- Email: [email protected]
Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model
Applied Mathematics and Computation 176 (2006) 431–441 www.elsevier.com/locate/amc
Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
Abstract A neutral multispecies Logarithmic population model is proposed in this paper. By using the contraction mapping principle and constructing a suitable Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) of the model. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Periodic solution; Almost periodic solution; Lyapunov functional; Global attractivity; Contraction mapping principle
1. Introduction The aim of this paper is to investigate the existence, uniqueness and global attractivity of the positive periodic solution (positive almost periodic solution) of the following neutral multispecies Logarithmic population model " n n X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ dt j¼1 j¼1 # Z t n n X X d ln N j ðt gij ðtÞÞ cij ðtÞ K ij ðt sÞ ln N j ðsÞ ds d ij ðtÞ ; ð1:1Þ dt 1 j¼1 j¼1 where i = 1, 2, . . . , n, ri(t), aijR(t), bij(t), cij(t), dij(t) 2 C(R, (0, +1)), sij(t), gij(t) 2 C(R, R+) are all continuous R þ1 þ1 functions. 0 K ij ðsÞ ds ¼ 1; 0 sK ij ðsÞ ds < þ1. We consider (1.1) together with the initial conditions N i ðtÞ ¼ /i ðtÞ P 0; /i 2 Cðð1; 0;
N_ i ðtÞ ¼ /_ i ðtÞ;
/i ð0Þ > 0;
1
½0; þ1ÞÞ \ C ðð1; 0; ½0; þ1ÞÞ.
For the ecological justification of Eq. (1.1), see [1–8].
E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.032
ð1:2Þ
432
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Gopalsamy [1] and Kirlinger [2] had proposed the following single species Logarithmic model: dN ðtÞ ¼ N ðtÞ½a b ln N ðtÞ c ln N ðt sÞ. dt
ð1:3Þ
System (1.3) is then generalized by Li [3] to the nonautonomous case dN ðtÞ ¼ N ðtÞ½aðtÞ bðtÞ ln N ðtÞ cðtÞ ln N ðt sðtÞÞ. dt
ð1:4Þ
In [3], by using the coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of system (1.4) are established. Also, sufficient conditions are obtained for the periodic solution attracting all other positive solutions. In [4], we generalized above system to the system with state dependent delays and investigated the existence of positive periodic solutions of the system. In [5], Liu proposed the following multispecies periodic Logarithmic population model: " # n n X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ ; ð1:5Þ dt j¼1 j¼1 where aij,bij 2 C(R, (0, +1)), ri, sij 2 C(R, R), i, j = 1, 2, . . . , n, are all continuous T-periodic functions. By using the coincidence degree theory and constructing the Lyapunov functional, some sufficient conditions which guarantee the existence, uniqueness and stability of the positive periodic solution of the system (1.5) are established. Recently, we [6] pointed out that the proof of Theorem 1 in [5] is incomplete, also, in [6], we proposed the following multispecies Logarithmic population model: " # Z t n n n X X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ cij ðtÞ K ij ðt sÞ ln N j ðsÞ ds ; dt 1 j¼1 j¼1 j¼1 R þ1
ð1:6Þ
where ri(t), aij, bij 2 C(R, (0, + 1)), sij 2 C(R, R) are all continuous functions. 0 K ij ðsÞds ¼ 1; R þ1 sK ij ðsÞ ds < þ1. By using the method of fixed point theory and constructing a suitable Lyapunov 0 functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) of the model (1.6). As was pointed out by Gopalsamy [1], in some case, the neutral delay population models are more realistic. There were many scholars done works on the periodic solution of neutral type Logistic model or Lotka– Volterra model (see [9–22] and the reference cited therein), seldom did scholars considered the neutral Logarithmic model (see [7,8]). Li [7] had studied the following single species neutral Logarithmic model: dN ðtÞ d ln N ðt sÞ ¼ N ðtÞ rðtÞ aðtÞ ln N ðt rÞ bðtÞ . ð1:7Þ dt dt Recently, Lu et al. [8] further investigated the following system: " # n m X X dN ðtÞ d ln N ðt sj ðtÞÞ ¼ N ðtÞ rðtÞ aj ðtÞ ln N ðt rj ðtÞÞ bj ðtÞ . dt dt j¼1 j¼1
ð1:8Þ
By employing an abstract continuous theorem of k-set contractive operator, some new results on the existence of positive periodic solution of system (1.8) are obtained. However, to this day, still no scholars investigate the multispecies neutral delay Logarithmic model. Specially, no scholars had done works on the existence, uniqueness and stability of positive periodic solution (positive almost periodic solution) of system (1.1). One could easily see that system (1.3)–(1.7) are all special cases of system (1.1). The outline of the paper is as follows. In Section 2, we first introducing a transformation, where some adjustable real parameters qi > 0 are introduced; After that, by using contraction mapping principle, some sufficient conditions which ensure the existence and uniqueness of positive periodic solution (positive almost periodic solution) of system (1.1) and (1.2) are established. In Section 3, by constructing a suitable Lyapunov
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
433
functional, we derive a set of easily verifiable criteria for the global attractivity of the positive periodic solution (almost periodic solution) of (1.1) and (1.2). We must point out, the idea of introducing parameters is stimulated by the recent works of Chen et al. [6,23] and Xie et al. [24]. However, to the best of the authors knowledge, this is the first time such a technique is applied to the neutral delays ecosystem. 2. Periodic solutions and almost periodic solutions We will investigate the existence and uniqueness of positive periodic solution (positive almost periodic solution) of system (1.1) in this section. Lemma 2.1. The domain Rnþ ¼ fðx1 ; . . . ; xn Þjxi > 0; i ¼ 1; 2; . . . ; ng is invariant with respect to (1.1) and (1.2). From now on, let T > 0 be a constant, CT = {xjx 2 C(Rn, R), x(t + T) x(t)} with the norm defined by kxk = maxt2[0,T]jx(t)j. Then under above norm CT is Banach space. Following, we will discuss the existence of positive periodic solution of system (1.1) and (1.2). To do so, we assume that: (H1) ri(t), aij(t), bij(t) are all continuous, real-valued T-periodic functions on R, dij(t) are all continuously differentiable T-periodic functions such that Z
Z
T
ri ðtÞ dt P 0; aii ðtÞ P 0; 0
T
aii ðtÞ dt > 0; aij ðtÞ P 0 ði 6¼ jÞ;
0
bij ðtÞ P 0;
cij ðtÞ P 0;
d ij ðtÞ P 0; for all i; j ¼ 1; 2; . . . ; n.
(H2) sij(t), gij(t) are all real-valued T-periodic functions on R such that sij(t), gij(t) 2 C2(R, R), s0ij ðtÞ < 1, g0ij ðtÞ < 1. Let Ni(t) = exp{qixi(t)}, i = 1, 2, . . ., n, then Eq. (1.1) can be rewrite in the form n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ dt qi qi qi j¼1 j¼1 j¼1
Z
j6¼i
t
K ij ðt sÞxj ðsÞ ds
1
n X qj d ij ðtÞð1 g0ij ðtÞÞx0j ðt gij ðtÞÞ þ q1 i ri ðtÞ. q i j¼1
ð2:1Þ
Obviously, the existence of unique positive periodic solution of system (1.1) is equivalent to the existence of unique periodic solution of system (2.1). Lemma 2.2. Assume that v(t), g(t) are all continuously R T differentiable T-periodic functions; a(t), b(t) are all nonnegative continuous T-periodic functions such that 0 aðtÞ dt > 0; then Z t Z t Rt Rt aðsÞds aðsÞ ds e s bðsÞv0 ðs gðsÞÞ ds ¼ cðtÞvðt gðtÞÞ e s ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds; 1
1
where c(s) = b(s)/(1 g 0 (s)). Proof Z
t
e 1 Z ¼
Rt s
aðsÞ ds
t
bðsÞv0 ðs gðsÞÞ ds
Rt
aðsÞ ds
cðsÞ dvðs gðsÞÞ t Z Rt aðsÞ ds ¼e s cðsÞvðs gðsÞÞ e
s
1
¼e
Rt s
aðsÞ ds
1 t
cðsÞvðs gðsÞÞ
1
t
vðs gðsÞÞdðe
1 Z t
e 1
Rt s
aðsÞ ds
Rt s
aðsÞ ds
cðsÞÞ
ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds.
ð2:2Þ
434
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
RT
RT aðsÞ ds Denote k ¼ e 0 , then from aðtÞ P 0; 0 aðtÞ dt > 0 it follows k < 1. Also, when t P s without loss of generality, we may assume s + nT 6 t 6 t + (n + 1)T, thus n1 R Rt P sþðjþ1ÞT Rt Rt aðsÞ ds aðsÞ ds sþjT sþnT aðsÞ ds aðsÞ ds je s cðsÞvðs gðsÞÞj 6 e s jjvjj jjcjj ¼ e j¼0 jjvjj jjcjj Rt aðsÞ ds ¼ k n e sþnT jjvjj jjcjj 6 k n jjvjj jjcjj. Therefore lim e
Rt s
aðsÞ ds
s!1
cðsÞvðs gðsÞÞ ¼ 0;
and so, from (2.2) it follows: Z t Z Rt aðsÞ ds 0 e s bðsÞv ðs gðsÞÞ ds ¼ cðtÞvðt gðtÞÞ 1
t
e
Rt s
aðsÞ ds
ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds.
1
The proof is complete.
h T
For vðtÞ ¼ ðv1 ðtÞ; . . . ; vn ðtÞÞ 2 C 1T , letÕs consider the equation n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞvj ðtÞ bij ðtÞvj ðt sij ðtÞÞ cij ðtÞ dt q q qi i i j¼1 j¼1 j¼1 j6¼i
Z
t
K ij ðt sÞvj ðsÞ ds
1
Since aii ðtÞ P 0;
RT 0
n X qj d ij ðtÞð1 g0ij ðtÞÞv0j ðt gij ðtÞÞ þ q1 i ri ðtÞ. qi j¼1
ð2:3Þ
aii ðtÞ dt > 0 , it follows that the linear system of system (2.3)
x_ i ðtÞ ¼ aii ðtÞxi ðtÞ; i ¼ 1; 2; . . . ; n
ð2:4Þ
admits exponential dichotomies on R, and so, system (2.3) has a unique continuous periodic solution xiv(t), which can be expressed as Z t Rt a ðsÞ ds xiv ðtÞ ¼ e s ii fiv ðsÞ ds; ð2:5Þ 1
where fiv ðsÞ ¼
Z t n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij ðsÞÞ cij ðsÞ K ij ðs sÞvj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj d ij ðsÞð1 g0ij ðsÞÞv0j ðs gij ðsÞÞ þ q1 i r i ðsÞ. qi j¼1
Now, by using Lemma 2.2, xiv(t) can also be expressed as Z t Rt n X qj a ðsÞ ds xiv ðtÞ ¼ d ij ðtÞvj ðt gij ðtÞÞ þ e s ii giv ðsÞ ds; q 1 i j¼1 where giv ðsÞ ¼
ð2:6Þ
Z t n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij ðsÞÞ cij ðsÞ K ij ðs sÞvj ðsÞds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj aii ðsÞd ij ðsÞ þ d 0ij ðsÞ vj ðs gij ðsÞÞ þ q1 þ i ri ðsÞ. qi j¼1
ð2:7Þ
Our main result on the global existence of a positive periodic solution of (1.1) and (1.2) is stated as follows. Theorem 2.1. In addition to (H1)–(H2), assume further that there exist positive constants qi, i = 1, 2, . . ., n such that
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
sup t2R
e
Rt s
aii ðsÞ ds
1
X n qj qi ðsÞ ds < 1 d ij ðtÞ; j¼1 qi
435
ð2:8Þ
where n n n X X X qj qj qj 0 qi ðsÞ ¼ aij ðsÞ þ bij ðsÞ þ cij ðsÞ þ ðaii ðsÞd ij ðsÞ þ jd ij ðsÞjÞ ; d ij ðtÞ q q q i i j¼1 j¼1 j¼1;j6¼i i ) ( X n qj ¼ max d ij ðtÞ . t2½0;T q i j¼1 def
ð2:9Þ
Then (1.1) and (1.2) has a unique T-periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . Proof. For v(t) = (v1(t), . . ., vn(t))T 2 CT, from (2.6) we know that Z t Rt n X qj a ðsÞ ds xiv ðtÞ ¼ d ij ðtÞvj ðt gij ðtÞÞ þ e s ii giv ðsÞ ds; q 1 i j¼1
ð2:10Þ
where giv(s) are defined by (2.7), is a continuous T-periodic function, and so xv(t) = (x1v(t), . . ., xnv(t))T 2 CT. Now define the mapping T:CT ! CT as follows: TvðtÞ ¼ xv ðtÞ
ð8v 2 C T Þ.
ð2:11Þ
Following we will prove the mapping T is a contraction mapping. In fact, for any u(t) = (u1(t), . . ., un(t))T and v(t) = (v1(t), . . ., vn(t))T, from (2.10), (2.11) and the conditions of theorem it follows: X n qj jjTu Tvjj 6 sup max c1j ðtÞðvj ðt g1j ðtÞÞ uj ðt g1j ðtÞÞÞ qi t2R j¼1 Z t Rt X n qj a11 ðsÞ ds þ e s jg1u ðsÞ g1v ðsÞjds; . . . ; cnj ðtÞðvj ðt gnj ðtÞÞ uj ðt gnj ðtÞÞÞ q 1 i j¼1 ! Z t Rt a ðsÞ ds þ e s nn jgnu ðsÞ gnv ðsÞjds 1
Z ! Rt X n t qj a11 ðsÞds < sup max c ðtÞ þ e s q1 ðsÞ ds jju vjj; . . . ; j¼1 qi 1j t2R 1 Z ! ! Rt X n t qj ann ðsÞds c ðtÞ þ e s qn ðsÞ ds jju vjj 6 jju vjj; j¼1 qi nj 1 where we use the fact jjgiu ðsÞ giv ðsÞjj 6
n n X X qj qj aij ðsÞjuj ðsÞ vj ðsÞj þ bij ðsÞjuj ðs sij ðsÞÞ vj ðs sij ðsÞÞj qi qi j¼1 j¼1 j6¼i
Z s n X qj cij ðsÞ K ij ðs sÞjuj ðsÞ vj ðsÞjds qi 1 j¼1 n X qj þ aii ðsÞd ij ðsÞ þ jd 0ij ðsÞj juj ðs gij ðsÞÞ vj ðs gij ðsÞÞj qi j¼1 þ
ð2:12Þ
436
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
6
n n n X X X qj qj qj aij ðsÞjju vjj þ bij ðsÞjju vjj þ cij ðsÞjju vjj q q qi i i j¼1 j¼1 j¼1 j6¼i
þ
n X qj aii ðsÞd ij ðsÞ þ jd 0ij ðsÞj jju vjj ¼ qi ðsÞjju vjj. qi j¼1
That is, jjTu Tvjj < jju vjj. This shows that T is a contraction mapping. x(t) = (x1(t), . . ., xn(t))T 2 CT such that Tx = x, that is
ð2:13Þ Hence,
there
exists
a
unique
fixed
point
Z t Rt n X qj a ðsÞ ds xi ðtÞ ¼ d ij ðtÞxj ðt gij ðtÞÞ þ e s ii q 1 i j¼1 " Z s n n n X X X qj qj qj aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ K ij ðs sÞxj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
# n X qj 0 1 aii ðsÞd ij ðsÞ þ d ij ðsÞ xj ðs gij ðsÞÞ þ qi ri ðsÞ ds. þ qi j¼1
ð2:14Þ
Following, we prove x(t) = (x1(t), . . ., xn(t))T 2 CT is the periodic solution of system (2.1). Noticing that (2.14) is equivalent to n X qj d ij ðtÞxj ðt gij ðtÞÞ xi ðtÞ þ qi j¼1 " Z t Rt n n n X X X qj qj qj aii ðsÞ ds e s aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ ¼ qi qi qi 1 j¼1 j¼1 j¼1 j6¼i # Z s n X qj K ij ðs sÞxj ðsÞ ds þ ðaii ðsÞd ij ðsÞ þ d 0ij ðsÞÞxj ðs gij ðsÞÞ þ q1 ð2:15Þ i r i ðsÞ ds. qi 1 j¼1 From the right-hand sides of (2.15), we know that xi ðtÞ þ
n X qj d ij ðtÞxj ðt gij ðtÞÞ qi j¼1
is differentiable. And so, from (2.15) it follows that n n X qj 0 qj dxi ðtÞ X þ d ij ðtÞxj ðt gij ðtÞÞ þ d ij ðtÞð1 g0ij ðtÞÞxj ðt gij ðtÞÞ dt q q i i j¼1 j¼1 ! n X qj d xi ðtÞ þ d ij ðtÞxj ðt gij ðtÞÞ ¼ dt qi j¼1 Z s n n n X X X qj qj qj ¼ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ K ij ðt sÞxj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
þ
n X qj ðaii ðtÞd ij ðtÞ þ d 0ij ðtÞÞxj ðt gij ðtÞÞ þ q1 i ri ðtÞ aii ðtÞ q i j¼1
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
e
1
Rt s
2 aii ðsÞds 6
4
437
n n n X X X qj qj qj aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ q q qi i i j¼1 j¼1 j¼1 j6¼i
3 n X qj 7 K ij ðs sÞxj ðsÞ ds þ ðaii ðsÞd ij ðsÞ þ d 0ij ðsÞÞxj ðs gij ðsÞÞ þ q1 i ri ðsÞ5ds q 1 i j¼1 Z
¼
s
n n n X X X qj qj qj aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ qi qi qi j¼1 j¼1 j¼1 j6¼i
Z
n X qj ðaii ðtÞd ij ðtÞ þ d 0ij ðtÞÞxj ðt gij ðtÞÞ qi 1 j¼1 " # n X qj 1 þ qi ri ðtÞ aii ðtÞ xi ðtÞ þ d ij ðtÞxj ðt gij ðtÞÞ . qi j¼1
s
K ij ðt sÞxj ðsÞ ds þ
(here using the equality (2.15) again). That is n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ dt q q qi i i j¼1 j¼1 j¼1
Z
t
K ij ðt sÞxj ðsÞ ds
1
j6¼i
n X j¼1
qj d ij ðtÞð1 g0ij ðtÞÞxj ðt gij ðtÞÞ þ q1 i ri ðtÞ. qi
This shows that x(t) = (x1(t), . . ., xn(t))T is continuously differentiable T-periodic function and satisfies Eq. (2.1). Therefore, x(t) = (x1(t), . . ., xn(t))T is the unique continuously differentiable T-periodic solution of system (2.1), and so, N(t) = (exp{d1x1(t)}, . . ., exp{dnxn(t)})T is the unique positive T-periodic solution of system (1.1). The proof is complete. h As a direct corollary of Theorem 2.1, one has Corollary 1. In addition to (H1)–(H2), Assume further that there exist positive constants qi, i = 1, 2, . . ., n such that ! X n n n X X qj qj qj def 0 aij ðtÞ þ bij ðtÞ þ cij ðtÞ þ ðaii ðtÞd ij ðtÞ þ jd ij ðtÞjÞ < aii ðtÞ 1 d ij ðtÞ . qi ðtÞ ¼ q q q i i i j¼1 j¼1;j6¼i j¼1;j6¼i Then (1.1) and (1.2) has unique positive T-periodic solution. Our next theorem concerned with the existence of unique positive almost periodic solution of system (1.1) and (1.2). To do so, we assume that: (H3) ri(t), aij(t), bij(t), cij(t) are all continuous, real-valued almost periodic functions on R, dij(t) are all continuously differentiable almost periodic functions such that ri ðtÞ P 0;
aii ðtÞ P 0;
mðaii ðtÞÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; R tþT where mðaii ðtÞÞ ¼ limT !þ1 T1 t aii ðsÞ ds; i; j ¼ 1; 2; . . . ; n; (H4) sij(t) sij, gij(t) gij are all nonnegative real numbers. Similarly to the proof of Lemma 2.2, we have
bij ðtÞ P 0;
d ij ðtÞ P 0;
Lemma 2.3. Assume that v(t) is continuously differentiable almost periodic function; a(t), b(t) are all nonnegative continuous almost periodic functions such that m(a(t)) dt > 0, g is positive number. Then
438
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
e
Rt s
aðsÞds
0
bðsÞv ðs gÞds ¼ bðtÞvðt gÞ
1
where mðaðtÞÞ ¼ limT !þ1
R tþT t
Z
t
e
Rt s
aðsÞ ds
ðaðsÞbðsÞ þ b0 ðsÞÞvðs gÞds;
1
aðtÞ dt.
Let Ni(t) = exp{qixi(t)}, i = 1, 2, . . ., n, Eq. (1.1) can be rewrite in the following form n n X X qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij Þ dt qi qi j¼1 j¼1 j6¼i
Z t n n X X qj qj cij ðtÞ K ij ðt sÞxj ðsÞds d ij ðtÞx0j ðt gij Þ þ q1 i r i ðtÞ. q q 1 i i j¼1 j¼1
ð2:16Þ
Obviously, the existence of unique positive almost periodic solution of system (1.1) is equivalent to the existence of unique almost periodic solution of system (2.16). Let v(t) = (v1(t), . . ., vn(t))T be any continuously differentiable almost periodic function, and consider equation n n X X qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞvj ðtÞ bij ðtÞvj ðt sij Þ dt qi qi j¼1 j¼1 j6¼i
Z t n n X X qj qj cij ðtÞ K ij ðt sÞvj ðsÞ ds d ij ðtÞv0j ðt gij Þ þ q1 i ri ðtÞ. q q 1 i i j¼1 j¼1 Since m(aii(t)) > 0, it follows that the linear system of system (2.17) x_ i ðtÞ ¼ aii ðtÞxi ðtÞ; i ¼ 1; 2; . . . ; n
ð2:17Þ
ð2:18Þ
admits exponential dichotomies on R, and so, system (2.17) has unique continuously almost periodic solution xiv(t), which can be expressed as Z t Rt a ðsÞ ds e s ii fiv ðsÞ ds; ð2:19Þ xiv ðtÞ ¼ 1
where fiv ðsÞ ¼
Z s n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij Þ cij ðsÞ K ij ðs sÞvj ðsÞds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj d ij ðsÞv0j ðs gij Þ þ q1 i ri ðsÞ. q i j¼1
Now, by using Lemma 2.3, xiv(t) can also be expressed as Z t Rt n X qj a ðsÞds xiv ðtÞ ¼ d ij ðtÞvj ðt gij Þ þ e s ii giv ðsÞ ds; q 1 i j¼1
ð2:20Þ
where
Z s n n n X X X qj qj qj giv ðsÞ ¼ aij ðsÞvj ðsÞ bij ðsÞvj ðs sij Þ cij ðsÞ K ij ðs sÞvj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj þ aii ðsÞd ij ðsÞ þ d 0ij ðsÞ vj ðs gij Þ þ q1 i ri ðsÞ. qi j¼1
ð2:21Þ
Then, we have Theorem 2.2. In addition to (H3)–(H4), assume that there exist positive constants qi, i = 1, 2, . . ., n such that Z t Rt X n qj aii ðsÞds ð2:22Þ sup e s qi ðsÞ ds < 1 d ij ðtÞ; qi t2R 1 j¼1
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
439
where n n X X qj qj aij ðsÞ þ bij ðsÞ þ cij ðsÞ þ ðaii ðsÞd ij ðsÞ þ jd 0ij ðsÞjÞ ; q qi j¼1 j¼1;j6¼i i ) ( X X n n qj qj d ij ðtÞ ¼ max d ij ðtÞ . t2R j¼1 qi j¼1 qi def
qi ðsÞ ¼
ð2:23Þ
Then (1.1) and (1.2) have a unique positive almost periodic solution with strictly positive components, say T N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞ . Proof. Set C = {u(t) = (u1(t), . . ., un(t))T—u:R ! Rn is continuous almost periodic function}, then under the norm kuk = sup{ku(t)k:t 2 R}, C is a Banach space. For any continuously almost periodic function v(t) = (v1(t), . . ., vn(t))T we know that xiv(t) defined by (2.20) is also a continuously almost periodic function. Now define the mapping F:C ! C as follows: FvðtÞ ¼ xv ðtÞ; ð8v 2 CÞ.
ð2:24Þ
Then similarly to the prove of Theorem 2.1, we could prove that under the assumptions of Theorem 2.2, the mapping F is a contract mapping, and so system (2.20) has a unique fixed point x(t) = (x1(t), . . ., xn(t))T. Also, similarly to the proof of Theorem 2.1, we could prove that N(t) = (exp{d1x1(t)}, . . ., exp{dnxn(t)})T is the unique positive almost periodic solution of system (1.1). The proof is complete. h As a direct corollary of Theorem 2.2, one has Corollary 2. In addition to (H3)–(H4), assume further that there exist positive constants qi, i = 1, 2, . . ., n such that ! X n n n X X qj qj qj def 0 qi ðtÞ ¼ aij ðtÞ þ bij ðtÞ þ cij ðtÞ þ ðaii ðtÞd ij ðtÞ þ jd ij ðtÞjÞ < aii ðtÞ 1 d ij ðtÞ . q q q i i j¼1 j¼1 j¼1;j6¼i i Then, (1.1) and (1.2) have a unique positive almost periodic solution. 3. Global asymptotic stability In this section, we devote ourselves to the study of the global attractivity of periodic solution (almost periodic solution) of system (1.1) and (1.2). Definition 3.1. Let N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT be a strictly positive periodic solution (almost periodic solution) of (1.1) and (1.2). We say N*(t) is globally attractive if any other solution Y(t) = (y1(t), . . ., yn(t)) of (1.1) and (1.2) has the property lim jN i ðtÞ y i ðtÞj ¼ 0;
t!þ1
i ¼ 1; 2; . . . ; n.
ð3:1Þ
Now, we state our main results of this section below. Theorem 3.1. Assume that the conditions in Theorem 2.1 (or Theorem 2.2) hold. Assume further that (H5) and (H6) hold, where (H5) cij(t)0,sij(t) sij, gij(t) gij are all nonnegative real numbers; (H6) There exist positive constant a,b such that aii ðtÞ P a > 0; X n n X qi aji ðtÞ qi bji ðt þ sij Þ P b > 0; aii ðtÞ q q j¼1;j6¼i
j
j¼1
j
440
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
for all t P 0 and i = 1, 2, . . ., n. Then system (1.1) and (1.2) has a unique periodic solution (almost periodic solution) which is globally attractive. Remark. P From the conditions of Theorem 2.1 (or Theorem 2.2) one could easily obtains that def q c ¼ max j nj¼1 qji d ij ðtÞj < 1. 16i6n t2R
T
Proof. Let N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞ be the unique positive periodic solution (almost periodic solution) of system (1.1) and (1.2), whose existence and uniqueness are guarantee by Theorem 2.1 (Theorem 2.2) , and y(t) = (y1(t), . . ., y2(t)) be any other solution of system 1.1,1.2. Let vi ðtÞ ¼ expfqi N i ðtÞg; xi ðtÞ ¼ expfqi y i ðtÞg; then n n X X qj qj aij ðtÞvj ðtÞ bij ðtÞvj ðt sij Þ v_ i ðtÞ ¼ aii ðtÞvi ðtÞ qi qi j¼1 j¼1 j6¼i
n X j¼1
qj d ij ðtÞv0j ðt gij Þ þ q1 i ri ðtÞ. qi
x_ i ðtÞ ¼ aii ðtÞxi ðtÞ
ð3:2Þ
n n X X qj qj aij ðtÞxj ðtÞ bij ðtÞxj ðt sij Þ q qi i j¼1 j¼1 j6¼i
n X j¼1
qj d ij ðtÞx0j ðt gij Þ þ q1 i ri ðtÞ. qi
Now we let wi(t) = vi(t) xi(t), then n n n X X X qj qj qj aij ðtÞwj ðtÞ bij ðtÞwj ðt sij Þ d ij ðtÞw0j ðt gij Þ. w_ i ðtÞ ¼ aii ðtÞwi ðtÞ qi q q i i j¼1 j¼1 j¼1
ð3:3Þ
ð3:4Þ
j6¼i
Then under the conditions of Theorem 3.1, all the conditions of Theorem 1 of [25] are satisfied. Now follows from the Theorem 1 of [25] one has n X lim jwj ðtÞj ¼ 0; t!þ1
j¼1
which is equivalent to n X lim jN j ðtÞ y j ðtÞj ¼ 0. t!þ1
j¼1
The proof of Theorem 3.1 is finished.
h
Finally, let us consider the following examples: Example 1.
dN 1 1 1 d ¼ N ðtÞ 2 þ sin t ln N ðtÞ ln N ðt j2 cos tjÞ ln N ðt j sin 4tjÞ . dt 4 4 4 dt
ð3:5Þ
One could easily see that all the conditions of Theorem 2.1 are satisfied for system (3.5), and so, system (3.5) has unique positive 2p-periodic solution. Example 2.
dN 1 1 1 d ¼ N ðtÞ 2 þ sin t ln N ðtÞ ln N ðt 2Þ ln N ðt 4Þ . dt 4 8 4 dt
ð3:6Þ
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
441
One could easily see that all the conditions of Theorem 3.1 are satisfied for system (3.6), and so, system (3.6) has unique globally attractive positive 2p-periodic solution. Acknowledgements This work is supported by the National Natural Science Foundation of China (10501007), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.amc.2005.09.032. References [1] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population DynamicsMathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. [2] G. Kirlinger, Permanence in Lotka–Volterra equations linked prey-predator systems, Math. Biosci. 82 (1986) 165–169. [3] Y.K. Li, Attractivity of a positive periodic solution for all other positive solution in a delay population model, Appl. Math.-JCU 12 (3) (1997) 279–282 (in Chinese). [4] F.D. Chen et al., Positive periodic solutions of state-dependent delay logarithm population model, J. Fuzhou University 31 (3) (2003) 1–4 (in Chinese). [5] Z.J. Liu, Positive periodic solutions for delay multispecies Logrithmic population model, J. Eng. Math. 19 (4) (2002) 11–16 (in Chinese). [6] F.D. Chen, Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model, Appl. Math. Comput., in press. [7] Y.K. Li, On a periodic neutral delay logarithmic population model, J. Sys. Sci. Math. Sci. 19 (1) (1999) 34–38. [8] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral logarithmic population model with multiple delays, J. Comput. Appl. Math. 166 (2) (2004) 371–383. [9] Y.K. Li, Positive periodic solution for neutral delay model, Acta Math. Sin. 39 (6) (1996) 789–795 (in Chinese). [10] H. Fang, J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl. 259 (2001) 8–17. [11] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral functional differential equations with deviating arguments, Appl. Math. J. Chinese Univ. Ser. B 17 (4) (2002) 382–390. [12] Z.H. Yang, J.D. Cao, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delays models, Appl. Math. Comput. 142 (1) (2003) 123–142. [13] Z.H. Yang, J.D. Cao, Positive periodic solutions of neutral Lotka-Volterra system with periodic delays, Appl. Math. Comput. 149 (3) (2004) 661–687. [14] Y.K. Li, On a periodic neutral delay Lotka–Volterra system, Nonlinear Anal. 39 (2000) 767–778. [15] H.F. Huo, W.T. Li, Existence of positive periodic solutions of a neutral Lotka-Volterra system with delays, Acta Math. Sinica 46 (6) (2003) 1199–1210 (in Chinese). [16] F.D. Chen, F.X. Lin, X.X. Chen, Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput. 158 (1) (2004) 45–68. [17] F.D. Chen, Positive periodic solutions of neutral Lotka–Volterra system with feedback control, Appl. Math. Comput. 162 (3) (2005) 1279–1302. [18] F.D. Chen, S.J. Lin, Periodicity in a Logistic type system with several delays, Comput. Math. Appl. 48 (1–2) (2004) 35–44. [19] F.D. Chen, D.X. Sun, F.X. Lin, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl. 288 (1) (2003) 132–142. [20] H. Fang, Positive periodic solutions of n-species neutral delay systems, Czechoslovak Math. J. 53 (3) (2003) 561–570. [21] Z.J. Liu, Positive periodic solution for a neutral delay competitive system, J. Math. Anal. Appl. 293 (1) (2004) 181–189. [22] Y.N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional delay 2003 (102) (2003) 1–7. [23] F.D. Chen et al., On the existence and uniqueness of periodic solutions of a kind of integro-differential equations, Acta Math. Sinica 47 (5) (2004) 973–985 (in Chinese). [24] H.Q. Xie, Q.Y. Wang, Exponential stability and periodic solution for cellular neural networks with time delay, J. Huaqiao University 25 (1) (2004) 22–26 (in Chinese). [25] K. Gopalsamy, A simple stability criterion for linear neutral differential systems, Funkcial Ekvac. 28 (1985) 33–38.
Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
Abstract A neutral multispecies Logarithmic population model is proposed in this paper. By using the contraction mapping principle and constructing a suitable Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) of the model. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Periodic solution; Almost periodic solution; Lyapunov functional; Global attractivity; Contraction mapping principle
1. Introduction The aim of this paper is to investigate the existence, uniqueness and global attractivity of the positive periodic solution (positive almost periodic solution) of the following neutral multispecies Logarithmic population model " n n X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ dt j¼1 j¼1 # Z t n n X X d ln N j ðt gij ðtÞÞ cij ðtÞ K ij ðt sÞ ln N j ðsÞ ds d ij ðtÞ ; ð1:1Þ dt 1 j¼1 j¼1 where i = 1, 2, . . . , n, ri(t), aijR(t), bij(t), cij(t), dij(t) 2 C(R, (0, +1)), sij(t), gij(t) 2 C(R, R+) are all continuous R þ1 þ1 functions. 0 K ij ðsÞ ds ¼ 1; 0 sK ij ðsÞ ds < þ1. We consider (1.1) together with the initial conditions N i ðtÞ ¼ /i ðtÞ P 0; /i 2 Cðð1; 0;
N_ i ðtÞ ¼ /_ i ðtÞ;
/i ð0Þ > 0;
1
½0; þ1ÞÞ \ C ðð1; 0; ½0; þ1ÞÞ.
For the ecological justification of Eq. (1.1), see [1–8].
E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.032
ð1:2Þ
432
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Gopalsamy [1] and Kirlinger [2] had proposed the following single species Logarithmic model: dN ðtÞ ¼ N ðtÞ½a b ln N ðtÞ c ln N ðt sÞ. dt
ð1:3Þ
System (1.3) is then generalized by Li [3] to the nonautonomous case dN ðtÞ ¼ N ðtÞ½aðtÞ bðtÞ ln N ðtÞ cðtÞ ln N ðt sðtÞÞ. dt
ð1:4Þ
In [3], by using the coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of system (1.4) are established. Also, sufficient conditions are obtained for the periodic solution attracting all other positive solutions. In [4], we generalized above system to the system with state dependent delays and investigated the existence of positive periodic solutions of the system. In [5], Liu proposed the following multispecies periodic Logarithmic population model: " # n n X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ ; ð1:5Þ dt j¼1 j¼1 where aij,bij 2 C(R, (0, +1)), ri, sij 2 C(R, R), i, j = 1, 2, . . . , n, are all continuous T-periodic functions. By using the coincidence degree theory and constructing the Lyapunov functional, some sufficient conditions which guarantee the existence, uniqueness and stability of the positive periodic solution of the system (1.5) are established. Recently, we [6] pointed out that the proof of Theorem 1 in [5] is incomplete, also, in [6], we proposed the following multispecies Logarithmic population model: " # Z t n n n X X X dN i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ cij ðtÞ K ij ðt sÞ ln N j ðsÞ ds ; dt 1 j¼1 j¼1 j¼1 R þ1
ð1:6Þ
where ri(t), aij, bij 2 C(R, (0, + 1)), sij 2 C(R, R) are all continuous functions. 0 K ij ðsÞds ¼ 1; R þ1 sK ij ðsÞ ds < þ1. By using the method of fixed point theory and constructing a suitable Lyapunov 0 functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) of the model (1.6). As was pointed out by Gopalsamy [1], in some case, the neutral delay population models are more realistic. There were many scholars done works on the periodic solution of neutral type Logistic model or Lotka– Volterra model (see [9–22] and the reference cited therein), seldom did scholars considered the neutral Logarithmic model (see [7,8]). Li [7] had studied the following single species neutral Logarithmic model: dN ðtÞ d ln N ðt sÞ ¼ N ðtÞ rðtÞ aðtÞ ln N ðt rÞ bðtÞ . ð1:7Þ dt dt Recently, Lu et al. [8] further investigated the following system: " # n m X X dN ðtÞ d ln N ðt sj ðtÞÞ ¼ N ðtÞ rðtÞ aj ðtÞ ln N ðt rj ðtÞÞ bj ðtÞ . dt dt j¼1 j¼1
ð1:8Þ
By employing an abstract continuous theorem of k-set contractive operator, some new results on the existence of positive periodic solution of system (1.8) are obtained. However, to this day, still no scholars investigate the multispecies neutral delay Logarithmic model. Specially, no scholars had done works on the existence, uniqueness and stability of positive periodic solution (positive almost periodic solution) of system (1.1). One could easily see that system (1.3)–(1.7) are all special cases of system (1.1). The outline of the paper is as follows. In Section 2, we first introducing a transformation, where some adjustable real parameters qi > 0 are introduced; After that, by using contraction mapping principle, some sufficient conditions which ensure the existence and uniqueness of positive periodic solution (positive almost periodic solution) of system (1.1) and (1.2) are established. In Section 3, by constructing a suitable Lyapunov
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
433
functional, we derive a set of easily verifiable criteria for the global attractivity of the positive periodic solution (almost periodic solution) of (1.1) and (1.2). We must point out, the idea of introducing parameters is stimulated by the recent works of Chen et al. [6,23] and Xie et al. [24]. However, to the best of the authors knowledge, this is the first time such a technique is applied to the neutral delays ecosystem. 2. Periodic solutions and almost periodic solutions We will investigate the existence and uniqueness of positive periodic solution (positive almost periodic solution) of system (1.1) in this section. Lemma 2.1. The domain Rnþ ¼ fðx1 ; . . . ; xn Þjxi > 0; i ¼ 1; 2; . . . ; ng is invariant with respect to (1.1) and (1.2). From now on, let T > 0 be a constant, CT = {xjx 2 C(Rn, R), x(t + T) x(t)} with the norm defined by kxk = maxt2[0,T]jx(t)j. Then under above norm CT is Banach space. Following, we will discuss the existence of positive periodic solution of system (1.1) and (1.2). To do so, we assume that: (H1) ri(t), aij(t), bij(t) are all continuous, real-valued T-periodic functions on R, dij(t) are all continuously differentiable T-periodic functions such that Z
Z
T
ri ðtÞ dt P 0; aii ðtÞ P 0; 0
T
aii ðtÞ dt > 0; aij ðtÞ P 0 ði 6¼ jÞ;
0
bij ðtÞ P 0;
cij ðtÞ P 0;
d ij ðtÞ P 0; for all i; j ¼ 1; 2; . . . ; n.
(H2) sij(t), gij(t) are all real-valued T-periodic functions on R such that sij(t), gij(t) 2 C2(R, R), s0ij ðtÞ < 1, g0ij ðtÞ < 1. Let Ni(t) = exp{qixi(t)}, i = 1, 2, . . ., n, then Eq. (1.1) can be rewrite in the form n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ dt qi qi qi j¼1 j¼1 j¼1
Z
j6¼i
t
K ij ðt sÞxj ðsÞ ds
1
n X qj d ij ðtÞð1 g0ij ðtÞÞx0j ðt gij ðtÞÞ þ q1 i ri ðtÞ. q i j¼1
ð2:1Þ
Obviously, the existence of unique positive periodic solution of system (1.1) is equivalent to the existence of unique periodic solution of system (2.1). Lemma 2.2. Assume that v(t), g(t) are all continuously R T differentiable T-periodic functions; a(t), b(t) are all nonnegative continuous T-periodic functions such that 0 aðtÞ dt > 0; then Z t Z t Rt Rt aðsÞds aðsÞ ds e s bðsÞv0 ðs gðsÞÞ ds ¼ cðtÞvðt gðtÞÞ e s ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds; 1
1
where c(s) = b(s)/(1 g 0 (s)). Proof Z
t
e 1 Z ¼
Rt s
aðsÞ ds
t
bðsÞv0 ðs gðsÞÞ ds
Rt
aðsÞ ds
cðsÞ dvðs gðsÞÞ t Z Rt aðsÞ ds ¼e s cðsÞvðs gðsÞÞ e
s
1
¼e
Rt s
aðsÞ ds
1 t
cðsÞvðs gðsÞÞ
1
t
vðs gðsÞÞdðe
1 Z t
e 1
Rt s
aðsÞ ds
Rt s
aðsÞ ds
cðsÞÞ
ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds.
ð2:2Þ
434
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
RT
RT aðsÞ ds Denote k ¼ e 0 , then from aðtÞ P 0; 0 aðtÞ dt > 0 it follows k < 1. Also, when t P s without loss of generality, we may assume s + nT 6 t 6 t + (n + 1)T, thus n1 R Rt P sþðjþ1ÞT Rt Rt aðsÞ ds aðsÞ ds sþjT sþnT aðsÞ ds aðsÞ ds je s cðsÞvðs gðsÞÞj 6 e s jjvjj jjcjj ¼ e j¼0 jjvjj jjcjj Rt aðsÞ ds ¼ k n e sþnT jjvjj jjcjj 6 k n jjvjj jjcjj. Therefore lim e
Rt s
aðsÞ ds
s!1
cðsÞvðs gðsÞÞ ¼ 0;
and so, from (2.2) it follows: Z t Z Rt aðsÞ ds 0 e s bðsÞv ðs gðsÞÞ ds ¼ cðtÞvðt gðtÞÞ 1
t
e
Rt s
aðsÞ ds
ðaðsÞcðsÞ þ c0 ðsÞÞvðs gðsÞÞ ds.
1
The proof is complete.
h T
For vðtÞ ¼ ðv1 ðtÞ; . . . ; vn ðtÞÞ 2 C 1T , letÕs consider the equation n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞvj ðtÞ bij ðtÞvj ðt sij ðtÞÞ cij ðtÞ dt q q qi i i j¼1 j¼1 j¼1 j6¼i
Z
t
K ij ðt sÞvj ðsÞ ds
1
Since aii ðtÞ P 0;
RT 0
n X qj d ij ðtÞð1 g0ij ðtÞÞv0j ðt gij ðtÞÞ þ q1 i ri ðtÞ. qi j¼1
ð2:3Þ
aii ðtÞ dt > 0 , it follows that the linear system of system (2.3)
x_ i ðtÞ ¼ aii ðtÞxi ðtÞ; i ¼ 1; 2; . . . ; n
ð2:4Þ
admits exponential dichotomies on R, and so, system (2.3) has a unique continuous periodic solution xiv(t), which can be expressed as Z t Rt a ðsÞ ds xiv ðtÞ ¼ e s ii fiv ðsÞ ds; ð2:5Þ 1
where fiv ðsÞ ¼
Z t n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij ðsÞÞ cij ðsÞ K ij ðs sÞvj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj d ij ðsÞð1 g0ij ðsÞÞv0j ðs gij ðsÞÞ þ q1 i r i ðsÞ. qi j¼1
Now, by using Lemma 2.2, xiv(t) can also be expressed as Z t Rt n X qj a ðsÞ ds xiv ðtÞ ¼ d ij ðtÞvj ðt gij ðtÞÞ þ e s ii giv ðsÞ ds; q 1 i j¼1 where giv ðsÞ ¼
ð2:6Þ
Z t n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij ðsÞÞ cij ðsÞ K ij ðs sÞvj ðsÞds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj aii ðsÞd ij ðsÞ þ d 0ij ðsÞ vj ðs gij ðsÞÞ þ q1 þ i ri ðsÞ. qi j¼1
ð2:7Þ
Our main result on the global existence of a positive periodic solution of (1.1) and (1.2) is stated as follows. Theorem 2.1. In addition to (H1)–(H2), assume further that there exist positive constants qi, i = 1, 2, . . ., n such that
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
sup t2R
e
Rt s
aii ðsÞ ds
1
X n qj qi ðsÞ ds < 1 d ij ðtÞ; j¼1 qi
435
ð2:8Þ
where n n n X X X qj qj qj 0 qi ðsÞ ¼ aij ðsÞ þ bij ðsÞ þ cij ðsÞ þ ðaii ðsÞd ij ðsÞ þ jd ij ðsÞjÞ ; d ij ðtÞ q q q i i j¼1 j¼1 j¼1;j6¼i i ) ( X n qj ¼ max d ij ðtÞ . t2½0;T q i j¼1 def
ð2:9Þ
Then (1.1) and (1.2) has a unique T-periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . Proof. For v(t) = (v1(t), . . ., vn(t))T 2 CT, from (2.6) we know that Z t Rt n X qj a ðsÞ ds xiv ðtÞ ¼ d ij ðtÞvj ðt gij ðtÞÞ þ e s ii giv ðsÞ ds; q 1 i j¼1
ð2:10Þ
where giv(s) are defined by (2.7), is a continuous T-periodic function, and so xv(t) = (x1v(t), . . ., xnv(t))T 2 CT. Now define the mapping T:CT ! CT as follows: TvðtÞ ¼ xv ðtÞ
ð8v 2 C T Þ.
ð2:11Þ
Following we will prove the mapping T is a contraction mapping. In fact, for any u(t) = (u1(t), . . ., un(t))T and v(t) = (v1(t), . . ., vn(t))T, from (2.10), (2.11) and the conditions of theorem it follows: X n qj jjTu Tvjj 6 sup max c1j ðtÞðvj ðt g1j ðtÞÞ uj ðt g1j ðtÞÞÞ qi t2R j¼1 Z t Rt X n qj a11 ðsÞ ds þ e s jg1u ðsÞ g1v ðsÞjds; . . . ; cnj ðtÞðvj ðt gnj ðtÞÞ uj ðt gnj ðtÞÞÞ q 1 i j¼1 ! Z t Rt a ðsÞ ds þ e s nn jgnu ðsÞ gnv ðsÞjds 1
Z ! Rt X n t qj a11 ðsÞds < sup max c ðtÞ þ e s q1 ðsÞ ds jju vjj; . . . ; j¼1 qi 1j t2R 1 Z ! ! Rt X n t qj ann ðsÞds c ðtÞ þ e s qn ðsÞ ds jju vjj 6 jju vjj; j¼1 qi nj 1 where we use the fact jjgiu ðsÞ giv ðsÞjj 6
n n X X qj qj aij ðsÞjuj ðsÞ vj ðsÞj þ bij ðsÞjuj ðs sij ðsÞÞ vj ðs sij ðsÞÞj qi qi j¼1 j¼1 j6¼i
Z s n X qj cij ðsÞ K ij ðs sÞjuj ðsÞ vj ðsÞjds qi 1 j¼1 n X qj þ aii ðsÞd ij ðsÞ þ jd 0ij ðsÞj juj ðs gij ðsÞÞ vj ðs gij ðsÞÞj qi j¼1 þ
ð2:12Þ
436
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
6
n n n X X X qj qj qj aij ðsÞjju vjj þ bij ðsÞjju vjj þ cij ðsÞjju vjj q q qi i i j¼1 j¼1 j¼1 j6¼i
þ
n X qj aii ðsÞd ij ðsÞ þ jd 0ij ðsÞj jju vjj ¼ qi ðsÞjju vjj. qi j¼1
That is, jjTu Tvjj < jju vjj. This shows that T is a contraction mapping. x(t) = (x1(t), . . ., xn(t))T 2 CT such that Tx = x, that is
ð2:13Þ Hence,
there
exists
a
unique
fixed
point
Z t Rt n X qj a ðsÞ ds xi ðtÞ ¼ d ij ðtÞxj ðt gij ðtÞÞ þ e s ii q 1 i j¼1 " Z s n n n X X X qj qj qj aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ K ij ðs sÞxj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
# n X qj 0 1 aii ðsÞd ij ðsÞ þ d ij ðsÞ xj ðs gij ðsÞÞ þ qi ri ðsÞ ds. þ qi j¼1
ð2:14Þ
Following, we prove x(t) = (x1(t), . . ., xn(t))T 2 CT is the periodic solution of system (2.1). Noticing that (2.14) is equivalent to n X qj d ij ðtÞxj ðt gij ðtÞÞ xi ðtÞ þ qi j¼1 " Z t Rt n n n X X X qj qj qj aii ðsÞ ds e s aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ ¼ qi qi qi 1 j¼1 j¼1 j¼1 j6¼i # Z s n X qj K ij ðs sÞxj ðsÞ ds þ ðaii ðsÞd ij ðsÞ þ d 0ij ðsÞÞxj ðs gij ðsÞÞ þ q1 ð2:15Þ i r i ðsÞ ds. qi 1 j¼1 From the right-hand sides of (2.15), we know that xi ðtÞ þ
n X qj d ij ðtÞxj ðt gij ðtÞÞ qi j¼1
is differentiable. And so, from (2.15) it follows that n n X qj 0 qj dxi ðtÞ X þ d ij ðtÞxj ðt gij ðtÞÞ þ d ij ðtÞð1 g0ij ðtÞÞxj ðt gij ðtÞÞ dt q q i i j¼1 j¼1 ! n X qj d xi ðtÞ þ d ij ðtÞxj ðt gij ðtÞÞ ¼ dt qi j¼1 Z s n n n X X X qj qj qj ¼ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ K ij ðt sÞxj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
þ
n X qj ðaii ðtÞd ij ðtÞ þ d 0ij ðtÞÞxj ðt gij ðtÞÞ þ q1 i ri ðtÞ aii ðtÞ q i j¼1
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
e
1
Rt s
2 aii ðsÞds 6
4
437
n n n X X X qj qj qj aij ðsÞxj ðsÞ bij ðsÞxj ðt sij ðsÞÞ cij ðsÞ q q qi i i j¼1 j¼1 j¼1 j6¼i
3 n X qj 7 K ij ðs sÞxj ðsÞ ds þ ðaii ðsÞd ij ðsÞ þ d 0ij ðsÞÞxj ðs gij ðsÞÞ þ q1 i ri ðsÞ5ds q 1 i j¼1 Z
¼
s
n n n X X X qj qj qj aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ qi qi qi j¼1 j¼1 j¼1 j6¼i
Z
n X qj ðaii ðtÞd ij ðtÞ þ d 0ij ðtÞÞxj ðt gij ðtÞÞ qi 1 j¼1 " # n X qj 1 þ qi ri ðtÞ aii ðtÞ xi ðtÞ þ d ij ðtÞxj ðt gij ðtÞÞ . qi j¼1
s
K ij ðt sÞxj ðsÞ ds þ
(here using the equality (2.15) again). That is n n n X X X qj qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ cij ðtÞ dt q q qi i i j¼1 j¼1 j¼1
Z
t
K ij ðt sÞxj ðsÞ ds
1
j6¼i
n X j¼1
qj d ij ðtÞð1 g0ij ðtÞÞxj ðt gij ðtÞÞ þ q1 i ri ðtÞ. qi
This shows that x(t) = (x1(t), . . ., xn(t))T is continuously differentiable T-periodic function and satisfies Eq. (2.1). Therefore, x(t) = (x1(t), . . ., xn(t))T is the unique continuously differentiable T-periodic solution of system (2.1), and so, N(t) = (exp{d1x1(t)}, . . ., exp{dnxn(t)})T is the unique positive T-periodic solution of system (1.1). The proof is complete. h As a direct corollary of Theorem 2.1, one has Corollary 1. In addition to (H1)–(H2), Assume further that there exist positive constants qi, i = 1, 2, . . ., n such that ! X n n n X X qj qj qj def 0 aij ðtÞ þ bij ðtÞ þ cij ðtÞ þ ðaii ðtÞd ij ðtÞ þ jd ij ðtÞjÞ < aii ðtÞ 1 d ij ðtÞ . qi ðtÞ ¼ q q q i i i j¼1 j¼1;j6¼i j¼1;j6¼i Then (1.1) and (1.2) has unique positive T-periodic solution. Our next theorem concerned with the existence of unique positive almost periodic solution of system (1.1) and (1.2). To do so, we assume that: (H3) ri(t), aij(t), bij(t), cij(t) are all continuous, real-valued almost periodic functions on R, dij(t) are all continuously differentiable almost periodic functions such that ri ðtÞ P 0;
aii ðtÞ P 0;
mðaii ðtÞÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; R tþT where mðaii ðtÞÞ ¼ limT !þ1 T1 t aii ðsÞ ds; i; j ¼ 1; 2; . . . ; n; (H4) sij(t) sij, gij(t) gij are all nonnegative real numbers. Similarly to the proof of Lemma 2.2, we have
bij ðtÞ P 0;
d ij ðtÞ P 0;
Lemma 2.3. Assume that v(t) is continuously differentiable almost periodic function; a(t), b(t) are all nonnegative continuous almost periodic functions such that m(a(t)) dt > 0, g is positive number. Then
438
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
Z
t
e
Rt s
aðsÞds
0
bðsÞv ðs gÞds ¼ bðtÞvðt gÞ
1
where mðaðtÞÞ ¼ limT !þ1
R tþT t
Z
t
e
Rt s
aðsÞ ds
ðaðsÞbðsÞ þ b0 ðsÞÞvðs gÞds;
1
aðtÞ dt.
Let Ni(t) = exp{qixi(t)}, i = 1, 2, . . ., n, Eq. (1.1) can be rewrite in the following form n n X X qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞxj ðtÞ bij ðtÞxj ðt sij Þ dt qi qi j¼1 j¼1 j6¼i
Z t n n X X qj qj cij ðtÞ K ij ðt sÞxj ðsÞds d ij ðtÞx0j ðt gij Þ þ q1 i r i ðtÞ. q q 1 i i j¼1 j¼1
ð2:16Þ
Obviously, the existence of unique positive almost periodic solution of system (1.1) is equivalent to the existence of unique almost periodic solution of system (2.16). Let v(t) = (v1(t), . . ., vn(t))T be any continuously differentiable almost periodic function, and consider equation n n X X qj qj dxi ðtÞ ¼ aii ðtÞxi ðtÞ aij ðtÞvj ðtÞ bij ðtÞvj ðt sij Þ dt qi qi j¼1 j¼1 j6¼i
Z t n n X X qj qj cij ðtÞ K ij ðt sÞvj ðsÞ ds d ij ðtÞv0j ðt gij Þ þ q1 i ri ðtÞ. q q 1 i i j¼1 j¼1 Since m(aii(t)) > 0, it follows that the linear system of system (2.17) x_ i ðtÞ ¼ aii ðtÞxi ðtÞ; i ¼ 1; 2; . . . ; n
ð2:17Þ
ð2:18Þ
admits exponential dichotomies on R, and so, system (2.17) has unique continuously almost periodic solution xiv(t), which can be expressed as Z t Rt a ðsÞ ds e s ii fiv ðsÞ ds; ð2:19Þ xiv ðtÞ ¼ 1
where fiv ðsÞ ¼
Z s n n n X X X qj qj qj aij ðsÞvj ðsÞ bij ðsÞvj ðs sij Þ cij ðsÞ K ij ðs sÞvj ðsÞds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj d ij ðsÞv0j ðs gij Þ þ q1 i ri ðsÞ. q i j¼1
Now, by using Lemma 2.3, xiv(t) can also be expressed as Z t Rt n X qj a ðsÞds xiv ðtÞ ¼ d ij ðtÞvj ðt gij Þ þ e s ii giv ðsÞ ds; q 1 i j¼1
ð2:20Þ
where
Z s n n n X X X qj qj qj giv ðsÞ ¼ aij ðsÞvj ðsÞ bij ðsÞvj ðs sij Þ cij ðsÞ K ij ðs sÞvj ðsÞ ds qi qi qi 1 j¼1 j¼1 j¼1 j6¼i
n X qj þ aii ðsÞd ij ðsÞ þ d 0ij ðsÞ vj ðs gij Þ þ q1 i ri ðsÞ. qi j¼1
ð2:21Þ
Then, we have Theorem 2.2. In addition to (H3)–(H4), assume that there exist positive constants qi, i = 1, 2, . . ., n such that Z t Rt X n qj aii ðsÞds ð2:22Þ sup e s qi ðsÞ ds < 1 d ij ðtÞ; qi t2R 1 j¼1
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
439
where n n X X qj qj aij ðsÞ þ bij ðsÞ þ cij ðsÞ þ ðaii ðsÞd ij ðsÞ þ jd 0ij ðsÞjÞ ; q qi j¼1 j¼1;j6¼i i ) ( X X n n qj qj d ij ðtÞ ¼ max d ij ðtÞ . t2R j¼1 qi j¼1 qi def
qi ðsÞ ¼
ð2:23Þ
Then (1.1) and (1.2) have a unique positive almost periodic solution with strictly positive components, say T N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞ . Proof. Set C = {u(t) = (u1(t), . . ., un(t))T—u:R ! Rn is continuous almost periodic function}, then under the norm kuk = sup{ku(t)k:t 2 R}, C is a Banach space. For any continuously almost periodic function v(t) = (v1(t), . . ., vn(t))T we know that xiv(t) defined by (2.20) is also a continuously almost periodic function. Now define the mapping F:C ! C as follows: FvðtÞ ¼ xv ðtÞ; ð8v 2 CÞ.
ð2:24Þ
Then similarly to the prove of Theorem 2.1, we could prove that under the assumptions of Theorem 2.2, the mapping F is a contract mapping, and so system (2.20) has a unique fixed point x(t) = (x1(t), . . ., xn(t))T. Also, similarly to the proof of Theorem 2.1, we could prove that N(t) = (exp{d1x1(t)}, . . ., exp{dnxn(t)})T is the unique positive almost periodic solution of system (1.1). The proof is complete. h As a direct corollary of Theorem 2.2, one has Corollary 2. In addition to (H3)–(H4), assume further that there exist positive constants qi, i = 1, 2, . . ., n such that ! X n n n X X qj qj qj def 0 qi ðtÞ ¼ aij ðtÞ þ bij ðtÞ þ cij ðtÞ þ ðaii ðtÞd ij ðtÞ þ jd ij ðtÞjÞ < aii ðtÞ 1 d ij ðtÞ . q q q i i j¼1 j¼1 j¼1;j6¼i i Then, (1.1) and (1.2) have a unique positive almost periodic solution. 3. Global asymptotic stability In this section, we devote ourselves to the study of the global attractivity of periodic solution (almost periodic solution) of system (1.1) and (1.2). Definition 3.1. Let N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT be a strictly positive periodic solution (almost periodic solution) of (1.1) and (1.2). We say N*(t) is globally attractive if any other solution Y(t) = (y1(t), . . ., yn(t)) of (1.1) and (1.2) has the property lim jN i ðtÞ y i ðtÞj ¼ 0;
t!þ1
i ¼ 1; 2; . . . ; n.
ð3:1Þ
Now, we state our main results of this section below. Theorem 3.1. Assume that the conditions in Theorem 2.1 (or Theorem 2.2) hold. Assume further that (H5) and (H6) hold, where (H5) cij(t)0,sij(t) sij, gij(t) gij are all nonnegative real numbers; (H6) There exist positive constant a,b such that aii ðtÞ P a > 0; X n n X qi aji ðtÞ qi bji ðt þ sij Þ P b > 0; aii ðtÞ q q j¼1;j6¼i
j
j¼1
j
440
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
for all t P 0 and i = 1, 2, . . ., n. Then system (1.1) and (1.2) has a unique periodic solution (almost periodic solution) which is globally attractive. Remark. P From the conditions of Theorem 2.1 (or Theorem 2.2) one could easily obtains that def q c ¼ max j nj¼1 qji d ij ðtÞj < 1. 16i6n t2R
T
Proof. Let N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞ be the unique positive periodic solution (almost periodic solution) of system (1.1) and (1.2), whose existence and uniqueness are guarantee by Theorem 2.1 (Theorem 2.2) , and y(t) = (y1(t), . . ., y2(t)) be any other solution of system 1.1,1.2. Let vi ðtÞ ¼ expfqi N i ðtÞg; xi ðtÞ ¼ expfqi y i ðtÞg; then n n X X qj qj aij ðtÞvj ðtÞ bij ðtÞvj ðt sij Þ v_ i ðtÞ ¼ aii ðtÞvi ðtÞ qi qi j¼1 j¼1 j6¼i
n X j¼1
qj d ij ðtÞv0j ðt gij Þ þ q1 i ri ðtÞ. qi
x_ i ðtÞ ¼ aii ðtÞxi ðtÞ
ð3:2Þ
n n X X qj qj aij ðtÞxj ðtÞ bij ðtÞxj ðt sij Þ q qi i j¼1 j¼1 j6¼i
n X j¼1
qj d ij ðtÞx0j ðt gij Þ þ q1 i ri ðtÞ. qi
Now we let wi(t) = vi(t) xi(t), then n n n X X X qj qj qj aij ðtÞwj ðtÞ bij ðtÞwj ðt sij Þ d ij ðtÞw0j ðt gij Þ. w_ i ðtÞ ¼ aii ðtÞwi ðtÞ qi q q i i j¼1 j¼1 j¼1
ð3:3Þ
ð3:4Þ
j6¼i
Then under the conditions of Theorem 3.1, all the conditions of Theorem 1 of [25] are satisfied. Now follows from the Theorem 1 of [25] one has n X lim jwj ðtÞj ¼ 0; t!þ1
j¼1
which is equivalent to n X lim jN j ðtÞ y j ðtÞj ¼ 0. t!þ1
j¼1
The proof of Theorem 3.1 is finished.
h
Finally, let us consider the following examples: Example 1.
dN 1 1 1 d ¼ N ðtÞ 2 þ sin t ln N ðtÞ ln N ðt j2 cos tjÞ ln N ðt j sin 4tjÞ . dt 4 4 4 dt
ð3:5Þ
One could easily see that all the conditions of Theorem 2.1 are satisfied for system (3.5), and so, system (3.5) has unique positive 2p-periodic solution. Example 2.
dN 1 1 1 d ¼ N ðtÞ 2 þ sin t ln N ðtÞ ln N ðt 2Þ ln N ðt 4Þ . dt 4 8 4 dt
ð3:6Þ
F. Chen / Applied Mathematics and Computation 176 (2006) 431–441
441
One could easily see that all the conditions of Theorem 3.1 are satisfied for system (3.6), and so, system (3.6) has unique globally attractive positive 2p-periodic solution. Acknowledgements This work is supported by the National Natural Science Foundation of China (10501007), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.amc.2005.09.032. References [1] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population DynamicsMathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. [2] G. Kirlinger, Permanence in Lotka–Volterra equations linked prey-predator systems, Math. Biosci. 82 (1986) 165–169. [3] Y.K. Li, Attractivity of a positive periodic solution for all other positive solution in a delay population model, Appl. Math.-JCU 12 (3) (1997) 279–282 (in Chinese). [4] F.D. Chen et al., Positive periodic solutions of state-dependent delay logarithm population model, J. Fuzhou University 31 (3) (2003) 1–4 (in Chinese). [5] Z.J. Liu, Positive periodic solutions for delay multispecies Logrithmic population model, J. Eng. Math. 19 (4) (2002) 11–16 (in Chinese). [6] F.D. Chen, Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model, Appl. Math. Comput., in press. [7] Y.K. Li, On a periodic neutral delay logarithmic population model, J. Sys. Sci. Math. Sci. 19 (1) (1999) 34–38. [8] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral logarithmic population model with multiple delays, J. Comput. Appl. Math. 166 (2) (2004) 371–383. [9] Y.K. Li, Positive periodic solution for neutral delay model, Acta Math. Sin. 39 (6) (1996) 789–795 (in Chinese). [10] H. Fang, J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl. 259 (2001) 8–17. [11] S.P. Lu, W.G. Ge, Existence of positive periodic solutions for neutral functional differential equations with deviating arguments, Appl. Math. J. Chinese Univ. Ser. B 17 (4) (2002) 382–390. [12] Z.H. Yang, J.D. Cao, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delays models, Appl. Math. Comput. 142 (1) (2003) 123–142. [13] Z.H. Yang, J.D. Cao, Positive periodic solutions of neutral Lotka-Volterra system with periodic delays, Appl. Math. Comput. 149 (3) (2004) 661–687. [14] Y.K. Li, On a periodic neutral delay Lotka–Volterra system, Nonlinear Anal. 39 (2000) 767–778. [15] H.F. Huo, W.T. Li, Existence of positive periodic solutions of a neutral Lotka-Volterra system with delays, Acta Math. Sinica 46 (6) (2003) 1199–1210 (in Chinese). [16] F.D. Chen, F.X. Lin, X.X. Chen, Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput. 158 (1) (2004) 45–68. [17] F.D. Chen, Positive periodic solutions of neutral Lotka–Volterra system with feedback control, Appl. Math. Comput. 162 (3) (2005) 1279–1302. [18] F.D. Chen, S.J. Lin, Periodicity in a Logistic type system with several delays, Comput. Math. Appl. 48 (1–2) (2004) 35–44. [19] F.D. Chen, D.X. Sun, F.X. Lin, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl. 288 (1) (2003) 132–142. [20] H. Fang, Positive periodic solutions of n-species neutral delay systems, Czechoslovak Math. J. 53 (3) (2003) 561–570. [21] Z.J. Liu, Positive periodic solution for a neutral delay competitive system, J. Math. Anal. Appl. 293 (1) (2004) 181–189. [22] Y.N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional delay 2003 (102) (2003) 1–7. [23] F.D. Chen et al., On the existence and uniqueness of periodic solutions of a kind of integro-differential equations, Acta Math. Sinica 47 (5) (2004) 973–985 (in Chinese). [24] H.Q. Xie, Q.Y. Wang, Exponential stability and periodic solution for cellular neural networks with time delay, J. Huaqiao University 25 (1) (2004) 22–26 (in Chinese). [25] K. Gopalsamy, A simple stability criterion for linear neutral differential systems, Funkcial Ekvac. 28 (1985) 33–38.