≥ xi (t )ed θ .

(45)

Then, from (26), (28), (30), (36), (37) and (45),



x0i (t ) ≤ xi (t ) d0 −

0

Z



xi (t + θ )dθ ηii (t , θ ) −τ 0

≤ xi (t )[d − α e−τ d xi (t )], 0

∀i ∈ IN , ∀t ≥ t0 + τ . d0 t

Dividing by −(xi (t ))2 and then multiplying by e

∀i ∈ IN , ∀t ≥ t0 + τ ,

e

d0 t

xi ( t )

(46)

, we reduce (46) to

!0 0 (t −τ )

≥ α ed

.

0 Then (41) follows from integrating the above inequality over [t0 + τ , t0 + t ] and then substituting xi (t0 + τ ) ≤ ϕi (0)ed τ . The rest follows from (41). 

For d0 , d0 , β , ρ0 defined by (35), (36), (38) and (39), fix a ρ > ρ0 and let 0 −N ρβ ed0 τ )

σ = d0 β −1 eτ (d0 −d

.

(47)

Lemma 2. For each ϕ ∈ C ([−τ , 0], RN+ ) with ϕ(0) 6= 0, there is a T 0 = T 0 (ϕ) ≥ 0 such that the solution x(t ) = x(t , t0 , ϕ) of (26) with (34) satisfies

∀t0 ∈ R0 , ∀t ≥ t0 + T 0 ,

y(t ) = x1 (t ) + · · · + xN (t ) ≥ σ > 0.

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Proof. From (44) we have 0 xi (t + θ ) ≤ xi (t − τ )ed τ .

∀i ∈ IN , ∀t ≥ t0 + τ , ∀θ ∈ [−τ , 0],

(48)

Then, by (26), (28), (31), (35)–(38) and (48), x satisfies

" xi (t ) ≥ xi (t ) d0 − 0

N X

# xj (t − τ )e

d0 τ

aij (t )

j =1

≥ xi (t )[d0 − β eτ d y(t − τ )], 0

∀i ∈ IN , ∀t ≥ t0 + τ

so y satisfies

∀t ≥ t 0 + τ ,

0 y0 (t ) ≥ y(t )[d0 − β ed τ y(t − τ )].

(49)

Note that the above differential inequality for y is independent of t0 . By Lemma 1, there is a T = T (ϕ) ≥ 0 such that

∀t0 ∈ R0 , ∀t ≥ t0 + T ,

y(t ) ≤ N ρ.

(50)

Case 1. If there is a T1 ≥ T + τ such that y(t ) < y1 = d0 e−d τ β −1 holds for all t ≥ t0 + T1 , then by (49), y0 (t ) > 0 so y(t ) is increasing for t ≥ t0 + T1 + τ . In this case, we must have limt →∞ y(t ) = y1 > σ so there is a T 0 ≥ T1 such that y(t ) ≥ σ holds for all t0 ∈ R0 and t ≥ t0 + T 0 . 0

Case 2. If there is a T 0 ≥ T + τ such that y(t ) ≥ y1 for all t ≥ t0 + T 0 , then this T 0 meets the requirement of the lemma. Case 3. There is a sequence {[sn , tn ]} such that for each n ≥ 1, y(t ) < y1 for t ∈ (sn , tn ) but y(sn ) = y1 = y(tn ) and y(t ) ≥ y1 for t 6∈ ∪∞ n=1 [sn , tn ]. If tn − sn ≤ τ , by (50) and (49) we have

∀t ∈ [sn , tn ],

d0 τ )(t −s ) n

y(t ) ≥ y(sn )e(d0 −N ρβ e

≥ σ.

If tn > sn + τ , then y0 (t ) > 0 so y(t ) is increasing for t ∈ (sn + τ , tn + τ ). Thus, min{y(t ) : t ∈ [sn , tn ]} = min{y(t ) : t ∈ [sn , sn + τ ]} ≥ σ . Take T 0 = s1 − t0 . Then y(t ) ≥ σ for all t0 ∈ R0 and t ≥ t0 + T 0 .



Remark 4. From Lemmas 1 and 2 we know that every one-dimensional subsystem of (26) is permanent. For any fixed ρ > ρ0 , define a set S0 ⊂ C ([−τ , 0], RN+ ) by S0 = {ϕ ∈ C ([−τ , 0], RN+ ) : ∀i ∈ IN , kϕi k ≤ ρ and (52) holds},

(51)

where

∀θ1 , θ2 ∈ [−τ , 0], θ1 < θ2 ,

0 (θ

ϕi (θ2 ) ≤ ϕi (θ1 )ed

2 −θ1 )

.

(52)

Lemma 3. For each ϕ ∈ C ([−τ , 0], RN+ ), there is a T = T (ϕ) ≥ 0 such that for all t0 ∈ R0 , the solution of (26) with (34) satisfies xt (t0 , ϕ) ∈ S0 for t ≥ t0 + T . Moreover, S0 is closed, bounded and the solution map xt (t0 , ·) (t ≥ t0 ∈ R0 ) defined by (26) with (34) maps S0 into itself. Proof. From (44) and (42) we know the existence of T = T (ϕ) ≥ 0 for every ϕ ∈ C ([−τ , 0], RN+ ) ensuring that xt (t0 , ϕ) ∈ S0 for all t0 ∈ R0 and t ≥ t0 + T . Thus, S0 6= ∅. Suppose there is a sequence {ϕ n } ⊂ S0 satisfying ϕ n → ϕ ∈ C ([−τ , 0], RN+ ) as n → ∞. Then, for all i ∈ IN and θ1 , θ2 ∈ [−τ , 0] with θ1 < θ2 , the ϕ n satisfy

kϕin k ≤ ρ,

0 (θ −θ ) 2 1

ϕin (θ2 ) ≤ ϕin (θ1 )ed

.

Since ϕ → ϕ implies kϕ k → kϕi k, ϕ (θ2 ) → ϕi (θ2 ) and ϕin (θ1 ) → ϕi (θ1 ) as n → ∞, the above inequalities imply n

kϕi k ≤ ρ,

n i

n i

0 (θ

ϕi (θ2 ) ≤ ϕi (θ1 )ed

2 −θ1 )

.

Thus, ϕ ∈ S0 and S0 is closed. The boundedness of S0 is obvious from (51). Now for each ϕ ∈ S0 , we show that xt (t0 , ϕ) ∈ S0 for all t0 ∈ R0 and t ≥ t0 . For any t2 ≥ t1 ≥ t0 − τ , if t1 ≥ t0 then

∀ i ∈ IN ,

0 xi (t2 ) ≤ xi (t1 )ed (t2 −t1 )

by (44). If t2 ≤ t0 then (53) follows from (xi )t0 = ϕi and (52). If t1 < t0 < t2 , then, by (44) and (52), 0 0 0 0 xi (t2 ) ≤ xi (t0 )ed (t2 −t0 ) ≤ xi (t1 )ed (t0 −t1 ) ed (t2 −t0 ) = xi (t1 )ed (t2 −t1 ) .

(53)

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Therefore, (53) holds for all t2 ≥ t1 ≥ t0 − τ . Replacing t2 by t and t1 by t + θ in (53), we obtain 0 xi (t + θ ) ≥ xi (t )ed θ

∀i ∈ IN , ∀t ≥ t0 , ∀θ ∈ [−τ , 0],

(54)

and, as an analogue of (46),

∀i ∈ IN , ∀t ≥ t0 ,

0 x0i (t ) ≤ xi (t )[d0 − α e−d τ xi (t )].

(55)

From (55) and (39) we see that, for each i ∈ IN , either ϕi (0) ≤ ρ0 implies xi (t ) ≤ ρ0 for all t ≥ t0 or ρ0 < ϕi (0) (≤ρ) implies xi (t , t0 , ϕ) < max{ϕi (0), ρ0 } = ϕi (0) ≤ ρ for all t > t0 . Therefore, xt (t0 , ϕ) ∈ S0 for all t ≥ t0 .



We further define S1 ⊂ S0 by S1 = {ϕ ∈ S0 : ∀i ∈ IN , ϕi satisfies (57)},

(56)

where

∀θ1 , θ2 ∈ [−τ , 0] with θ1 < θ2 ,

ϕi (θ2 ) ≥ ϕi (θ1 )e(d0 −N ρβ)(θ2 −θ1 ) .

(57)

Lemma 4. For each ϕ ∈ C ([−τ , 0], RN+ ), there is a T 0 = T 0 (ϕ) ≥ 0 such that for every t0 ∈ R0 , the solution of (26) with (34) satisfies xt (t0 , ϕ) ∈ S1 for all t ≥ t0 + T 0 . Moreover, S1 is compact and the solution map xt (t0 , ·) (t ≥ t0 ∈ R0 ) defined by (26) with (34) maps S1 into itself. Proof. For ϕ ∈ S0 , by Lemma 3 and the assumptions on (26), the solution of (26) with (34) satisfies

∀i ∈ IN , ∀t ≥ t0 ,

x0i (t ) ≥ xi (t )[d0 − N ρβ]

so xi (t2 ) ≥ xi (t1 )e(d0 −N ρβ)(t2 −t1 ) .

∀i ∈ IN , ∀t2 ≥ t1 ≥ t0 ,

(58)

Replacing t2 by t + θ2 and t1 by t + θ1 in (58), we have

∀i ∈ IN , ∀t ≥ t0 + τ ,

(xi )t (θ2 ) ≥ (xi )t (θ1 )e(d0 −N ρβ)(θ2 −θ1 )

for all θ1 , θ2 ∈ [−τ , 0] with θ1 < θ2 . Hence, ϕ ∈ S0 implies xt (t0 , ϕ) ∈ S1 for all t0 ∈ R0 and t ≥ t0 + τ . By Lemma 3, ϕ ∈ C ([−τ , 0], RN+ ) implies the existence of T 0 = T 0 (ϕ) ≥ 0 such that xt (t0 , ϕ) ∈ S1 for all t0 ∈ R0 and t ≥ t0 + T 0 . For each ϕ ∈ S1 , i ∈ IN and θ1 , θ2 ∈ [−τ , 0] with θ1 < θ2 , from (51) and (52) we have 0 (θ

ϕi (θ2 ) − ϕi (θ1 ) ≤ ϕi (θ1 )(ed

2 −θ1 )

− 1) 0 (θ

≤ ϕi (θ1 )d (θ2 − θ1 )ed 0

2 −θ1 )

≤ ρ d0 ed τ (θ2 − θ1 ). 0

(59)

From (56) and (57),

ϕi (θ2 ) − ϕi (θ1 ) ≥ ϕi (θ1 )(e(d0 −N ρβ)(θ2 −θ1 ) − 1) ≥ ϕi (θ1 )(d0 − N ρβ)(θ2 − θ1 ) ≥ −ρ|d0 − N ρβ|(θ2 − θ1 ). 0 d0 τ

Taking M0 = ρ max{d e

(60)

, |d0 − N ρβ|} and combining (59) with (60), we obtain

|ϕi (θ2 ) − ϕi (θ1 )| ≤ M0 |θ2 − θ1 |.

(61)

This shows that the functions in S1 are equi-continuous over [−τ , 0]. As S1 is bounded, by Ascoli’s theorem S1 is relatively compact. The closedness of S1 follows from that of S0 and the fact that (57) is retained if ϕ is a limit of a sequence in S1 . Therefore, S1 is compact. Next we show that ϕ ∈ S1 implies xt (t0 , ϕ) ∈ S1 for all t0 ∈ R0 and t ≥ t0 . By Lemma 3 and the definition of S1 , we need only show that xt (t0 , ϕ) satisfies (57) for all t ≥ t0 . For i ∈ IN and t2 ≥ t1 ≥ t0 − τ , if t2 ≤ t0 then xi (t2 ) ≥ xi (t1 )e(d0 −N ρβ)(t2 −t1 ) holds by (57) as xt0 = ϕ . If t1 ≥ t0 , (62) holds by (58). If t2 > t0 > t1 , by (57) and (58) we have xi (t2 ) ≥ xi (t0 )e(d0 −N ρβ)(t2 −t0 )

≥ xi (t1 )e(d0 −N ρβ)(t0 −t1 ) e(d0 −N ρβ)(t2 −t0 ) = xi (t1 )e(d0 −N ρβ)(t2 −t1 ) .

(62)

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This shows that (62) holds for all t2 ≥ t1 ≥ t0 − τ . Replacing t2 by t + θ2 and t1 by t + θ1 in (62) gives (xi )t (θ2 ) ≥ (xi )t (θ1 )e(d0 −N ρβ)(θ2 −θ1 ) . Therefore, xt (t0 , ϕ) ∈ S1 for all t0 ∈ R0 and t ≥ t0 .  5. Proofs of the ‘‘only if’’ part of Theorems 3 and 4 Lemma 5. Assume that (25) with (27), (29) and (15) are permanent. Then (20) has an equilibrium x∗ ∈ int RN+ . If x∗ is the unique equilibrium of (20) in int RN+ , then for each t0 ∈ R0 , every solution of (25) with (34) in int RN+ on [t0 , ∞) satisfies

∀ i ∈ IN ,

lim m(xi , t0 , t ) = x∗i .

(63)

t →∞

Proof. By assumption, there are δ1 > 0 and δ2 > δ1 such that every solution of (25) in int RN+ satisfies

∀ i ∈ IN ,

δ1 ≤ xi (t ) ≤ δ2

(64)

for sufficiently large t. Dividing (25) by xi (t ) and integrating both sides from t0 to t, we have ln

xi (t ) xi (t0 )

N X

t

Z

ri (s)ds −

= t0

Z t Z

0

(65)

−τ

t0

j =1



xj (s + θ )dξij (θ ) ds.

aij

Since (64) implies

∀ i ∈ IN ,

lim

t →∞

1

ln

t − t0

xi (t ) xi (t0 )

= 0,

(66)

from (29), (65) and (66) we obtain

∀ i ∈ IN ,

lim

t →∞

N X

aij

j =1

t − t0

Z t Z

0

 xj (s + θ )dξij (θ ) ds = bi .

(67)

−τ

t0

By (15),

Z t Z t0

0



xj (s + θ )dξij (θ ) ds =

Z

−τ

0

Z

−τ

Z

t



xj (s + θ )ds dξij (θ ) t0

0

Z

t +θ

−τ

Z

t0 +θ

t

xj (s)ds +

=



xj (s)ds dξij (θ )

=

Z

0

−τ

t0

Z

t0

xj (s)ds −

t0 +θ

Z

t



xj (s)ds dξij (θ ). t +θ

Bearing in mind that (64) implies lim

t →∞

Z

1 t − t0

0

Z

−τ

t0

xj (s)ds −

t0 +θ

Z

t



xj (s)ds dξij (θ ) = 0, t +θ

we can rewrite (67) as

∀ i ∈ IN ,

lim

t →∞

N X

aij m(xj , t0 , t ) = bi .

(68)

j =1

From (64) again,

∀ j ∈ IN ,

δ1 ≤ lim inf m(xj , t0 , t ) ≤ lim sup m(xj , t0 , t ) ≤ δ2 . t →∞

(69)

t →∞

Then there exist x∗1 ∈ [δ1 , δ2 ] and an increasing sequence {tn } with tn → ∞ as n → ∞ such that lim m(x1 , t0 , tn ) = lim inf m(x1 , t0 , t ) = x∗1 .

n→∞

t →∞

(70)

By choosing subsequences of {tn } if necessary, without loss of generality we may assume that

∀ j ∈ IN ,

lim m(xj , t0 , tn ) = x∗j ∈ [δ1 , δ2 ].

n→∞

(71)

Then, after the replacement of t by tn in (68), substitution of (71) into (68) gives Ax∗ = b. This shows that x∗ is an equilibrium of (20) in int RN+ .

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If limt →∞ m(xi , t0 , t ) 6= x∗i for some i ∈ IN , by the same reasoning as above we can find a sequence {tn0 } such that limn→∞ m(x, t0 , tn0 ) exists and is another equilibrium of (20) in int RN+ . Hence, if x∗ is the unique equilibrium of (20) in int RN+ then we must have (63).  Lemma 6. If (14) with (15) is permanent, then (14) has a unique equilibrium x∗ in int RN+ and (63) holds for all t0 ∈ R0 and every solution in int RN+ on [t0 , ∞). Proof. By Lemma 5, (14) has at least one equilibrium x∗ in int RN+ . If (14) has more than one equilibrium in int RN+ , then T ∗ N N N i∈IN γi contains at least a line segment connecting x to ∂ R+ = R+ \ int R+ . Since this line segment consists of constant solutions of (14), this contradicts the permanence of (14). Thus, x∗ is the unique equilibrium of (14) in int RN+ and (63) holds by Lemma 5. 

Proof of the ‘‘only if’’ part of Theorem 4. Since all subsystems of (25) and its small perturbations on A are permanent, application of Lemma 5 to every subsystem results in the existence of an equilibrium x∗I ∈ CI0 of (20) for each I ⊂ IN . To show that b and A satisfy the (I − J )-conditions, by Remark 1 we need only show that, when I is a proper subset of IN , x∗I is the unique equilibrium of (20) in CI0 and satisfies (23). For each i0 ∈ IN and I = IN \ {i0 }, the permanence of each one-dimensional subsystem of (25) implies the uniqueness of the equilibrium x∗I of (20) in CI0 given by

(x∗I )i0 = bi0 /ai0 i0 > 0,

∀j ∈ IN \ {i0 }, (x∗I )j = 0.

If, for some proper I ⊂ IN , x∗I is not the unique equilibrium of (20) in CI0 then there is a proper J ⊂ IN with I ⊂ J but I 6= J such that

! x∗I x∗J ⊂

\

! \

πi

\

i∈I

γj ,

(72)

j∈IN \I

where x∗I x∗J denotes the line segment from x∗I to x∗J . Since Ai x∗J = bi for i ∈ IN \ J and Aj x∗J = bj for j ∈ J \ I by (72), x∗J does not satisfy (23). If x∗J is not the unique equilibrium of (20) in CJ0 , we repeat the above process to find an equilibrium which has more zero components than x∗J has and fails to satisfy (23). Now suppose b and A do not satisfy the (I − J )-conditions. Then the existence of a proper I ⊂ IN is shown in the last paragraph for which x∗I is the unique equilibrium of (20) in CI0 but x∗I fails to satisfy (23). Thus,

∀ i ∈ IN \ I ,

∃i0 ∈ I ,

Ai x∗I = bi ;

Ai0 x∗I ≥ bi0 .

(73)

∗

If Ai0 xI = bi0 , since all subsystems of (25) and its small perturbations on A are permanent, we can replace Ai0 by A(ε)i0 = (ai0 1 + ε, . . . , ai0 N + ε) and A by A(ε) for a sufficiently small ε > 0 to get A(ε)i0 x∗I > bi0 , here A(ε)i = Ai for all i ∈ IN \ {i0 }. Without loss of generality, we assume that

∀ i ∈ IN \ I ,

∃i0 ∈ I ,

Ai x∗I = bi ;

Ai0 x∗I > bi0

(74)

and let

γ0 = Ai0 x∗I − bi0 > 0.

(75)

Let y(t ) be a solution of (25) on [t0 , ∞) with

∀i ∈ I ,

yi (t ) = 0;

∀j ∈ IN \ I ,

yj (t ) > 0

(76)

for all t ≥ t0 − τ . Then, applying Lemma 5 to the subsystem of (25) corresponding to this I, we have limt →∞ m(yj , t0 , t ) = (x∗I )j for all j ∈ IN so that

∀ j ∈ IN ,

lim

T →∞

1 T

Z

t0 + T

Z

0



yj (t + θ )dξij (θ ) dt = (x∗I )j .

(77)

−τ

t0

By (29), (75) and (77), there is a T1 > t0 such that

X j∈IN \I

ai0 j

1 t − t0

Z t Z t0

0



yj (s + θ )dξi0 j (θ ) ds − m(ri0 , t0 , t ) ≥

−τ

1 2

γ0

(78)

for all t ≥ T1 . Since all subsystems of (25) are permanent, there are δ1 > 0 and δ2 > δ1 such that for each t0 ∈ R0 and i ∈ IN , every solution of (25) on [t0 , ∞) satisfies xi (t0 ) > 0 H⇒ δ1 ≤ xi (t ) ≤ δ2

for large enough t ≥ t0 .

(79)

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Put

Φ = {ϕ ∈ C ([−τ , 0], RN+ ) : ∀i ∈ IN \ {i0 }, ϕi = (yt0 )i , ∀h ∈ [0, δ1 /2], ϕi0 (θ ) ≡ h}

(80)

and consider the set U of solutions of (25) on [t0 , ∞) defined by U = {x(t , t0 , ϕ) : ϕ ∈ Φ }.

(81)

Since x(t , t0 , ϕ) is continuous in (t , t0 , ϕ), Φ defined by (80) is compact and yi (t0 ) > 0 for i ∈ IN \ I, by (79) there are u > 0 and v > u satisfying

∀i ∈ IN \ I , ∀ϕ ∈ Φ , ∀t ≥ t0 ,

u ≤ xi (t , t0 , ϕ) ≤ v.

(82)

For each integer k ≥ 2 and Tk = T1 + k, by continuous dependence of the solution on ϕ there is an hk ∈ (0, δ1 /2) such that every x(t , t0 , ϕ) in U with ϕi0 (0) ≤ hk satisfies, for each i ∈ (IN \ I ) ∪ {i0 },

∀t ∈ [t0 − τ , Tk ],

|xi (t , t0 , ϕ) − yi (t )| <

γ0 4k(N − |I | + 1)a0

,

(83)

where a0 = max{aij : i, j ∈ IN }. Take ϕ k ∈ Φ with ϕik0 (θ ) ≡ hk . Then, from (25), (78) and (83) we have 1

xi0 (Tk , t0 , ϕ k )

ln

Tk − t0

hk

= m(ri0 , t0 , Tk ) −

j∈(IN \I )∪{i0 }

≤ m(ri0 , t0 , Tk ) − +

j∈(IN \I )∪{i0 }

1

1

2 1

4k

≤ − γ0 +

t0

Tk

Z

Z

Tk − t0

t0

Tk

0

Z

ai 0 j

X

Tk − t0

Tk

Tk − t0

ai0 j

X j∈IN \I

Z

ai 0 j

X

Z

Z

0



xj (t + θ , t0 , ϕ )dξi0 j (θ ) dt k

−τ 0



yj (t + θ )dξi0 j (θ ) dt −τ

 |xj (t + θ , t0 , ϕ ) − yj (t + θ )|dξi0 j (θ ) dt k

−τ

t0

γ0

< − γ0 . 4

Thus, xi0 (Tk , t0 , ϕ k ) < hk . From (79) and the continuity of x(t , t0 , ϕ) there is a tk > Tk satisfying

∀t ∈ [Tk , tk ),

xi0 (t , t0 , ϕ k ) < hk = xi0 (tk , t0 , ϕ k ).

(84)

This, together with (83) and yi0 (t ) ≡ 0, gives

∀t ∈ [t0 − τ , tk ],

0 < xi0 (t , t0 , ϕ k ) <

γ0 4k(N − |I | + 1)a0

.

(85)

From (25) we see that each solution x(t , t0 , ϕ k ) on [t0 , tk ] satisfies 1 tk − t0

xi (tk , t0 , ϕ k )

ln

= m(ri , t0 , tk ) −

xi (t0 )

aij

X t j∈(IN \I )∪{i0 } k

− t0

Z

tk t0

Z

0

 xj (t + θ , t0 , ϕ )dξij (θ ) dt k

(86)

−τ

for i ∈ (IN \ I ) ∪ {i0 }. Since tk − t0 > T1 + k − t0 → ∞ as k → ∞ and ln xi (tk , t0 , ϕ k ) − ln xi (t0 ) is bounded by (82) and (84), the left-hand side of (86) vanishes as k → ∞, i.e.

∀i ∈ (IN \ I ) ∪ {i0 },

lim

k→∞ tk

1

− t0

ln

xi (tk , t0 , ϕ k ) xi (t0 )

= 0.

(87)

By (85), for all i ∈ (IN \ I ) ∪ {i0 },

− t0

k→∞ tk

tk

Z

1

lim

Z

0



xi0 (t + θ , t0 , ϕ k )dξii0 (θ ) dt = 0.

(88)

−τ

t0

Since (29) leads to

∀ i ∈ IN ,

lim m(ri , t0 , tk ) = bi

(89)

k→∞

and, as a consequence of (15), (82) and the technique of changing the order of integrals, 1 tk − t0

Z

tk t0

Z

0

−τ

 xj (t + θ , t0 , ϕ )dξij (θ ) dt = m(xj (·, t0 , ϕ k ), t0 , tk ) + o(1) (k → ∞) k

(90)

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holds for all j ∈ (IN \ I ) ∪ {i0 }, from (87)–(90) we can rewrite (86) as

∀i ∈ (IN \ I ) ∪ {i0 },

X

lim

k→∞

aij

j∈IN \I

1

Z

tk − t0

tk

xj (t , t0 , ϕ k )dt = bi .

(91)

t0

From (82) we know that each of the sequences



1 tk − t0

tk

Z



xj (t , t0 , ϕ )dt , k

j ∈ IN \ I

(92)

t0

has convergent subsequences. By choosing convergent subsequences one after another N − |I | times, we find a subsequence {nk } ⊂ {k} with nk → ∞ as k → ∞ such that all of the sequences in (92) after the replacement of k by nk are convergent. Without loss of generality, we assume that

∀ j ∈ IN \ I ,

lim

k→∞ tk

tk

Z

1

− t0

xj (t , t0 , ϕ k )dt = zj ∈ [u, v].

(93)

t0

Then it follows from (91) and (93) that

∀i ∈ (IN \ I ) ∪ {i0 },

X

aij zj = bi .

(94)

j∈IN \I

As x∗I is the unique equilibrium of (20) in CI0 , from (93) and (94) we obtain zj = (x∗I )j for all j ∈ IN \ I so (94) can be written as

∀i ∈ (IN \ I ) ∪ {i0 },

Ai x∗I = bi ,

a contradiction to (74). This contradiction shows that b and A satisfy the (I − J )-conditions.



Proof of the ‘‘only if’’ part of Theorem 3. By Lemma 6, every xI is the unique equilibrium of (20) in CI0 . If b and A do not satisfy the (I − J )-conditions, then there is a proper I ⊂ IN such that x∗I satisfies (73). If Ai0 x∗I = bi0 , then, with J = I \ {i0 }, x∗J is the unique equilibrium of (20) in CJ0 . But since ∗

! ∗ ∗

xI xJ ⊂

\

πj

! \

j∈J

\

γi ,

i∈IN \J

every point in x∗I x∗J \ {x∗I } is also an equilibrium of (20) in CJ0 , a contradiction to the uniqueness of the equilibrium x∗J in CJ0 . Thus, (73) implies (74). The rest of the proof for Theorem 4 is still valid here.  6. Proof of Theorem 5 In this section, we shall prove that, when b and A satisfy the (I − J )-conditions, all subsystems of system (26) as an ε-perturbation of (25) are permanent for sufficiently small ε > 0. The method of [11] is adopted here. Suppose b and A satisfy the (I − J )-conditions. By Remark 1, (20) satisfies Condition (C). As each equilibrium x∗I of (20) in 0 CI depends on b linearly, there is a small ε0 > 0 with ε0 < min{b1 , b2 , . . . , bN } such that for each b˜ in B(b, ε0 ) = {x ∈ int RN+ : ∀i ∈ IN , |xi − bi | ≤ ε0 }, (20) with the replacement of b by b˜ also satisfies Condition (C) so b˜ and A also satisfy the (I −J )-conditions. By [11, Lemma 2.7], A−1 B(b, ε0 ) ⊂ int RN+ . For this ε0 > 0 and r (t ) satisfying (27) and (29), take ε1 satisfying 0 < ε1 < min{ε0 , r1L , . . . , rNL },

(95)

E (r , ε1 ) = {e ∈ C (R0 , RN+ ) : ∀i ∈ IN , ∀t ∈ R0 , |ei (t ) − ri (t )| ≤ ε1 },

(96)

set and consider the system

" xi (t ) = xi (t ) ei (t ) − 0

N X j =1

Z

#

0

xj (t + θ )dξij (θ ) ,

aij

i ∈ IN ,

(97)

−τ

where e ∈ E (r , ε1 ) and the aij and ξij are the same as in (25). By (27), (29), (95) and (96), each e(t ) in E (r , ε1 ) satisfies

∀ i ∈ IN , ∀ i ∈ IN ,

0 < riL − ε1 ≤ ei (t ) ≤ riM + ε1 ,

(98)

0 < bi − ε0 < bi − ε1 ≤ lim inf m(ei , t0 , t ),

(99)

∀ i ∈ IN ,

lim sup m(ei , t0 , t ) ≤ bi + ε1 < bi + ε0 .

t →∞

t →∞

(100)

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To apply Lemmas 1–4 to (97), we replace (35)–(38) by d0 = min{riL : i ∈ IN } − ε1 ,

(101)

d = max{riM : i ∈ IN } + ε1 ,

(102)

α = min{aii : i ∈ IN }, β = max{aij : i, j ∈ IN }.

(103)

0

(104)

Lemma 7. Assume that b and A satisfy the (I − J )-conditions. Then, for each e ∈ E (r , ε1 ), every solution of (97) in RN+ satisfies

∀i ∈ IN , ∀t0 ∈ R0 ,

xi (t0 ) > 0 implies lim inf xi (t ) > 0. t →∞

Proof. Suppose the conclusion is not true. Then there exist an e ∈ E (r , ε1 ), a t0 ∈ R0 , an i0 ∈ IN and a ϕ ∈ S1 with ϕi0 (0) > 0 such that the solution x(t ) of (97) with (34) satisfies lim inft →∞ xi0 (t ) = 0. Let J0 ⊂ IN such that ϕi (0) > 0 if and only if i ∈ J0 . Then, for all t ≥ t0 , xi (t ) > 0 for i ∈ J0 and xj (t ) ≡ 0 for j ∈ IN \ J0 . By Lemma 2, x1 (t ) + · · · + xN (t ) ≥ σ (>0) holds for sufficiently large t. Hence, there is a proper subset J ⊂ IN with i0 ∈ J such that lim inf max{xj (t ) : j ∈ J } = 0,

(105)

∀ i ∈ IN \ J ,

(106)

t →∞

lim inf max{xj (t ) : j ∈ J ∪ {i}} > 0. t →∞

From (106) and the definition of J0 we have

ρ1 = inf{max{xj (t ) : j ∈ J ∪ {i}} : t ≥ t0 , i ∈ IN \ J } > 0, ρ2 =

1

(107)

min{ρ1 , xi (t0 ) : i ∈ J0 } > 0.

2

(108)

By (105) there is an increasing sequence {tn } satisfying

∀n ≥ 1 ,

max{xj (tn ) : j ∈ J } <

ρ2 n

e−nN ρβ .

(109)

Then, as (108) implies max{xj (t0 ) : j ∈ J } ≥ 2ρ2 , by continuity there are sn ∈ (t0 , tn ) and jn ∈ J such that

∀n ≥ 1 ,

max{xj (sn ) : j ∈ J } = xjn (sn ) =

∀n ≥ 1, ∀t ∈ (sn , tn ],

ρ2 n

,

(110)

ρ2

max{xj (t ) : j ∈ J } <

n

.

(111)

From (107), (108), (111) and Lemma 4 we deduce that

∀i ∈ IN \ J , ∀t ∈ [sn , tn ],

ρ1 ≤ xi (t ) ≤ ρ.

(112)

Since integration of (97) gives xjn (tn ) = xjn (sn ) exp

Z

tn

" ejn (t ) −

N X

sn

!

xj (t + θ )dξjn j (θ ) dt

aj n j −τ

j=1

> xjn (sn )e−N ρβ(tn −sn ) ,

#

0

Z

∀n ≥ 1,

this together with (109) and (110) leads to

ρ2

∀n ≥ 1 ,

n

e−nN ρβ >

ρ2 n

e−N ρβ(tn −sn ) ,

which is equivalent to

∀n ≥ 1 ,

tn − sn > n.

(113)

For i ∈ IN \ J, from (97) we have 1 tn − sn

ln

xi (tn ) xi (sn )

= m(ei , sn , tn ) −

N X

aij m(xj , sn , tn ) + εi (n),

(114)

j =1

where

εi (n) =

N X

aij

j =1

tn − sn

Z

0

−τ

Z

tn tn +θ

xj (t )dt −

Z

sn sn +θ

xj (t )dt



dξij (θ ).

(115)

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Z. Hou / Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301

Since (110) and (111) imply limn→∞ m(xj , sn , tn ) = 0 for all j ∈ J and an immediate consequence of (112) and (113) is

∀ i ∈ IN \ J ,

1

lim

n→∞ tn

− sn

ln

xi (tn ) xi (sn )

= 0 = lim εi (n), n→∞

we can rewrite (114) as

X

∀ i ∈ IN \ J ,

aij m(xj , sn , tn ) = m(ei , sn , tn ) + o(1) (n → ∞).

(116)

j∈IN \J

By (112), (98)–(100), there is a subsequence {n` } ⊂ {n} such that

∀` ≥ 1, jn` = j0 ∈ J , ∀i ∈ IN \ J , lim m(xi , sn` , tn` ) = x∗i ∈ [ρ1 , ρ], `→∞

∀ i ∈ In ,

lim m(ei , sn` , tn` ) = e∗i ∈ [bi − ε1 , bi + ε1 ].

`→∞

Replacing n by n` in (116) and letting ` → ∞, we obtain

X

∀ i ∈ IN \ J ,

aij x∗j = e∗j .

(117)

j∈IN \J

Since (109) and (110) imply xjn (tn ) < xjn (sn ), from this and (114) with the replacement of i by jn follows N X

ajn j m(xj , sn , tn ) > m(ejn , sn , tn ) + εjn (n).

j =1

Replacing n by n` and letting ` → ∞, we further obtain

X

aj0 j x∗j ≥ e∗j0 .

(118)

j∈IN \J

By the choice of ε1 in (95), e∗ ∈ B(b, ε0 ) so e∗ and A satisfy the (I − J )-conditions. Then (117) should imply

X

∀i ∈ J ,

aij x∗j < e∗i ,

j∈IN \J

which contradicts (118). This contradiction shows the truth of the conclusion.



Lemma 8. Assume that b and A satisfy the (I − J )-conditions. Then there is a µ0 > 0 such that, for all e ∈ E (r , ε1 ), every solution of (97) in int RN+ satisfies

∀ i ∈ IN ,

lim sup xi (t ) ≥ µ0 .

(119)

t →∞

Proof. For each e ∈ E (r , ε1 ), t0 ∈ R0 and every solution of (97) on [t0 , ∞), let Fi (t ) =

N X

aij m(xj , t0 , t ),

t ≥ t0 , i ∈ IN .

(120)

j =1

Then, from the equations obtained from (97) similar to (114), we have

∀ i ∈ IN ,

Fi (t ) = m(ei , t0 , t ) −

1 t − t0

ln

xi (t ) xi (t0 )

+ εi (t ),

where

εi ( t ) =

N X

aij

j =1

t − t0

Z

0

−τ

Z

t

xj (s)ds −

Z

t +θ

t0



xj (s)ds dξij (θ ).

t0 +θ

As limt →∞ εi (t ) = 0 follows from Lemma 1, by Lemma 7,

∀ i ∈ IN ,

lim [Fi (t ) − m(ei , t0 , t )] = 0.

t →∞

Take ε 0 = (ε0 + ε1 )/2. Then ε1 < such that F (t ) ∈ B(b, ε 0 ) for all t A−1 B(b, ε 0 ) ⊂ A−1 B(b, ε0 ) ⊂ int RN+

(121)

ε 0 < ε0 so B(b, ε1 ) ⊂ B(b, ε0 ) ⊂ B(b, ε0 ). By (99), (100) and (121), there is a T > t0 ≥ T . Hence, from (120), m(x, t0 , t ) = A−1 F (t ) ∈ A−1 B(b, ε0 ) for all t ≥ T . Since and both B(b, ε 0 ) and A−1 B(b, ε 0 ) are compact, with

µ0 = min{zi : ∀z ∈ A−1 B(b, ε0 ), ∀i ∈ IN },

(122)

Z. Hou / Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301

4299

we have

∀ i ∈ IN , ∀ t ≥ T , This results in (119).

m(xi , t0 , t ) ≥ µ0 > 0. 

Lemma 9. Assume that b and A satisfy the (I − J )-conditions. Then there exists a µ1 > 0 such that for all e ∈ E (r , ε1 ), every solution of (97) in int RN+ satisfies

∀ i ∈ IN ,

µ1 ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ ρ. t →∞

(123)

t →∞

Proof. The upper bound in (123) follows from Lemma 1. Suppose the conclusion of the lemma is not true and we aim to derive a contradiction. By Lemma 7, for each e ∈ E (r , ε1 ), each component of every solution of (97) in intRN+ has a positive lower limit as t → ∞. Then the nonexistence of µ1 > 0 for (123) means the existence of {en } ⊂ E (r , ε1 ), {Tn } ⊂ R0 , {ϕ n } ⊂ S1 and {αn } ⊂ R+ such that, for each n ≥ 1, yn (t ) = x(t , en , Tn , ϕ n ) is the solution of (97) with (yn )Tn = ϕ n after the substitution e(t ) = en (t ) and satisfies

n

min lim inf yni (t ) : i ∈ IN

o

t →∞

= αn ↓ 0 (n → ∞).

(124)

By (124), for each n ≥ 1, there is an i0 ∈ IN such that

n

min lim inf yni (t ) : i ∈ IN t →∞

o

= lim inf yni0 (t ) = αn .

(125)

t →∞

As IN is a finite set, by choosing a subsequence of {n} if necessary, without loss of generality we assume that this fixed i0 ∈ IN in (125) suits all n ≥ 1. By Lemma 2,





inf lim inf max{ (t ) : i ∈ IN } : n ≥ 1 yni

t →∞

 ≥ inf lim inf t →∞

N 1 X

N i =1

yni



(t ) : n ≥ 1 > 0.

This along with (124) and (125) justifies the existence of a proper subset J ⊂ IN with i0 ∈ J fulfilling



inf lim inf max{ynj (t ) : j ∈ J } : n ≥ 1



t →∞

∀ i ∈ IN \ J ,



= 0,

(126)



inf lim inf max{ (t ) : j ∈ J ∪ {i}} : n ≥ 1 ynj

t →∞

> 0.

(127)

By (127) there exist a σ0 > 0 and a sequence {Tn0 } with Tn0 ≥ Tn such that

∀n ≥ 1, ∀i ∈ IN \ J , ∀t ≥ Tn0 ,

max{ynj (t ) : j ∈ J ∪ {i}} ≥ σ0 .

(128)

By Lemma 8, there is a sequence {Tn00 } with Tn00 ≥ Tn0 that satisfies

∀n ≥ 1 ,

max{ynj (Tn00 ) : j ∈ J } ≥ µ0 /2.

(129)

Note that (126) implies the existence of a subsequence {nk } ⊂ {n} meeting the requirement of

∀k ≥ 1,

n

lim inf max{yj k (t ) : j ∈ J } < t →∞

min{µ0 , σ0 } −kN ρβ e . 4k

(130)

By (130) we can choose a sequence {tk } with tk > Tn00k to satisfy

∀k ≥ 1,

n

max{yj k (tk ) : j ∈ J } <

min{µ0 , σ0 } −kN ρβ e . 4k

(131)

Then, by (129), (131) and continuity, we can choose {sk } with Tn00k < sk < tk and {jk } with jk ∈ J such that

∀k ≥ 1,

n

n

max{yj k (sk ) : j ∈ J } = yjkk (sk ) =

∀k ≥ 1, ∀t ∈ (sk , tk ],

n

min{µ0 , σ0 }

max{yj k (t ) : j ∈ J } <

,

4k min{µ0 , σ0 } 4k

(132)

.

(133)

It then follows from (128), (132) and (133) that

∀k ≥ 1, ∀i ∈ IN \ J , ∀t ∈ [sk , tk ],

n

σ0 ≤ yi k (t ) ≤ ρ.

(134)

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Z. Hou / Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301

From (97) with the replacements of e(t ) by enk (t ) and x(t ) by ynk (t ) we have yj k (tk ) ≥ yj k (sk )e−N ρβ(tk −sk ) . n

n

∀k ≥ 1 , ∀j ∈ J ,

Combining this with (131) and (132) we obtain

∀k ≥ 1 ,

tk − sk > k.

(135)

As jk ∈ J for all k ≥ 1, there exist a subsequence {km } ⊂ {k} and a j0 ∈ J such that jkm = j0 for all m ≥ 1. Without loss of generality, we view {k} as {km } so that jk = j0 for all k ≥ 1 and (132) becomes n

n

max{yj k (sk ) : j ∈ J } = yj0k (sk ) =

∀k ≥ 1 ,

min{µ0 , σ0 } 4k

.

(136)

n

n

As yj0k (tk ) < yj0k (sk ) by (133) and (136), using (133)–(136) and the same technique as that used in the proof of Lemma 7, we deduce that

X

∀ i ∈ IN \ J ,

X

aij x∗j = b∗i ,

j∈IN \J

aj0 j x∗j ≥ b∗j0 ,

(137)

j∈IN \J

where b∗ ∈ B(b, ε1 ) and x∗ ∈ CJ0 . This shows that b∗ and A do not satisfy the (I − J )-conditions, a contradiction to the fact that every vector in B(b, ε1 ) and A still satisfy the (I − J )-conditions. Therefore, the conclusion of the lemma must be true.  Proof of Theorem 5. Take ε = ε1 satisfying (95) and suppose (26) is an ε -perturbation of (25). Then, by Definition 2, for each nonzero solution x˜ (t ) of (26) in RN+ there is a T ∈ R0 such that, for all i ∈ IN and t ≥ T ,

N Z 0 X x˜ j (t + θ )dθ (aij ξij (θ ) − ηij (t , θ )) ≤ ε1 . r˜i (t ) − ri (t ) + j=1 −τ

(138)

For each i ∈ IN , define ei ∈ C (R0 , R+ ) by ei (t ) =

    

r˜i (t ) +

N Z X

0

x˜ j (t + θ )dθ (aij ξij (θ ) − ηij (t , θ )),

t ≥ T,

−τ

j =1

ri (t ) + ei (T ) − ri (T ),

t ∈ R0 \ [T , ∞).

Then e ∈ E (r , ε1 ) and the solution x˜ (t ) of (26) is also a solution of (97) on [T , ∞). The conclusion then follows from the application of Lemma 9 to all subsystems of (97).  7. An example Consider the three-dimensional system

" xi (t ) = xi (t ) r˜i (t ) − 0

3 X

# aij (t )xj (t − τij (t )) ,

i ∈ I3 ,

(139)

j =1

where the aij (t ) and τij (t ) are continuous, bounded and nonnegative on [0, ∞) and each aii (t ) has a positive lower bound. Assume that lim τij (t ) = τij0 ,

∀i, j ∈ I3 ,

t →∞

lim (aij (t )) = A =

t →∞

ri ( t ) =

t +1

+

2 1 1

1

t +2 i+4 Consider also the system

" x0i (t ) = xi (t ) ri (t ) −

1 3 2

(140) 2 2 , 4

sin(it ),

3 X

!

(141)

∀i ∈ I3 , ∀t ≥ 0.

(142)

# aij xj (t − τij0 ) ,

i ∈ I3 .

(143)

j =1

Then, for b ∈ int R3+ with b1 = b2 = b3 = 1, r (t ) satisfies

∀ i ∈ I3 ,

lim m(ri , t0 , t ) = bi

t →∞

uniformly for t0 ≥ 0.

It can be checked that b = (1, 1, 1)T and A given by (141) satisfy the (I − J )-conditions. So, by Theorem 4, all subsystems of (143) are permanent. Assume also that r˜ (t ) satisfies

∀ i ∈ I3 ,

lim (˜ri (t ) − ri (t )) = 0.

t →∞

(144)

Z. Hou / Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301

4301

Then, by Lemma 1, every solution of (139) in R3+ is bounded so the x0i (t ) are also bounded. For each solution x(t ) of (139), the assumption (140), (141) and (144) imply that, for all i ∈ I3 ,

3 X 0 lim sup r˜i (t ) − ri (t ) + [aij xj (t − τij ) − aij (t )xj (t − τij (t ))] = 0. t →∞ j =1 Thus, (139) is an ε -perturbation of (143) for any ε > 0. By Theorem 5, all subsystems of (139) are permanent. Acknowledgements The author is grateful to the referees for their corrections and suggestions for improvement of the paper adopted in this final version. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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