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Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model
Applied Mathematics and Computation 171 (2005) 760–770 www.elsevier.com/locate/amc
Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model Fengde Chen School of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, People’s Republic of China
Abstract By using the method of fixed point theory and Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic (almost periodic solution) of a delay multispecies Logarithmic population model. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Periodic solution; Almost periodic solution; Lyapunov functional; Stability; Contraction principle
1. Introduction The aim of this paper is to investigate the existence, uniqueness and global attractivity of the positive periodic solution (almost periodic solution) of the following delay multispecies Logarithmic population model
E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.085
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
" N_ i ðtÞ ¼ N i ðtÞ ri ðtÞ
n X
aij ðtÞ ln N j ðtÞ
j¼1
n X
cij ðtÞ
Z
n X
761
bij ðtÞ ln N j ðt sij ðtÞÞ
j¼1
t
#
K ij ðt sÞ ln N j ðsÞ ds ;
ð1:1Þ
1
j¼1
ri(t), aij,bij R2 C(R, (0, +1)), sij 2 C(R, R) are all continuous functions. Rwhere þ1 þ1 K sK ij ðsÞ ds < þ1: We consider (1.1) together with the ij ðsÞ ds ¼ 1; 0 0 following initial conditions N i ðhÞ ¼ /i ðhÞ P 0; h 2 ð1; 0;
/i ð0Þ > 0:
ð1:2Þ
For the ecological justification of Eq. (1.1), see [1–5]. Recently, by using MawhinÕs continuation theorem in [6], Liu [1] studied the existence of positive periodic solution of the following periodic Logarithmic population model " # n n X X N_ i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ ð1:3Þ j¼1
j¼1
together with the initial condition N i ðhÞ ¼ /i ðhÞ P 0; h 2 ðs ; 0;
/i ð0Þ > 0;
ð1:4Þ
where aij,bij 2 C (R, (0, +1)),ri, sij 2 C(R, R) are all continuous T-periodic function, s* = max16i,j6nmaxt2[0,T]{jsij(t)j}. By introducing the change of variable N i ðtÞ ¼ exi ðtÞ ;
i ¼ 1; 2; . . . ; n;
then system (1.3) is equal to x_ i ðtÞ ¼ ri ðtÞ
n X
aij ðtÞxj ðtÞ
j¼1
n X
bij ðtÞxj ðt sij ðtÞÞ:
ð1:5Þ
j¼1
And the existence of positive periodic solution of system (1.3) is equal to the existence of periodic solution of system (1.5). Noticing that for system (1.5), corresponding to the operator equation Lx = kNx,k 2 (0, 1), one has " # n n X X aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ : ð1:6Þ x_ i ðtÞ ¼ k ri ðtÞ j¼1
j¼1
T
Let x(t) = (x1(t), . . . , xn(t)) be any periodic solution of system (1.6), and let xi ðni Þ ¼ min xi ðtÞ; t2½0;T
i ¼ 1; 2; . . . ; n:
762
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Then in proving the theorem, Liu [1] had used the following inequality # Z T "X n n X ri T P aij ðtÞxj ðnj Þ þ bij ðtÞxj ðnj Þ dt 0
¼
n X
j¼1
ð aij þ bij Þxj ðnj ÞT > ð aii þ bii Þxi ðni ÞT ;
j¼1
RT where ri ¼
0
ri ðtÞ dt T
j¼1
RT ; aij ¼
0
aij ðtÞ dt T
; bij ¼
RT 0
bij ðtÞ dt T
ð1:7Þ
; and so
ri : xi ðni Þ 6 bii aii þ We point out that the positivity of Ni(t) could not guarantee the positivity of xi(t) and so, xj(nj) might be negative, in this case, inequality (1.7) may not hold, and the proof of Theorem 1 of [1] is incomplete. The aim of this paper is to give a set of new conditions to guarantee the existence, uniqueness and stability of the periodic solution (almost periodic solution) of system (1.1) and (1.2). The outline of the paper is as follows. In Section 2, we first introducing a new transformation, where some new adjustable real parameter di > 0 are introduced; After that, by using contraction principle, some new results on the existence of positive periodic solutions (almost periodic solutions) of system (1.1) and (1.2) are established. In Section 3, by constructing a suitable Lyapunov functional, we derive a set of easily verifiable criteria for the global asymptotic stability of positive periodic solutions (almost periodic solution) of (1.1) and (1.2). We must point out, the idea of introducing parameter is stimulate by the works of Chen et al. [7] and Xie and Wang [8]. However, to the best of the authors knowledge, this is the first time such a technique is applied to the ecosystem. 2. Periodic solutions and almost periodic solutions A very basic and important ecological problem associated with the study of multispecies population interactions in a periodic environment (almost periodic environment) is the global existence of positive periodic solution (almost periodic solution) which plays the role played by the equilibrium of the autonomous models. It is reasonable to ask for conditions under which the resulting periodic (almost periodic) nonautonomous system would have a periodic solution (almost periodic solution). To our knowledge, no such work has been done on the global existence of positive periodic solution (almost periodic solutions) of (1.1) and (1.2). 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).
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
763
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), sij(t), cij(t) are continuous, real-valued T-periodic functions on R such that Z T ri ðtÞ dt P 0; aii ðtÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; bij ðtÞ P 0; cij ðtÞ P 0: 0
Also, s0ij ðtÞ < 1: Our main result on the global existence of a positive periodic solution of (1.1) and (1.2) is stated in the following theorem. Theorem 2.1. In addition to (H1), there are positive constants di, i = 1,2, . . . , n such that n n X X d j aij ðtÞ þ d 1 d j ðbij ðtÞ þ cij ðtÞÞ: ð2:1Þ aii ðtÞ > d 1 i i j¼1 j6¼i
j¼1
Then (1.1) and (1.2) has unique T-periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . Proof. Make the change of variables N i ðtÞ ¼ expfd i xi ðtÞg;
i ¼ 1; 2; . . . ; n;
ð2:2Þ
then (1.1) can be reformulated as n n X X 1 x_ i ðtÞ ¼ aii ðtÞxi ðtÞ d 1 d a ðtÞx ðtÞ d d j bij ðtÞxj ðt sij ðtÞÞ j ij j i i d 1 i
n X
j¼1 j6¼i t
d j cij ðtÞ
Z
1
j¼1
j¼1
K ij ðt sÞxj ðsÞ ds þ d 1 i ri ðtÞ:
ð2:3Þ
Let B ¼ fuðtÞju : R ! Rn is continuousT —periodic functiong; then under the norm kuk = sup{ku(t)k:t 2 [0, T]}, B is a Banach space. For any u(t) = (u1 (t), . . . , un(t))T 2 B, we consider the periodic solution xu(t) of periodic differential equation n n X X 1 d a ðtÞu ðtÞ d d j bij ðtÞuj ðt sij ðtÞÞ x_ i ðtÞ ¼ aii ðtÞxi ðtÞ d 1 j ij j i i d 1 i
n X j¼1
j¼1 j6¼i t
d j cij ðtÞ
Z
1
j¼1
K ij ðt sÞuj ðsÞ ds þ d 1 i r i ðtÞ;
i ¼ 1; 2; . . . ; n: ð2:4Þ
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F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Since aii(t) > 0, we know that the linear system of system (2.4) x_ i ðtÞ ¼ aii ðtÞxi ðtÞ;
i ¼ 1; 2; . . . ; n
ð2:5Þ
admits exponential dichotomies on R, and so, system (2.4) has unique periodic solution xu(t), which can be expressed as T
xu ðtÞ ¼ ðx1u ðtÞ; . . . ; xnu ðtÞÞ Z t Z t ¼ exp a11 ðsÞ ds h1u ðsÞ ds; . . . ; Z
1 t
s
Z t T exp ann ðsÞ ds hnu ðsÞ ds ;
1
s
where hiu ðsÞ ¼ d 1 i
d 1 i
n X j¼1 j6¼i n X
d j aij ðsÞuj ðsÞ d 1 i
n X
d j bij ðsÞuj ðs sij ðsÞÞ
j¼1
d j cij ðsÞ
Z
s 1
j¼1
K ij ðs vÞuj ðvÞdv þ d 1 i ri ðsÞ;
i ¼ 1; 2; . . . ; n:
Now we define mapping T:B ! B, Tu(t) = xu(t). Following we will prove T is a contraction mapping. In fact, in view of the condition of Theorem 2.1, for any u(t) = (u1 (t), . . . , un(t))T and v(t) = (v1(t), . . . , vn(t))T, we have Z t Z t jjTu Tvjj 6 sup max exp a11 ðsÞ ds jh1u ðsÞ h1v ðsÞj ds; . . . ; t2R 1 Zs t Z t exp ann ðsÞ ds jhnu ðsÞ hnv ðsÞj ds Z 1 Zst t < sup max exp a11 ðsÞ ds a11 ðsÞjju vjj ds; . . . ; t2R 1 Zs t Z t exp ann ðsÞ ds ann ðsÞjju vjj ds 1
s
¼ jju vjj;
ð2:6Þ
where we had use the fact jjhiu ðsÞ hiv ðsÞj 6 d 1 i
n X
d j aij ðsÞjuj ðsÞ vj ðsÞj þ d 1 i
j¼1 j6¼i
n X
juj ðs sij ðsÞÞ vj ðs sij ðsÞÞj þ d 1 i
Z
d j bij ðsÞ
j¼1 n X j¼1
s
K ij ðs sÞjuj ðsÞ vj ðsÞjds 1
d j cij ðsÞ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
6 d 1 i
n X
d j aij ðsÞjju vjj þ d 1 i
j¼1 j6¼i
þ d 1 i 0 B 1 ¼B @d i
n X
n X
765
d j bij ðsÞjju vjj
j¼1
d j cij ðsÞjju vjj
j¼1 n X
1 d j aij ðsÞ þ d 1 i
j¼1 j6¼i
n X
d j bij ðsÞ þ d 1 i
j¼1
n X j¼1
C d j cij ðsÞC A
jju vjj < aii ðsÞjju vjj: That is, jjTu Tvjj < jju vjj:
ð2:7Þ
This shows that T is a contraction mapping. Hence, there exists a unique fixed T point x ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ 2 B, that is Tx* = x*. Therefore, x*(t) is the unique periodic solution of system (2.4). From (2.2) we know that T N ðtÞ ¼ ðN 1 ðtÞ; . . . ; N n ðtÞÞT ¼ expfd 1 x1 ðtÞg; . . . ; expfd n xn ðtÞg is the unique positive periodic solution of system (1.1). This finish the proof of Theorem 2.1. Our next theorem concerned with the almost periodic solution of system (1.1) and (1.2). To do so, we assume that: (H2) ri(t),aij(t),bij(t),cij(t) are continuous, real-valued almost periodic functions on R such that Z 1 T ri ðtÞ dt > 0; aii ðtÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; bij ðtÞ mðri ðtÞÞ ¼ lim t!þ1 T 0 P 0; cij ðtÞ P 0: sij(t)sij are positive constants.
h
Then we have Theorem 2.2. In addition to (H2), there are positive constants di,i = 1,2, . . . , n such that aii ðtÞ > d 1 i
n X j¼1 j6¼i
d j aij ðtÞ þ d 1 i
n X
d j ðbij ðtÞ þ cij ðtÞÞ:
ð2:8Þ
j¼1
Then (1.1) and (1.2) has unique almost periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT .
766
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Proof. Set C ¼ fuðtÞju : R ! Rn is continuous almost periodic functiong; then under the norm kuk = sup{ku(t)k:t 2 R}, C is a Banach space. Under the condition of Theorem 2.2, one could easily see that system (2.4) has a unique almost periodic solution which can be expressed as T
xu ðtÞ ¼ ðx1u ðtÞ; . . . ; xnu ðtÞÞ Z t Z t ¼ exp a11 ðsÞ ds h01u ðsÞ ds; . . . ; 1
Z
t
1
s
Z t T exp ann ðsÞ ds h0nu ðsÞ ds ; s
where h0iu ðsÞ ¼ d 1 i
n X
d j aij ðsÞuj ðsÞ d 1 i
j¼1 j6¼i
d 1 i
n X
n X
d j bij ðsÞuj ðs sij Þ
j¼1
d j cij ðsÞ
j¼1
Z
s
1
K ij ðs vÞuj ðvÞdv þ d 1 i ri ðsÞ; i ¼ 1; 2; . . . ; n:
The rest of the proof is similar to the proof of Theorem 2.1, and we omit the detail here. h
3. Global asymptotic stability In this section, we devote ourselves to the study of the global asymptotic stability of periodic solution (almost periodic solution) of system (1.1) and (1.2). Our method involves the construction of a suitable Lyapunov functional, which is based on an essential modification of Lyapunov functional introduced by Fan et al. [9]. 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 asymptotically stable 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:
Now we state our main results of this section below.
ð3:1Þ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
767
Theorem 3.1. Assume that the conditions in Theorem 2.1 (or Theorem 2.2) hold. Moreover, if there are positive constants ki > 0 such that 8 > > < n n X X bji ðn1 ji ðtÞÞ inf k i aii ðtÞ k j aji ðtÞ kj 0 t2½0;þ1Þ > 1 sji ðn1 > j¼1 j¼1 ji ðtÞÞ : j6¼i 9 > > Z n = þ1 X ð3:2Þ kj K ji ðsÞcji ðt þ sÞ ds > 0; i ¼ 1; 2; . . . ; n; > 0 > j¼1 ; where n1 ji ðtÞ are the inverse function of nji(t) = t sji(t), i, j = 1,2, . . . , n, respectively. Then system (1.1) and (1.2) has a unique globally attractive periodic solution (almost periodic solution). Proof. We only proof the periodic case, since the almost periodic case is similar to that of periodic case. By Theorem 2.1, there exists a strictly positive periodic solution of (1.1) and (1.2), say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . To complete the proof, we only need to show that N*(t) is globally attractive. Let Y(t) = (y1 (t), . . . , yn(t))T be any solution of (1.1) and (1.2). Consider a Lyapunov functional V(t) = V(t, N*(t),Y(t)) defined by V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ; for t P 0, where n X k i j ln N i ðtÞ ln y i ðtÞj; V 1 ðtÞ ¼ i¼1
V 2 ðtÞ ¼
n X
ki
i¼1
V 3 ðtÞ ¼
n X
n Z X j¼1
ki
i¼1
tsij ðtÞ
n Z X j¼1
bij ðn1 ij ðsÞÞ
t
0
1 s0ij ðn1 ij ðsÞÞ
þ1
K ij ðsÞ
Z
j ln N j ðsÞ ln y j ðsÞj ds;
t
cij ðh þ sÞj ln N j ðhÞ ln y j ðhÞj dh ds:
ts
From the definition of V(t), it is not difficult to show that V ð0Þ < þ1;
ð3:3Þ
and V ðtÞ P
n X i¼1
k i j ln N i ðtÞ ln y i ðtÞj;
t P 0:
ð3:4Þ
768
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Calculating the upper right derivative D+V(t) of V(t) along the solution of (1.1) and (1.2), by computation, one could obtain n n n X X X k i aii ðtÞj ln N i ðtÞ ln y i ðtÞj þ ki aij ðtÞj ln N j ðtÞ Dþ V ðtÞ 6 i¼1
i¼1 n X
ln y j ðtÞj þ
ki
n X
i¼1 n X
þ
n X
ki
i¼1 n X
þ
i¼1 n X
i¼1 n X
þ
n X
n X
bij ðn1 ij ðtÞÞ
j¼1
1 s0ij ðn1 ij ðtÞÞ
j ln N j ðtÞ ln y j ðtÞj
bij ðtÞj ln N j ðt sij ðtÞÞ ln y j ðt sij ðtÞÞj
ki
þ1
K ij ðsÞcij ðt þ sÞj ln N j ðtÞ ln y j ðtÞj ds 0
j¼1 n Z X
ki
þ1
K ij ðsÞcij ðtÞj ln N j ðt sÞ ln y j ðt sÞj ds 0
j¼1
(
n X
n X
k i aii ðtÞ
k j aji ðtÞ
j¼1;j6¼i
i¼1
K ij ðt sÞj ln N j ðsÞ ln y j ðsÞj ds
n X
n Z X
i¼1
6
t
j¼1
i¼1
cij ðtÞ
1
n X
ki
bij ðtÞj ln N j ðt sij ðtÞÞ ln y j ðt sij ðtÞÞj
j¼1
Z
j¼1
ki
j¼1;j6¼i
Z
n X
kj
j¼1
bji ðn1 ji ðtÞÞ 1 s0ji ðn1 ji ðtÞÞ
)
þ1
K ji ðsÞcji ðt þ sÞ ds j ln N i ðtÞ ln y i ðtÞj:
kj 0
j¼1
From (3.2), it follows that there exists a constant K > 0 and T > 0 such that for all t P T, k i aii ðtÞ
n X
k j aji ðtÞ
j¼1;j6¼i
n X j¼1
Z kj
n X j¼1
kj
bji ðn1 ji ðtÞÞ 1 s0ji ðn1 ji ðtÞÞ
þ1
K ji ðsÞcji ðt þ sÞ ds > K:
0
Hence, for t P T, it follows that n X j ln N i ðtÞ ln y i ðtÞj: Dþ V ðtÞ 6 K i¼1
ð3:5Þ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
769
Then, by using (3.3) and (3.5), similar to the analysis of [9, p. 816], one could obtain n X lim j ln N i ðtÞ ln y i ðtÞj ¼ 0: t!þ1
i¼1
From this, one could easily obtain lim j ln N i ðtÞ ln y i ðtÞj ¼ 0;
t!þ1
which means N*(t) is globally attractive. This completes the proof.
h
Finally, to show the feasibility of our results, letÕs consider the following two species competition Logarithmic model. Example x_ 1 ðtÞ ¼ x1 ðtÞ½3 ð12 þ cos2 tÞ ln x1 ðtÞ 44 ln x2 ðtÞ; x_ 2 ðtÞ ¼ x2 ðtÞ½1 þ 2 sin t 12 ln x1 ðtÞ ð50 sin tÞ ln x2 ðtÞ:
ð3:6Þ
In this case, we have r1(t) = 3, r2(t) = 1 + 2 sin t, a11 ðtÞ ¼ 12 þ cos2 t, a12(t) = 44, a21(t) = 12, a22(t) = 50 sin t. Now we take d1 = 4,d2 = 1 then a11 ðtÞ ¼ 12 þ
cos t 1 1 > 44 ¼ d 2 a12 ðtÞ; 2 4 d1
a22 ðtÞ ¼ 50 sin t > 4 12 ¼
1 d 1 a21 ðtÞ: d2
Also, we could choose k1 = 1.1, k2 = 1 such that cos t k 1 a11 ðtÞ k 2 a21 ðtÞ ¼ 1:1 12 þ 1 12 > 0:5; 2 k 2 a22 ðtÞ k 1 a12 ðtÞ ¼ 1 ð50 sin tÞ 1:1 44 > 0:5:
8
6
4
2
0
2
4
6
8
10 t
12
14
16
18
20
Fig. 1. Dynamics of system (3.6) with initial values x1(0) = 1.4, x2(0) = 1.6 and t 2 [0,20].
770
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
This shows that all the conditions of Theorems 2.1 and 3.1 are hold, and so, T system (3.6) has a unique periodic solution x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ ; which is globally attractive (see Fig. 1).
Acknowledgements This work is supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (10426010), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156) and the Foundation of Developing Science and Technology of Fuzhou University (2003QX-21).
References [1] Z.J. Liu, Positive periodic solutions for delay multispecies Logrithmic population model, Journal of Engineering Mathematics 19 (4) (2002) 11–16 (in Chinese). [2] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Mathematics and Its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. [3] Y.K. Li, Attractivity of a positive periodic solution for all other positive solution in a delay population model, Applied Mathematics—JCU 12 (3) (1997) 279–282 (in Chinese). [4] G. Kirlinger, Permanence in Lotka–Volterra equations linked prey-predator systems, Mathematical Bioscience 82 (1986) 165–169. [5] F.D. Chen, et al., Positive periodic solutions of state-dependent delay logarithm population model, Journal of Fuzhou University 31 (3) (2003) 1–4 (in Chinese). [6] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations [M], Springer-Verlag, Berlin, 1977. [7] F.D. Chen, et al., On the existence and uniqueness of periodic solutions of a kind of integrodifferential equations, Acta Mathematics Sinica 46 (5) (2004) 1–12 (in Chinese). [8] H.Q. Xie, Q.Y. Wang, Exponential stability and periodic solution for cellular neural networks with time delay 25 (1) (2004) 22–26 (in Chinese). [9] M. Fan, P.J.Y. Wong, Ravi P. Agarwal, Periodicity and stability in Periodic n-species Lotka– Volterra competition system with feedback controls and deviating arguments, Acta Mathematics Sinica 19 (4) (2003) 801–822.
Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model Fengde Chen School of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, People’s Republic of China
Abstract By using the method of fixed point theory and Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic (almost periodic solution) of a delay multispecies Logarithmic population model. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Periodic solution; Almost periodic solution; Lyapunov functional; Stability; Contraction principle
1. Introduction The aim of this paper is to investigate the existence, uniqueness and global attractivity of the positive periodic solution (almost periodic solution) of the following delay multispecies Logarithmic population model
E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.085
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
" N_ i ðtÞ ¼ N i ðtÞ ri ðtÞ
n X
aij ðtÞ ln N j ðtÞ
j¼1
n X
cij ðtÞ
Z
n X
761
bij ðtÞ ln N j ðt sij ðtÞÞ
j¼1
t
#
K ij ðt sÞ ln N j ðsÞ ds ;
ð1:1Þ
1
j¼1
ri(t), aij,bij R2 C(R, (0, +1)), sij 2 C(R, R) are all continuous functions. Rwhere þ1 þ1 K sK ij ðsÞ ds < þ1: We consider (1.1) together with the ij ðsÞ ds ¼ 1; 0 0 following initial conditions N i ðhÞ ¼ /i ðhÞ P 0; h 2 ð1; 0;
/i ð0Þ > 0:
ð1:2Þ
For the ecological justification of Eq. (1.1), see [1–5]. Recently, by using MawhinÕs continuation theorem in [6], Liu [1] studied the existence of positive periodic solution of the following periodic Logarithmic population model " # n n X X N_ i ðtÞ ¼ N i ðtÞ ri ðtÞ aij ðtÞ ln N j ðtÞ bij ðtÞ ln N j ðt sij ðtÞÞ ð1:3Þ j¼1
j¼1
together with the initial condition N i ðhÞ ¼ /i ðhÞ P 0; h 2 ðs ; 0;
/i ð0Þ > 0;
ð1:4Þ
where aij,bij 2 C (R, (0, +1)),ri, sij 2 C(R, R) are all continuous T-periodic function, s* = max16i,j6nmaxt2[0,T]{jsij(t)j}. By introducing the change of variable N i ðtÞ ¼ exi ðtÞ ;
i ¼ 1; 2; . . . ; n;
then system (1.3) is equal to x_ i ðtÞ ¼ ri ðtÞ
n X
aij ðtÞxj ðtÞ
j¼1
n X
bij ðtÞxj ðt sij ðtÞÞ:
ð1:5Þ
j¼1
And the existence of positive periodic solution of system (1.3) is equal to the existence of periodic solution of system (1.5). Noticing that for system (1.5), corresponding to the operator equation Lx = kNx,k 2 (0, 1), one has " # n n X X aij ðtÞxj ðtÞ bij ðtÞxj ðt sij ðtÞÞ : ð1:6Þ x_ i ðtÞ ¼ k ri ðtÞ j¼1
j¼1
T
Let x(t) = (x1(t), . . . , xn(t)) be any periodic solution of system (1.6), and let xi ðni Þ ¼ min xi ðtÞ; t2½0;T
i ¼ 1; 2; . . . ; n:
762
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Then in proving the theorem, Liu [1] had used the following inequality # Z T "X n n X ri T P aij ðtÞxj ðnj Þ þ bij ðtÞxj ðnj Þ dt 0
¼
n X
j¼1
ð aij þ bij Þxj ðnj ÞT > ð aii þ bii Þxi ðni ÞT ;
j¼1
RT where ri ¼
0
ri ðtÞ dt T
j¼1
RT ; aij ¼
0
aij ðtÞ dt T
; bij ¼
RT 0
bij ðtÞ dt T
ð1:7Þ
; and so
ri : xi ðni Þ 6 bii aii þ We point out that the positivity of Ni(t) could not guarantee the positivity of xi(t) and so, xj(nj) might be negative, in this case, inequality (1.7) may not hold, and the proof of Theorem 1 of [1] is incomplete. The aim of this paper is to give a set of new conditions to guarantee the existence, uniqueness and stability of the periodic solution (almost periodic solution) of system (1.1) and (1.2). The outline of the paper is as follows. In Section 2, we first introducing a new transformation, where some new adjustable real parameter di > 0 are introduced; After that, by using contraction principle, some new results on the existence of positive periodic solutions (almost periodic solutions) of system (1.1) and (1.2) are established. In Section 3, by constructing a suitable Lyapunov functional, we derive a set of easily verifiable criteria for the global asymptotic stability of positive periodic solutions (almost periodic solution) of (1.1) and (1.2). We must point out, the idea of introducing parameter is stimulate by the works of Chen et al. [7] and Xie and Wang [8]. However, to the best of the authors knowledge, this is the first time such a technique is applied to the ecosystem. 2. Periodic solutions and almost periodic solutions A very basic and important ecological problem associated with the study of multispecies population interactions in a periodic environment (almost periodic environment) is the global existence of positive periodic solution (almost periodic solution) which plays the role played by the equilibrium of the autonomous models. It is reasonable to ask for conditions under which the resulting periodic (almost periodic) nonautonomous system would have a periodic solution (almost periodic solution). To our knowledge, no such work has been done on the global existence of positive periodic solution (almost periodic solutions) of (1.1) and (1.2). 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).
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
763
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), sij(t), cij(t) are continuous, real-valued T-periodic functions on R such that Z T ri ðtÞ dt P 0; aii ðtÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; bij ðtÞ P 0; cij ðtÞ P 0: 0
Also, s0ij ðtÞ < 1: Our main result on the global existence of a positive periodic solution of (1.1) and (1.2) is stated in the following theorem. Theorem 2.1. In addition to (H1), there are positive constants di, i = 1,2, . . . , n such that n n X X d j aij ðtÞ þ d 1 d j ðbij ðtÞ þ cij ðtÞÞ: ð2:1Þ aii ðtÞ > d 1 i i j¼1 j6¼i
j¼1
Then (1.1) and (1.2) has unique T-periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . Proof. Make the change of variables N i ðtÞ ¼ expfd i xi ðtÞg;
i ¼ 1; 2; . . . ; n;
ð2:2Þ
then (1.1) can be reformulated as n n X X 1 x_ i ðtÞ ¼ aii ðtÞxi ðtÞ d 1 d a ðtÞx ðtÞ d d j bij ðtÞxj ðt sij ðtÞÞ j ij j i i d 1 i
n X
j¼1 j6¼i t
d j cij ðtÞ
Z
1
j¼1
j¼1
K ij ðt sÞxj ðsÞ ds þ d 1 i ri ðtÞ:
ð2:3Þ
Let B ¼ fuðtÞju : R ! Rn is continuousT —periodic functiong; then under the norm kuk = sup{ku(t)k:t 2 [0, T]}, B is a Banach space. For any u(t) = (u1 (t), . . . , un(t))T 2 B, we consider the periodic solution xu(t) of periodic differential equation n n X X 1 d a ðtÞu ðtÞ d d j bij ðtÞuj ðt sij ðtÞÞ x_ i ðtÞ ¼ aii ðtÞxi ðtÞ d 1 j ij j i i d 1 i
n X j¼1
j¼1 j6¼i t
d j cij ðtÞ
Z
1
j¼1
K ij ðt sÞuj ðsÞ ds þ d 1 i r i ðtÞ;
i ¼ 1; 2; . . . ; n: ð2:4Þ
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F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Since aii(t) > 0, we know that the linear system of system (2.4) x_ i ðtÞ ¼ aii ðtÞxi ðtÞ;
i ¼ 1; 2; . . . ; n
ð2:5Þ
admits exponential dichotomies on R, and so, system (2.4) has unique periodic solution xu(t), which can be expressed as T
xu ðtÞ ¼ ðx1u ðtÞ; . . . ; xnu ðtÞÞ Z t Z t ¼ exp a11 ðsÞ ds h1u ðsÞ ds; . . . ; Z
1 t
s
Z t T exp ann ðsÞ ds hnu ðsÞ ds ;
1
s
where hiu ðsÞ ¼ d 1 i
d 1 i
n X j¼1 j6¼i n X
d j aij ðsÞuj ðsÞ d 1 i
n X
d j bij ðsÞuj ðs sij ðsÞÞ
j¼1
d j cij ðsÞ
Z
s 1
j¼1
K ij ðs vÞuj ðvÞdv þ d 1 i ri ðsÞ;
i ¼ 1; 2; . . . ; n:
Now we define mapping T:B ! B, Tu(t) = xu(t). Following we will prove T is a contraction mapping. In fact, in view of the condition of Theorem 2.1, for any u(t) = (u1 (t), . . . , un(t))T and v(t) = (v1(t), . . . , vn(t))T, we have Z t Z t jjTu Tvjj 6 sup max exp a11 ðsÞ ds jh1u ðsÞ h1v ðsÞj ds; . . . ; t2R 1 Zs t Z t exp ann ðsÞ ds jhnu ðsÞ hnv ðsÞj ds Z 1 Zst t < sup max exp a11 ðsÞ ds a11 ðsÞjju vjj ds; . . . ; t2R 1 Zs t Z t exp ann ðsÞ ds ann ðsÞjju vjj ds 1
s
¼ jju vjj;
ð2:6Þ
where we had use the fact jjhiu ðsÞ hiv ðsÞj 6 d 1 i
n X
d j aij ðsÞjuj ðsÞ vj ðsÞj þ d 1 i
j¼1 j6¼i
n X
juj ðs sij ðsÞÞ vj ðs sij ðsÞÞj þ d 1 i
Z
d j bij ðsÞ
j¼1 n X j¼1
s
K ij ðs sÞjuj ðsÞ vj ðsÞjds 1
d j cij ðsÞ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
6 d 1 i
n X
d j aij ðsÞjju vjj þ d 1 i
j¼1 j6¼i
þ d 1 i 0 B 1 ¼B @d i
n X
n X
765
d j bij ðsÞjju vjj
j¼1
d j cij ðsÞjju vjj
j¼1 n X
1 d j aij ðsÞ þ d 1 i
j¼1 j6¼i
n X
d j bij ðsÞ þ d 1 i
j¼1
n X j¼1
C d j cij ðsÞC A
jju vjj < aii ðsÞjju vjj: That is, jjTu Tvjj < jju vjj:
ð2:7Þ
This shows that T is a contraction mapping. Hence, there exists a unique fixed T point x ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ 2 B, that is Tx* = x*. Therefore, x*(t) is the unique periodic solution of system (2.4). From (2.2) we know that T N ðtÞ ¼ ðN 1 ðtÞ; . . . ; N n ðtÞÞT ¼ expfd 1 x1 ðtÞg; . . . ; expfd n xn ðtÞg is the unique positive periodic solution of system (1.1). This finish the proof of Theorem 2.1. Our next theorem concerned with the almost periodic solution of system (1.1) and (1.2). To do so, we assume that: (H2) ri(t),aij(t),bij(t),cij(t) are continuous, real-valued almost periodic functions on R such that Z 1 T ri ðtÞ dt > 0; aii ðtÞ > 0; aij ðtÞ P 0ði 6¼ jÞ; bij ðtÞ mðri ðtÞÞ ¼ lim t!þ1 T 0 P 0; cij ðtÞ P 0: sij(t)sij are positive constants.
h
Then we have Theorem 2.2. In addition to (H2), there are positive constants di,i = 1,2, . . . , n such that aii ðtÞ > d 1 i
n X j¼1 j6¼i
d j aij ðtÞ þ d 1 i
n X
d j ðbij ðtÞ þ cij ðtÞÞ:
ð2:8Þ
j¼1
Then (1.1) and (1.2) has unique almost periodic solution with strictly positive components, say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT .
766
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Proof. Set C ¼ fuðtÞju : R ! Rn is continuous almost periodic functiong; then under the norm kuk = sup{ku(t)k:t 2 R}, C is a Banach space. Under the condition of Theorem 2.2, one could easily see that system (2.4) has a unique almost periodic solution which can be expressed as T
xu ðtÞ ¼ ðx1u ðtÞ; . . . ; xnu ðtÞÞ Z t Z t ¼ exp a11 ðsÞ ds h01u ðsÞ ds; . . . ; 1
Z
t
1
s
Z t T exp ann ðsÞ ds h0nu ðsÞ ds ; s
where h0iu ðsÞ ¼ d 1 i
n X
d j aij ðsÞuj ðsÞ d 1 i
j¼1 j6¼i
d 1 i
n X
n X
d j bij ðsÞuj ðs sij Þ
j¼1
d j cij ðsÞ
j¼1
Z
s
1
K ij ðs vÞuj ðvÞdv þ d 1 i ri ðsÞ; i ¼ 1; 2; . . . ; n:
The rest of the proof is similar to the proof of Theorem 2.1, and we omit the detail here. h
3. Global asymptotic stability In this section, we devote ourselves to the study of the global asymptotic stability of periodic solution (almost periodic solution) of system (1.1) and (1.2). Our method involves the construction of a suitable Lyapunov functional, which is based on an essential modification of Lyapunov functional introduced by Fan et al. [9]. 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 asymptotically stable 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:
Now we state our main results of this section below.
ð3:1Þ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
767
Theorem 3.1. Assume that the conditions in Theorem 2.1 (or Theorem 2.2) hold. Moreover, if there are positive constants ki > 0 such that 8 > > < n n X X bji ðn1 ji ðtÞÞ inf k i aii ðtÞ k j aji ðtÞ kj 0 t2½0;þ1Þ > 1 sji ðn1 > j¼1 j¼1 ji ðtÞÞ : j6¼i 9 > > Z n = þ1 X ð3:2Þ kj K ji ðsÞcji ðt þ sÞ ds > 0; i ¼ 1; 2; . . . ; n; > 0 > j¼1 ; where n1 ji ðtÞ are the inverse function of nji(t) = t sji(t), i, j = 1,2, . . . , n, respectively. Then system (1.1) and (1.2) has a unique globally attractive periodic solution (almost periodic solution). Proof. We only proof the periodic case, since the almost periodic case is similar to that of periodic case. By Theorem 2.1, there exists a strictly positive periodic solution of (1.1) and (1.2), say N ðtÞ ¼ ðN 1 ðtÞ; N 2 ðtÞ; . . . ; N n ðtÞÞT . To complete the proof, we only need to show that N*(t) is globally attractive. Let Y(t) = (y1 (t), . . . , yn(t))T be any solution of (1.1) and (1.2). Consider a Lyapunov functional V(t) = V(t, N*(t),Y(t)) defined by V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ; for t P 0, where n X k i j ln N i ðtÞ ln y i ðtÞj; V 1 ðtÞ ¼ i¼1
V 2 ðtÞ ¼
n X
ki
i¼1
V 3 ðtÞ ¼
n X
n Z X j¼1
ki
i¼1
tsij ðtÞ
n Z X j¼1
bij ðn1 ij ðsÞÞ
t
0
1 s0ij ðn1 ij ðsÞÞ
þ1
K ij ðsÞ
Z
j ln N j ðsÞ ln y j ðsÞj ds;
t
cij ðh þ sÞj ln N j ðhÞ ln y j ðhÞj dh ds:
ts
From the definition of V(t), it is not difficult to show that V ð0Þ < þ1;
ð3:3Þ
and V ðtÞ P
n X i¼1
k i j ln N i ðtÞ ln y i ðtÞj;
t P 0:
ð3:4Þ
768
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
Calculating the upper right derivative D+V(t) of V(t) along the solution of (1.1) and (1.2), by computation, one could obtain n n n X X X k i aii ðtÞj ln N i ðtÞ ln y i ðtÞj þ ki aij ðtÞj ln N j ðtÞ Dþ V ðtÞ 6 i¼1
i¼1 n X
ln y j ðtÞj þ
ki
n X
i¼1 n X
þ
n X
ki
i¼1 n X
þ
i¼1 n X
i¼1 n X
þ
n X
n X
bij ðn1 ij ðtÞÞ
j¼1
1 s0ij ðn1 ij ðtÞÞ
j ln N j ðtÞ ln y j ðtÞj
bij ðtÞj ln N j ðt sij ðtÞÞ ln y j ðt sij ðtÞÞj
ki
þ1
K ij ðsÞcij ðt þ sÞj ln N j ðtÞ ln y j ðtÞj ds 0
j¼1 n Z X
ki
þ1
K ij ðsÞcij ðtÞj ln N j ðt sÞ ln y j ðt sÞj ds 0
j¼1
(
n X
n X
k i aii ðtÞ
k j aji ðtÞ
j¼1;j6¼i
i¼1
K ij ðt sÞj ln N j ðsÞ ln y j ðsÞj ds
n X
n Z X
i¼1
6
t
j¼1
i¼1
cij ðtÞ
1
n X
ki
bij ðtÞj ln N j ðt sij ðtÞÞ ln y j ðt sij ðtÞÞj
j¼1
Z
j¼1
ki
j¼1;j6¼i
Z
n X
kj
j¼1
bji ðn1 ji ðtÞÞ 1 s0ji ðn1 ji ðtÞÞ
)
þ1
K ji ðsÞcji ðt þ sÞ ds j ln N i ðtÞ ln y i ðtÞj:
kj 0
j¼1
From (3.2), it follows that there exists a constant K > 0 and T > 0 such that for all t P T, k i aii ðtÞ
n X
k j aji ðtÞ
j¼1;j6¼i
n X j¼1
Z kj
n X j¼1
kj
bji ðn1 ji ðtÞÞ 1 s0ji ðn1 ji ðtÞÞ
þ1
K ji ðsÞcji ðt þ sÞ ds > K:
0
Hence, for t P T, it follows that n X j ln N i ðtÞ ln y i ðtÞj: Dþ V ðtÞ 6 K i¼1
ð3:5Þ
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
769
Then, by using (3.3) and (3.5), similar to the analysis of [9, p. 816], one could obtain n X lim j ln N i ðtÞ ln y i ðtÞj ¼ 0: t!þ1
i¼1
From this, one could easily obtain lim j ln N i ðtÞ ln y i ðtÞj ¼ 0;
t!þ1
which means N*(t) is globally attractive. This completes the proof.
h
Finally, to show the feasibility of our results, letÕs consider the following two species competition Logarithmic model. Example x_ 1 ðtÞ ¼ x1 ðtÞ½3 ð12 þ cos2 tÞ ln x1 ðtÞ 44 ln x2 ðtÞ; x_ 2 ðtÞ ¼ x2 ðtÞ½1 þ 2 sin t 12 ln x1 ðtÞ ð50 sin tÞ ln x2 ðtÞ:
ð3:6Þ
In this case, we have r1(t) = 3, r2(t) = 1 + 2 sin t, a11 ðtÞ ¼ 12 þ cos2 t, a12(t) = 44, a21(t) = 12, a22(t) = 50 sin t. Now we take d1 = 4,d2 = 1 then a11 ðtÞ ¼ 12 þ
cos t 1 1 > 44 ¼ d 2 a12 ðtÞ; 2 4 d1
a22 ðtÞ ¼ 50 sin t > 4 12 ¼
1 d 1 a21 ðtÞ: d2
Also, we could choose k1 = 1.1, k2 = 1 such that cos t k 1 a11 ðtÞ k 2 a21 ðtÞ ¼ 1:1 12 þ 1 12 > 0:5; 2 k 2 a22 ðtÞ k 1 a12 ðtÞ ¼ 1 ð50 sin tÞ 1:1 44 > 0:5:
8
6
4
2
0
2
4
6
8
10 t
12
14
16
18
20
Fig. 1. Dynamics of system (3.6) with initial values x1(0) = 1.4, x2(0) = 1.6 and t 2 [0,20].
770
F. Chen / Appl. Math. Comput. 171 (2005) 760–770
This shows that all the conditions of Theorems 2.1 and 3.1 are hold, and so, T system (3.6) has a unique periodic solution x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ ; which is globally attractive (see Fig. 1).
Acknowledgements This work is supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (10426010), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156) and the Foundation of Developing Science and Technology of Fuzhou University (2003QX-21).
References [1] Z.J. Liu, Positive periodic solutions for delay multispecies Logrithmic population model, Journal of Engineering Mathematics 19 (4) (2002) 11–16 (in Chinese). [2] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Mathematics and Its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. [3] Y.K. Li, Attractivity of a positive periodic solution for all other positive solution in a delay population model, Applied Mathematics—JCU 12 (3) (1997) 279–282 (in Chinese). [4] G. Kirlinger, Permanence in Lotka–Volterra equations linked prey-predator systems, Mathematical Bioscience 82 (1986) 165–169. [5] F.D. Chen, et al., Positive periodic solutions of state-dependent delay logarithm population model, Journal of Fuzhou University 31 (3) (2003) 1–4 (in Chinese). [6] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations [M], Springer-Verlag, Berlin, 1977. [7] F.D. Chen, et al., On the existence and uniqueness of periodic solutions of a kind of integrodifferential equations, Acta Mathematics Sinica 46 (5) (2004) 1–12 (in Chinese). [8] H.Q. Xie, Q.Y. Wang, Exponential stability and periodic solution for cellular neural networks with time delay 25 (1) (2004) 22–26 (in Chinese). [9] M. Fan, P.J.Y. Wong, Ravi P. Agarwal, Periodicity and stability in Periodic n-species Lotka– Volterra competition system with feedback controls and deviating arguments, Acta Mathematics Sinica 19 (4) (2003) 801–822.