Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

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

On a periodic predator–prey system with time delays on time scales q

Jie Liu, Yongkun Li *, Lili ZhaoDepartment of Mathematics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China

a r t i c l e i n f o

Article history:Received 2 July 2008Received in revised form 7 December 2008Accepted 7 December 2008Available online 24 December 2008

PACS:02.30.Ks05.45.-a87.23.Cc

Keywords:Predator–prey systemTime scalesPeriodic solutionFrodholm operator

1007-5704/$ - see front matter � 2008 Elsevier B.Vdoi:10.1016/j.cnsns.2008.12.008

q This work is supported by the National NaturalProvince, China.

* Corresponding author.E-mail address: yklie@ynu.edu.cn (Y. Li).

a b s t r a c t

With the help of a continuation theorem based on Gaines and Mawhin’s coincidencedegree theory, easily verifiable criteria are established for the global existence of positiveperiodic solutions of the predator–prey system with time delays on time scales

. All righ

Science

xD1 ðtÞ ¼ a1ðtÞ � b1ðtÞ expfx1ðt � s1ðtÞÞg � cðtÞ expfx2ðt�s2ðtÞÞg

1þm expfx1ðtÞg;

xD2 ðtÞ ¼ �a2ðtÞ þ b2ðtÞ expfx1ðt�s2ðtÞÞg

1þm expfx1ðt�s2ðtÞÞg;

8<:

where ai; bi; c; si 2 CðT;RþÞ; i ¼ 1;2 are T-periodic functions.� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The predator–prey system involving delays modelled by He [1] as follows:

dN1ðtÞdt ¼ N1ðtÞ½a1 � b1N1ðt � sÞ � c1N2ðt � rÞ�;

dN2ðtÞdt ¼ N2ðtÞ½�a2 � b2N2ðtÞ þ c2N1ðt � rÞ�;

(ð1:1Þ

with initial conditions

N1ðtÞ ¼ /1ðtÞP 0; t 2 ½�C; 0�; 0 < /1ð0Þ;N2ðtÞ ¼ /2ðtÞP 0; t 2 ½�C; 0�; 0 < /2ð0Þ; C ¼maxfs;rg;

�

where N1(t) and N2(t) denote the densities (per square unit of the habitat) of prey and predator population, respectively.Since a very basic and important problem in the study of a population growth model with a periodic environment is the

global existence and stability of a positive periodic solution, which plays a similar role as a globally stable equilibrium doesin an autonomous model, in the past years, predator–prey systems with or without time delay have been of great interest toboth applied mathematicians and ecologists [2–8]. In [8], the author by the theory coincidence degree, investigated the exis-tence of on a periodic predator–prey system with time delays

ts reserved.

s Foundation of People’s Republic of China and the Natural Sciences Foundation of Yunnan

J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438 3433

_N1ðtÞ ¼ N1ðtÞ a1ðtÞ � b1ðtÞR t�1 kðt � uÞN1ðuÞdu� cðtÞN2ðt�sðtÞÞ

1þmN1ðtÞ

h i;

_N2ðtÞ ¼ N2ðtÞ �a2ðtÞ þ b2ðtÞN1ðt�sðtÞÞ1þmN1ðt�sðtÞÞ

h i;

8><>: ð1:2Þ

where Ni (i = 1,2) stand for the prey’s and the predator’s density at time t, respectively; m is a positive constant that denotesthe half capturing saturation constant; a1 2 CðR;RÞ; a2; bi ði ¼ 1;2Þ; c; s 2 CðR;RþÞ; Rþ ¼ ½0;þ1Þ are T-periodic functions;k : Rþ ! Rþ is a measurable, T-periodic, normalized function such that

Rþ10 kðsÞds ¼ 1, corresponding to a delay kernel or a

weighting factor, which says how much emphases should be given to the size of the prey population at earlier times to deter-mine the present effect on resource availability, and also formally yields

R t�1 d0ðt � uÞxðuÞdu ¼ xðtÞ if k(s) = d0(s) andR t

�1 d0ðt � s� uÞxðuÞdu ¼ xðt � sÞ if k(s) = d0(s � s), where d0(s) is the Dirac delta function at s = 0.Recently, in order to unify differential and difference equations, people have done a lot of research about dynamic equa-

tions on time scales. Moreover, many results on this issue have been well documented in the monographs [9–12]. And, infact, continuous and discrete systems are very important in implementing and applications. But it is troublesome to studythe existence and stability of periodic solutions for continuous and discrete systems, respectively. therefore, it is meaningfulto study that on time scale which can unify the continuous and discrete situations.

The main purpose of this paper is by using Gaines and Mawhin’s coincidence degree [13, p. 40], to establish verifiablecriteria to guarantee the global existence of positive periodic solutions of the following predator–prey system with time de-lays on time scales:

xD1 ðtÞ ¼ a1ðtÞ � b1ðtÞ expfx1ðt � s1ðtÞÞg � cðtÞ expfx2ðt�s2ðtÞÞg

1þm expfx1ðtÞg;

xD2 ðtÞ ¼ �a2ðtÞ þ b2ðtÞ expfx1ðt�s2ðtÞÞg

1þm expfx1ðt�s2ðtÞÞg;

8<: ð1:3Þ

where x1(t) and x2(t) stand for the prey’s and the predator’s density at time t, respectively; m is a positive constant that de-notes the half capturing saturation constant; ai; bi; c; si 2 CðT;RþÞ; i ¼ 1;2; Rþ ¼ ½0;þ1Þ are T-periodic functions; T > 0 isa constant. The symbol D stands for the delta-derivative. T is a periodic time scale which has the subspace topology inheritedfrom the standard topology on R.

Remark 1.1. Let N1(t) = exp{x1(t)} and N2(t) = exp{x2(t)}. If T ¼ R, then (1.3) reduces to the Lotka–Volterra type predator–prey model with Michaelis–Menten or Holling Type II functional response [14]

dN1ðtÞdt ¼ N1ðtÞ a1ðtÞ � b1ðtÞN1ðt � s1ðtÞÞ � cðtÞN2ðt�s2ðtÞÞ

1þmN1ðtÞ

h i;

dN2ðtÞdt ¼ N2ðtÞ �a2ðtÞ þ b2ðtÞN1ðt�s2ðtÞÞ

1þmN1ðt�s2ðtÞÞ

h i:

8><>: ð1:4Þ

If T ¼ Z, then (1.3) is reformulated as

N1ðkþ 1Þ ¼ N1ðkÞ exp a1ðkÞ � b1ðkÞN1ðk� s1ðkÞÞ � cðkÞN2ðk�s2ðkÞÞ1þmN1ðkÞ

n o;

N1ðkþ 1Þ ¼ N1ðkÞ exp �a2ðkÞ þ b2ðkÞN1ðt�s2ðkÞÞ1þmN1ðk�s2ðkÞÞ

n o:

8><>: ð1:5Þ

2. Preliminaries

In this section, we first recall some basic definitions, lemmas on time scales which are used in what follows.Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators r;q : T! T and the

graininess l : T! Rþ are defined, respectively, by

rðtÞ ¼ inffs 2 T : s > tg; qðtÞ ¼ supfs 2 T : s < tg and lðtÞ ¼ rðtÞ � t:

A point t 2 T is called left-dense if t > inf T and q(t) = t, left-scattered if q(t) < t, right-dense if t < sup T and r(t) = t, andright-scattered if r(t) > t. If T has a left-scattered maximum m, then Tk ¼ T n fmg; otherwise Tk ¼ T. If T has a right-scat-tered minimum m, then Tk ¼ T n fmg; otherwise Tk ¼ T.

Let x 2 R; x > 0, T is an x-periodic time scale if T is a nonempty closed subset of R such that t þx 2 T andl(t) = l(t + x) whenever t 2 T.

A function f : T! R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limitsexist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be acontinuous function on T. We define CðJ;RÞ ¼ fuðtÞ is continuous on J}.

For y : T! R and t 2 Tk, we define the delta derivative of y(t), yD(t), to be the number (if it exists) with the property thatfor a given e > 0, there exists a neighborhood U of t such that

j ½yðrðtÞÞ � yðsÞ� � yDðtÞ½rðtÞ � s� j< e j rðtÞ � s j

for all s 2 U.If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.

3434 J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438

Let y be right-dense continuous. If YD(t) = y(t), then we define the delta integral by

Z t

ayðsÞDs ¼ YðtÞ � YðaÞ:

Definition 2.1. ([9]) We say that a time scale T is periodic if there exists p > 0 such that if t 2 T, then t � p 2 T. For T–R, thesmallest positive p is called the period of the time scale.

Definition 2.2. ([9]) Let T–R be a periodic time scale with period p. We say that the function f : T! R is periodic with per-iod x if there exists a natural number n such that x = np, f(t + x) = f(t) for all t 2 T and x is the smallest number such thatf(t + x) = f(t).

If T ¼ R, we say that f is periodic with period x > 0 if x is the smallest positive number such that f(t + x) = f(t) for allt 2 T.

Definition 2.3. ([9]) A function f : T! R is said to be rd-continuous if it is continuous at right-dense points in T and its left-sides limits exist (finite) at left-dense points in T. The set of rd-continuous functions is denoted by Crd ¼ CrdðTÞ ¼ CrdðT;RÞ.

Lemma 2.1. Every rd-continuous function has an antiderivative.

Lemma 2.2. If a; b 2 T, a; b 2 R and f ; g 2 CðT;RÞ, then

(i)R b

a ½af ðtÞ þ bgðtÞ�Dt ¼ aR b

a f ðtÞDt þ bR b

a gðtÞDt;

(ii) if f(t) P 0 for all a 6 t < b, thenR b

a f ðtÞDt P 0;

(iii) if jf(t)j 6 g(t) on ½a; bÞ :¼ ft 2 T : a 6 t < bg, then jR b

a f ðtÞDt j6R b

a gðtÞDt.

For convenience, we introduce the notation

j ¼minf½0;1Þ \ Tg; IT ¼ ½j;jþ T� \ T;

G ¼ 1T

ZIT

gðsÞDs ¼ 1T

Z jþT

jgðsÞDs; G ¼ 1

T

ZIT

j gðsÞ j Ds ¼ 1T

Z jþT

jj gðsÞ j Ds;

where g 2 CðT;RÞ is a T-periodic real function, i.e., g(t + T) = g(t) for all t 2 T.Next, let us recall the continuation theorem in coincidence degree theory. To do so, we need to introduce the following

notation.Let X,Y be normed Banach spaces, L :DomL � X ? dimY be a linear mapping, and N :X ? Y be a continuous mapping. The

mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +1 and ImL is closed in Y. If L is aFredholm mapping of index zero and there exist continuous projector P :X ? X and Q :Y ? Y such that ImP = KerL, Ker-Q = Im(I � Q), it follows that mapping LjDom L\Ker P : (I � P)X ? ImL is invertible. We denote the inverse of that mapping byKP. If X is an open bounded subset of X, the mapping N will be called L-compact on X if QNðXÞ is bounded andKPðI � QÞN : X! X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ ? KerL.

Here, we state the Gaines–Mawhin theorem [13, p. 40], which is a main tool in the proof of our main result.

Lemma 2.3. (Continuation Theorem) Let X � X be an open bounded set and N:X ? Y be a continuous operator which is L-compact on X. Assume

(i) for each k 2 (0,1), x 2 oX \ DomL, Lx – kNx;(ii) for each x 2 @ \ KerL, QNx – 0, and deg{JQN,X \ KerL,0} – 0.

Then Lx = Nx has at least one solution in �X \ DomL.

In the proof of our main result, we will use the following lemma which can be found in [12,15].

Lemma 2.4. Let t1, t2 2 IT and t 2 T. If f : T! R is T-periodic, then

f ðtÞ 6 f ðt1Þ þZ

IT

j f DðsÞ j Ds

and

f ðtÞP f ðt2Þ �Z

IT

j f DðsÞ j Ds:

J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438 3435

Lemma 2.5. ([16]) Assume that {fn}n2N is a function on J such that

(i) {fn}n2N is uniformly bounded on J;(ii) ff D

n gn2N is uniformly bounded on J.

Then there is a subsequence of {fn}n2N converges uniformly on J.

3. Existence of positive periodic solution

We are now in a position to state and prove our main result of this paper.

Theorem 3.1. Assume that in system (1.3)

(i) A1 > 0,(ii) A2B1 expfA1 þ A1g < A1ðB2 �mA2Þ and B2 > mA2,

where Ai ¼ ð1=TÞR

ITj aðtÞ j Dt; Ai ¼ ð1=TÞ

RIT

aðtÞDt; Bi ¼ ð1=TÞR

ITj bðtÞ j Dt; Bi ¼ ð1=TÞ

RIT

bðtÞDt; i ¼ 1;2; C ¼ ð1=TÞR

ITcðtÞDt.

Then system (1.3)has at least one positive T-periodic solution.

Proof. In order to apply Lemma 2.3 to system (1.3), let

X ¼ Z ¼ fðx1; x2ÞT : xi 2 Crd and xiðt þ TÞ ¼ xiðtÞ; i ¼ 1;2g;kðx1; x2Þk ¼max

t2IT

j x1ðtÞ j þmaxt2IT

j x2ðtÞ j; ðx1; x2Þ 2 Xðor ZÞ;

Then X and Z are both Banach spaces, if they are endowed with the above norm k�k (see [11]).

For x1x2

� �2 X, we define� � � �

Nx1

x2¼ a1ðtÞ � b1ðtÞ expfx1ðt � s1ðtÞÞg �

cðtÞ expfx2ðt � s2ðtÞÞg1þm expfx1ðtÞg

� a2ðtÞ þb2ðtÞ expfx1ðt � s2ðtÞÞg

1þm expfx1ðt � s2ðtÞÞg;

Lx1

x2

� �¼

xD1

xD2

" #; P

x1

x2

� �¼ Q

x1

x2

� �¼

1T

RIT

x1ðtÞDt1T

RIT

x2ðtÞDt

" #:

Then it follows that

KerL ¼ fðx1; x2ÞT 2 X : ðx1ðtÞ; x2ðtÞÞT ¼ ðh1;h2Þ 2 R2 for t 2 Tg;

ImL ¼ ðx1; x2ÞT 2 Z :

Z jþT

jx1ðtÞDt ¼ 0;

Z jþT

jx2ðtÞDt ¼ 0 for t 2 T

� �

and

dimKerL ¼ 2 ¼ codim Im L:

Since ImL is closed in Z, then L is a Fredholm mapping of zero. It is easy to show that P,Q are continuous projectors suchthat

ImP ¼ KerL; ImL ¼ KerQ ¼ ImðI � QÞ:

Furthermore, the generalized inverse (to L) KP:ImL ? DomL \ KerP exists and is given by

KPx1

x2

� �¼

R tj x1ðsÞDs� 1

T

R jþTj

R tj x1ðsÞDsDtR t

j x2ðsÞDs� 1T

R jþTj

R tj x2ðsÞDsDt

" #:

Thus,

QNx1

x2

� �¼

1T

R jþTj a1ðsÞ � b1ðsÞ expfx1ðs� s1ðsÞÞg � cðtÞ expfx2ðs�s2ðsÞÞg

1þm expfx1ðsÞg

h iDs

1T

R jþTj �a2ðsÞ þ b2ðsÞ expfx1ðs�s2ðsÞÞg

1þm expfx1ðs�s2ðsÞÞg

h iDs

264

375

and

KPðI � QÞNx1

x2

� �¼

R tj n1ðsÞDs� 1

T

R jþTj

R tj n1ðsÞDsDt � t � j� 1

T

R jþTj ðt � jÞDt

� �N1R t

j n2ðsÞDs� 1T

R jþTj

R tj n2ðsÞDsDt � t � j� 1

T

R jþTj ðt � jÞDt

� �N2

264

375:

3436 J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438

Obviously, QN and KP(I � Q)N are continuous. It is not difficult to show that KPðI � QÞNð�XÞ is compact for any openbounded set X � X by using Lemma 2.5. Moreover, QNðXÞ is clearly bounded. Thus, N is L-compact on X with any openbounded set X � X (see [11]).

Now we reach the position to search for an appropriate open, bounded subset X for the application of continuationtheorem (Lemma 2.3). Corresponding to the equation

Lðx1; x2ÞT ¼ kNðx1; x2ÞT ; k 2 ð0;1Þ;

we have

xD1 ðtÞ ¼ k a1ðtÞ � b1ðtÞ expfx1ðt � s1ðtÞÞg � cðtÞ expfx2ðt�s2ðtÞÞg

1þm expfx1ðtÞg

n o;

xD2 ðtÞ ¼ k �a2ðtÞ þ b2ðtÞ expfx1ðt�s2ðtÞÞg

1þm expfx1ðt�s2ðtÞÞg

n o:

8><>: ð3:1Þ

Assume that (x1(t),x2(t))T 2 X is a solution of (3.1) for a certain k 2 (0,1). By integrating (3.1) over the interval IT, we obtain

A1T ¼Z

IT

a1ðtÞDt ¼Z

IT

b1ðtÞ expfx1ðt � s1ðtÞÞgDt þZ

IT

cðtÞ expfx2ðt � s2ðtÞÞg1þm expfx1ðtÞg

Dt ð3:2Þ

and

A2T ¼Z

IT

a2ðtÞDt ¼Z

IT

b2ðtÞ expfx1ðt � s2ðtÞÞg1þm expfx1ðt � s2ðtÞÞg

Dt: ð3:3Þ

From (3.1)–(3.3), it follows that

ZIT

j xD1 ðtÞ j Dt 6 k

ZIT

j a1ðtÞ j Dt þZ

IT

b1ðtÞ expfx1ðt � s1ðtÞÞg½�

þ cðtÞ expfx2ðt � s2ðtÞÞg1þm expfx1ðtÞg

�Dt�< ðA1 þ A1ÞT ð3:4Þ

and

ZIT

j xD2 ðtÞ j Dt 6 k

ZIT

j a2ðtÞ j Dt þZ

IT

b2ðtÞ expfx1ðt � s2ðtÞÞg1þm expfx1ðt � s2ðtÞÞg

Dt� �

< ðA2 þ A2ÞT: ð3:5Þ

Noting that (x1(t),x2(t))T 2 X, then there exist ni,gi 2 IT, i = 1,2, such that

xiðniÞ ¼mint2IT

xiðtÞ; xiðgiÞ ¼maxt2IT

xiðtÞ; i ¼ 1;2: ð3:6Þ

From (3.3) and (3.6), we have

A2T Pexpfx1ðn1Þg

1þm expfx1ðn1Þg

ZIT

b2ðtÞDt;

that is

x1ðn1Þ 6 lnA2

B2 �mA2

� :

Consequently,

x1ðtÞ 6 x1ðn1Þ þZ

IT

j xD1 ðtÞ j Dt 6 ln

A2

B2 �mA2

� þ ðA1 þ A1ÞT: ð3:7Þ

Similarly, we get

A2T 6expfx1ðg1Þg

1þm expfx1ðg1Þg

ZIT

b2ðtÞDt;

which yields

x1ðg1ÞP lnA2

B2 �mA2

� :

Thus,

x1ðtÞP x1ðg1Þ �Z

IT

j xD1 ðtÞ j Dt 6 ln

A2

B2 �mA2

� � ðA1 þ A1ÞT: ð3:8Þ

It follows from (3.7) and (3.8) that

maxt2IT

j x1ðtÞ j6 max lnA2

B2 �mA2

� þ ðA1 þ A1ÞT

; ln

A2

B2 �mA2

� � ðA1 þ A1ÞT

� �¼: M1:

J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438 3437

On the other hand,

A1T PZ

IT

cðtÞ expfx2ðn2Þg1þm expfx1ðg1Þg

Dt ¼ CT expfx2ðn2Þg1þm expfx1ðg1Þg

;

that is

expfx2ðn2Þg 6A1

Cð1þm expfx1ðg1ÞgÞ 6

A1

Cð1þm expfM1gÞ;

which yields

x2ðn2Þ 6 lnA1

Cð1þm expfM1gÞ

� :

Thus,

x2ðtÞ 6 x2ðn2Þ þZ

IT

j xD2 ðtÞ j Dt 6 ln

A1

Cð1þm expfM1gÞ

� þ ðA1 þ A1ÞT ¼: M2: ð3:9Þ

Furthermore, it follows from (3.3)–(3.8) that

A1T 6Z

IT

½b1ðtÞ expfx1ðt � s1ðtÞÞg þ cðtÞ expfx2ðt � s2ðtÞÞg�Dt

6 B1T expfx1ðtÞg þ CT expfx2ðg2Þg

6B1A2T

B2 �mA2expfðA1 þ A1ÞTg þ CT expfx2ðg2Þg:

Then it follows that

expfx2ðg2ÞgPA1

C� B1A2

CðB2 �mA2ÞexpfðA1 þ A1ÞTg;

that is

x2ðg2ÞP lnA1

C� B1A2

CðB2 �mA2ÞexpfðA1 þ A1ÞTg

� �:

Hence,

x2ðtÞP x2ðg2Þ �Z

IT

j xD2 ðtÞ j Dt 6 ln

A1

C� B1A2

CðB2 �mA2ÞexpfðA1 þ A1ÞTg

� � ðA2 þ A2ÞT ¼: M3: ð3:10Þ

In view of (3.9) and (3.10), we obtain

maxt2IT

j x2ðtÞ j6 maxfM2;M3g ¼: M4:

Clearly, Mi (i = 1,2,3,4) are independent of the choice of k. Under the assumptions in Theorem 3.1 it is easy to show thatthe system of algebraic equations

A1 � B1v1 �Cv2

1þmv1¼ 0; A2 �

B2v1

1þmv1¼ 0

has a unique solution ðv�1;v�2ÞT 2 R2 with v�i > 0; i ¼ 1;2. Take M = M1 + M4 + M5, where M5 > 0 is taken sufficiently large such

that kðlnfv�1g; lnfv�2gÞTk ¼j lnfv�1g j þ j lnfv�2g j< M5 and define

X ¼ fxðtÞ ¼ ðx1ðtÞ; x2ðtÞÞT 2 X : kxk < Mg:

Therefore L x – kNx for any k 2 (0,1], x 2 DomL \ oX. On the other hand, when x 2 @X \ KerL ¼ @X \ R2, x is a constantvector in R2 with kxk = M, then

QNx ¼ 1T

A1 � B1 expfx1g �C expfx2g

1þm expfx1g

� ;

1T

A2 � B2expfx1g

1þm expfx1g

� � T

–0:

Furthermore, in view of the assumptions in Theorem 3.1, it is easy to see that

degfJQN;X \ KerL;0g–0;

where J is the identity mapping since ImP = KerL. Thus, we proved that X satisfies all the requirements in Lemma 2.3. Hence,system (1.3) has at least one T-periodic solution. The proof is complete. h

Remark 3.1. By Remark 1.1, we know that (1.4) and (1.5) has at least one periodic solution if (1.3) has at least one periodicsolution.

3438 J. Liu et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3432–3438

References

[1] He HZ. Stability and delays in a predator–prey system. J Math Anal Appl 1996;198:335–70.[2] Li Y. Periodic solutions of a periodic delay predator–prey system. Proc Amer Math Soc 1999;127:1331–5.[3] Xu R, Ma Z. Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure. Chaos, Solitons and Fractals

2008;38:669–84.[4] Zhang Z, Hou Z, Wang L. Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking. Nonlinear Anal

2008;68:2608–22.[5] Zhang W, Zhu D, Bi P. Multiple positive periodic solutions of a delayed discrete predator–prey system with type IV functional responses. Appl Math Lett

2007;20:1031–8.[6] Zhu H, Wang K, Li X. Existence and global stability of positive periodic solutions for predator–prey system with infinite delay and diffusion. Nonlinear

Anal: Real World Appl 2007;8:872–86.[7] Guo H, Chen L. The effects of impulsive harvest on a predator–prey system with distributed time delay. Commun Nonlinear Sci Numer Simulat

2009;14:2301–9.[8] Zhao CJ. On a periodic predator–prey system with time delays. J Math Anal Appl 2007;331:978–85.[9] Bohner M, Peterson A. Dynamic equation on time scales. An introduction with applications. Boston: Birkhauser; 2001.

[10] Kaufmann ER, Raffoul YN. Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J Math Anal Appl 2006;319:315–25.[11] Bohner M, Fan M, Zhang J. Existence of periodic solutions in predator–prey and competition dynamic systems. Nonlinear Anal: Real World Appl

2006;7:1193–204.[12] Agarwal R, Bohner M, Peterson A. Inequalities on time scales: a survey. Math Ineq Appl 2001;4(4):535–57.[13] Gaines, Mawhin JL. Coincidence degree and nonlinear differential equations. Berlin: Springer; 1977.[14] Agiza HN, ELabbasy EM, EL-Metwally H, Elsadany AA. Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlinear Anal.: Real

World Appl. 2009;10:116–29.[15] Wang FH, Yeh CC, Yu SL, Hong CH. Youngs inequality and related results on time scales. Appl Math Lett 2005;18:983–98.[16] Xing Y, Han M, Zheng G. Initial value problem for first-order integro-differential equation of Volterra type on time scale. Nonlinear Anal

2005;60:429–42.