Existence and exponential stability of almost periodic solutions for a neutral multi-species Logarithmic population model

Existence and exponential stability of almost periodic solutions for a neutral multi-species Logarithmic population model

Applied Mathematics and Computation 218 (2012) 5346–5356 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation jour...

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Applied Mathematics and Computation 218 (2012) 5346–5356

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Existence and exponential stability of almost periodic solutions for a neutral multi-species Logarithmic population model Yumei Zhao, Ling Wang, Hongyong Zhao ⇑ Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

a r t i c l e

i n f o

Keywords: Almost periodic solution Globally exponential stability Fixed point Differential inequality Multi-species neutral Logarithmic population model

a b s t r a c t In this paper, we study the almost periodic solution for a neutral multi-species Logarithmic population model. By employing Banach’s fixed point theorem and using differential inequality technique, we present some sufficient conditions ensuring the existence, uniqueness and globally exponential stability of almost periodic solution for the model. The results obtained extend and improve the earlier publications. Finally, two examples are provided to show the correctness of our analysis.  2011 Elsevier Inc. All rights reserved.

1. Introduction The dynamical analysis of multi-species Logarithmic population model has attracted a great attention of many mathematicians and biologists in recent years. As is well known, non-autonomous phenomena often occur in many realistic systems. Particularly, when the environment is varying periodically with time, the parameters of the system usually will change along with time. So it is reasonable to study periodic solution for multi-species Logarithmic population model with periodic coefficients. There exist some results on the existence of periodic solution for the class of model, see for example [1–4]. As was pointed out by Gopalsamy [5], in some case, the neutral delay population model are more realistic. Recently, there have been some nice results on the periodicity for neutral multi-species Logarithmic population model. We refer the reader to [6–8], and the references cited therein. However, in practice, almost periodic solution is more accordant with fact. To the best of the authors knowledge, few authors discuss almost periodic solution for neutral multi-species Logarithmic population model with variable coefficients [9]. Motivated by the above discussion, in this paper, we will study further the neutral multi-species Logarithmic population model, and improve the results of [9] by relaxing the constraints on delays being constants and the differentiability of coefficient of neutral term. By employing Banach’s fixed point theorem and using inequality technique, we will give some new conditions ensuring the existence, uniqueness and globally exponential stability of almost periodic solution for the neutral multi-species Logarithmic population model. The results obtained extend and improved the earlier publications. Two examples are provided to show the correctness of our analysis. 2. Preliminary Throughout the paper, Rn denotes the n-dimensional Euclidean. C[X, Y] denotes a continuous mapping set and C1[X, Y] denotes a continuous differentiable mapping set separately from topological space X to the topological space Y. For a continuous function k(t): R ? R, we define k+ = supt2Rjk(t)j, and k = inft2Rjk(t)j, respectively. ⇑ Corresponding author. E-mail address: [email protected] (H. Zhao). 0096-3003/$ - see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.018

Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

5347

Consider the following neutral multi-species Logarithmic population model

8 n n P P > > N0i ðtÞ ¼ Ni ðtÞ½r i ðtÞ  aij ðtÞlnNj ðtÞ  bij ðtÞlnN j ðt  sij ðtÞÞ > > > j¼1 j¼1 > > > > n > Rt P > <  cij ðtÞ 1 K ij ðt  sÞlnN j ðsÞds > > > > > > > > > > > :

j¼1



n P j¼1

0

dij ðtÞðlnN j ðt  gij ðtÞÞÞ ;

Ni ðtÞ ¼ ui ðtÞ;

N0i ðtÞ ¼ u0i ðtÞ;

ð1Þ

t P 0;

t 2 ð1; 0;

where i = 1, . . . , n, ri(t), aij(t), bij(t), cij(t), dij(t), sij(t), gij(t) are all continuous almost periodic functions. In addition, ii > 0; ui ð0Þ > 0; ui 2 C 1 ðð1; 0; ½0; þ1ÞÞ. r i ðtÞ; aij ðtÞði – jÞ; bij ðtÞ; cij ðtÞ; dij ðtÞ 2 CðR; ½0; þ1ÞÞ; aii ðtÞ 2 CðR; ð0; þ1ÞÞ; a n For any x = col{xi} 2 R , we define the norm kx(t)k1 = max{kx(t)k, kx0 (t)k}, where kxðtÞk ¼ max16i6n jxi ðtÞj; kx0 ðtÞk ¼ max16i6n jx0i ðtÞj. For any u = col{ui} 2 C1((1, 0] ; Rn), we define the norm kuk1 = max{kuk0, ku0 k0}, where kuk0 ¼ supt60 max16i6n jui ðtÞj; ku0 k0 ¼ supt60 max16i6n ju0i ðtÞj. Definition 1 ([10,11]). The continuous function x(t): R ? Rn which is called almost periodic on R, if for any e > 0, it is possible to find a real number l = l(e) > 0 such that, for any interval with length l, there is a number s = s(e) in this interval such that jx(t + s)  x(t)j < e, for any t 2 R.

Definition 2 ([10,11]). Let y 2 Rn and Q(t) be a n  n continuous matrix defined on R. The linear system

y0 ðtÞ ¼ Q ðtÞyðtÞ is said to be an exponential dichotomy on R, if there exist constants k, l > 0, projection S and the fundamental matrix Y(t) satisfying lðtsÞ

kYðtÞSY 1 ðsÞk 6 ke

;

for t P s;

lðtsÞ

kYðtÞðI  SÞY 1 ðsÞk 6 ke

for t 6 s:

;

 T Definition 3. Let N  ðtÞ ¼ N 1 ðtÞ; . . . ; N n ðtÞ be a continuously differentiable almost periodic solution of system (1) with ini  T tial value u ðtÞ ¼ u1 ðtÞ; . . . ; un ðtÞ . If there exist constants k > 0 and M P 1 such that for every solution N(t) = (N1(t), . . . , Nn(t))T of system (1) with any initial value u(t) = (u1(t), . . . , un(t))T,

kNðtÞ  N ðtÞk1 6 Mku  u k1 ekt ;

8t P 0:



Then N (t) is said to be globally exponential stable. _ Lemma 1 ([10,11]). If the linear system yðtÞ ¼ PðtÞyðtÞ has an exponential dichotomy, then almost periodic system

y0 ðtÞ ¼ PðtÞyðtÞ þ gðtÞ has a unique almost periodic solution y(t) which can be expressed as follows:

yðtÞ ¼

Z

t

YðtÞSY 1 ðsÞgðsÞds 

Z

1

YðtÞðI  SÞY 1 ðsÞgðsÞds:

t

1

Lemma 2 ([10,11]). Assume that ci(t) is an almost periodic function on R and

M½ci  ¼ lim

T!þ1

Z

tþT

ci ðsÞds > 0;

i ¼ 1; . . . ; n:

t

Then the linear system y0 (t) = c(t)y(t) admits an exponential dichotomy, where c(t) = diag{c1(t), . . . , cn(t)}. Throughout this paper, we assume that

R þ1 R þ1 (H1) 0 K ij ðsÞds ¼ 1 and there exists k0 > 0 such that 0 K ij ðsÞek0 s ds < þ1; i; j ¼ 1; . . . ; n. (H2) There exist constants s > 0, g > 0, L > 0 such that 0 < sij ðtÞ 6 s; 0 < gij ðtÞ 6 g; j1  g0ij ðtÞj 6 L; i; j ¼ 1; . . . ; n. (H3) There exist a set of positive constants q1, . . . , qn, such that

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

( " : r ¼ max max a1 16i6n n P

max 16i6n

j¼1

n P

ii

qj þ qi aij

j¼1;j–i

þ

n P j¼1

qj þ qi aij

qj þ qi bij

n P

þ

j¼1 n P

þ

j¼1

qj þ qi bij

qj þ qi cij

þ

þ

n P j¼1

n P j¼1

qj þ qi cij

n P

þ

!)

j¼1

qj þ qi dij L

qj þ qi dij L

!# ;

< 1:

Let Ni(t) = exp(qixi(t)), qi > 0, i = 1, . . . , n, then (1) can be rewrite in the form

8 n n P P qj > > x0i ðtÞ ¼ aii ðtÞxi ðtÞ  > qi aij ðtÞxj ðtÞ  > > j¼1;j–i j¼1 > > > > n Rt > P qj > <  q cij ðtÞ 1 K ij ðt  sÞxj ðsÞds > > > > > > > > > > > :

 sij ðtÞÞ

i

j¼1



qj qi bij ðtÞxj ðt

n P

qj qi dij ðtÞ

j¼1

  1  g0ij ðtÞ x0j ðt  gij ðtÞÞ þ q1 i r i ðtÞ;

 0 xi ðtÞ ¼ wi ðtÞ; x0i ðtÞ ¼ wi ðtÞ ;

ð2Þ tP0

t 2 ð1; 0:

Obviously, system (1) has a unique almost periodic solution which is globally exponentially stable if and only if system (2) has a unique almost periodic solution which is globally exponentially stable. 3. Existence and uniqueness of almost periodic solution Theorem 1. If the conditions (H1)–(H3) hold, then there exists exactly one almost periodic solution of system (2) in the region rI k/  /0 kX 6 1r , where

    þ  r þi : 1 þ þ ri ; max ; I ¼ max max q1 q r þ a i i i ii 16i6n aii 16i6n aii Z t T Z t Rt Rt :  a ðuÞdu 1  a ðuÞdu 1 /0 ¼ e s 11 q1 r1 ðsÞds; . . . ; e s nn qn rn ðsÞds : 1

1

Proof. Let X = {/(t)j/(t) = (/1(t), . . . , /n(t))T}, where /i(t): R ? R is a continuously differentiable almost periodic function. For any /(t) 2 X, define

k/kX ¼ maxfk/k0 ; k/0 k0 g ¼ maxfsup max j/i ðtÞj; sup max j/0i ðtÞjg: t2R 16i6n

t2R 16i6n

Obviously, (X, k  kX) is a Banach space. For any /(t) 2 X, consider the following system:

x0i ðtÞ ¼ aii ðtÞxi ðtÞ  

n P

qj qi dij ðtÞ

j¼1

n P j¼1;j–i

n P qj qj qi aij ðtÞ/j ðtÞ  qi bij ðtÞ/j ðt

 sij ðtÞÞ 

j¼1

n P j¼1

Rt qj qi cij ðtÞ 1 K ij ðt

 sÞ/j ðsÞds ð3Þ

  1  g0ij ðtÞ /0j ðt  gij ðtÞÞ þ q1 i r i ðtÞ;

where i = 1, . . . , n. From aii(t) > 0, we have

M½aii  ¼ lim

Z

T!þ1

tþT

aii ðsÞds > 0:

t

By Lemma 1, system (3) has a unique almost periodic solution X/(t) which can be expressed as follows

 T X / ðtÞ ¼ x/1 ðtÞ; . . . ; x/n ðtÞ ¼ Z

Z

t

e

1



Rt s

" a11 ðuÞdu



n X qj j¼2

q1

a1j ðsÞ/j ðsÞ 

n X qj

n X qj

j¼1

j¼1

b ðsÞ/j ðs  s1j ðsÞÞ  q1 1j #

q1

c1j ðsÞ

n X qj K 1j ðs  uÞ/j ðuÞdu  d1j ðsÞð1  g01j ðsÞÞ/0j ðs  g1j ðsÞÞ þ q1 1 r 1 ðsÞ ds; . . . ; q 1 1 j¼1 " Z t Z s Rt n1 n n X X X qj qj qj  a ðuÞdu  e s nn  anj ðsÞ/j ðsÞ  bnj ðsÞ/j ðs  snj ðsÞÞ  cnj ðsÞ K nj ðs  uÞ/j ðuÞdu



t

1



n X j¼1

j¼1

qn

j¼1

qn

# !T   qj 0 0 1 dnj ðsÞ 1  gnj ðsÞ /j ðs  gnj ðsÞÞ þ qn r n ðsÞ ds :

qn

j¼1

qn

1

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

Now define a mapping T: X ? X by (T/)(t) = x/(t), "/ 2 X. By the definition of k  kX, we have



k/0 kX ¼ max k/0 k0 ; k/00 k0

Z t Rt  aii ðuÞdu s q1 dsj ; ¼ maxfsup max j i r i ðsÞe t2R 16i6n

1



r þi ; 6 max max q1 i  16i6n aii

Z sup max jq1 i r i ðtÞ  t2R 16i6n



t

1

  rþ max q1 ¼ I: r þi þ aþii i i 16i6n aii

 q1 i aii ðtÞr i ðsÞe

Rt s

aii ðuÞdu

dsj



: rI So, for any /ðtÞ 2 X 0 ¼ /ðtÞj/ðtÞ 2 X; k/  /0 kX 6 1r , one has

k/kX 6 k/  /0 kX þ k/0 kX 6

rI I þI ¼ : 1r 1r

In the following, we show T maps the set X0 into itself. In fact, for any /(t) 2 X0, we have

( Z kT/  /0 k0 ¼ sup max j t2R 16i6n

t

e

Rt s

"

aii ðuÞdu

t2R 16i6n

j¼1;j–i

n X qj j¼1

(Z

t

e

qi

n X qj

a ðtsÞ ii

j¼1;j–i

n X qj

1 6 max  16i6n aii

j¼1;j–i

qi

n X qj

aij ðsÞ/j ðsÞ 

qi

bij ðsÞ/j ðs  sij ðsÞÞ 

qi

j¼1

n X qj j¼1

# )   dij ðsÞ 1  g0ij ðsÞ /0j ðs  gij ðsÞÞ dsj

1

(

n X qj



1

 uÞ/j ðuÞdu 6 sup max



aþij

þ

qi

n X qj

qi

j¼1

aþij

þ

n X qj j¼1

þ bij

þ

qi

n X qj j¼1

qi

þ bij

þ

n X qj j¼1

cþij

þ

qi

þ

þ dij L

qi

þ dij L

qi

j¼1

Z

s

K ij ðs 1

! )

n X qj

!)

n X qj j¼1

cþij

qi

cij ðsÞ

ds k/kX

k/kX

and

( 0

kðT/  /0 Þ k0 ¼ sup max j  t2R 16i6n



n X qj j¼1

"  

qi

j¼1

qi

qi

aij ðtÞ/j ðtÞ 

n X qj j¼1

qi

aij ðsÞ/j ðsÞ 

n X qj

j¼1;j–i

qi

aþij

þ

qi

j¼1 t



aii ðtÞe

Rt s

cij ðtÞ

Z

t

K ij ðt  uÞ/j ðuÞdu

1

aii ðuÞdu

j¼1

qi

n X qj

bij ðsÞ/j ðs  sij ðsÞÞ 

j¼1

qi

cij ðsÞ

Z

s

K ij ðs  uÞ/j ðuÞdu

1

# )   0 0 dij ðsÞ 1  gij ðsÞ /j ðs  gij ðsÞÞ dsj n X qj j¼1;j–i

qi

aþij

þ

n X qj j¼1

qi

Moreover, by (H3), we have n X qj

n X qj

1

(  aþ 1 þ ii 6 max 16i6n aii   aþ 1 þ ii aii

qi

bij ðtÞ/j ðt  sij ðtÞÞ 

Z   dij ðtÞ 1  g0ij ðtÞ /0j ðt  gij ðtÞÞ 

j¼1;j–i



j¼1;j–i

n X qj

n X qj

n X qj

n X qj j¼1

qi

þ bij

þ

n X qj j¼1

qi

cþij

þ

n X qj j¼1

qi

þ bij

þ

n X qj j¼1

! þ dij L

¼

qi

cþij

n X qj

þ

¼

qi

aþii aii

aþij þ

j¼1;j–i

qi

aþij þ

So, we obtain

rI ; 1r

which implies (T/)(t) 2 X0. So, the mapping T is self-mapping from X0 to X0.

k/kX :

n X qj

n X qj

qi

qj þ a þ qi ij j¼1;j–i n X qj

!) þ dij L

j¼1

n X

j¼1

kT/  /0 kX ¼ maxfkT/  /0 k0 ; kðT/  /0 Þ0 k0 g 6 rk/kX 6

qi

j¼1

j¼1;j–i

<

þ

n X qj

qi

n X qj j¼1

j¼1

qi

qj þ b þ qi ij

n X qj

qi

n X qj

qi

j¼1

aþij þ

n X

j¼1

þ

bij þ

þ

bij þ

þ

bij þ

n X qj

j¼1

qi

qj þ c þ qi ij

n X qj

qi

n X qj j¼1

j¼1

n X

j¼1

cþij þ

cþij þ

cþij þ

n X qj j¼1

n X j¼1

qi

! þ

dij L

qj þ d L þ aþii qi ij

n X qj j¼1

þ

dij L

qi

qi

þ

dij L:

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

Finally, we prove that T is a contraction mapping. For any /(t), w(t) 2 X0, one has

(

n X qj

1 aii

kT/  Twk0 6 max 16i6n

j¼1;j–i

qi

aþij þ

n X qj

þ

bij þ

qi

j¼1

n X qj

qi

j¼1

!)

n X qj

cþij þ

j¼1

þ

dij L

qi

k/  wkX 6 rk/  wkX

and

(  aþ kðT/  TwÞ k0 6 max 1 þ ii 16i6n aii

n X qj

0

qi

j¼1;j–i

aþij

þ

n X qj

þ bij

qi

j¼1

þ

n X qj j¼1

qi

cþij

þ

n X qj j¼1

qi

!) þ dij L

k/  wkX 6 rk/  wkX :

Thus, one have

kT/  TwkX 6 rk/  wkX : Noting that 0 < r < 1, which means that T is a contraction mapping. By Banach’s fixed point theorem, there exists a unique fixed solution /(t) 2 X0 such that (T/)(t) = /(t), which implies that system (2) has a unique almost periodic solution. This completes the proof. h

4. Globally exponential stability of almost periodic solution Theorem 2. Assume that (H1)–(H3) hold, then system (2) has a unique almost periodic solution which is globally exponentially stable. Proof. It follows from Theorem 1 that system (2) has a unique almost periodic solution z(t) = (z1(t), . . . , zn(t))T 2 X with initial value l(t) = (l1(t), . . . , ln(t))T. Let x(t) = (x1(t), . . . , xn(t))T be an arbitrary solution of system (2) with initial value  T w ðtÞ ¼ w1 ðtÞ; . . . ; wn ðtÞ . Let yi ðtÞ ¼ xi ðtÞ  zi ðtÞ; wi ðtÞ ¼ wi ðtÞ  li ðtÞ; i ¼ 1; . . . ; n. Then n X qj

y0i ðtÞ ¼ aii ðtÞyi ðtÞ 

j¼1;j–i

qi

aij ðtÞyj ðtÞ 

n X qj

qi

j¼1

bij ðtÞyj ðt  sij ðtÞÞ 

n X qj

qi

j¼1

cij ðtÞ

Z

t

K ij ðt  sÞyj ðsÞds 

1

j¼1

  qj  dij ðtÞ 1  g0ij ðtÞ y0j ðt  gij ðtÞÞ;

ð4Þ

qi

where i = 1, . . . , n. Let Fi() and Gi() be defined by

"

F i ð xÞ ¼

aii

n X qj

x

j¼1;j–i

Gi ðxÞ ¼ 1 

" n X qj j¼1

qi

þ

n X qj

qi

j¼1

n X qj

aþij þ

qi

aþij

qi

j¼1

þ bij exs

þ

n X qj

qi

j¼1

þ

bij exs þ

n X

n X qj j¼1

qi

cþij

Z

cþij

þ1

Z

þ1

xu

K ij ðuÞe du þ

0

n X qj j¼1

K ij ðuÞexu du þ

0

n X qj j¼1

qi

qi

# þ dij Lexg

;

# dij Lexg ; þ

where i = 1, . . . , n, x 2 [0, +1). Then

" F i ð0Þ ¼ aii 

n X qj

j¼1;j–i

Gi ð0Þ ¼ 1 

" n X qj j¼1

qi

qi aþij

aþij þ

n X qj j¼1

þ

n X j¼1

qi

þ

bij þ

n X qj

qi

j¼1

qj þ xs b e þ qi ij

n X j¼1

cþij þ

n X qj

qi

j¼1

qj þ c þ qi ij

n X j¼1

# þ

dij L > 0; #

qj þ d L > 0; qi ij

where i = 1, . . . , n. Since Fi() and Gi() are continuous on [0, + 1) and Fi(x), Gi(x) ? 1 as x ? +1, so there exist gi > 0; ei > 0 such that

    F i gi ¼ Gi ei ¼ 0:



By choosing n ¼ min g1 ; . . . ; gn ; e1 ; . . . ; en , we obtain Fi(n) P 0, Gi(n) P 0, i = 1, . . . , n. So, we can choose a positive constant 

0 < k < min a11 ; . . . ; a such that Fi(k) > 0, Gi(k) > 0, which implies that nn ; n; k0

1 aii  k

n X qj j¼1;j–i

qi

aþij þ

n X qj j¼1

qi

þ

bij eks þ

n X qj j¼1

qi

cþij

Z

þ1

K ij ðuÞeku du þ

0

n X qj j¼1

qi

!

dij Lekg þ

<1

ð5Þ

and n X qj j¼1

qi

aþij þ

n X qj j¼1

qi

þ

bij eks þ

n X qj j¼1

qi

cþij

Z 0

þ1

K ij ðuÞeku du þ

n X qj j¼1

qi

dij Lekg < 1: þ

ð6Þ

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

By (4), we have 

yi ðtÞ ¼ wi ð0Þe

Rt 0

aii ðuÞdu

Z

þ

t

e

Rt



s

" aii ðuÞdu



0 n X qj

 uÞyj ðuÞdu

qi

j¼1

n X qj j¼1;j–i

qi

aij ðsÞyj ðsÞ 

n X qj j¼1

#   0 0 dij ðsÞ 1  gij ðsÞ yj ðs  gij ðsÞÞ ds:

qi

bij ðsÞyj ðs  sij ðsÞÞ 

n X qj j¼1

qi

cij ðsÞ

Z

s

K ij ðs

1

ð7Þ

Let M ¼ 1r , by (H3) we have M > 1. Thus

kyðtÞk1 ¼ kwðtÞk1 6 kwk1 6 Mkwk1 ekt ;

8t 2 ð1; 0:

In the following, we shall prove that

kyðtÞk1 6 Mkwk1 ekt ;

t P 0:

ð8Þ

To prove (8), we first show, for any p > 1, the following inequality holds

kyðtÞk1 < pMkwk1 ekt ;

t P 0:

ð9Þ

If (9) is false, then there must be some t1 > 0, such that

kyðt 1 Þk1 ¼ pMkwk1 ekt1

ð10Þ

and

kyðtÞk1 6 pMkwk1 ekt ;

8t 2 ð1; t 1 :

ð11Þ

For all i = 1, . . . , n, by (5) and (11), we have

jyi ðt 1 Þj 6 jwi ð0Þje þ

n X qj j¼1

qi



R t1

aii ðuÞdu

0

þ

Z

t1



e

R t1 s

"

0

cþij

Z

n X qj

aii ðuÞdu

j¼1;j–i

s

K ij ðs  uÞjyj ðuÞjdu þ

n X qj

1

qi

j¼1

qi

aþij jyj ðsÞj þ

n X qj j¼1

þ

bij jyj ðs  sij ðsÞÞj

qi

#

þ

dij j1  g0ij ðsÞjjy0j ðs  gij ðsÞÞj ds:

Then

kyðt 1 Þk 6 e

a t ii 1

þ

n X qj j¼1

6e

kwk1 þ

a t ii 1

qi

cþij

Z

Z

t1

e



"

R t1

aii ðuÞdu

s

0

j¼1;j–i

¼e

qi

aþij pMkwk1 eks þ

0

K ij ðuÞpMkwk1 e

kðsuÞ

du þ

n X qj j¼1

kwk1 þ pMkwk1

Z

kwk1 þ pMkwk1

t1

" ðt 1 sÞa ks ii

e

0 n X qj

qi j¼1;j–i

n X qj

j¼1;j–i

aþij

þ

n X qj j¼1

þ1

" a t ii 1

n X qj

n X qj j¼1

qi

qi

qi aþij

þ bij eks

þ

bij pMkwk1 ekðssij ðsÞÞ

qi

# þ dij LpMkwk1 ekðsgij ðsÞÞ

þ

n X qj j¼1

þ

qi

n X qj j¼1

qi

þ bij eks

þ

n X qj j¼1

cþij

Z

ds

qi

cþij

Z

þ1

K ij ðuÞe du þ

0

K ij ðuÞe du þ

n X qj

0

þ1 ku

ku

j¼1 n X qj j¼1

qi

qi

# þ dij Lekg

# þ dij Lekg



ð1  eðkaii Þt1 Þ

"

n n n X  1 qj þ X qj þ ks X qj þ Z þ1 1 ðka Þt1   e ii þ 1  eðkaii Þt1  aij þ bij e þ c K ij ðuÞeku du pM aii  k j¼1;j–i qi qi qi ij 0 j¼1 j¼1 !# n X    qj þ kg   < pMkwk1 ekt1 eðkaii Þt1 þ 1  eðkaii Þt1 ¼ pMkwk1 ekt1 : dij Le þ

¼ pMkwk1 ekt1

j¼1

qi

Let y0i ðtÞ ¼ Z i ðtÞ. By (4), we obtain

Z i ðtÞ ¼ aii ðtÞyi ðtÞ 

n X qj j¼1;j–i

qi

aij ðtÞyj ðtÞ 

n X qj j¼1

  qj  dij ðtÞ 1  g0ij ðtÞ Z j ðt  gij ðtÞÞ:

qi

qi

bij ðtÞyj ðt  sij ðtÞÞ 

n X qj j¼1

qi

cij ðtÞ

Z

t

1

K ij ðt  sÞyj ðsÞds 

n X j¼1

ds

ekt1 aii  k

5352

Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

Then, we have

jZ i ðt1 Þj 6 aþii jyi ðt 1 Þj þ

n X qj

qi

j¼1;j–i

n X qj

þ

j¼1

qi

n X qj j¼1

n X qj j¼1

qi

þ

bij jyj ðt 1  sij ðt 1 ÞÞj þ

qi

cþij

¼ pMkwk1 e

Z 0

kt 1

n X qj j¼1;j–i

qi

n X qj

aþij pMkwk1 ekt1 þ

þ1

K ij ðuÞpMkwk1 ekðt1 uÞ du þ

j¼1 n X j¼1

n X qj j¼1

qi

aþij

þ

n X qj j¼1

qi

þ bij eks

þ

n X qj j¼1

  þ dij j 1  g0ij ðt 1 Þ jjZ j ðt1  gij ðt1 ÞÞj;

kZðt1 Þk 6 aþii pMkwk1 ekt1 þ þ

aþij jyj ðt1 Þj þ

n X qj j¼1

qi

qi

cþij

Z

t1

K ij ðt 1  sÞjyj ðsÞjds 1

þ

b pMkwk ekðt1 sij ðt1 ÞÞ

1 qi ij qj þ d LpMkwk1 ekðt1 gij ðt1 ÞÞ qi ij

cþij

Z

þ1 ku

K ij ðuÞe du þ

0

n X qj j¼1

qi

! þ dij Lekg

< pMkwk1 ekt1 :

Thus

kyðt 1 Þk1 ¼ maxfkyðt 1 Þk; ky0 ðt 1 Þkg < pMkwk1 ekt1 ; which contradicts the equality (10), so (9) holds. Letting p ? 1, then (8) holds. This completes the proof.

h

Remark 1. In Ref.[9], Chen stated system (1) had a unique and globally asymptotically stable almost periodic solution provided that the delays are constants and the coefficients of neutral term are differentiable. However, these hypothesis are not necessary in the paper, and globally exponential stability implies globally asymptotic stability. Thus, we extend and improve the main results of [9]. When dij(t) = 0, i, j = 1, . . . , n, then system (1) becomes the following system without the neutral term

" 8 n n > P P > 0 > Ni ðtÞ ¼ N i ðtÞ r i ðtÞ  aij ðtÞlnNj ðtÞ  bij ðtÞlnNj ðt  sij ðtÞÞ > > > j¼1 j¼1 > < # n R P t > >  cij ðtÞ 1 K ij ðt  sÞlnN j ðsÞds ; t P 0; > > > j¼1 > > : Ni ðtÞ ¼ ui ðtÞ; t 2 ð1; 0:

ð12Þ

For system (12), (H2) and (H3) convert into the following form, respectively:     H2 There exists a constant s > 0 such that 0 < sij(t) 6 s, i, j = 1, . . . , n. Item H3 There exist a set of positive constants q1, . . . , qn, such that

"

1 max  16i6n a ii

n X qj j¼1;j–i

qi

aþij

þ

n X qj j¼1

qi

þ bij

þ

n X qj j¼1

qi

!# cþij

< 1:

By Theorems 1 and 2, we have the following result.     Corollary 1. If ðH1 Þ; H2 and H3 hold, then system (12) has a unique almost periodic solution which is globally exponential stability. Remark 2. It should be noted that Chen [4] derived the existence and global attractivity of the almost periodic solution for system (12) under the assumptions that sij(t) is differentiable and 1  sij(t) is reversible. However, in the above Corollary, these hypothesis are not necessary, and globally exponential stability implies global attractivity. Thus, the Corollary improves the main results of [4]. 5. Examples Example 1. Let n = 2. Consider the following neutral multi-species Logarithmic population model

8 0 N ðtÞ ¼ N1 ðtÞ½r1 ðtÞ  a11 ðtÞlnN1 ðtÞ  b11 ðtÞlnN1 ðt  s11 ðtÞÞ > > > 1 Rt > > > b12 ðtÞlnN2 ðt  s12 ðtÞÞ  c11 ðtÞ 1 K 11 ðt  sÞlnN 1 ðsÞds > > > < d11 ðtÞðlnN 1 ðt  g11 ðtÞÞÞ0 ; 0 > N2 ðtÞ ¼ N2 ðtÞ½r2 ðtÞ  a22 ðtÞlnN2 ðtÞ  b21 ðtÞlnN1 ðt  s21 ðtÞÞ > > > Rt > > > b22 ðtÞlnN2 ðt  s22 ðtÞÞ  c22 ðtÞ 1 K 22 ðt  sÞlnN 2 ðsÞds > > : d22 ðtÞðlnN 2 ðt  g22 ðtÞÞÞ0 ;

ð13Þ

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 1 1 1 where a11 ðtÞ ¼ 12  16 cosð3tÞ; a22 ðtÞ ¼ 12  15 cosð 4tÞ; b11 ðtÞ ¼ 12 sin ð 2tÞ; b12 ðtÞ ¼ 12 cos2 ð 2tÞ; c11 ðtÞ ¼ 12 cos2 ð 7tÞ; pffiffiffi pffiffiffi pffiffiffi 2 2 1 1 1 1 1 sin ð 8tÞ; b22 ðtÞ ¼ 12 cos2 ð 8tÞ; c22 ðtÞ ¼ 12 sin ð 7tÞ; d11 ðtÞ ¼ 50 sinj50tj; d22 ðtÞ ¼ 50 cosj50tj; s11 ðtÞ ¼ s12 ðtÞ b21 ðtÞ ¼ 12 pffiffiffi pffiffiffi pffiffiffi 1 1 1 1 g11 ðtÞ ¼ 1  4 cost; g22 ðtÞ ¼ 1  5 cost; r1 ðtÞ ¼ 23 þ 16 cosð 3tÞ; r2 ðtÞ ¼ ¼ 1 þ 2 sinð 2tÞ; s21 ðtÞ ¼ s22 ðtÞ ¼ 1 þ 2 cosð 2tÞ; 5 þ 56 cosð2tÞ; K 11 ðtÞ ¼ K 22 ðtÞ ¼ et :Obviously, 3



5 ; 4

aþ11 ¼

2 ; 3

a11 ¼

1 ; 3

aþ22 ¼

7 ; 10

a22 ¼

3 ; 10

þ

þ

þ

þ

b11 ¼ b12 ¼ b21 ¼ b22 ¼ cþ11 ¼ cþ22 ¼

1 ; 12

4

2

3

1 d(N1(t))/dt

N1(t)

Take q1 = 6, q2 = 5.

2

1

−1

0

20

40 60 time t

80

−2

100

40

40

30

20 d(N2(t))/dt

N2(t)

0

0

20

10

0

0

20

40 60 time t

80

100

0

−20

0

20

40 60 time t

80

−40

100

0

20

40 60 time t

80

100

Fig. 1. The trajectory of N1(t), d(N1(t))/dt and N2(t), d(N2(t))/dt versus time in Example 1.

40

35

30

N (t) 2

25

20

15

10

5

0 0.5

1

1.5

2 N1(t)

2.5

Fig. 2. The trajectory of N1(t) and N2(t) in Example 1.

3

3.5

þ

þ

d11 ¼ d22 ¼

1 : 50

5354

Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356 40

30

20

d(N (t))/dt 2

10

0

−10

−20

−30

−40 −2

−1.5

−1

−0.5

0 d(N1(t))/dt

0.5

1

1.5

2

Fig. 3. The trajectory of d(N1(t))/dt and d(N2(t))/dt in Example 1.

By simple calculation, we have

  1 q2 þ 47 þ þ þ < 1; b þ b þ c þ d L ¼ 11 11 a11 11 q1 12 60 q þ 167 þ þ aþ11 þ b11 þ 2 b12 þ cþ11 þ d11 L ¼ < 1; 180 q1

  1 q1 þ 35 þ þ þ < 1; b þ b þ c þ d L ¼ 22 22 22 a22 q2 21 36 q þ 119 þ þ aþ22 þ 1 b21 þ b22 þ cþ22 þ d22 L ¼ < 1: 120 q2

4

2

3

1 d(N1(t))/dt

N1(t)

It is easy to see that (H1)–(H3) hold. By Theorems 1 and 2, system (13) has a unique almost periodic solution which is globally exponentially stable. The dynamical behavior of Example 1 is shown in Figs. 1–3.

2

1

−1

0

20

40 60 time t

80

−2

100

40

40

30

20 d(N2(t))/dt

N2(t)

0

0

20

10

0

0

20

40 60 time t

80

100

0

20

40 60 time t

80

100

0

−20

0

20

40 60 time t

80

100

−40

Fig. 4. The trajectory of N1(t), d(N1(t))/dt and N2(t), d(N2(t))/dt versus time in Example 2.

5355

Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

1 Example 2. In system (13), taking sij(t) = 1, gij(t) = 1 and dii ðtÞ ¼ 50 ðsinð50tÞÞ2 ; i; j ¼ 1; 2, the rest of the parameters are the same as Example 1. Thus, system (13) has a unique almost periodic solution which is globally exponentially stable. However, by simple calculation, we have

t

e



Rt s

aii ðsÞds

qi ðsÞds þ k

1

n X qj j¼1

qi



dij ðtÞk P

dii > 1; aii

i ¼ 1; 2;

40

35

30

25

N2(t)

t2R

Z

20

15

10

5

0 0.5

1

1.5

2 N (t)

2.5

3

3.5

1

Fig. 5. The trajectory of N1(t) and N2(t) in Example 2.

40

30

20

10 d(N (t))/dt 2

sup

0

−10

−20

−30

−40 −2

−1.5

−1

−0.5

0 d(N1(t))/dt

0.5

1

Fig. 6. The trajectory of d(N1(t))/dt and d(N2(t))/dt in Example 2.

1.5

2

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Y. Zhao et al. / Applied Mathematics and Computation 218 (2012) 5346–5356

where n X qj

qi ðsÞ ¼

j¼1;j–i

qi

aij ðsÞ þ

  ( )  n n n   X X X qj  qj qj   0 bij ðsÞ þ cij ðsÞ þ aii ðsÞdij ðsÞ þ jdij ðsÞj ;  dij ðtÞ ¼ max dij ðtÞ ;   j¼1 qi t2R qi qi j¼1 j¼1

which implies that the condition of Theorem 2.2 in [9] is not satisfied. Thus, we improve the result in [9]. The dynamical behavior of Example 2 is shown in Figs. 4–6.

6. Conclusion In this paper, a neutral multi-species Logarithmic population model is investigated. For the model, we have given some sufficient conditions ensuring the existence, uniqueness and globally exponential stability of almost periodic solution by applying Banach’s fixed point theorem and using differential inequality technique. These obtained results complement previous results. Moreover, two simple examples are given to illustrate the effectiveness of the new results. Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grants 11032009, 61174155. We wish to thank the referees for their valuable suggestion and comments. References [1] Hongyong Zhao, Nan Ding, Existence and global attractivity of positive periodic solution for competition-predator system with variable delay 29, 2006, pp. 162–170. [2] Weirui Zhao, New results of existence and stability of periodic solution for a delay multispecies Logarithmic population model, Nonlinear Analysis: Real Word Applications 10 (1) (2009) 544–553. [3] Changzhong Wang, Jinlin Shi, Periodic solution for a delay multispecies Logarithmic population model with feedback control, Applied Mathematics and Computation 193 (1) (2007) 257–265. [4] Fengde Chen, Periodic solutions and almost periodic solutions of a delay multispecies Logarithmic population model, Applied Mathematics and Computation 171 (2) (2005) 760–770. [5] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. [6] Q. Wang, Y. Wang, B. Dai, Existence and uniqueness of positive periodic solutions for a neutral Logarithmic population model, Applied Mathematics and Computation 213 (1) (2009) 137–147. [7] S. Lu, W. Ge, Existence and positive periodic solutions for neutral Logarithmic population model with multiple delays, Journal of Computational and Applied Mathematics 166 (2) (2004) 371–383. [8] Xinge Liu, Lanmei Tang, Periodic solution for a class of delay neutral population, Journal of Jiangxi Normal University (Natural sciences edition) 30 (5) (2006) 475–477 (in Chinese). [9] F. Chen, Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model, Applied Mathematics and Computation 176 (2) (2006) 431–441. [10] A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [11] C.Y. He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992 (in Chinese).