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Permanence and existence of periodic solutions for a generalized system with feedback control
Applied Mathematics and Computation 216 (2010) 902–910
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Permanence and existence of periodic solutions for a generalized system with feedback control Lin-Lin Wang a,*, Yong-Hong Fan a,b a b
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, People’s Republic of China Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t Sufficient conditions are obtained for the permanence and the existence of positive periodic solutions of the delay differential system with feedback control
Keywords: Delay differential equations Feedback control Positive periodic solutions Permanence
(
dx dt du dt
¼ xðtÞ½Fðt; xðt s1 ðtÞÞ; . . . ; xðt sn ðtÞÞÞ cðtÞuðt dðtÞÞ; ¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ:
ð0:1Þ
The method involves the application of estimation for uniform upper and lower bounds of solutions. When these results are applied to some special population models with multiple delays, some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in Chen et al. [F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587] and Huo and Li [H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46]. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction For dynamical systems, if the environment is not temporally constant (e.g., seasonal effects of weather, food supplies, mating habits, etc.), then the parameters become time dependent. It has been suggested by Nicholson [1] that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. Pianka [2] discussed the relevance of periodic environment to evolutionary theory. On the other hand, ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. The control variables discussed in most literatures are constants or time dependent (see [3,4]). Recently, logistic model with several delays and feedback control (see [5])
8 n P > < dxðtÞ ¼ xðtÞ rðtÞ ai ðtÞxðt si ðtÞÞ cðtÞuðt dðtÞÞ ; dt
> : duðtÞ dt
i¼1
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
* Corresponding author. E-mail addresses: [email protected] (L.-L. Wang), [email protected] (Y.-H. Fan). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.103
ð1:1Þ
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
903
the mutualism delay model with feedback control (see [6])
8 Pn > < dxðtÞ ¼ rðtÞxðtÞ KðtÞþPn i¼1 ai ðtÞxðtsi ðtÞÞ xðt aðtÞÞ cðtÞuðt dðtÞÞ ; dt
1þ
> : duðtÞ dt
s
c ðtÞxðt i ðtÞÞ i¼1 i
ð1:2Þ
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
the so-called Michaelis–Menton single-species growth model with feedback control (see [5])
8 n P > < dxðtÞ ¼ rðtÞxðtÞ 1 dt
> : duðtÞ dt
i¼1
ai ðtÞxðtsi ðtÞÞ 1þci ðtÞxðtsi ðtÞÞ
cðtÞuðt dðtÞÞ ;
ð1:3Þ
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
have been studied by several authors. The coincidence degree theory [7] has been used to establish the existence of periodic solutions. For more works related to the study of feedback control systems, we refer the reader to [8–17,19,20] and the references cited therein. Motivated by the above literature, in this paper, we consider the following general nonlinear nonautonomous delay differential system with feedback control
(
dx dt du dt
¼ xðtÞ½Fðt; xðt s1 ðtÞÞ; . . . ; xðt sn ðtÞÞÞ cðtÞuðt dðtÞÞ; ¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
ð1:4Þ
where Fðt; z1 ; z2 ; . . . ; zn ; znþ1 Þ 2 CðRnþ2 ; ½0; 1ÞÞ; si ðtÞ ði ¼ 1; 2; . . . ; nÞ; dðtÞ; rðtÞ; cðtÞ; gðtÞ; aðtÞ 2 CðR; ð0; 1ÞÞ, we further assume that all of the above functions are x-periodic in t and x > 0 is a constant. By employing the differential inequality developed by Chen et al. [6] and Fan and Wang [18], we obtain sufficient conditions of the permanence and the existence of periodic solutions for system (1.4). Then we apply the obtained criteria to some population models with multiple delays and feedback control, such as the periodic logistic equation with several delays (1.1)–(1.3). Some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in [6,5]. For biological meaning, we only consider (1.4) with the following initial conditions
xðtÞ ¼ uðtÞ P 0;
t 2 ½s; 0; uð0Þ > 0;
uðtÞ ¼ wðtÞ P 0;
t 2 ½s; 0; wð0Þ > 0;
where u and w are continuous on ½s; 0. Here
s ¼ max max si ðtÞ ði ¼ 1; 2; . . . ; nÞ; max dðtÞ; max rðtÞ : t2½0;x
t2½0;x
t2½0;x
By the fundamental theory of functional differential equations [21], it is clear that the solution of (1.4) exists and remains nonnegative on its maximal existence interval. 2. Main results In this section, we establish the permanence and existence of positive periodic solutions for system (1.4) by applying some useful lemmas developed by Chen et al. [6] and Fan and Wang [18]. For basic concepts of permanence for functional differential equations, one can refer to [22]. For the sake of convenience, we introduce some notations and definitions. Denote R and Rþ as the sets of all real numbers and nonnegative real numbers, respectively. Let C denote the set of all bounded continuous function f : R ! R; C þ is the set of all f 2 C such that f > 0, and C x ¼ ff 2 C þ jf ðt þ xÞ ¼ f ðtÞ; t 2 Rg. Define
f M ¼ sup f ðtÞ;
f L ¼ inf f ðtÞ;
t2R
t2R
for any f 2 C. Of cause, if f is an x-periodic function, then
f M ¼ max f ðtÞ; t2½0;x
f L ¼ min ðtÞ: t2½0;x
We also define
f ¼ 1
x
Z
x
f ðtÞ dt; 0
if f is an x-periodic function. Definition 2.1. System (1.4) is said to be permanent if there exist two positive constants k1 ; k2 such that
k1 6 lim inf Ni ðtÞ 6 lim sup Ni ðtÞ 6 k2 ; t!1
t!1
i ¼ 1; 2;
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for any solution ðN 1 ðtÞ; N 2 ðtÞÞ of (1.4). The following lemma will be useful to establish the main result, which can be seen as a corollary of Lemma 2.1 in [23]. Lemma 2.1. The problem
x0 ¼ x½aðtÞ bðtÞx;
ð2:1Þ
> 0, moreover, has only one positive x-periodic solution U if b 2 C þ ; a 2 C; a; b are both continuous x-periodic functions and a the following properties hold: (a) U is constant if a=b is constant, in this case, U ¼ a=b; (b) uðtÞ UðtÞ ! 0 as t ! þ1, for any positive solution uðtÞ of Eq. (2.1); þ jajÞxg 6 U 6 a expfða þ jajÞxg, and if a 2 C x , then (c) ab expfða b
a a expfaxg 6 U 6 b expfaxg: b Proof. Notice that (b) is a direct conclusion of Lemma 2.1 in [23], this implies that the positive x-periodic solution is unique. On the other hand, (a) is a direct conclusion of (c). So we only need to prove (c). According to the positivity of U, let
UðtÞ ¼ expfVðtÞg;
ð2:2Þ
then
V 0 ðtÞ ¼ aðtÞ bðtÞ expfVðtÞg: Notice that VðtÞ is also an x-periodic function, thus
0¼
1
Z
x
x
V 0 ðtÞ dt ¼
0
1
x
Z
x
½aðtÞ bðtÞ expfVðtÞg dt:
0
By the integral mean value theorem, the above equality implies that there exists a n 2 ½0; x such that
a expfVðnÞg ¼ ; b therefore
Z t Z t a VðtÞ ¼ VðnÞ þ V 0 ðsÞ ds ¼ ln þ ½aðsÞ bðsÞ expfVðsÞg ds b n n Z x a 6 ln þ ½jaðsÞj þ bðsÞ expfVðsÞg ds b 0 a þ jajÞx ¼ ln þ ða b
and
VðtÞ ¼ VðnÞ þ
Z
t
n
Z
a V 0 ðsÞ ds ¼ ln þ b
Z
t
½aðsÞ bðsÞ expfVðsÞg ds
n
x
a P ln ½jaðsÞj þ bðsÞ expfVðsÞg ds b 0 a þ jajÞx; ¼ ln ða b via the transformation (2.2), we obtain
a a þ jajÞxg 6 U 6 expfða þ jajÞxg: expfða b b If a 2 C x , then
VðtÞ ¼ VðnÞ þ a 6 ln þ b
Z n t
Z
t
a V 0 ðsÞ ds ¼ ln þ b
Z
t
½aðsÞ bðsÞ expfVðsÞg ds
n
aðsÞ ds for n 6 t 6 x
n
and
a VðtÞ 6 ln b
Z n
t
bðsÞ expfVðsÞg ds for 0 6 t 6 n;
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
905
thus
a VðtÞ 6 ln þ b
Z
x
aðsÞ ds for any t 2 ½0; x;
0
which yields
a xg: U 6 expfa b Similarly, we can obtain
a xg: U P expfa b This complete the proof. h Lemma 2.2. Consider the inequality problem
x0 ðtÞ 6 xðtÞ½aðtÞ bðtÞxðtÞ:
ð2:3Þ
> 0, then every positive solution xðtÞ of (2.3) satisfies If b 2 C x ; a 2 C and a is an x-periodic function with a
lim sup xðtÞ 6 H1 ; t!1
where
a þ jajÞxg: H1 ¼ expfða b Moreover, if a 2 C x , then
lim sup xðtÞ 6 H2 ; t!1
where
a xg: H2 ¼ expfa b Proof. Constructing the following auxiliary equation
z0 ðtÞ ¼ zðtÞ½aðtÞ bðtÞzðtÞ;
ð2:4Þ
by Lemma 2.1, we only need to prove the former inequality, the latter can be proved similarly. Eq. (2.4) has one positive xperiodic solution, denote it as z ðtÞ and
z ðtÞ 6 H1 : Let
xðtÞ ¼ expfu1 ðtÞg;
z ðtÞ ¼ expfu2 ðtÞg;
ð2:5Þ
then
u01 ðtÞ 6 aðtÞ bðtÞ expfu1 ðtÞg and
u02 ðtÞ ¼ aðtÞ bðtÞ expfu2 ðtÞg: Make the transformation uðtÞ ¼ u1 ðtÞ u2 ðtÞ, we can obtain
u0 ðtÞ 6 bðtÞz ðtÞ½expfuðtÞg 1:
ð2:6Þ
Now we divide the proof into two cases according to the oscillating property of uðtÞ. First we assume that uðtÞ does not oscillate about zero, then uðtÞ will be either eventually positive or eventually negative. If the latter holds, i.e., u1 ðtÞ < u2 ðtÞ, we have
xðtÞ < z ðtÞ 6 H1 :
ð2:7Þ 0
Either if the former holds, then by (2.6), we know u ðtÞ < 0, which means that uðtÞ is eventually decreasing, also in terms of its positivity, we know that limt!1 uðtÞ exists. We claim that
lim uðtÞ ¼ 0:
t!1
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L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
If this is not true, then limt!1 uðtÞ ¼ a > 0, thus there exists a sufficiently large T (fix it) such that
uðtÞ P a for t P T: By (2.6),
u0 ðtÞ 6 bðtÞz ðtÞ½expfag 1: Integrate the above inequality from T to þ1, we have
a uðTÞ 6 ½expfag 1
Z
þ1
bðtÞz ðtÞ dt ¼ 1 ðnotice that b; z 2 C x Þ: T
This contradiction shows our claim is true. Which leads to
lim sup xðtÞ 6 H1 :
ð2:8Þ
t!1
Now we assume that uðtÞ oscillates about zero, let t n > 0 such that
uðt n Þ ¼ 0;
uðtÞ P 0 for t 2 ½t2k1 ; t 2k and uðtÞ 6 0 for t 2 ½t2k ; t2kþ1 ;
k ¼ 1; 2; . . . ;
and let nk 2 ½t 2k1 ; t2k ðk ¼ 1; 2; . . .Þ such that
uðnk Þ ¼
max uðtÞ:
t2½t2k1 ;t2k
Notice that uðtÞ 2 C 1 ð0; þ1Þ, then by Fermat Theorem, either u0 ðnk Þ ¼ 0 or nk ¼ t 2k1 or nk ¼ t2k , however, if u0 ðnk Þ ¼ 0, then (2.6) implies
0 ¼ u0 ðnk Þ 6 bðnk Þz ðnk Þ½expfuðnk Þg 1 6 0; which implies that uðnk Þ ¼ 0. That is, in any case, uðnk Þ ¼ 0, this yields
uðtÞ 6 0 for t P t 1 : Therefore,
lim sup xðtÞ 6 H1 :
t!1
By Lemma 2.2, we have Corollary 2.1. Any positive solution of the inequality problem (2.3) satisfies
lim sup xðtÞ 6 t!1
aM b
L
;
where a 2 C; aM > 0 and b 2 C þ . Especially, when aðtÞ a > 0 and bðtÞ b > 0,
a lim sup xðtÞ 6 : b t!1 Remark. The last conclusion of Corollary 2.1 is the same as that in [6]. Similarly, we can get Lemma 2.3. Any positive solution xðtÞ of the inequality problem
x0 ðtÞ P xðtÞ½aðtÞ bðtÞxðtÞ satisfies
lim inf xðtÞ P H3 ; t!1
where
a Þxg; H3 ¼ expfðjaj þ a b > 0. Moreover, if a 2 C x , then if b 2 C x ; a 2 C and a is an x-periodic sequence with a
lim inf xðtÞ P H4 ; t!1
where
ð2:9Þ
907
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
a xg: H4 ¼ expfa b From Lemma 2.3, we can easily obtain Corollary 2.2. Any positive solution xðtÞ of the inequality problem (2.9) satisfies
lim inf xðtÞ P t!1
aL b
M
;
if a 2 C þ and b 2 C þ . Moreover, if aðtÞ a > 0 and bðtÞ b > 0, then
a lim inf xðtÞ P : t!1 b Remark. The last conclusion of Corollary 2.2 is the same as that in [6]. Now we are in the position to give the main result. Theorem 2.1. Suppose that for any positive solution ðxðtÞ; uðtÞÞ of (1.4) , there exist x-periodic functions a1 ðtÞ; b1 ðtÞ; a2 ðtÞ; b2 ðtÞ such that (i) ai ðtÞ 2 C and ai > 0; i ¼ 1; 2; (ii) bi ðtÞ 2 C þ ; i ¼ 1; 2; (iii) the following differential inequality holds
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ cðtÞuðt dðtÞÞ;
ð2:10Þ
(iv) if xðtÞ and uðtÞ are both bounded above, then the following differential inequality holds
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ: Then system (1.4) is permanent. Proof. Firstly, by (2.10), we know that any solution ðxðtÞ; uðtÞÞ of system (1.4) globally exists and satisfies
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ; thus from (i), (ii) and Lemma 2.2, we have
lim sup xðtÞ 6 t!1
a1 b1
expfðja1 j þ a1 Þxg :¼ K 1 :
ð2:11Þ
Notice that the second equation of (1.4) implies that
Z t Z t Z uðtÞ ¼ uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp
Z
0
n
gðsÞ ds aðnÞxðn rðnÞÞ dn ;
ð2:12Þ
0
by (2.11), we have
Z t Z t Z uðtÞ 6 uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp 0
Z 0
n
Z t Z t M a gðsÞ ds þ K 1 1 exp gðsÞ ds ; 6 uð0Þ exp
g
0
gðsÞ ds gðnÞ
aðnÞ K 1 dn gðnÞ
0
which shows that
lim sup uðtÞ 6 K 1
M a
t!1
g
:¼ K 2 :
ð2:13Þ
Note that (2.10), (2.11) and (2.13), we have
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞK 1 cðtÞK 2 ; for simplicity, we set
dðtÞ ¼ a2 ðtÞ b2 ðtÞK 1 cðtÞK 2 : Integrate both sides of (2.14) from n to t ðn 6 tÞ, we have
ln xðtÞ ln xðnÞ P
Z n
t
dðtÞ dt;
ð2:14Þ
908
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
this yields
Z t xðnÞ 6 exp dðsÞ ds xðtÞ; n
Notice that the second equation of (1.4) also implies that for any s P 0,
Z uðtÞ ¼ exp
t
Z
ts
Z
t
gðhÞ dh uðt sÞ þ exp
t
gð#Þ d#
0
exp
Z
ts
n
gðlÞ dl aðnÞxðn rðnÞÞ dn ;
0
thus
Z uðtÞ 6 exp
Z t "Z gðhÞ dh uðt sÞ þ exp gð#Þ d#
t
ts
Z 6 exp
0
t
ts
exp
Z
ts
Z t aðnÞ exp jdðsÞj ds dn M ts (Z tsMr ) Z
gðhÞ dh K 2 þ xðtÞ
Z s ¼ exp gðhÞ dh K 2 þ xðtÞ
t
Z
0
n
( Z gðlÞ dl aðnÞ exp
t
)
#
dðsÞ ds xðtÞ dn
nrðnÞ
0
t
sþr
s
aðnÞ exp
0
jdðsÞj ds dn:
0
From
Z s lim exp gðhÞ dh K 2 ¼ 0;
s!þ1
0
we know that there exists a constant K such that
Z exp 0
K
a
gðhÞ dh K 2 < 2 ; 2c
therefore if we fix such a K, then for t > K,
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ " Z K Z gðhÞ dh K 2 b2 ðtÞ þ P xðtÞ a2 ðtÞ cðtÞ exp 0
K
aðnÞ exp
(Z
)
KþrM
!#
jdðsÞj ds dn
xðtÞ;
0
0
since
1
x
x
Z a2 ðtÞ cðtÞ exp
Z
K
gðhÞ dh K 2 dt >
0
0
a2 ; 2
then by Lemma 2.3, we can obtain
lim inf xðtÞ P k1 ;
ð2:15Þ
t!1
where
a2 nR o expfða2 þ ja2 jÞxg: k1 ¼ RK KþrM 2 b2 þ 0 aðnÞ exp 0 jdðsÞj ds dn Similarly, from (2.12) and (2.15), we have
Z t Z t Z uðtÞ P uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp
Z
0
0
n
gðsÞ ds
aðnÞ k1 dn ; gðnÞ
thus
lim inf uðtÞ P k1 t!1
L a
g
ð2:16Þ
:
From (2.11), (2.13), (2.15) and (2.16), we complete the proof.
h
Remark. The models (1.1)–(1.3) all satisfy the conditions in Theorem 2.1 (in the last section of this paper, we will give a detailed proof). As a direct conclusion of Teng [24], we get
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
909
Corollary 2.3. If the conditions in Theorem 2.1 hold, then system (1.4) has at least one positive x-periodic solution. By similar analysis as above, we can obtain Theorem 2.2. If the functions si ðtÞ ði ¼ 1; 2; . . . ; nÞ; dðtÞ; rðtÞ; cðtÞ; gðtÞ; aðtÞ in (1.4) are all bounded, further suppose that for any positive solution ðxðtÞ; uðtÞÞ of (1.4) , there exist bounded functions a1 ðtÞ; b1 ðtÞ; a2 ðtÞ; b2 ðtÞ such that (i) ai ðtÞ 2 C and aLi > 0; i ¼ 1; 2; (ii) bi ðtÞ 2 C þ ; i ¼ 1; 2; (iii) the following differential inequality holds
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ cðtÞuðt dðtÞÞ; (iv) if xðtÞ and uðtÞ are both bounded above by a positive constant, then the following differential inequality holds
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ: Then system (1.4) is permanent.
3. Applications In this section, we give some applications of Theorem 2.1, Corollary 2.1 and Theorem 2.2, when our main results are used to the models (1.1)–(1.3), some known results are generalized. Theorem 3.1. Let rðtÞ 2 C be a continuous x-periodic function with r > 0; ai ðtÞ; si ðtÞ ði ¼ 1; 2; . . . ; nÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; rðtÞ 2 C x , then system (1.1) is permanent and has at least one positive x-periodic solution. Proof. Notice that any positive solution of (1.1) satisfies
x0 ðtÞ 6 rðtÞxðtÞ; which implies that
Z
t
tsi ðtÞ
x0 ðtÞ dt 6 xðtÞ
Z
t
rðsÞ ds;
tsi ðtÞ
that is
xðt si ðtÞÞ P xðtÞ exp
(Z
t
) rðsÞ ds ;
tsi ðtÞ
then
" (Z ) # n r X dxðtÞ ai ðtÞ exp ðsÞ ds xðtÞ cðtÞuðt dðtÞÞ : 6 xðtÞ rðtÞ dt tsi ðtÞ i¼1 Notice that r > 0, then the above inequality implies that (iii) in Theorem 2.1 holds. If xðtÞ and uðtÞ are both bounded above, then by similar analysis as above, (iv) in Theorem 2.1 also holds. Thus all the conditions in Theorem 2.1 hold. Also in view of Corollary 2.1, we complete the proof. h Remark. If rL < 0, while r > 0, for the permanence of (1.1), we can only use Theorem 2.1, but cannot use Theorem 2.2. Similarly, we can prove the following two conclusions. Theorem 3.2. Let rðtÞ; ai ðtÞ; si ðtÞ; ci ðtÞ ði ¼ 1; 2; . . . ; nÞ; KðtÞ; aðtÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; permanent and has at least one positive x-periodic solution. Theorem 3.3. Let rðtÞ; ai ðtÞ; si ðtÞ; ci ðtÞ ði ¼ 1; 2; . . . ; nÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; has at least one positive x-periodic solution.
rðtÞ 2 C x , then system (1.2) is
rðtÞ 2 C x , then system (1.3) is permanent and
Obviously, the existence results of Theorems 3.1 and 3.3 are the same as that in [5], and the permanence results are new. And the conditions of Theorem 3.2 is the same as that in [6]. So our Theorem 2.1 generalizes the main theorem in [6]. Acknowledgements Supported by NNSF of China (10771032), the Natural Science Foundation of Ludong University (24070301, 24070302, 24200301), Program for Innovative Research Team in Ludong University and China Postdoctoral Science Foundation funded project.
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L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
A.J. Nicholson, The balance of animal population, J. Animal Ecol. 2 (1933) 132–178. E.R. Pianka, Evolutionary Ecology, Harper and Row, New York, 1974. J. La Salle, S. Lefschetz, Stability by Lyapunov’s Direct Method, Academic Press, New York, 1961. S. Lefschetz, Stability of Nonlinear Control Systems, Academic Press, New York, 1965. H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46. F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587. R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977. K. Gopalsamy, P.X. Weng, Global attractivity in a competition system with feedback controls, Comput. Math. Appl. 45 (2003) 665–676. F.Q. Yin, Y.K. Li, Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput. 153 (2004) 475–484. M. Fan, K. Wang, P.J.Y. Wong, R.P. Agarwal, Periodicity and stability in Periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica 19 (2003) 801–822. L.F. Nie, J.G. Peng, Z.D. Teng, Permanence and stability in multi-species non-autonomous Lotka–Volterra competitive systems with delays and feedback controls, Math. Comput. Model. 49 (1–2) (2009) 295–306. H.X. Hu, Z.D. Teng, S.J. Gao, Extinction in nonautonomous Lotka–Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal.: Real World Appl. 10 (4) (2009) 2508–2520. H. X Hu, Z. D Teng, H.J. Jiang, Permanence of the nonautonomous competitive systems with infinite delay and feedback controls, Nonlinear Anal.: Real World Appl. 10 (4) (2009) 2420–2433. W.T. Li, L.L. Wang, Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control, J. Comput. Appl. Math. 180 (2005) 293–309. Q. Liu, R. Xu, Persistence and global stability for a delayed nonautonomous single-species model with dispersal and feedback control, Differ. Eq. Dynam. Syst. 11 (3–4) (2003) 353–367. Y.H. Xia, J.D. Cao, H.Y. Zhang, F.D. Chen, Almost periodic solutions of n-species competitive system with feedback controls, J. Math. Anal. Appl. 294 (2004) 503–522. J.X. Li, A.M. Zhao, J.R. Yan, The permanence and global attractivity of a Kolmogorov system with feedback controls, Nonlinear Anal.: Real World Appl. 10 (1) (2009) 506–518. Y.H. Fan, L.L. Wang, Permanence for a discrete model with feedback control and delay, Discrete Dyn. Nat. Soc. 2008 (2008) 8 pages (Article ID 945109). K. Gopalsamy, Stability and Oscillations in Delay Different Equations of Population Dynamics, Kluwer Academic, Dordrecht/Norwell, MA, 1992. Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993. J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. H.I. Freedman, S. Ruan, Uniform persistence in functional differential equations, J. Differ. Eq. 115 (1995) 173–192. Y.H. Fan, W.T. Li, Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses, Nonlinear Anal.: Real World Appl. 8 (2007) 424–434. Z.D. Teng, L.S. Chen, The positive solutions in periodic Kolmogorov type systems with delay, Acta Math. Appl. Sinica 22 (1999) 446–456 (in Chinese).
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Permanence and existence of periodic solutions for a generalized system with feedback control Lin-Lin Wang a,*, Yong-Hong Fan a,b a b
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, People’s Republic of China Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t Sufficient conditions are obtained for the permanence and the existence of positive periodic solutions of the delay differential system with feedback control
Keywords: Delay differential equations Feedback control Positive periodic solutions Permanence
(
dx dt du dt
¼ xðtÞ½Fðt; xðt s1 ðtÞÞ; . . . ; xðt sn ðtÞÞÞ cðtÞuðt dðtÞÞ; ¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ:
ð0:1Þ
The method involves the application of estimation for uniform upper and lower bounds of solutions. When these results are applied to some special population models with multiple delays, some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in Chen et al. [F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587] and Huo and Li [H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46]. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction For dynamical systems, if the environment is not temporally constant (e.g., seasonal effects of weather, food supplies, mating habits, etc.), then the parameters become time dependent. It has been suggested by Nicholson [1] that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. Pianka [2] discussed the relevance of periodic environment to evolutionary theory. On the other hand, ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. The control variables discussed in most literatures are constants or time dependent (see [3,4]). Recently, logistic model with several delays and feedback control (see [5])
8 n P > < dxðtÞ ¼ xðtÞ rðtÞ ai ðtÞxðt si ðtÞÞ cðtÞuðt dðtÞÞ ; dt
> : duðtÞ dt
i¼1
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
* Corresponding author. E-mail addresses: [email protected] (L.-L. Wang), [email protected] (Y.-H. Fan). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.103
ð1:1Þ
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
903
the mutualism delay model with feedback control (see [6])
8 Pn > < dxðtÞ ¼ rðtÞxðtÞ KðtÞþPn i¼1 ai ðtÞxðtsi ðtÞÞ xðt aðtÞÞ cðtÞuðt dðtÞÞ ; dt
1þ
> : duðtÞ dt
s
c ðtÞxðt i ðtÞÞ i¼1 i
ð1:2Þ
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
the so-called Michaelis–Menton single-species growth model with feedback control (see [5])
8 n P > < dxðtÞ ¼ rðtÞxðtÞ 1 dt
> : duðtÞ dt
i¼1
ai ðtÞxðtsi ðtÞÞ 1þci ðtÞxðtsi ðtÞÞ
cðtÞuðt dðtÞÞ ;
ð1:3Þ
¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
have been studied by several authors. The coincidence degree theory [7] has been used to establish the existence of periodic solutions. For more works related to the study of feedback control systems, we refer the reader to [8–17,19,20] and the references cited therein. Motivated by the above literature, in this paper, we consider the following general nonlinear nonautonomous delay differential system with feedback control
(
dx dt du dt
¼ xðtÞ½Fðt; xðt s1 ðtÞÞ; . . . ; xðt sn ðtÞÞÞ cðtÞuðt dðtÞÞ; ¼ gðtÞuðtÞ þ aðtÞxðt rðtÞÞ;
ð1:4Þ
where Fðt; z1 ; z2 ; . . . ; zn ; znþ1 Þ 2 CðRnþ2 ; ½0; 1ÞÞ; si ðtÞ ði ¼ 1; 2; . . . ; nÞ; dðtÞ; rðtÞ; cðtÞ; gðtÞ; aðtÞ 2 CðR; ð0; 1ÞÞ, we further assume that all of the above functions are x-periodic in t and x > 0 is a constant. By employing the differential inequality developed by Chen et al. [6] and Fan and Wang [18], we obtain sufficient conditions of the permanence and the existence of periodic solutions for system (1.4). Then we apply the obtained criteria to some population models with multiple delays and feedback control, such as the periodic logistic equation with several delays (1.1)–(1.3). Some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in [6,5]. For biological meaning, we only consider (1.4) with the following initial conditions
xðtÞ ¼ uðtÞ P 0;
t 2 ½s; 0; uð0Þ > 0;
uðtÞ ¼ wðtÞ P 0;
t 2 ½s; 0; wð0Þ > 0;
where u and w are continuous on ½s; 0. Here
s ¼ max max si ðtÞ ði ¼ 1; 2; . . . ; nÞ; max dðtÞ; max rðtÞ : t2½0;x
t2½0;x
t2½0;x
By the fundamental theory of functional differential equations [21], it is clear that the solution of (1.4) exists and remains nonnegative on its maximal existence interval. 2. Main results In this section, we establish the permanence and existence of positive periodic solutions for system (1.4) by applying some useful lemmas developed by Chen et al. [6] and Fan and Wang [18]. For basic concepts of permanence for functional differential equations, one can refer to [22]. For the sake of convenience, we introduce some notations and definitions. Denote R and Rþ as the sets of all real numbers and nonnegative real numbers, respectively. Let C denote the set of all bounded continuous function f : R ! R; C þ is the set of all f 2 C such that f > 0, and C x ¼ ff 2 C þ jf ðt þ xÞ ¼ f ðtÞ; t 2 Rg. Define
f M ¼ sup f ðtÞ;
f L ¼ inf f ðtÞ;
t2R
t2R
for any f 2 C. Of cause, if f is an x-periodic function, then
f M ¼ max f ðtÞ; t2½0;x
f L ¼ min ðtÞ: t2½0;x
We also define
f ¼ 1
x
Z
x
f ðtÞ dt; 0
if f is an x-periodic function. Definition 2.1. System (1.4) is said to be permanent if there exist two positive constants k1 ; k2 such that
k1 6 lim inf Ni ðtÞ 6 lim sup Ni ðtÞ 6 k2 ; t!1
t!1
i ¼ 1; 2;
904
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
for any solution ðN 1 ðtÞ; N 2 ðtÞÞ of (1.4). The following lemma will be useful to establish the main result, which can be seen as a corollary of Lemma 2.1 in [23]. Lemma 2.1. The problem
x0 ¼ x½aðtÞ bðtÞx;
ð2:1Þ
> 0, moreover, has only one positive x-periodic solution U if b 2 C þ ; a 2 C; a; b are both continuous x-periodic functions and a the following properties hold: (a) U is constant if a=b is constant, in this case, U ¼ a=b; (b) uðtÞ UðtÞ ! 0 as t ! þ1, for any positive solution uðtÞ of Eq. (2.1); þ jajÞxg 6 U 6 a expfða þ jajÞxg, and if a 2 C x , then (c) ab expfða b
a a expfaxg 6 U 6 b expfaxg: b Proof. Notice that (b) is a direct conclusion of Lemma 2.1 in [23], this implies that the positive x-periodic solution is unique. On the other hand, (a) is a direct conclusion of (c). So we only need to prove (c). According to the positivity of U, let
UðtÞ ¼ expfVðtÞg;
ð2:2Þ
then
V 0 ðtÞ ¼ aðtÞ bðtÞ expfVðtÞg: Notice that VðtÞ is also an x-periodic function, thus
0¼
1
Z
x
x
V 0 ðtÞ dt ¼
0
1
x
Z
x
½aðtÞ bðtÞ expfVðtÞg dt:
0
By the integral mean value theorem, the above equality implies that there exists a n 2 ½0; x such that
a expfVðnÞg ¼ ; b therefore
Z t Z t a VðtÞ ¼ VðnÞ þ V 0 ðsÞ ds ¼ ln þ ½aðsÞ bðsÞ expfVðsÞg ds b n n Z x a 6 ln þ ½jaðsÞj þ bðsÞ expfVðsÞg ds b 0 a þ jajÞx ¼ ln þ ða b
and
VðtÞ ¼ VðnÞ þ
Z
t
n
Z
a V 0 ðsÞ ds ¼ ln þ b
Z
t
½aðsÞ bðsÞ expfVðsÞg ds
n
x
a P ln ½jaðsÞj þ bðsÞ expfVðsÞg ds b 0 a þ jajÞx; ¼ ln ða b via the transformation (2.2), we obtain
a a þ jajÞxg 6 U 6 expfða þ jajÞxg: expfða b b If a 2 C x , then
VðtÞ ¼ VðnÞ þ a 6 ln þ b
Z n t
Z
t
a V 0 ðsÞ ds ¼ ln þ b
Z
t
½aðsÞ bðsÞ expfVðsÞg ds
n
aðsÞ ds for n 6 t 6 x
n
and
a VðtÞ 6 ln b
Z n
t
bðsÞ expfVðsÞg ds for 0 6 t 6 n;
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
905
thus
a VðtÞ 6 ln þ b
Z
x
aðsÞ ds for any t 2 ½0; x;
0
which yields
a xg: U 6 expfa b Similarly, we can obtain
a xg: U P expfa b This complete the proof. h Lemma 2.2. Consider the inequality problem
x0 ðtÞ 6 xðtÞ½aðtÞ bðtÞxðtÞ:
ð2:3Þ
> 0, then every positive solution xðtÞ of (2.3) satisfies If b 2 C x ; a 2 C and a is an x-periodic function with a
lim sup xðtÞ 6 H1 ; t!1
where
a þ jajÞxg: H1 ¼ expfða b Moreover, if a 2 C x , then
lim sup xðtÞ 6 H2 ; t!1
where
a xg: H2 ¼ expfa b Proof. Constructing the following auxiliary equation
z0 ðtÞ ¼ zðtÞ½aðtÞ bðtÞzðtÞ;
ð2:4Þ
by Lemma 2.1, we only need to prove the former inequality, the latter can be proved similarly. Eq. (2.4) has one positive xperiodic solution, denote it as z ðtÞ and
z ðtÞ 6 H1 : Let
xðtÞ ¼ expfu1 ðtÞg;
z ðtÞ ¼ expfu2 ðtÞg;
ð2:5Þ
then
u01 ðtÞ 6 aðtÞ bðtÞ expfu1 ðtÞg and
u02 ðtÞ ¼ aðtÞ bðtÞ expfu2 ðtÞg: Make the transformation uðtÞ ¼ u1 ðtÞ u2 ðtÞ, we can obtain
u0 ðtÞ 6 bðtÞz ðtÞ½expfuðtÞg 1:
ð2:6Þ
Now we divide the proof into two cases according to the oscillating property of uðtÞ. First we assume that uðtÞ does not oscillate about zero, then uðtÞ will be either eventually positive or eventually negative. If the latter holds, i.e., u1 ðtÞ < u2 ðtÞ, we have
xðtÞ < z ðtÞ 6 H1 :
ð2:7Þ 0
Either if the former holds, then by (2.6), we know u ðtÞ < 0, which means that uðtÞ is eventually decreasing, also in terms of its positivity, we know that limt!1 uðtÞ exists. We claim that
lim uðtÞ ¼ 0:
t!1
906
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
If this is not true, then limt!1 uðtÞ ¼ a > 0, thus there exists a sufficiently large T (fix it) such that
uðtÞ P a for t P T: By (2.6),
u0 ðtÞ 6 bðtÞz ðtÞ½expfag 1: Integrate the above inequality from T to þ1, we have
a uðTÞ 6 ½expfag 1
Z
þ1
bðtÞz ðtÞ dt ¼ 1 ðnotice that b; z 2 C x Þ: T
This contradiction shows our claim is true. Which leads to
lim sup xðtÞ 6 H1 :
ð2:8Þ
t!1
Now we assume that uðtÞ oscillates about zero, let t n > 0 such that
uðt n Þ ¼ 0;
uðtÞ P 0 for t 2 ½t2k1 ; t 2k and uðtÞ 6 0 for t 2 ½t2k ; t2kþ1 ;
k ¼ 1; 2; . . . ;
and let nk 2 ½t 2k1 ; t2k ðk ¼ 1; 2; . . .Þ such that
uðnk Þ ¼
max uðtÞ:
t2½t2k1 ;t2k
Notice that uðtÞ 2 C 1 ð0; þ1Þ, then by Fermat Theorem, either u0 ðnk Þ ¼ 0 or nk ¼ t 2k1 or nk ¼ t2k , however, if u0 ðnk Þ ¼ 0, then (2.6) implies
0 ¼ u0 ðnk Þ 6 bðnk Þz ðnk Þ½expfuðnk Þg 1 6 0; which implies that uðnk Þ ¼ 0. That is, in any case, uðnk Þ ¼ 0, this yields
uðtÞ 6 0 for t P t 1 : Therefore,
lim sup xðtÞ 6 H1 :
t!1
By Lemma 2.2, we have Corollary 2.1. Any positive solution of the inequality problem (2.3) satisfies
lim sup xðtÞ 6 t!1
aM b
L
;
where a 2 C; aM > 0 and b 2 C þ . Especially, when aðtÞ a > 0 and bðtÞ b > 0,
a lim sup xðtÞ 6 : b t!1 Remark. The last conclusion of Corollary 2.1 is the same as that in [6]. Similarly, we can get Lemma 2.3. Any positive solution xðtÞ of the inequality problem
x0 ðtÞ P xðtÞ½aðtÞ bðtÞxðtÞ satisfies
lim inf xðtÞ P H3 ; t!1
where
a Þxg; H3 ¼ expfðjaj þ a b > 0. Moreover, if a 2 C x , then if b 2 C x ; a 2 C and a is an x-periodic sequence with a
lim inf xðtÞ P H4 ; t!1
where
ð2:9Þ
907
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
a xg: H4 ¼ expfa b From Lemma 2.3, we can easily obtain Corollary 2.2. Any positive solution xðtÞ of the inequality problem (2.9) satisfies
lim inf xðtÞ P t!1
aL b
M
;
if a 2 C þ and b 2 C þ . Moreover, if aðtÞ a > 0 and bðtÞ b > 0, then
a lim inf xðtÞ P : t!1 b Remark. The last conclusion of Corollary 2.2 is the same as that in [6]. Now we are in the position to give the main result. Theorem 2.1. Suppose that for any positive solution ðxðtÞ; uðtÞÞ of (1.4) , there exist x-periodic functions a1 ðtÞ; b1 ðtÞ; a2 ðtÞ; b2 ðtÞ such that (i) ai ðtÞ 2 C and ai > 0; i ¼ 1; 2; (ii) bi ðtÞ 2 C þ ; i ¼ 1; 2; (iii) the following differential inequality holds
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ cðtÞuðt dðtÞÞ;
ð2:10Þ
(iv) if xðtÞ and uðtÞ are both bounded above, then the following differential inequality holds
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ: Then system (1.4) is permanent. Proof. Firstly, by (2.10), we know that any solution ðxðtÞ; uðtÞÞ of system (1.4) globally exists and satisfies
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ; thus from (i), (ii) and Lemma 2.2, we have
lim sup xðtÞ 6 t!1
a1 b1
expfðja1 j þ a1 Þxg :¼ K 1 :
ð2:11Þ
Notice that the second equation of (1.4) implies that
Z t Z t Z uðtÞ ¼ uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp
Z
0
n
gðsÞ ds aðnÞxðn rðnÞÞ dn ;
ð2:12Þ
0
by (2.11), we have
Z t Z t Z uðtÞ 6 uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp 0
Z 0
n
Z t Z t M a gðsÞ ds þ K 1 1 exp gðsÞ ds ; 6 uð0Þ exp
g
0
gðsÞ ds gðnÞ
aðnÞ K 1 dn gðnÞ
0
which shows that
lim sup uðtÞ 6 K 1
M a
t!1
g
:¼ K 2 :
ð2:13Þ
Note that (2.10), (2.11) and (2.13), we have
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞK 1 cðtÞK 2 ; for simplicity, we set
dðtÞ ¼ a2 ðtÞ b2 ðtÞK 1 cðtÞK 2 : Integrate both sides of (2.14) from n to t ðn 6 tÞ, we have
ln xðtÞ ln xðnÞ P
Z n
t
dðtÞ dt;
ð2:14Þ
908
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
this yields
Z t xðnÞ 6 exp dðsÞ ds xðtÞ; n
Notice that the second equation of (1.4) also implies that for any s P 0,
Z uðtÞ ¼ exp
t
Z
ts
Z
t
gðhÞ dh uðt sÞ þ exp
t
gð#Þ d#
0
exp
Z
ts
n
gðlÞ dl aðnÞxðn rðnÞÞ dn ;
0
thus
Z uðtÞ 6 exp
Z t "Z gðhÞ dh uðt sÞ þ exp gð#Þ d#
t
ts
Z 6 exp
0
t
ts
exp
Z
ts
Z t aðnÞ exp jdðsÞj ds dn M ts (Z tsMr ) Z
gðhÞ dh K 2 þ xðtÞ
Z s ¼ exp gðhÞ dh K 2 þ xðtÞ
t
Z
0
n
( Z gðlÞ dl aðnÞ exp
t
)
#
dðsÞ ds xðtÞ dn
nrðnÞ
0
t
sþr
s
aðnÞ exp
0
jdðsÞj ds dn:
0
From
Z s lim exp gðhÞ dh K 2 ¼ 0;
s!þ1
0
we know that there exists a constant K such that
Z exp 0
K
a
gðhÞ dh K 2 < 2 ; 2c
therefore if we fix such a K, then for t > K,
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ " Z K Z gðhÞ dh K 2 b2 ðtÞ þ P xðtÞ a2 ðtÞ cðtÞ exp 0
K
aðnÞ exp
(Z
)
KþrM
!#
jdðsÞj ds dn
xðtÞ;
0
0
since
1
x
x
Z a2 ðtÞ cðtÞ exp
Z
K
gðhÞ dh K 2 dt >
0
0
a2 ; 2
then by Lemma 2.3, we can obtain
lim inf xðtÞ P k1 ;
ð2:15Þ
t!1
where
a2 nR o expfða2 þ ja2 jÞxg: k1 ¼ RK KþrM 2 b2 þ 0 aðnÞ exp 0 jdðsÞj ds dn Similarly, from (2.12) and (2.15), we have
Z t Z t Z uðtÞ P uð0Þ exp gðsÞ ds þ exp gðsÞ ds 0
0
t
exp
Z
0
0
n
gðsÞ ds
aðnÞ k1 dn ; gðnÞ
thus
lim inf uðtÞ P k1 t!1
L a
g
ð2:16Þ
:
From (2.11), (2.13), (2.15) and (2.16), we complete the proof.
h
Remark. The models (1.1)–(1.3) all satisfy the conditions in Theorem 2.1 (in the last section of this paper, we will give a detailed proof). As a direct conclusion of Teng [24], we get
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
909
Corollary 2.3. If the conditions in Theorem 2.1 hold, then system (1.4) has at least one positive x-periodic solution. By similar analysis as above, we can obtain Theorem 2.2. If the functions si ðtÞ ði ¼ 1; 2; . . . ; nÞ; dðtÞ; rðtÞ; cðtÞ; gðtÞ; aðtÞ in (1.4) are all bounded, further suppose that for any positive solution ðxðtÞ; uðtÞÞ of (1.4) , there exist bounded functions a1 ðtÞ; b1 ðtÞ; a2 ðtÞ; b2 ðtÞ such that (i) ai ðtÞ 2 C and aLi > 0; i ¼ 1; 2; (ii) bi ðtÞ 2 C þ ; i ¼ 1; 2; (iii) the following differential inequality holds
x0 ðtÞ 6 xðtÞ½a1 ðtÞ b1 ðtÞxðtÞ cðtÞuðt dðtÞÞ; (iv) if xðtÞ and uðtÞ are both bounded above by a positive constant, then the following differential inequality holds
x0 ðtÞ P xðtÞ½a2 ðtÞ b2 ðtÞxðtÞ cðtÞuðt dðtÞÞ: Then system (1.4) is permanent.
3. Applications In this section, we give some applications of Theorem 2.1, Corollary 2.1 and Theorem 2.2, when our main results are used to the models (1.1)–(1.3), some known results are generalized. Theorem 3.1. Let rðtÞ 2 C be a continuous x-periodic function with r > 0; ai ðtÞ; si ðtÞ ði ¼ 1; 2; . . . ; nÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; rðtÞ 2 C x , then system (1.1) is permanent and has at least one positive x-periodic solution. Proof. Notice that any positive solution of (1.1) satisfies
x0 ðtÞ 6 rðtÞxðtÞ; which implies that
Z
t
tsi ðtÞ
x0 ðtÞ dt 6 xðtÞ
Z
t
rðsÞ ds;
tsi ðtÞ
that is
xðt si ðtÞÞ P xðtÞ exp
(Z
t
) rðsÞ ds ;
tsi ðtÞ
then
" (Z ) # n r X dxðtÞ ai ðtÞ exp ðsÞ ds xðtÞ cðtÞuðt dðtÞÞ : 6 xðtÞ rðtÞ dt tsi ðtÞ i¼1 Notice that r > 0, then the above inequality implies that (iii) in Theorem 2.1 holds. If xðtÞ and uðtÞ are both bounded above, then by similar analysis as above, (iv) in Theorem 2.1 also holds. Thus all the conditions in Theorem 2.1 hold. Also in view of Corollary 2.1, we complete the proof. h Remark. If rL < 0, while r > 0, for the permanence of (1.1), we can only use Theorem 2.1, but cannot use Theorem 2.2. Similarly, we can prove the following two conclusions. Theorem 3.2. Let rðtÞ; ai ðtÞ; si ðtÞ; ci ðtÞ ði ¼ 1; 2; . . . ; nÞ; KðtÞ; aðtÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; permanent and has at least one positive x-periodic solution. Theorem 3.3. Let rðtÞ; ai ðtÞ; si ðtÞ; ci ðtÞ ði ¼ 1; 2; . . . ; nÞ; cðtÞ; dðtÞ; gðtÞ; aðtÞ; has at least one positive x-periodic solution.
rðtÞ 2 C x , then system (1.2) is
rðtÞ 2 C x , then system (1.3) is permanent and
Obviously, the existence results of Theorems 3.1 and 3.3 are the same as that in [5], and the permanence results are new. And the conditions of Theorem 3.2 is the same as that in [6]. So our Theorem 2.1 generalizes the main theorem in [6]. Acknowledgements Supported by NNSF of China (10771032), the Natural Science Foundation of Ludong University (24070301, 24070302, 24200301), Program for Innovative Research Team in Ludong University and China Postdoctoral Science Foundation funded project.
910
L.-L. Wang, Y.-H. Fan / Applied Mathematics and Computation 216 (2010) 902–910
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
A.J. Nicholson, The balance of animal population, J. Animal Ecol. 2 (1933) 132–178. E.R. Pianka, Evolutionary Ecology, Harper and Row, New York, 1974. J. La Salle, S. Lefschetz, Stability by Lyapunov’s Direct Method, Academic Press, New York, 1961. S. Lefschetz, Stability of Nonlinear Control Systems, Academic Press, New York, 1965. H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46. F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587. R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977. K. Gopalsamy, P.X. Weng, Global attractivity in a competition system with feedback controls, Comput. Math. Appl. 45 (2003) 665–676. F.Q. Yin, Y.K. Li, Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput. 153 (2004) 475–484. M. Fan, K. Wang, P.J.Y. Wong, R.P. Agarwal, Periodicity and stability in Periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica 19 (2003) 801–822. L.F. Nie, J.G. Peng, Z.D. Teng, Permanence and stability in multi-species non-autonomous Lotka–Volterra competitive systems with delays and feedback controls, Math. Comput. Model. 49 (1–2) (2009) 295–306. H.X. Hu, Z.D. Teng, S.J. Gao, Extinction in nonautonomous Lotka–Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal.: Real World Appl. 10 (4) (2009) 2508–2520. H. X Hu, Z. D Teng, H.J. Jiang, Permanence of the nonautonomous competitive systems with infinite delay and feedback controls, Nonlinear Anal.: Real World Appl. 10 (4) (2009) 2420–2433. W.T. Li, L.L. Wang, Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control, J. Comput. Appl. Math. 180 (2005) 293–309. Q. Liu, R. Xu, Persistence and global stability for a delayed nonautonomous single-species model with dispersal and feedback control, Differ. Eq. Dynam. Syst. 11 (3–4) (2003) 353–367. Y.H. Xia, J.D. Cao, H.Y. Zhang, F.D. Chen, Almost periodic solutions of n-species competitive system with feedback controls, J. Math. Anal. Appl. 294 (2004) 503–522. J.X. Li, A.M. Zhao, J.R. Yan, The permanence and global attractivity of a Kolmogorov system with feedback controls, Nonlinear Anal.: Real World Appl. 10 (1) (2009) 506–518. Y.H. Fan, L.L. Wang, Permanence for a discrete model with feedback control and delay, Discrete Dyn. Nat. Soc. 2008 (2008) 8 pages (Article ID 945109). K. Gopalsamy, Stability and Oscillations in Delay Different Equations of Population Dynamics, Kluwer Academic, Dordrecht/Norwell, MA, 1992. Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993. J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. H.I. Freedman, S. Ruan, Uniform persistence in functional differential equations, J. Differ. Eq. 115 (1995) 173–192. Y.H. Fan, W.T. Li, Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses, Nonlinear Anal.: Real World Appl. 8 (2007) 424–434. Z.D. Teng, L.S. Chen, The positive solutions in periodic Kolmogorov type systems with delay, Acta Math. Appl. Sinica 22 (1999) 446–456 (in Chinese).