global positive periodic solutions for periodic two species competitive systems with multiple delays and impulses

Research ArticleGlobal Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses Zhenguo Luo,1,2Liping Luo,1Liu Yang,1and Yunhui Zeng1 1

Research Article

Global Positive Periodic Solutions for Periodic Two-Species

Competitive Systems with Multiple Delays and Impulses

Zhenguo Luo,1,2Liping Luo,1Liu Yang,1and Yunhui Zeng1

1 Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

2 Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Correspondence should be addressed to Zhenguo Luo; robert186@163.com

Received 19 November 2013; Accepted 25 February 2014; Published 3 April 2014

Academic Editor: Francisco J S Lozano

Copyright © 2014 Zhenguo Luo et al This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodicsolutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques Thisimproves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously

being defined by|𝑥|0= max𝑡∈[0,𝜔]|𝑥(𝑡)|;

𝑃𝐶1𝜔= {𝑥 | 𝑥 ∈ 𝑃𝐶1, 𝑥(𝑡 + 𝜔) = 𝑥(𝑡)}, with the norm

being defined by‖𝑥‖ = max𝑡∈[0,𝜔]{|𝑥|0, |𝑥󸀠|0};

then those spaces are all Banach spaces We also denote that

Abstract and Applied Analysis

Volume 2014, Article ID 785653, 23 pages

http://dx.doi.org/10.1155/2014/785653

where 𝑎1(𝑡), 𝑎2(𝑡), 𝑏1𝑖(𝑡), 𝑏2𝑗(𝑡), 𝑐1𝑗(𝑡), and 𝑐2𝑖(𝑡) are all in

𝑃𝐶𝜔 Also 𝜏𝑖(𝑡), 𝛿𝑗(𝑡), 𝜂𝑗(𝑡), and 𝜎𝑖(𝑡) are all in 𝑃𝐶1

intrinsic growth rates𝑟1(𝑡), 𝑟2(𝑡) ∈ 𝑃𝐶𝜔are with∫0𝜔𝑟𝑙(𝑡)𝑑𝑡 >

0, (𝑙 = 1, 2) For the ecological justification of (2) and (3) and

similar types refer to [1–10]

single-species population growth models with periodic delay:

𝑦󸀠(𝑡) = 𝑦 (𝑡) [𝑟 (𝑡) − 𝑎 (𝑡) 𝑦 (𝑡) + 𝑏 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))] (5)

differentiable𝜔-periodic functions, and 𝑟(𝑡) > 0, 𝑎(𝑡) > 0,𝑏(𝑡) ≥ 0, and 𝜏(𝑡) ≥ 0 for 𝑡 ∈ 𝑅 The positive feedback

has a positive time delay (the sign of the time delay term ispositive), which is a delay due to gestation (see [1,2]) Theyhad established sufficient conditions which guarantee that

asymptotically stable

single-species population growth models with periodic delay:

𝑦󸀠(𝑡) = 𝑦 (𝑡) [𝑟 (𝑡) − 𝑎 (𝑡) 𝑦 (𝑡) − 𝑏 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))] (6)

differentiable𝜔-periodic functions, and 𝑟(𝑡) > 0, 𝑎(𝑡) > 0,𝑏(𝑡) ≥ 0, and 𝜏(𝑡) ≥ 0 for 𝑡 ∈ 𝑅 The negative feedback

has a negative time delay (the sign of the time delay term

is negative), which can be regarded as the deleterious effect

had derived sufficient conditions for the existence and globalattractivity of positive periodic solutions of system (6) Butthe discussion of global attractivity is only confined to thespecial case when the periodic delay is constant

following two-species competitive system without delay:

𝑦1󸀠(𝑡) = 𝑦1(𝑡) [𝑟1(𝑡) − 𝑎1(𝑡) 𝑦1(𝑡) − 𝑐1(𝑡) 𝑦2(𝑡)] ,

𝑦2󸀠(𝑡) = 𝑦2(𝑡) [𝑟2(𝑡) − 𝑎2(𝑡) 𝑦2(𝑡) − 𝑐2(𝑡) 𝑦1(𝑡)] (7)They had derived sufficient conditions for the existenceand global attractivity of positive periodic solutions of system(7) by using differential inequalities and topological degree,respectively In fact, in many practical situations the timedelay occurs so often A more realistic model should includesome of the past states of the system Therefore, in [10], Liu

et al considered two corresponding periodic Lotka-Volterracompetitive systems involving multiple delays:

𝑐2𝑖(𝑡) ∈ 𝐶(𝑅, [0, +∞)), 𝜏𝑖(𝑡), 𝜌𝑗(𝑡), 𝜂𝑗(𝑡), and 𝜎𝑖(𝑡) ∈

𝐶1(𝑅, [0, +∞)) (𝑖 = 1, 2, , 𝑛; 𝑗 = 1, 2, , 𝑚) are

𝐶(𝑅, 𝑅) are 𝜔-periodic functions with ∫0𝜔𝑟𝑘(𝑡)𝑑𝑡 > 0 (𝑘 =

1, 2) They had derived the same criteria for the existence and

globally asymptotic stability of positive periodic solutions

of the above two competitive systems by using Gaines and

Mawhin’s continuation theorem of coincidence degree theory

and by means of a suitable Lyapunov functional

However, the ecological system is often deeply perturbed

by human exploitation activities such as planting, harvesting,

and so on, which makes it unsuitable to be considered

continually For having a more accurate description of such

a system, we need to consider the impulsive differential

equations The theory of impulsive differential equations not

only is richer than the corresponding theory of differential

equations without impulses, but also represents a more

natu-ral framework for mathematical modeling of many real world

been recently introduced in population dynamics in relation

treat-ment [27,28] However, to the best of the authors’ knowledge,

to this day, few scholars have done works on the existence,

uniqueness, and global stability of positive periodic solution

of (2) and (4) One could easily see that systems (5)–(9) are

all special cases of systems (2) and (3) Therefore, we propose

and study the systems (2) and (3) in this paper

For the sake of generality and convenience, we always

make the following fundamental assumptions

lim𝑘 → ∞𝑡𝑘 = +∞, 𝜃𝑙𝑘 (𝑖 = 1, 2) are constants, and

there exists a positive integer𝑞 > 0 such that 𝑡𝑘+𝑞 =

𝑡𝑘+ 𝜔, 𝜃𝑙(𝑘+𝑞)= 𝜃𝑙𝑘 Without loss of generality, we can

assume that𝑡𝑘 ̸= 0 and [0, 𝜔]∩{𝑡𝑘} = 𝑡1, 𝑡2, , 𝑡𝑚, and

(𝐻3) {𝜃𝑙𝑘} is a real sequence such that 𝜃𝑙𝑘 + 1 > 0,

∏0<𝑡𝑘<𝑡(1 + 𝜃𝑙𝑘), 𝑙 = 1, 2 is an 𝜔-periodic function

Definition 1 A function𝑥𝑙 : 𝑅 → (0, +∞), 𝑙 = 1, 2 issaid to be a positive solution of (2) and (3), if the followingconditions are satisfied:

(a)𝑥𝑙(𝑡) is absolutely continuous on each (𝑡𝑘, 𝑡𝑘+1);(b) for each𝑘 ∈ 𝑍+,𝑥𝑙(𝑡+𝑘) and 𝑥𝑙(𝑡−𝑘) exist, and 𝑥𝑙(𝑡−𝑘) =

𝑥𝑙(𝑡𝑘);

(c)𝑥𝑙(𝑡) satisfies the first equation of (2) and (3) foralmost everywhere (for short a.e.) in[0, ∞] \ {𝑡𝑘} andsatisfies𝑥𝑙(𝑡+

𝑘) = (1 + 𝜃𝑙𝑘)𝑥𝑙(𝑡𝑘) for 𝑡 = 𝑡𝑘,𝑘 ∈ 𝑍+ ={1, 2, }

following nonimpulsive delay differential equation:

with the initial conditions

The following lemmas will be used in the proofs of our

Proof. (1) It is easy to see that 𝑥𝑙(𝑡) = ∏0<𝑡𝑘<𝑡(1+𝜃𝑙𝑘)𝑦𝑙(𝑡) (𝑙 =

1, 2) is absolutely continuous on every interval (𝑡𝑘, 𝑡𝑘+1],

1, 2, , and in view of (15), it follows that for any𝑘 = 1, 2, ,

(1 + 𝜃1𝑘)−1𝑥1(𝑡−𝑘) = 𝑦1(𝑡𝑘) ,

(17)which implies that𝑦1(𝑡) is continuous on [−𝜏, +∞) It is easy

𝑦𝑙(𝑡) = ∏0<𝑡𝑘<𝑡(1 + 𝜃𝑙𝑘)−1𝑥𝑙(𝑡) (𝑙 = 1, 2) are solutions of (10)–

(12) on [−𝜏, +∞) If 𝑥(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡))𝑇 is a solution of

𝑦𝑙(𝑡) = ∏0<𝑡𝑘<𝑡(1 + 𝜃𝑙𝑘)−1𝑥𝑙(𝑡) (𝑙 = 1, 2) are solutions of (10)–

FromLemma 2, if we want to discuss the existence and

global asymptotic stability of positive periodic solutions of

systems (2)–(4), we only discuss the existence of the existence

and global asymptotic stability of positive periodic solutions

of systems (10)–(12)

The organization of this paper is as follows InSection 2,

we introduce several useful definitions and lemmas In

Section 3, first, we study the existence of at least one periodic

investigate the global asymptotic stability of positive

peri-odic solutions of the above systems by using the method

been investigated extensively in the references mentioned

previously

2 Preliminaries

In this section, we will introduce some concepts and some

important lemmas which are useful for the next section

𝑍 be a linear mapping, and let 𝑁 : 𝑋 → 𝑍 be a continuous

of index zero if dimKer𝐿 = condimIm𝐿 < +∞ and Im 𝐿 is

𝑍 such that Im 𝑃 = Ker 𝐿, Ker 𝑄 = Im 𝐿 = Im(𝐼 − 𝑄), it

of the first kind at point𝑡 = 𝑡𝑘 We also denote that𝑃𝐶1𝜔 =

{Ψ ∈ 𝑃𝐶𝜔: Ψ󸀠∈ 𝑃𝐶𝜔}

Definition 3 (see [11]) The set 𝐹 ∈ 𝑃𝐶𝜔 is said to be

𝛿 > 0 such that if 𝑥 ∈ 𝐹, 𝑘 ∈ 𝑁+,𝑡1,𝑡2 ∈ (𝑡𝑘−1, 𝑡𝑘) ∩ [0, 𝜔],and|𝑡1− 𝑡2| < 𝛿, then |𝑥(𝑡1) − 𝑥(𝑡2)| < 𝜖

Definition 4 Let𝑥∗(𝑡) = (𝑥∗

1(𝑡), 𝑥∗

2(𝑡))𝑇be a strictly positiveperiodic solution of (2)–(4) One says that𝑥∗(𝑡) is globallyattractive if any other solution𝑥(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡))𝑇of (2)–(4) has the property lim𝑡 → +∞|𝑥∗𝑖(𝑡) − 𝑥𝑖(𝑡)| = 0, 𝑖 = 1, 2

Lemma 5 The region 𝑅2+ = {(𝑥1, 𝑥2) : 𝑥1(0) > 0, 𝑥2(0) > 0}

is the positive invariable region of the systems (2 )–(4).

Proof By the definition of𝑥𝑙(𝑡) (𝑙 = 1, 2) we have 𝑥𝑙(0) > 0

In view of having

𝑥1(𝑡) = 𝑥1(0) exp {∫𝑡

0[𝑟1(𝜉) − 𝑎1(𝜉) 𝑥1(𝜉)+∑𝑛

𝑡 ∈ [0, 𝑡1] ,

𝑥1(𝑡) = 𝑥1(𝑡𝑘) exp {∫𝑡

0[𝑟1(𝜉) − 𝑎1(𝜉) 𝑥1(𝜉)+∑𝑛

𝑡 ∈ [0, 𝜔] Then the function 𝑡 − 𝜎(𝑡) has a unique inverse 𝜇(𝑡)

satisfying 𝜇 ∈ 𝐶(𝑅, 𝑅) with 𝜇(𝑎 + 𝜔) = 𝜇(𝑎) + 𝜔 ∀𝑎 ∈ 𝑅, and

if𝑔 ∈ 𝑃𝐶𝜔, 𝜏󸀠(𝑡) < 1, 𝑡 ∈ [0, 𝜔], then 𝑔(𝜇(𝑡)) ∈ 𝑃𝐶𝜔 Proof Since𝜎󸀠(𝑡) < 1, 𝑡 ∈ [0, 𝜔], and 𝑡 − 𝜎(𝑡) is continuous

𝜇(𝑡) ∈ 𝐶(𝑅, 𝑅) on 𝑅 Hence, it suffices to show that 𝜇(𝑎 +𝜔) = 𝜇(𝑎) + 𝜔, ∀𝑎 ∈ 𝑅 For any 𝑎 ∈ 𝑅, by the condition

𝜎󸀠(𝑡) < 1, one can find that 𝑡 − 𝜎(𝑡) = 𝑎 exists as a unique

𝑡1; that is,𝑡0 − 𝜎(𝑡0) = 𝑎 and 𝑡1− 𝜎(𝑡1) = 𝑎 + 𝜔; that is,𝜇(𝑎) = 𝑡0= 𝜎(𝑡0) + 𝑎 and 𝜇(𝑎 + 𝜔) = 𝑡1

completed

Lemma 7 (see [9]) Let 𝑋 and 𝑍 be two Banach spaces, and

zero Ω ⊂ 𝑋 is an open bounded set, and let 𝑁 : Ω → 𝑍 be

L-compact on Ω Suppose that

Lemma 9 (see [30]) Assume that 𝑓(𝑡), 𝑔(𝑡) are continuous

nonnegative functions defined on the interval [𝛼, 𝛽]; then there

exists 𝜉 ∈ [𝛼, 𝛽] such that ∫𝛼𝛽𝑓(𝑡)𝑔(𝑡)𝑑𝑡 = 𝑓(𝜉) ∫𝛼𝛽𝑔(𝑡)𝑑𝑡.

Lemma 10 (see [20, 31]) Suppose that 𝜙(𝑡) is a differently

continuous 𝜔-periodic function on 𝑅 with (𝜔 > 0); then, for

any𝑡∗∈ 𝑅, the following inequality holds:

max

𝑡∈[𝑡 ∗ ,𝑡 ∗ +𝜔]Φ (𝑡) ≤ 󵄨󵄨󵄨󵄨Φ (𝑡∗)󵄨󵄨󵄨󵄨 +12[∫𝜔

0 󵄨󵄨󵄨󵄨󵄨Φ󸀠(𝑡)󵄨󵄨󵄨󵄨󵄨 𝑑𝑡] (21)

Lemma 11 (see Barbalat’s Lemma [32]) Let 𝑓(𝑡) be a

nonneg-ative function defined on [0, +∞) such that 𝑓(𝑡) is integrable

and uniformly continuous on [0, +∞); then lim𝑡 → +∞𝑓(𝑡) = 0.

In the following section, we only discuss the existence and

global asymptotic stability of positive periodic solutions of

Lemma 6, we see that all𝑡 − 𝜏𝑖(𝑡) have their inverse functions

Throughout the following part, we set𝛼𝑖(𝑡), 𝛽𝑖(𝑡), 𝜇𝑗(𝑡), and

]𝑗(𝑡) to represent the inverse function of 𝑡−𝜏𝑖(𝑡), 𝑡−𝜎𝑖(𝑡), 𝑡−

𝛿𝑗(𝑡), and 𝑡−𝜂𝑗(𝑡), respectively Obviously, 𝛼𝑖(𝑡), 𝛽𝑖(𝑡), 𝜇𝑗(𝑡),

Theorem 12 In addition to (𝐻1)–(𝐻3), assume that one of the

following conditions hold:

periodic solution, where 𝐹∗

2 (𝑡))𝑇 is a positive 𝜔-periodic solution of system (2)

𝑋 = 𝑍 = {𝑢 (𝑡) = (𝑢1(𝑡) , 𝑢2(𝑡))𝑇 | 𝑢𝑖(𝑡)

∈ 𝐶 (𝑅, 𝑅2) : 𝑢𝑖(𝑡 + 𝜔) = 𝑢𝑖(𝑡) , 𝑙 = 1, 2} (24)and define

𝑁𝑢 =

[[[[

and dimKer𝐿 = 2 = codimIm𝐿 So, Im 𝐿 is closed in 𝑍, and 𝐿

is a Fredholm mapping of index zero It is trivial to show that

𝑃, 𝑄 are continuous projectors such that

1

𝜔 0

[[

1

𝜔 0

[[

0𝑓2(𝜉) 𝑑𝜉

reach the position to search for an appropriate open bounded

Since𝑢(𝑡) = (𝑢1(𝑡), 𝑢2(𝑡))𝑇is a𝜔-periodic function, we need

only to prove the result in the interval[0, 𝜔] Integrating (33)

over the interval[0, 𝜔] leads to the following:

∫𝜔

0

[[

𝑖(𝑡) < 1, we can let 𝑠 = 𝑡 − 𝜏𝑖(𝑡), that is, 𝑡 = 𝛼𝑖(𝑠) (𝑖 =

According to Lemma 7, we know that ((𝐵1𝑖(𝛼𝑖(𝑠)))/(1 −

2(𝑠), 𝐺1(𝑠), and 𝐺2(𝑠) are defined by (22) On

the other hand, byLemma 7, we can see that𝛼𝑖(𝜔) = 𝛼𝑖(0)+𝜔,

(46)

that there are𝜆1,𝜆2, 𝜌1, and 𝜌2∈ [0, 𝜔] such that

lnΓ1≤ 𝑢1(𝜉1) , 𝑢1(𝜁1) ≤ ln 𝑟2

𝐺𝐿 2

.(52)

From the first equation of (32), we get

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

∫𝜔

0 󵄨󵄨󵄨󵄨󵄨𝑢󸀠

1(𝑡)󵄨󵄨󵄨󵄨󵄨 𝑑𝑡 ≤ 𝑅1𝜔 + 𝐹1∗𝑀𝑟2𝜔

𝐺𝐿 2

+ 𝐺𝑀1 𝑟1𝜔

𝐺𝐿 1

+ 𝐺𝑀2 𝑟2𝜔

𝐺𝐿 2

where𝑅2 = (1/𝜔) ∫0𝜔|𝑟2(𝑡)|𝑑𝑡, 𝐹∗

2(𝑠), 𝐺2(𝑠) are defined by(22) From (52), (54), and (55) andLemma 10, it follows that

𝑡∈[0,𝜔]󵄨󵄨󵄨󵄨𝑢2(𝑡)󵄨󵄨󵄨󵄨 ≤ 𝑅2 (58)Clearly,Γ𝑙, Δ𝑙,𝑅𝑙 (𝑙 = 1, 2) are independent of 𝜆, respectively

which deduces that

Brouwer degree, a straightforward calculation shows that

= sign{{{

̸= 0.(65)

∏0<𝑡𝑘<𝑡(1 + 𝜃1𝑘)𝑦1∗(𝑡), 𝑥2∗(𝑡) = ∏0<𝑡𝑘<𝑡(1 + 𝜃2𝑘)𝑦2∗(𝑡), then(𝑥∗1(𝑡), 𝑥∗2(𝑡))𝑇 has at least one positive𝜔-periodic solution

of systems (3) and (4) If(𝐻5) holds, similarly, we can prove

that systems (2) and (4) have at least one positive𝜔-periodic

We now proceed to the discussion on the uniqueness

Theorem 12 It is immediate that if𝑥∗(𝑡) is globally

asymp-totically stable, then𝑥∗(𝑡) is unique in fact

Theorem 13 In addition to (𝐻1)–(𝐻3), assume further that

Then systems (3 ) and (4) have a unique positive

𝜔-periodic solution 𝑥∗(𝑡) = (𝑥∗1(𝑡), 𝑥∗2(𝑡))𝑇 which is globally

(𝑦1(𝑡), 𝑦2(𝑡))𝑇be any positive solution of system (11) with the

initial conditions (12) It follows fromTheorem 12that there

exist positive constants𝑇, 𝑟𝑙,𝑅𝑙, such that, for all𝑡 ≥ 𝑇,

1 𝐹∗𝐿

2 >

𝐺𝑀1 𝐺𝑀2 , and then there exist constants𝛼1> 0, 𝛼2> 0; we can

Calculating the upper right derivative of𝑉2(𝑡) along solutions

(74)

where

𝑉 (0) = 𝛼1󵄨󵄨󵄨󵄨ln𝑦1(0) − ln 𝑦1∗(0)󵄨󵄨󵄨󵄨 + 𝛼2󵄨󵄨󵄨󵄨ln𝑦2(0) − ln 𝑦∗2(0)󵄨󵄨󵄨󵄨+∑𝑛

which implies that

On the other hand, we know that

From (66) and (79), it follows that𝑦𝑙(𝑡) (𝑙 = 1, 2) is bounded

for 𝑡 ≥ 0 Hence, 𝑦1(𝑡) − 𝑦1∗(𝑡), 𝑦2(𝑡) − 𝑦2∗(𝑡), and their

derivatives remain bounded on[0, +∞) So |𝑦1(𝑡) − 𝑦1∗(𝑡)|,

Theorem 14 In addition to (𝐻1)–(𝐻3), assume that one of the

following conditions holds:

periodic solution, where 𝐹1(𝑡), 𝐹2(𝑡), 𝐺1(𝑡), and 𝐺2(𝑡) are

𝑁𝑢 =

[[[[

= [𝑔1(𝑡)

and dimKer𝐿 = 2 = codimIm𝐿 So, Im 𝐿 is closed in 𝑍,

1

𝜔

0[[

1

𝜔 0

[[

0𝑔2(𝜉) 𝑑𝜉)

(89)

reach the position to search for an appropriate open bounded

Since𝑢(𝑡) = (𝑢1(𝑡), 𝑢2(𝑡))𝑇is a𝜔-periodic function, we need

only to prove the result in the interval[0, 𝜔] Integrating (90)

over the interval[0, 𝜔] leads to the following:

∫𝜔

0

[[

∫𝜔

0

[[

Noting that𝑢(𝑡) = (𝑢1(𝑡), 𝑢2(𝑡)) ∈ 𝑋, then there exists 𝜁𝑙,𝜉𝑙∈

[0, 𝜔] (𝑙 = 1, 2) such that

𝑢𝑙(𝜁𝑙) = inf

𝑡∈[0,𝜔]𝑢𝑙(𝑡) , 𝑢𝑙(𝜉𝑙) = sup

𝑡∈[0,𝜔]𝑢𝑙(𝑡) , 𝑙 = 1, 2 (93)Since𝜏𝑖󸀠(𝑡) < 1, we can let 𝑠 = 𝑡 − 𝜏𝑖(𝑡), that is, 𝑡 = 𝛼𝑖(𝑠),

,

∫𝜔

0 𝑒𝑢 1 (𝑠)𝑑𝑠 ≤ 𝑟2𝜔

𝐺𝐿 2

(103)

that there is𝜆1, 𝜆2,𝜌1, and𝜌2∈ [0, 𝜔] such that

lnΓ3≤ 𝑢1(𝜉1) , 𝑢1(𝜁1) ≤ ln 𝑟2

𝐺𝐿 2

.(109)

From the first equation of (90), we get

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

0 󵄨󵄨󵄨󵄨𝑟1(𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡+ ∫𝜔

∫𝜔

0 󵄨󵄨󵄨󵄨󵄨𝑢󸀠

1(𝑡)󵄨󵄨󵄨󵄨󵄨 𝑑𝑡 ≤ 𝑅1𝜔 + 𝐹1∗𝑀𝑟2𝜔

𝐺𝐿 2

+ 𝐺𝑀1 𝑟1𝜔

𝐺𝐿 1

+ 𝐺𝑀2 𝑟2𝜔

𝐺𝐿 2

where𝑅2 = (1/𝜔) ∫0𝜔|𝑟2(𝑡)|𝑑𝑡, 𝐹∗

2(𝑠), 𝐺2(𝑠) are defined by(22) From (109), (111), and (112) andLemma 10, it follows that

which deduces that

Brouwer degree, a straightforward calculation shows that

= sign{{{

̸= 0.(123)

1(𝑡), 𝑦∗

1(𝑡), 𝑥∗

of systems (2) and (4) If(𝐻8) holds, similarly we can prove

that systems (2) and (4) have at least one positive𝜔-periodic

We now proceed to the discussion on the uniqueness

Theorem 14 It is immediate that if𝑥∗(𝑡) is globally

asymp-totically stable, then𝑥∗(𝑡) is unique in fact

Theorem 15 In addition to (𝐻1)–(𝐻3), assume further that

periodic solution 𝑥∗(𝑡) = (𝑥∗1(𝑡), 𝑥∗2(𝑡))𝑇 which is globally

and (12), and let 𝑦𝑙(𝑡) = (𝑦1(𝑡), 𝑦2(𝑡))𝑇 be any positive

solution of system (10) with the initial conditions (12) It

𝑇, 𝑟𝑙, 𝑅𝑙, such that for all𝑡 ≥ 𝑇

2 ; then there exist constants𝛼3 > 0, 𝛼4 > 0; we can

×{{{

Calculating the upper right derivative of𝑉2(𝑡) along solutions

(132)

where

𝑉 (0) = 𝛼3󵄨󵄨󵄨󵄨ln𝑦1(0) − ln 𝑦1∗(0)󵄨󵄨󵄨󵄨 + 𝛼4󵄨󵄨󵄨󵄨ln𝑦2(0) − ln 𝑦2∗(0)󵄨󵄨󵄨󵄨+∑𝑛

which implies that

From (124) and (137), it follows that 𝑦𝑙(𝑡) (𝑙 = 1, 2) are

positive solution𝑥∗(𝑡) = (𝑥∗1(𝑡), 𝑥∗2(𝑡))𝑇is uniformly

4 Applications

In this section, for some applications of our main results, wewill consider some special cases of systems (2) and (3), whichhave been investigated extensively in [10]

Application 1 consider the following equations:

Theorem 16 In addition to (𝐻1), assume that the following

𝑥∗(𝑡) = (𝑥∗1(𝑡), 𝑥∗2(𝑡))𝑇which is globally asymptotically stable, where𝐹1(𝑡), 𝐹2(𝑡), 𝐺1(𝑡), and 𝐺2(𝑡) are defined in (22).

Proof It is similar to the proof of Theorems12and13, so weomit the details here

Theorem 17 In addition to (𝐻1), assume further that

(𝐻11) 𝑟1𝐹1∗𝐿> 𝑟2𝐺𝑀2 ,𝑟2𝐹2∗𝐿> 𝑟1𝐺𝑀1 .