Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model

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 mult...

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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

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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.