Permanence and almost periodic solution of a Lotka–Volterra model with mutual interference and...

Applied Mathematical Modelling 37 (2013) 1054–1068

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Permanence and almost periodic solution of a Lotka–Volterra modelwith mutual interference and time delays q

Zengji Du ⇑, Yansen LvSchool of Mathematical Sciences, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China

a r t i c l e i n f o

Article history:Received 10 July 2011Received in revised form 3 March 2012Accepted 12 March 2012Available online 22 March 2012

Keywords:Lotka–Volterra modelAlmost periodic solutionMutual interferenceTime delaysPermanenceGlobal attractivity

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.03.022

q Sponsored by the Natural Science Foundation ofEducation Department of Jiangsu Province (11KJB11⇑ Corresponding author. Tel./fax: +86 516 834031

E-mail addresses: [email protected] (Z. Du), lvy

a b s t r a c t

In this paper, a Volterra model with mutual interference and time delays is investigated. Byapplying the comparison theorem of the differential equations and constructing a suitableLyapunov functional, sufficient conditions which guarantee the permanence and existenceof a unique globally attractive positive almost periodic solution of the system are obtained.Two suitable examples together with their numeric simulations are given to illustrate ourresults by using MatLab.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Recently, predator–prey models with periodic or almost periodic coefficients have been studied extensively, see papers[1–7], etc. There are few papers considering the mutual interference between the predators and preys, which was introducedby Hassell in 1971, during research on capture behavior among hosts and parasites. He found that the hosts or parasites hadthe tendency to leave each other when they met, which interfered with host capture effects. It is obvious that the mutualinterference will be stronger while the size of parasite becomes larger. From this observation, Hassell introduced the conceptof mutual interference constant mð0 < m � 1Þ and established a Volterra model with mutual interference as follows [8,9]

_y ¼ xgðxÞ � wðxÞym;

_y ¼ yð�dþ kuðxÞym�1 � qðyÞÞ:

�

Wang and Zhu [10] considered a Volterra model with mutual interference and Holling II type functional response. Guoand Chen [11] discussed a more general Volterra model with mutual interference and Beddington–DeAngelis functionalresponse which extended the results in Ref. [10]. Wang et al. [12] studied a Volterra model with mutual interference andHolling III type functional response

_x ¼ xðtÞðr1ðtÞ � b1ðtÞxðtÞÞ � c1ðtÞx2ðtÞx2ðtÞþk2 ymðtÞ;

_y ¼ yðtÞð�r2ðtÞ � b2ðtÞyðtÞÞ þ c2ðtÞx2ðtÞx2ðtÞþk2 ymðtÞ;

8<: ð0 < m < 1Þ: ð1:1Þ

. All rights reserved.

China (11071205, 11101349), the Natural Science Foundation of Jiangsu Province (BK2011042), NSF of0013), Qing Lan Project and Jiangsu Province postgraduate training [email protected] (Y. Lv).

http://dx.doi.org/10.1016/j.apm.2012.03.022

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Z. Du, Y. Lv / Applied Mathematical Modelling 37 (2013) 1054–1068 1055

Under the assumption that the coefficients of system (1.1) are all continuous T-periodic functions, by applying Mawhin con-tinuation theorem and constructing a suitable Lyapunov function, they obtained sufficient conditions which guarantee theexistence of a unique globally attractive positive T-periodic solution of system (1.1). It is well known that the assumption ofalmost periodicity of the coefficients in (1.1) is a way of incorporating the time-dependent variability of the environment,especially when the various components of the environment are periodic with not necessary commensurate periods (e.g.seasonal effects of weather, food supplies, mating habits, and harvesting). Furthermore, as we know, time delay plays animportant role in many biological dynamical systems, being particularly relevant in ecology. Therefore, it is interestingand important to study the almost periodic solutions of population dynamics models with time delay.

Chen [3] considered the almost periodic solution of a non-autonomous two-species competitive model with stage struc-ture. Wang and Dai [7] discussed the almost periodic solution for n-species Lotka–Volterra competitive system with delayand feedback controls. Alzabut et al. [13] studied the existence and exponential stability of positive almost periodic solutionsfor a model of hematopoiesis. Lin and Chen [14] considered almost periodic solution of a Volterra model with mutual inter-ference and Beddington–DeAngelis functional response.

Considering that the delay may occur in the competition among preys [15–17], Wang [18] considered a non-autonomouspredator–prey model with Hassell–Varley type functional response and a delay in the prey specific growth term as follows

_N1ðtÞ ¼ N1 aðtÞ � bðtÞN1ðt � sðtÞÞ � cðtÞN2ðtÞmNc

2ðtÞþN1ðtÞ

� �;

_N2ðtÞ ¼ N2ðtÞ �dðtÞ þ rðtÞN1ðtÞmNc

2ðtÞþN1ðtÞ

� �;

8>>><>>>:

ð0 < c < 1Þ;

where N1ðtÞ is the size of the prey population and N2ðtÞ is the size of the predator population.Motivated by the above papers, the aim of this paper is to obtain sufficient conditions for the existence of a unique glob-

ally attractive almost periodic solution of the following delayed modified Volterra model with mutual interference and Hol-ling III type functional response

_x ¼ xðtÞðr1ðtÞ � b1ðtÞxðt � sðtÞÞÞ � c1ðtÞx2ðtÞx2ðtÞþk2 ymðtÞ;

_y ¼ yðtÞð�r2ðtÞ � b2ðtÞyðtÞÞ þ c2ðtÞx2ðtÞx2ðtÞþk2 ymðtÞ;

8<: ð1:2Þ

with initial conditions

xðhÞ ¼ /ðhÞ; h 2 ½�s;0�; /ðhÞ 2 Cð½�s; 0�;RþÞ;yðhÞ ¼ wðhÞ; h 2 ½�s; 0�; wðhÞ 2 Cð½�s;0�;RþÞ;

ð1:3Þ

where xðtÞ is the size of the prey population and yðtÞ is the size of the predator population; 0 < m < 1; k > 0 is a constant; sðtÞ isnonnegative and continuously differentiable almost periodic function on R, and s ¼maxt2RfsðtÞg; mint2Rf1� _sðtÞg > 0;

r1ðtÞ; b1ðtÞ; r2ðtÞ; b2ðtÞ; c1ðtÞ and c2ðtÞ are continuous positive almost periodic functions with ecological meaning as follows:

r1ðtÞ (the prey population grows in the absence of predators),r2ðtÞ (the predator population decays in the absence of preys),b1ðtÞ (the prey population decays in the competition among the preys),b2ðtÞ (the predator population decays in the competition among the predators),c1ðtÞ (the prey is fed upon by the predators),c2ðtÞ (the coefficient of transformation from preys to predators).

The aim of this paper is to obtain sufficient conditions for the existence of a unique globally attractive almost periodicsolution of systems (1.2) and (1.3), by utilizing the comparison theorem of the differential equation and constructing a suit-able Lyapunov functional and applying the analysis technique of papers [1,5–7,11–14]. The mutual interference m of Theo-rem 4.1 in this paper, is allowed to be any real-valued number in interval (0,1), which improves the corresponding results inpapers [10,12,14] (where the mutual constant m needs to be a rational number).

The remaining part of this paper is organized as follows: In Section 2, we will state several definitions and lemmas whichwill be useful in the proving of main results of this paper. In Section 3, by applying the theory of differential inequality, wepresent the permanence results for systems (1.2) and (1.3). In Section 4, by constructing a suitable Lyapunov function, a setof sufficient conditions which ensure the global attractivity of systems (1.2) and (1.3) are obtained. In Section 5, we presentsome sufficient conditions which guarantee the existence and uniqueness of almost periodic solution of systems (1.2) and(1.3). In the end, two suitable examples together with their numeric simulations are given to illustrate our results.

2. Definitions and lemmas

Now let us state several definitions and lemmas which will be useful in the proving of main results of this paper.Let f be a continuous bounded function on R and we set

1056 Z. Du, Y. Lv / Applied Mathematical Modelling 37 (2013) 1054–1068

f u ¼ supt2R

f ðtÞ; f l ¼ inft2R

f ðtÞ:

From the properties of continuous positive almost periodic functions, we know that the coefficients of the almost periodicsystem (1.2) satisfy

mini¼1;2

rli; c

li; b

li

n o> 0; max

i¼1;2ru

i ; cui ; b

ui

� �< þ1:

Definition 2.1 [12]. If ð�xðtÞ; �yðtÞÞT is a positive solution of systems (1.2) and (1.3) and ðxðtÞ; yðtÞÞT is any positive solution of(1.2) and (1.3) satisfying

limt!þ1

j�xðtÞ � xðtÞj ¼ 0; limt!þ1

j�yðtÞ � yðtÞj ¼ 0; ð2:1Þ

then we say ð�xðtÞ; �yðtÞÞT is globally attractive.

Definition 2.2 [3]. A function f ðt; xÞ, where f is an m-vector, t is a real scalar and x is an n-vector, is said to be almost periodicin t uniformly with respect to x 2 X � Rn, if f ðt; xÞ is continuous in t 2 R and x 2 X, and if for any e > 0, it is possible to find aconstant lðeÞ > 0 such that in any interval of length lðeÞ there exists a s such that the inequality

kf ðt þ s; xÞ � f ðt; xÞk ¼Pmi¼1jfiðt þ s; xÞ � fiðt; xÞj < e

is satisfied for all t 2 R; x 2 X: The number s is called an e-translation number of f ðt; xÞ.

Definition 2.3 [7]. A function f : R! R is said to be asymptotically almost periodic function if there exists an almost-peri-odic function qðtÞ and a continuous function rðtÞ such that

f ðtÞ ¼ qðtÞ þ rðtÞ; t 2 R and rðtÞ ! 0 as t !1:

We refer to [13,19,20] for the relevant definitions and the properties of almost periodic functions.

Lemma 2.1 [17]. If function f is nonnegative, integral and uniformly continuous on ½0;þ1Þ, then limt!þ1f ðtÞ ¼ 0.

Lemma 2.2. fðxðtÞ; yðtÞÞT 2 R2jxðt0Þ > 0; yðt0Þ > 0; for some t0 2 Rg is positive invariant with respect to systems (1.2) and (1.3).

Proof. For xðt0Þ > 0; yðt0Þ > 0 then we get

xðtÞ ¼ xðt0Þ expZ t

t0

ðr1ðsÞ � b1ðsÞxðs� sðsÞÞÞ � c1ðsÞx2ðsÞx2ðsÞ þ k2 ymðsÞ

" #ds

( )> 0;

yðtÞ ¼ yðt0Þ expZ t

t0

ð�r2ðsÞ � b2ðsÞyðsÞÞ þc2ðsÞx2ðsÞx2ðsÞ þ k2 ymðsÞ

" #ds

( )> 0:

Thus, we prove Lemma 2.2. h

Lemma 2.3 [4]. If a > 0; b > 0, and _x P ð6Þxðb� axaÞ, where a is a positive constant, then

limt!1

inf xðtÞP ba

� �1a

; limt!1

sup xðtÞ 6 ba

� �1a

!: ð2:2Þ

Lemma 2.4 [21]. If a > 0; b > 0; and _zðtÞP ð6ÞzmðtÞðb� az1�mðtÞÞ; zð0Þ > 0; then

zðtÞP ð6Þ baþ z1�mð0Þ � b

a

� �e�að1�mÞt

� � 11�m

; 8t P 0: ð2:3Þ

Lemma 2.5 (Brouwer [22]). Suppose that the continuous operator A maps the closed and bounded convex set Q � Rn onto itself,then the operator A has at least one fixed point in set Q.


3. Permanence

In the following, we denote

m1 ¼rl

1

bu1

� cu1Mm

2

2kbu1

!exp rl

1 � bu1M1 �

cu1Mm

2

2k

� �s

� ; M1 ¼

ru1

bl1e�ru

1s;

m2 ¼cl

2m21

2 k2 þM22

�ru

2 þ bu2M2

� 24

35

11�m

; M2 ¼3cu

2

2rl2

� � 11�m

;

AðtÞ ¼ b1ðtÞ �c1ðtÞM2

1Mm2

k2 k2 þm21

�� r1ðtÞ þ b1ðtÞM1 þc1ðtÞM1

m21 þ k2 Mm

2

!Z u�1ðtÞ

tb1ðuÞdu

�c1ðtÞM1Mm

2 M21 þ k2

�k2 k2 þm2

1

� Z u�1ðtÞ

tb1ðuÞdu�M1b1ðu�1ðtÞÞ

1� _sðu�1ðtÞÞ

Z u�1ðu�1ðtÞÞ

u�1ðtÞb1ðuÞdu� 2M1c2ðtÞMm�1

2

k2 þm21

;

BðtÞ ¼ b2ðtÞ �c1ðtÞM1

m1�m2 k2 þm2

1

�� c1ðtÞRu�1ðtÞ

t b1ðuÞdum1�m

2

;

in which u�1 is the inverse functions of uðtÞ ¼ t � sðtÞ.

Theorem 3.1. Suppose that systems (1.2) and (1.3) satisfy the following condition

½H1� rl1 �

cu1Mm

2

2k> 0:

Then systems (1.2) and (1.3) are permanent, i.e., there exists a constant T > 0, when t > T, any solution ðxðtÞ; yðtÞÞT of systems(1.2) and (1.3) satisfying

0 < m1 6 xðtÞ 6 M1; 0 < m2 6 yðtÞ 6 M2:

Proof. From the first equation of system (1.2) it follows that

_xðtÞ 6 xðtÞr1ðtÞ: ð3:1Þ

From (3.1), one has

xðtÞ 6 xðt � sÞeR t

t�sr1ðhÞdh

6 xðt � sÞeru1s; t > s;

i.e.,

xðtÞe�ru1s 6 xðt � sÞ; t > s: ð3:2Þ

Substituting (3.2) into the first equation of system (1.2), we obtain

_xðtÞ 6 xðtÞ ru1 � bl

1xðtÞe�ru1s

h i; t > s: ð3:3Þ

By applying Lemma 2.3 to (3.3) leads to

limt!þ1

sup xðtÞ 6 ru1

bl1e�ru

1s� M1: ð3:4Þ

From (3.4), we know that there exists a T1 > s enough large such that

xðtÞ 6 M1; t P T1: ð3:5Þ

In view of (3.5), we know that there exists a T2 ¼ T1 þ s such that

xðt � sÞ 6 M1; t P T2: ð3:6Þ

From the second equation of system (1.2), we have

_yðtÞ 6 yðtÞ �rl2 þ cu

2ym�1ðtÞ�

6 ymðtÞ cu2 � rl

2y1�mðtÞ�

; t P 0: ð3:7Þ

Applying Lemma 2.4 to (3.7) leads to


yðtÞ 6 cu2

rl2

þ y1�mð0Þ � cu2

rl2

� �e�rl

2ð1�mÞt� � 1

1�m

; t P 0: ð3:8Þ

Therefore, there exists a T3 > 0 such that

yðtÞ 6 3cu2

2rl2

� � 11�m

� M2; t > T3: ð3:9Þ

From (3.9) and the first equation of system (1.2), we know that

_xðtÞP xðtÞ rl1 � bu

1xðt � sðtÞÞ � cu1Mm

2

2k

� �; t > T3: ð3:10Þ

Let xð�tÞ be any local minimal value of xðtÞ, then from (3.10) we get that

0 ¼ _xð�tÞP xð�tÞ rl1 � bu

1xð�t � sð�tÞÞ � cu1Mm

2

2k

� �: ð3:11Þ

In view of (3.11), one has

xð�t � sð�tÞÞPrl

1

bu1

� cu1Mm

2

2kbu1

: ð3:12Þ

Integrating the inequality (3.10) from �t � sð�tÞ to �t, noticing that rl1 � bu

1xð�t � sð�tÞÞ � cu1Mm

22k 6 0, we have

lnxð�tÞ

xð�t � sð�tÞÞ

� �PZ �t

�t�sð�tÞrl

1 � bu1xðt � sðtÞÞ � cu

1Mm2

2k

� �dt P rl

1 � bu1M1 �

cu1Mm

2

2k

� �s: ð3:13Þ

From (3.12) and (3.13), we have

xð�tÞP rl1

bu1

� cu1Mm

2

2kbu1

!exp rl

1 � bu1M1 �

cu1Mm

2

2k

� �s

� :

Thus there exists a T4 > 0; for t > T4; and

xðtÞP xð�tÞP rl1

bu1

� cu1Mm

2

2kbu1

!exp rl

1 � bu1M1 �

cu1Mm

2

2k

� �s

� � m1: ð3:14Þ

From (3.9), (3.14) and the second equation of system (1.2), we know that there exists a T5 P maxfT3; T4g > 0, for t > T5

_yðtÞP yðtÞ �ru2 � bu

2M2 þcl

2m21

k2 þM22

ym�1ðtÞ !

¼ ymðtÞ cl2m2

1

k2 þM22

� ru2 þ bu

2M2�

y1�mðtÞ !

: ð3:15Þ

It follows from Lemma 2.4 that there exists a T6 > 0 such that

y1�mðtÞP cl2m2

1

2 k2 þM22

�ru

2 þ bu2M2

� ; t > T6;

so we get

yðtÞP cl2m2

1

2 k2 þM22

�ru

2 þ bu2M2

� 24

35

11�m

� m2; t > T6:

Let T P maxfT2; T5; T6g > 0, for t > T , the following hold,

m1 6 xðtÞ 6 M1; m2 6 yðtÞ 6 M2:

We know that under the assumption of Theorem 3.1, systems (1.2) and (1.3) are permanent. The proof of Theorem 3.1 iscompleted. h

Next we will prove for t P 0, the above conclusions hold. We denote the set X as followX ¼ fðxðtÞ; yðtÞÞT 2 R2jðxðtÞ; yðtÞÞT is the solution of systems (1.2) and (1.3), satisfying m1 6 xðtÞ 6 M2; m2 6 yðtÞ

6 M2; t 2 R}.


Theorem 3.2. The set X – ;; i.e., systems (1.2) and (1.3) have at least one bounded positive solution.

Proof. From properties of almost periodic function, there exists a sequence ftng; tn !1 as n! þ1, such that

sðt þ tnÞ ! sðtÞ; riðt þ tnÞ ! riðtÞ; biðt þ tnÞ ! biðtÞ; ciðt þ tnÞ ! ciðtÞ; i ¼ 1;2;

as n! þ1 uniformly on R. According to Lemma 2.5 systems (1.2) and (1.3) have at least one solution zðtÞ satisfyingm1 6 xðtÞ 6 M1; m2 6 yðtÞ 6 M2; for t > T. Clearly, the sequence zðt þ tnÞ is uniformly bounded and equicontinuous on eachbounded subset of R. Therefore by Ascoli theorem we know that there exists a subsequence zðt þ tlÞ, which converges to a

continuous function pðtÞ ¼ ðp1ðtÞ; p2ðtÞÞT as l! þ1 uniformly on each bounded subset of R. Let T1 2 R be given. We may as-

sume that tl þ T1 P T for all l. For t P 0, we have

xðtþ tlþT1Þ�xðtlþT1Þ¼Z tþT1

T1

xðsþ tlÞ r1ðsþ tlÞ�b1ðsþ tlÞxðsþ tl�sðsþ tlÞÞ�c1ðsþ tlÞxðsþ tlÞ

k2þx2ðsþ tlÞymðsþ tlÞ

" #ds; ð3:16Þ

yðt þ tl þ T1Þ � yðtl þ T1Þ ¼Z tþT1

T1

yðsþ tlÞ �r2ðsþ tlÞ � b2ðsþ tlÞyðsþ tlÞ þc2ðsþ tlÞx2ðsþ tlÞ

k2 þ x2ðsþ tlÞym�1ðsþ tlÞ

" #ds: ð3:17Þ

Applying Lebesgue dominated convergence theorem, and letting l! þ1 in (3.16) and (3.17), we obtain

p1ðt þ T1Þ � p1ðT1Þ ¼Z tþT1

T1

p1ðsÞ r1ðsÞ � b1ðsÞp1ðs� sðsÞÞ � c1ðsÞp1ðsÞk2 þ p2

1ðsÞpm

2 ðsÞ" #

ds;

p2ðt þ T1Þ � p2ðT1Þ ¼Z tþT1

T1

p2ðsÞ �r2ðsÞ � b2ðsÞp2ðsÞ þc2ðsÞp2

1ðsÞk2 þ p2

1ðsÞpm�1

2 ðsÞ" #

ds;

for all t P 0. Since T1 2 R is arbitrarily given, pðtÞ ¼ ðp1ðtÞ; p2ðtÞÞT is a solution of systems (1.2) and (1.3) on R: It is clear that

mi 6 piðtÞ 6 Miði ¼ 1;2Þ; for t 2 R: Thus pðtÞ 2 X; i.e., systems (1.2) and (1.3) have at least one bounded positive solution. h

4. Global attractivity

Theorem 4.1. Assume that the coefficients of systems (1.2) and (1.3) satisfy condition ½H1� and the following condition

½H2� lim inft!þ1

AðtÞ > 0; lim inft!þ1

BðtÞ > 0;

where AðtÞ;BðtÞ are given in Section 3. Then the solution of systems (1.2) and (1.3) is globally attractive.

Proof. From Theorem 3.2, we know that systems (1.2) and (1.3) have at least one positive solution ð�xðtÞ; �yðtÞÞT and

m1 6 �xðtÞ 6 M1; m2 6 �yðtÞ 6 M2:

Set g ¼ minf�yðtÞg, then g P m2 > 0 and let x�ðtÞ ¼ xðtÞ; y�ðtÞ ¼ yðtÞg , it follows from (1.2) and (1.3), we have

_x�ðtÞ ¼ x�ðtÞðr1ðtÞ � b1ðtÞx�ðt � sðtÞÞÞ � gmc1ðtÞx2� ðtÞ

x2� ðtÞþk2 ym

� ðtÞ;

_y�ðtÞ ¼ y�ðtÞð�r2ðtÞ � gb2ðtÞy�ðtÞÞ þgm�1c2ðtÞx2

� ðtÞx2� ðtÞþk2 ym

� ðtÞ;

8<: ð0 < m < 1Þ; ð4:1Þ

with initial conditions

x�ðhÞ ¼ xðhÞ ¼ /ðhÞ; h 2 ½�s; 0�; /ðhÞ 2 Cð½�s;0�;RþÞ;

y�ðhÞ ¼yðhÞg¼ wðhÞ

g; h 2 ½�s;0�; wðhÞ 2 Cð½�s; 0�;RþÞ:

ð4:2Þ

So we see that systems (4.1) and (4.2) have at least one positive solution ð�x�ðtÞ; �y�ðtÞÞT with

m1 6 �x�ðtÞ 6 M1;m2

g6 �y�ðtÞ 6

M2

g: ð4:3Þ

Suppose ðx�ðtÞ; y�ðtÞÞT is any positive solution of (4.1) and (4.2).

Let

V11ðtÞ ¼ j ln x�ðtÞ � ln �x�ðtÞj

and calculate its Dini derivative along system (4.1), we get


DþV11ðtÞ ¼ sgnðx�ðtÞ � �x�ðtÞÞ_x�ðtÞx�ðtÞ

�_�x�ðtÞ�x�ðtÞ

!

¼ sgnðx�ðtÞ � �x�ðtÞÞ �b1ðtÞ½x�ðt � sðtÞÞ � �x�ðt � sðtÞÞ� � gmc1ðtÞx�ðtÞym

� ðtÞk2 þ x2

� ðtÞ�

�x�ðtÞ�ym� ðtÞ

k2 þ �x2� ðtÞ

!" #

¼ sgnðx�ðtÞ � �x�ðtÞÞ½�b1ðtÞ½x�ðt � sðtÞÞ � �x�ðt � sðtÞÞ��

þ gmc1ðtÞsgnðx�ðtÞ � �x�ðtÞÞ�x�ðtÞ�ym

� ðtÞk2 þ �x2

� ðtÞ�

�x�ðtÞym� ðtÞ

k2 þ �x2� ðtÞþ

�x�ðtÞym� ðtÞ

k2 þ �x2� ðtÞ� x�ðtÞym

� ðtÞk2 þ x2

� ðtÞ

!6 �b1ðtÞjx�ðtÞ

� �x�ðtÞj þ b1ðtÞZ t

t�sðtÞð _x�ðsÞ � _�x�ðsÞÞds

��þ gmc1ðtÞ�x�ðtÞ


�ym� ðtÞ � ym

� ðtÞ�� þ sgnðx�ðtÞ

� �x�ðtÞÞgmc1ðtÞym

� ðtÞk2

k2 þ x2� ðtÞ

�k2 þ �x2

� ðtÞ � ð�x�ðtÞ � x�ðtÞÞ þ gmc1ðtÞsgnðx�ðtÞ � �x�ðtÞÞym

� ðtÞx�ðtÞ�x�ðtÞ

k2 k2 þ x2� ðtÞ

� ðx�ðtÞ � �x�ðtÞÞ

6 �b1ðtÞjx�ðtÞ � �x�ðtÞj þ b1ðtÞZ t

t�sðtÞð _x�ðsÞ � _�x�ðsÞÞds

��þ gmc1ðtÞ�x�ðtÞ



� ðtÞ��

þ gmc1ðtÞx�ðtÞ�x�ðtÞym� ðtÞ

k2 k2 þ x2� ðtÞ

� jx�ðtÞ � �x�ðtÞj: ð4:4Þ

On substituting (4.1) into (4.4) yields

DþV11ðtÞ6�b1ðtÞjx�ðtÞ��x�ðtÞjþb1ðtÞZ t

t�sðtÞx�ðsÞ r1ðsÞ�b1ðsÞx�ðs�sðsÞÞ�gmc1ðsÞx�ðsÞ

x2� ðsÞþk2 ym

� ðsÞ !(��

��x�ðsÞ r1ðsÞ�b1ðsÞ�x�ðs�sðsÞÞ�gmc1ðsÞ�x�ðsÞ�x2� ðsÞþk2

�ym� ðsÞ

!)ds

��þgmc1ðtÞ�x�ðtÞk2þ�x2

� ðtÞ�ym� ðtÞ�ym

� ðtÞ��

þgmc1ðtÞx�ðtÞ�x�ðtÞym� ðtÞ

k2 k2þx2� ðtÞ

� jx�ðtÞ��x�ðtÞj6�b1ðtÞjx�ðtÞ��x�ðtÞj

þb1ðtÞZ t

t�sðtÞðr1ðsÞ�b1ðsÞx�ðs�sðsÞÞ�gmc1ðsÞx�ðsÞ

x2� ðsÞþk2 ym

� ðsÞÞðx�ðsÞ��x�ðsÞÞ��x�ðsÞb1ðsÞðx�ðs�sðsÞÞ��x�ðs�sðsÞÞÞ(��

��x�ðsÞgmc1ðsÞx�ðsÞ

x2� ðsÞþk2 ym

� ðsÞ�gmc1ðsÞ�x�ðsÞ

�x2� ðsÞþk2

�ym� ðsÞ

!)ds

��þgmc1ðtÞ�x�ðtÞk2þ�x2


� ðtÞ��

þgmc1ðtÞx�ðtÞ�x�ðtÞym� ðtÞ

k2 k2þx2� ðtÞ

� jx�ðtÞ��x�ðtÞj6�b1ðtÞjx�ðtÞ��x�ðtÞj

þb1ðtÞZ t

t�sðtÞr1ðsÞþb1ðsÞx�ðs�sðsÞÞþgmc1ðsÞx�ðsÞ

x2� ðsÞþk2 ym

� ðsÞ !

jx�ðsÞ��x�ðsÞjþ�x�ðsÞb1ðsÞjx�ðs�sðsÞÞ��x�ðs�sðsÞÞj(

þgmc1ðsÞ�x2� ðsÞ

k2þ�x2� ðsÞ

�ym� ðsÞ�ym

� ðsÞ�� þgmc1ðsÞ�x�ðsÞym

� ðsÞðx�ðsÞ�x�ðsÞþk2Þk2 k2þx2

� ðsÞ � jx�ðsÞ��x�ðsÞj

9=;ds

þgmc1ðtÞ�x�ðtÞk2þ�x2


� ðtÞ�� þgmc1ðtÞx�ðtÞ�x�ðtÞym

� ðtÞk2 k2þx2

� ðtÞ � jx�ðtÞ��x�ðtÞj: ð4:5Þ

It follows from (4.3) and (4.5) that for t P T þ s

DþV11ðtÞ 6 �b1ðtÞjx�ðtÞ � �x�ðtÞj þ b1ðtÞZ t

t�sðtÞr1ðsÞ þ b1ðsÞM1 þ

c1ðsÞM1

m21 þ k2 Mm

2

!jx�ðsÞ � �x�ðsÞj þM1b1ðsÞjx�ðs� sðsÞÞ

(

��x�ðs� sðsÞÞj þ gmc1ðsÞ �ym� ðsÞ � ym

� ðsÞ�� þ c1ðsÞM1Mm

2 ðM21 þ k2Þ

k2 k2 þm21

� jx�ðsÞ � �x�ðsÞj

9=;ds

þ gmc1ðtÞM1

k2 þm21


� ðtÞ�� þ c1ðtÞM2

1Mm2

k2 k2 þm21

� jx�ðtÞ � �x�ðtÞj � �b1ðtÞjx�ðtÞ � �x�ðtÞj

þ b1ðtÞZ t

t�sðtÞGðsÞdsþ gmc1ðtÞM1

k2 þm21



1Mm2

k2 k2 þm21

� jx�ðtÞ � �x�ðtÞj; ð4:6Þ


where

GðsÞ ¼ r1ðsÞ þ b1ðsÞM1 þc1ðsÞM1

m21 þ k2 Mm

2

!jx�ðsÞ � �x�ðsÞj þM1b1ðsÞjx�ðs� sðsÞÞ � �x�ðs� sðsÞÞj þ gmc1ðsÞ �ym

� ðsÞ � ym� ðsÞ

�� þ c1ðsÞM1Mm

2 ðM21 þ k2Þ

k2 k2 þm21

� jx�ðsÞ � �x�ðsÞj:

Define

V12ðtÞ ¼Z u�1ðtÞ

t

Z t

sðuÞb1ðuÞGðsÞdsdu: ð4:7Þ

We obtain from (4.6) and (4.7) that for t P T þ s

DþV11ðtÞ þ _V12ðtÞ 6 �b1ðtÞjx�ðtÞ � �x�ðtÞj þgmc1ðtÞM1

k2 þm21



1Mm2

k2 k2 þm21

� jx�ðtÞ � �x�ðtÞj þZ u�1ðtÞ

tb1ðuÞduGðtÞ:

ð4:8Þ

Next, define

V1ðtÞ ¼ V11ðtÞ þ V12ðtÞ þ V13ðtÞ; ð4:9Þ

where

V13ðtÞ ¼ M1

Z t

t�sðtÞ

Z u�1ðu�1ðlÞÞ

u�1ðlÞ

b1ðuÞb1ðu�1ðlÞÞ1� _sðu�1ðlÞÞ jx�ðlÞ �

�x�ðlÞjdudl: ð4:10Þ

It then follows from (4.8)–(4.10) that for t P T þ s

DþV1ðtÞ 6 �b1ðtÞjx�ðtÞ � �x�ðtÞj þgmc1ðtÞM1

k2 þm21



1Mm2

k2 k2 þm21

� jx�ðtÞ � �x�ðtÞj

þ r1ðtÞ þ b1ðtÞM1 þc1ðtÞM1

m21 þ k2 Mm

2

" #Z u�1ðtÞ

tb1ðuÞdujx�ðtÞ � �x�ðtÞj þ gmc1ðtÞ

�Z u�1ðtÞ

tb1ðuÞdu �ym

� ðtÞ � ym� ðtÞ

�� þ c1ðtÞM1Mm2 ðM

21 þ k2Þ

k2 k2 þm21

� Z u�1ðtÞ

tb1ðuÞdujx�ðtÞ � �x�ðtÞj

þM1b1ðu�1ðtÞÞ1� _sðu�1ðtÞÞ


u�1ðtÞb1ðuÞdujx�ðtÞ � �x�ðtÞj: ð4:11Þ

Since when a P b; a P 1; y ¼ ax � bx is an increasing function. From �y� ¼ �yg P 1; 0 < m < 1; we have


� ðtÞ�� 6 �y�ðtÞ � y�ðtÞj j: ð4:12Þ

From (4.12), we have

DþV1ðtÞ 6 �b1ðtÞjx�ðtÞ � �x�ðtÞj þgmc1ðtÞM1

k2 þm21

�y�ðtÞ � y�ðtÞj j þ c1ðtÞM21Mm

2

k2 k2 þm21


þ r1ðtÞ þ b1ðtÞM1 þc1ðtÞM1

m21 þ k2 Mm

2

" #Z u�1ðtÞ

tb1ðuÞdujx�ðtÞ � �x�ðtÞj þ gmc1ðtÞ

�Z u�1ðtÞ

tb1ðuÞdu �y�ðtÞ � y�ðtÞj j þ

c1ðtÞM1Mm2 M2

1 þ k2 �

k2 k2 þm21

� Z u�1ðtÞ

tb1ðuÞdujx�ðtÞ � �x�ðtÞj

þM1b1ðu�1ðtÞÞ1� _sðu�1ðtÞÞ


u�1ðtÞb1ðuÞdujx�ðtÞ � �x�ðtÞj: ð4:13Þ

Define

V2ðtÞ ¼ j ln y�ðtÞ � ln �y�ðtÞj: ð4:14Þ

Calculate its Dini derivative along system (4.1), we have


DþV2ðtÞ ¼ sgnðy�ðtÞ � �y�ðtÞÞ_y�ðtÞy�ðtÞ

�_�y�ðtÞ�y�ðtÞ

!

¼ sgnðy�ðtÞ � �y�ðtÞÞ �gb2ðtÞðy�ðtÞ � �y�ðtÞÞ þ gm�1c2ðtÞx2� ðtÞym�1

� ðtÞk2 þ x2

� ðtÞ�

�x2� ðtÞ�ym�1


� ðtÞ

!" #

¼ �gb2ðtÞjy�ðtÞ � �y�ðtÞj þ gm�1c2ðtÞsgnðy�ðtÞ � �y�ðtÞÞx2� ðtÞym�1

� ðtÞk2 þ x2

� ðtÞ�



� ðtÞ

!

¼ �gb2ðtÞjy�ðtÞ � �y�ðtÞj þ gm�1c2ðtÞsgnðy�ðtÞ

� �y�ðtÞÞx2� ðtÞym�1

� ðtÞk2 þ x2

� ðtÞ� x2

� ðtÞ�ym�1� ðtÞ

k2 þ x2� ðtÞ

þ x2� ðtÞ�ym�1

� ðtÞk2 þ x2

� ðtÞ�



� ðtÞ

" #

¼ �gb2ðtÞjy�ðtÞ � �y�ðtÞj þ gm�1c2ðtÞsgnðy�ðtÞ

� �y�ðtÞÞx2� ðtÞ

k2ðtÞ þ x2� ðtÞ

ym�1� ðtÞ � �ym�1

� ðtÞ�

þ k2ðx�ðtÞ þ �x�ðtÞÞ�ym�1� ðtÞ

k2 þ x2� ðtÞ

�k2 þ �x2

� ðtÞ � ðx�ðtÞ � �x�ðtÞÞ

24

35

6 �gb2ðtÞjy�ðtÞ � �y�ðtÞj þgm�1c2ðtÞk2ðx�ðtÞ þ �x�ðtÞÞ�ym�1

� ðtÞk2 þ x2

� ðtÞ �



6 �gb2ðtÞjy�ðtÞ � �y�ðtÞj þgm�1c2ðtÞðx�ðtÞ þ �x�ðtÞÞ�ym�1

� ðtÞk2 þ x2

� ðtÞjx�ðtÞ � �x�ðtÞj

6 �gb2ðtÞjy�ðtÞ � �y�ðtÞj þ2M1c2ðtÞMm�1

2

k2 þm21

jx�ðtÞ � �x�ðtÞj: ð4:15Þ

Define Lyapunov functional VðtÞ as

VðtÞ ¼ V1ðtÞ þ V2ðtÞ: ð4:16Þ

It follows from (4.13)–(4.15) that for t P T þ s

DþVðtÞ 6 �AðtÞjx�ðtÞ � �x�ðtÞj � gBðtÞjy�ðtÞ � �y�ðtÞj; ð4:17Þ

where AðtÞ;BðtÞ are defined in Theorem 4.1.From condition ½H2�, we know that there must be two positive constants a and b and T0 P T þ s such that

DþVðtÞ 6 �ajx�ðtÞ � �x�ðtÞj � bjy�ðtÞ � �y�ðtÞj; for 8t P T0:

Thus, VðtÞ is non-increasing on ½T0;þ1Þ. Integrating the above inequality from T0 to t, we obtain

VðtÞ þ aZ t

T0

jx�ðsÞ � �x�ðsÞjdsþ bZ t

T0

jy�ðsÞ � �y�ðsÞjds 6 VðT0Þ < þ1; for 8t > T0:

Applying Lemma 2.1, we have

limt!þ1

jx�ðtÞ � �x�ðtÞj ¼ 0; limt!þ1

jy�ðtÞ � �y�ðtÞj ¼ 0: ð4:18Þ

So we also have

limt!þ1

jxðtÞ � �xðtÞj ¼ 0; limt!þ1

jyðtÞ � �yðtÞj ¼ 0: ð4:19Þ

Then the solution of systems (1.2) and (1.3) is globally attractive. h

5. Existence and uniqueness of almost periodic solution

Theorem 5.1. Suppose ½H1� and ½H2� hold, then there exists a unique almost periodic solution of systems (1.2) and (1.3).

Proof. From Theorem 3.2, there exists a bounded positive solution ðxðtÞ; yðtÞÞT ; t 2 R: Then there exists a sequenceft0lg; t0l !1 as l! þ1 such that ðxðt þ t0lÞ; yðt þ t0lÞÞ

T is a solution of the following system

_x ¼ xðtÞðr1ðt þ t0lÞ � b1ðt þ t0lÞxðt � sðtÞÞÞ � c1ðtþt0lÞx2ðtÞ

x2ðtÞþk2 ymðtÞ;

_y ¼ yðtÞð�r2ðt þ t0lÞ � b2ðt þ t0lÞyðtÞÞ þc2ðtþt0

lÞx2ðtÞ

x2ðtÞþk2 ymðtÞ:

8><>: ð5:1Þ


From above and Theorem 3.1, we have that not only ðxðt þ t0lÞ; yðt þ t0lÞÞT but also ð _xðt þ t0lÞ; _yðt þ t0lÞÞ

T are uniformly bounded,thus ðxðt þ t0lÞ; yðt þ t0lÞÞ

T are uniformly bounded and equi-continuous. By Ascoli theorem there exists a uniformly convergentsubsequence fðxðt þ tlÞ; yðt þ tlÞÞTg# fðxðt þ t0lÞ; yðt þ t0lÞÞ

Tg such that for 8e > 0, there exists a l0ðeÞ > 0 with the property thatif j; l > l0ðeÞ then

jxðt þ tjÞ � xðt þ tlÞj < e; jyðt þ tjÞ � yðt þ tlÞj < e;

which shows that ðxðt þ tlÞ; yðt þ tlÞÞT is asymptotically almost periodic function, there exist p1ðtÞ; p2ðtÞ; q1ðtÞ; q2ðtÞ; such that

xðtÞ ¼ p1ðtÞ þ q1ðtÞ; yðtÞ ¼ p2ðtÞ þ q2ðtÞ; t 2 R;

where

liml!þ1

p1ðt þ tlÞ ¼ p1ðtÞ; liml!þ1

q1ðt þ tlÞ ¼ 0; liml!þ1

p2ðt þ tlÞ ¼ p2ðtÞ; liml!þ1

q2ðt þ tlÞ ¼ 0;

p1ðtÞ; p2ðtÞ are almost periodic functions. It means that

liml!þ1

xðt þ tlÞ ¼ p1ðtÞ; liml!þ1

yðt þ tlÞ ¼ p2ðtÞ:

On the other hand,

liml!þ1

_xðt þ tlÞ ¼ liml!þ1

limh!0

xðt þ tl þ hÞ � xðt þ tlÞh

¼ limh!0

liml!þ1

xðt þ tl þ hÞ � xðt þ tlÞh

¼ limh!0

p1ðt þ hÞ � p1ðtÞh

; ð5:2Þ

liml!þ1

_yðt þ tlÞ ¼ liml!þ1

limh!0

yðt þ tl þ hÞ � yðt þ tlÞh

¼ limh!0

liml!þ1

yðt þ tl þ hÞ � yðt þ tlÞh

¼ limh!0

p2ðt þ hÞ � p2ðtÞh

; ð5:3Þ

so the _p1ðtÞ; _p2ðtÞ exist. Now we will prove that pðtÞ ¼ ðp1ðtÞ; p2ðtÞÞT is an almost periodic solution of systems (1.2) and (1.3).

From properties of almost periodic function, there exists a sequence ftng; tn !1 as n! þ1, such that

sðt þ tnÞ ! sðtÞ; riðt þ tnÞ ! riðtÞ; biðt þ tnÞ ! biðtÞ; ciðt þ tnÞ ! ciðtÞ; i ¼ 1;2;

as n! þ1 uniformly on R. It is not difficult to know that

limn!þ1

xðt þ tnÞ ¼ p1ðtÞ; limn!þ1

yðt þ tnÞ ¼ p2ðtÞ;

then we have

_p1ðtÞ ¼ limn!þ1

_xðt þ tnÞ ¼ limn!þ1

xðt þ tnÞðr1ðt þ tnÞ � b1ðt þ tnÞxðt þ tn � sðt þ tnÞÞÞ �c1ðt þ tnÞx2ðt þ tnÞ

x2ðt þ tnÞ þ k2 ymðt þ tnÞ

¼ p1ðtÞðr1ðtÞ � b1ðtÞp1ðt � sðtÞÞÞ � c1ðtÞp21ðtÞ

p21ðtÞ þ k2 pm

2 ðtÞ;

_p2ðtÞ ¼ limn!þ1

_yðt þ tnÞ ¼ limn!þ1

yðt þ tnÞð�r2ðt þ tnÞ � b2ðt þ tnÞyðt þ tnÞÞ þc2ðt þ tnÞx2ðt þ tnÞ

x2ðt þ tnÞ þ k2 ymðt þ tnÞ

¼ p2ðtÞð�r2ðtÞ � b2ðtÞp2ðtÞÞ þc2ðtÞp2

1ðtÞp2

1ðtÞ þ k2 pm2 ðtÞ:

This prove that pðtÞ ¼ ðp1ðtÞ; p2ðtÞÞT satisfies systems (1.2) and (1.3), and pðtÞ is a positive almost periodic solution.

Lastly, we show that there is only one positive almost periodic solution of systems (1.2) and (1.3). For any two positivealmost periodic solutions

pðtÞ ¼ ðp1ðtÞ;p2ðtÞÞT; �pðtÞ ¼ ð�p1ðtÞ; �p2ðtÞÞT ;

of systems (1.2) and (1.3), we claim that p1ðtÞ � �p1ðtÞ and p2ðtÞ � �p2ðtÞ for all t 2 R. Otherwise there must be at least one n 2 Rsuch that p1ðnÞ – �p1ðnÞ, i.e., jp1ðnÞ � �p1ðnÞj :¼ d > 0: So we can easily know that

d ¼ jp1ðnÞ � �p1ðnÞj ¼ limn!þ1

jxðnþ tnÞ � �xðnþ tnÞj ¼ limt!þ1

jxðtÞ � �xðtÞj > 0;

which is a contradiction to (4.19). Thus p1ðtÞ � �p1ðtÞ; 8t 2 R holds. Similarly, we can prove p2ðtÞ � �p2ðtÞ; 8t 2 R. The proof iscompleted. h

Remark 5.1. If sðtÞ � s, where s is a nonnegative constant, then assumption ½H2� can be simplified. Therefore, we have thefollowing result.

Fig. 1. The integral curve of prey-time.

Fig. 2. The integral curve of predator-time.


Corollary 5.1. Let sðtÞ � s, where s is a nonnegative constant. In addition to ½H1� assume further that

lim inft!þ1

b1ðtÞ �c1ðtÞM2

1Mm2

k2 k2 þm21

�� r1ðtÞ þ b1ðtÞM1 þc1ðtÞM1

m21 þ k2 Mm

2

!Z tþs

tb1ðuÞdu� c1ðtÞM1Mm

2 ðM21 þ k2Þ

k2 k2 þm21

� Z tþs

tb1ðuÞdu

8<:

�M1b1ðt þ sÞZ tþ2s

tþsb1ðuÞdu� 2M1c2ðtÞMm�1

2

k2 þm21

)> 0;

lim inft!þ1

b2ðtÞ �c1ðtÞM1

m1�m2 k2 þm2

1

�� c1ðtÞR tþs

t b1ðuÞdum1�m

2

8<:

9=; > 0:

Then the systems (1.2) and (1.3) have a unique positive almost periodic solution which is globally attractive.

Fig. 3. The orbit of predator–prey.

Fig. 4. The orbit of predator–prey-time.


6. Examples and numerical simulation

Example 6.1. As application, we consider the following system:

_x ¼ xðtÞðð3þ 0:1sinffiffiffi7p

tÞ � ð2� 0:1sintÞxðt � 0:01ÞÞ � ð0:05�0:01sintÞx2ðtÞx2ðtÞþ1 y0:2ðtÞ;

_y ¼ yðtÞð�ðð0:02þ 0:01sinffiffiffi3p

tÞ � ð0:05� 0:01sintÞyðtÞÞ þ ð0:2þ0:01sintÞx2ðtÞx2ðtÞþ1 y0:2ðtÞ;

8<: ð6:1Þ

with initial conditions /1ðhÞ � 1; y1ð0Þ ¼ 1; /2ðhÞ � 2:5; y2ð0Þ ¼ 2.It is not difficult to verify that the coefficients of system (6.1) satisfy the conditions in Theorem 3.1 and Corollary 5.1, we

see that system (6.1) has a unique positive almost periodic solution which is globally attractive. Its integral curves and orbitsare shown in Figs. 1–4, respectively.

From Figs. 1–4, we see that there is only one positive almost periodic solution of system (6.1), which is globally attractive.Also we can easily see that the predator-y and prey-x are persistent.

Example 6.2. When m is an irrational number, we consider the following system:

_x ¼ xðtÞðð4þ 0:1sinffiffiffi5p

tÞ � ð2� 0:1sintÞxðt � 0:01ÞÞ � ð0:05�0:01sintÞx2ðtÞx2ðtÞþ1 y

ffiffi6p

10 ðtÞ;

_y ¼ yðtÞð�ð0:02þ 0:01sinffiffiffi3p

tÞ � ð0:06� 0:01costÞyðtÞÞ þ ð0:2þ0:01sintÞx2ðtÞx2ðtÞþ1 y

ffiffi6p

10 ðtÞ;

8<: ð6:2Þ

Fig. 5. The integral curve of prey-time.

Fig. 6. The integral curve of predator-time.


with initial conditions /1ðhÞ � 1:2; y1ð0Þ ¼ 0:8; /2ðhÞ � 2; y2ð0Þ ¼ 2.It is not difficult to verify that the coefficients of system (6.2) satisfy the conditions in Theorem 3.1 and Corollary 5.1, we

see that system (6.2) has a unique positive almost periodic solution which is globally attractive. Its integral curves and orbitsare shown in Figs. 5–8, respectively.

From Figs. 5–8, we see that there is only one positive almost periodic solution of system (6.2), which is globally attractive.Also we can easily see that the predator-y and prey-x are persistent.

7. Conclusion

In this paper, we consider a Lotka–Volterra model with mutual interference and time delay. By constructing a suitableLyapunov function and using the comparison theorem of the differential equation, we obtain sufficient conditions for the

Fig. 8. The orbits of predator–prey-time.

Fig. 7. The orbit of predator–prey.


existence of a unique globally attractive almost periodic solution of systems (1.2) and (1.3). Obviously, our results improvethe corresponding results in papers [10,12,14] (where the mutual constant m needs to be a rational number). However, thereare still many interesting and challenging questions that need to be studied for systems (1.2) and (1.3). For example, whetherour results can be simplified by constructing a more appropriate Lyapunov function? We will leave this for future work.

References

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