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Positive periodic solutions for neutral delay ratio-dependent predator–prey model with Holling type III functional response q Guirong Liu ⇑, Jurang Yan School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China

a r t i c l e

i n f o

Keywords: Predator–prey model Ratio-dependent Periodic solution Neutral Coincidence degree

a b s t r a c t By using a continuation theorem based on coincidence degree theory, some new and interesting sufﬁcient conditions are obtained for the existence of positive periodic solutions for neutral delay ratio-dependent predator–prey model with Holling type III functional response

8 2 ðtÞ < x0 ðtÞ ¼ xðtÞ½rðtÞ aðtÞxðt r1 Þ qx0 ðt r2 Þ m2 cðtÞx yðtÞ; y2 ðtÞþx2 ðtÞ h i hðtÞx2 ðtsðtÞÞ : y0 ðtÞ ¼ yðtÞ dðtÞ þ : m2 y2 ðtsðtÞÞþx2 ðtsðtÞÞ An example is represented to illustrate the feasibility of our main results. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The dynamic relationship between predator and its prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator–prey models have been studied extensively (see, for example, [1–5] and references cited therein). Recently, there is a growing explicit biological and physiological evidence [6–8] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator–prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous ﬁeld and laboratory experiments and observations [9,10]. Based on the Michaelis–Menten or Holling type-II function, Arditi and Ginzburg [11] proposed a ratio-dependent function of the form

P

cðyxÞ x cx ¼ ¼ y m þ ðyxÞ my þ x

and the following ratio-dependent predator–prey model:

8 0 cxy ; < x ¼ xða bxÞ myþx : y0 ¼ y d þ fx : myþx

ð1:1Þ

q This work was supported by the Natural Science Foundation of China (No. 11001157), Tianyuan Mathematics Fund of China (No. 10826080) and the Youth Science Foundation of Shanxi Province (Nos. 2009021001-1, 2010021001-1). ⇑ Corresponding author. E-mail addresses: [email protected] (G. Liu), [email protected] (J. Yan).

0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.009

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Here xðtÞ and yðtÞ represent the densities of the prey and the predator at time t, respectively. a=b is the carrying capacity, d > 0 is the death rate of the predator, and a; c; m and f =c are positive constants that stand for the intrinsic growth rate of the prey, capturing rate, half saturation constant and conversion rate of the predator, respectively. Later on, Beretta and Kuang [12], Hsu et al. [6], Jost et al. [7], and Kuang and Beretta [8] investigated system (1.1). The ratio-dependent predator–prey models with or without time delays have been studied by many researchers recently and very rich dynamics have been observed (see, for example, [13–24] and references cited therein). In view of periodicity of the actual environment, Fan and Wang [24] established veriﬁable criteria for the global existence of positive periodic solutions of a more general delayed ratio-dependent predator–prey model with periodic coefﬁcients of the form

8 h i Rt cðtÞxðtÞyðtÞ > < x0 ðtÞ ¼ xðtÞ aðtÞ bðtÞ 1 kðt sÞxðsÞds myðtÞþxðtÞ ; h i f ðtÞxðt s ðtÞÞ > 0 : y ðtÞ ¼ yðtÞ dðtÞ þ myðtsðtÞÞþxðtsðtÞÞ :

ð1:2Þ

By substituting Holling type III functional response for Holling type II functional response in system (1.2), Wang and Li [23] studied a delayed ratio-dependent predator–prey model with Holling type III functional response of the form

8 h i Rt 2 ðtÞyðtÞ > < x0 ðtÞ ¼ xðtÞ aðtÞ bðtÞ 1 kðt sÞxðsÞds mcðtÞx 2 y2 ðtÞþx2 ðtÞ ; h i 2 > : y0 ðtÞ ¼ yðtÞ dðtÞ þ 2 2f ðtÞx ðt2sÞ : m y ðtsÞþx ðtsÞ

ð1:3Þ

In 1991, Kuang [25] studied the local stability and oscillation of the following neutral delay Gause-type predator–prey system

(

h i 0 x0 ðtÞ ¼ rxðtÞ 1 xðtsÞþKqx ðtsÞ yðtÞpðxðtÞÞ;

ð1:4Þ

y0 ðtÞ ¼ yðtÞ½a þ bpðxðt rÞÞ:

As pointed out by Freedman and Wu [26] and Kuang [27], it would be of interest to study the existence of periodic solutions for periodic systems with time delay. The periodic solutions play the same role played by the equilibria of autonomous systems. In addition, in view of the fact that many predator–prey systems display sustained ﬂuctuations, it is thus desirable to construct predator–prey models capable of producing periodic solutions. In this paper, motivated by the above work, we shall study the existence of positive periodic solutions for the following neutral delay ratio-dependent predator–prey model with Holling type III functional response

8 2 ðtÞ < x0 ðtÞ ¼ xðtÞ½rðtÞ aðtÞxðt r1 Þ qx0 ðt r2 Þ m2 cðtÞx yðtÞ; y2 ðtÞþx2 ðtÞ h i hðtÞx2 ðtsðtÞÞ : y0 ðtÞ ¼ yðtÞ dðtÞ þ : m2 y2 ðtsðtÞÞþx2 ðtsðtÞÞ

ð1:5Þ

For convenience, we will use the notations:

jf j0 ¼ max fjf ðtÞjg; t2½0;x

f ¼ 1

x

Z

x

f ðtÞdt;

0

^f ¼ 1

x

Z

x

jf ðtÞjdt;

0

where f ðtÞ is a continuous x–periodic function. In this paper, we always make the following assumptions for system (1.5). (H1) q > 0; m > 0; r1 ; r2 are four constants. sðtÞ; rðtÞ; dðtÞ; aðtÞ; cðtÞ; hðtÞ are continuous x-periodic functions. In addi > 0; aðtÞ > 0; cðtÞ > 0; hðtÞ > 0 for any t 2 ½0; x; tion, r > 0; d n o 2r . (H2) qeB < 1, where B ¼ ln A þ qA þ ð^r þ rÞx and A ¼ maxt2½0;x aðtÞ (H3) c < 2mr. < h. (H ) d 4

Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [28], to derive a set of easily veriﬁable sufﬁcient conditions for the existence of positive periodic solutions of system (1.5). 2. Existence of positive periodic solution In this section, we shall study the existence of at least one positive periodic solution of system (1.5). The method to be used in this paper involves the applications of the continuation theorem of coincidence degree. For the readers’ convenience, we introduce a few concepts and results about the coincidence degree as follows. Let X; Z be real Banach spaces, L : DomL X ! Z be a linear mapping, and N : X ! Z be a continuous mapping. The mapping L is said to be a Fredholm mapping of index zero, if dimKerL ¼ codimImL < þ1 and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X ! X and Q : Z ! Z , such that ImP ¼ KerL; KerQ ¼ ImL ¼ ImðI Q Þ. It follows that the restriction LP of L to DomL \ KerP : ðI PÞX ! ImL is invertible. Denote the inverse of LP by K P .

G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

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The mapping N is said to be L-compact on X, if X is an open bounded subset of X; QNðXÞ is bounded and K P ðI Q ÞN : X ! X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ ! KerL. Lemma 2.1 (Continuation theorem [28]). Let X X be an open bounded set, L be a Fredholm mapping of index zero and N be Lcompact on X. Suppose (i) for each k 2 ð0; 1Þ; x 2 @ X \ DomL; Lx – kNx; (ii) for each x 2 @ X \ KerL; QNx – 0; (iii) degfJQN; X \ KerL; 0g – 0. Then Lx ¼ Nx has at least one solution in X \ DomL. Lemma 2.2. Assume that f ðtÞ; gðtÞ are continuous and nonnegative functions deﬁned on the interval ½a; b. Then there exists n 2 ½a; b such that

Z

b

f ðtÞgðtÞdt ¼ f ðnÞ

a

Z

b

gðtÞdt:

a

We are now in a position to state and prove our main result. Theorem 2.1. Assume that ðH1 Þ–ðH4 Þ hold. Then system (1.5) has at least one x -periodic solution with strictly positive components. Proof. Consider the following system:

8 < u01 ðtÞ ¼ rðtÞ aðtÞeu1 ðtr1 Þ qeu1 ðtr2 Þ u01 ðt r2 Þ : u0 ðtÞ ¼ dðtÞ þ 2

hðtÞe2u1 ðtsðtÞÞ m2 e2u2 ðtsðtÞÞ þe2u1 ðtsðtÞÞ

cðtÞeu1 ðtÞ eu2 ðtÞ m2 e2u2 ðtÞ þe2u1 ðtÞ

;

ð2:1Þ

;

where all functions are deﬁned as ones in system (1.5). It is easy to see that if system (2.1) has one x-periodic solution ðu1 ðtÞ; u2 ðtÞÞT , then ðx ðtÞ; y ðtÞÞT ¼ ðeu1 ðtÞ ; eu2 ðtÞ ÞT is a positive x-periodic solution of system (1.5). Therefore, to complete the proof it sufﬁces to show that system (2.1) has one x-periodic solution. Take

n o X ¼ u ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 C 1 ðR; R2 Þ : ui ðt þ xÞ ¼ ui ðtÞ; t 2 R; i ¼ 1; 2 and

n o Z ¼ u ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 CðR; R2 Þ : ui ðt þ xÞ ¼ ui ðtÞ; t 2 R; i ¼ 1; 2 and denote

juj1 ¼ max fju1 ðtÞj þ ju2 ðtÞjg; t2½0;x

kuk ¼ juj1 þ ju0 j1 :

Then X and Z are Banach spaces when they are endowed with the norms k k and j j1 , respectively. Let L : X ! Z and N : X ! Z be

Lðu1 ðtÞ; u2 ðtÞÞT ¼ ðu01 ðtÞ; u02 ðtÞÞT and

N

u1 ðtÞ u2 ðtÞ

2 ¼4

u1 ðtÞ u2 ðtÞ

e rðtÞ aðtÞeu1 ðtr1 Þ qeu1 ðtr2 Þ u01 ðt r2 Þ mcðtÞe 2 e2u2 ðtÞ þe2u1 ðtÞ 2u1 ðtsðtÞÞ

dðtÞ þ m2 e2uhðtÞe 2 ðtsðtÞÞ þe2u1 ðtsðtÞÞ

3 5:

With these notations system (2.1) can be written in the form

Lu ¼ Nu;

u 2 X:

n o Rx Obviously, KerL ¼ R2 ; ImL ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 Z : 0 ui ðtÞdt ¼ 0; i ¼ 1; 2 is closed in Z, and dimKerL ¼ codimImL ¼ 2. Therefore L is a Fredholm mapping of index zero. Now deﬁne two projectors P : X ! X and Q : Z ! Z as

1 ; u 2 ÞT ; ðu1 ðtÞ; u2 ðtÞÞT 2 X Pðu1 ðtÞ; u2 ðtÞÞT ¼ ðu

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G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

and

1 ; u 2 ÞT ; ðu1 ðtÞ; u2 ðtÞÞT 2 Z: Qðu1 ðtÞ; u2 ðtÞÞT ¼ ðu Then P and Q are continuous projectors such that

ImP ¼ KerL;

KerQ ¼ ImL ¼ ImðI QÞ:

Furthermore, the generalized inverse (to L) K P : ImL ! KerP \ DomL has the form

K P ðuÞ ¼

Z

t

uðsÞds

0

1

x

Z

x

Z

0

t

uðsÞdsdt:

0

Then QN : X ! Z and K P ðI Q ÞN : X ! X read

2

1

6x ðQNÞu ¼ 4 1

x

i 3 cðtÞeu1 ðtÞ eu2 ðtÞ u1 ðtr1 Þ rðtÞ aðtÞe dt 2u ðtÞ 2u ðtÞ 2 0 m e 2 þe 1 7 i 5 Rx h hðtÞe2u1 ðtsðtÞÞ dðtÞ þ dt 0 m2 e2u2 ðtsðtÞÞ þe2u1 ðtsðtÞÞ

Rx h

and

2R h

i 3 u1 ðsÞ u2 ðsÞ t e rðsÞ aðsÞeu1 ðsr1 Þ mcðsÞe ds q eu1 ðtr2 Þ eu1 ðr2 Þ 2 e2u2 ðsÞ þe2u1 ðsÞ 6 0 7 i ðK P ðI Q ÞNÞu ¼ 4 R h 5 t hðsÞe2u1 ðssðsÞÞ dðsÞ þ m2 e2u2 ðssðsÞÞ þe2u1 ðssðsÞÞ ds 0 2 R R h i 3 Rx u1 ðsÞ u2 ðsÞ x t e 1 dsdt x1 q 0 eu1 ðtr2 Þ eu1 ðr2 Þ dt rðsÞ aðsÞeu1 ðsr1 Þ mcðsÞe 2 e2u2 ðsÞ þe2u1 ðsÞ 0 0 x 6 7 i 4 R R h 5 x t hðsÞe2u1 ðssðsÞÞ 1 dðsÞ þ dsdt 0 x 0 m2 e2u2 ðssðsÞÞ þe2u1 ðssðsÞÞ 2

i 3 Rx h u1 ðsÞ u2 ðsÞ e t ds 1 rðsÞ aðsÞeu1 ðsr1 Þ mcðsÞe 2 e2u2 ðsÞ þe2u1 ðsÞ 6 x 2 0 7 h i 4

5: R 2u ðs s ðsÞÞ x hðsÞe 1 t 1 x 2 0 dðsÞ þ m2 e2u2 ðssðsÞÞ þe2u1 ðssðsÞÞ ds It is obvious that QN and K P ðI Q ÞN are continuous by the Lebesgue Theorem, and using the Arzela–Ascoli Theorem it is not difﬁcult to show that QNðXÞ is bounded, K P ðI Q ÞNðXÞ is compact for any open bounded set X X. Hence N is L-compact on X for any open bounded set X X. In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset X X. Corresponding to the operator equation Lu ¼ kNu; k 2 ð0; 1Þ, we have

8 h i u1 ðtÞ u2 ðtÞ e > < u01 ðtÞ ¼ k rðtÞ aðtÞeu1 ðtr1 Þ qeu1 ðtr2 Þ u01 ðt r2 Þ cðtÞe ; m2 e2u2 ðtÞ þe2u1 ðtÞ h i 2u1 ðtsðtÞÞ > : u02 ðtÞ ¼ k dðtÞ þ 2 2uhðtÞe : m e 2 ðtsðtÞÞ þe2u1 ðtsðtÞÞ

ð2:2Þ

Suppose that ðu1 ðtÞ; u2 ðtÞÞT 2 X is a solution of (2.2) for a certain k 2 ð0; 1Þ. Integrating (2.2) over the interval ½0; x leads to

Z

x

rðtÞ aðtÞeu1 ðtr1 Þ

0

cðtÞeu1 ðtÞ eu2 ðtÞ dt ¼ 0 m2 e2u2 ðtÞ þ e2u1 ðtÞ

ð2:3Þ

and

Z

x

dðtÞ þ

0

hðtÞe2u1 ðtsðtÞÞ dt ¼ 0: m2 e2u2 ðtsðtÞÞ þ e2u1 ðtsðtÞÞ

That is

Z

x

aðtÞeu1 ðtr1 Þ þ

0

cðtÞeu1 ðtÞ eu2 ðtÞ dt ¼ r x 2 2u ðtÞ 2u ðtÞ m e 2 þe 1

ð2:4Þ

and

Z 0

x

hðtÞe2u1 ðtsðtÞÞ x: dt ¼ d þ e2u1 ðtsðtÞÞ

m2 e2u2 ðtsðtÞÞ

By (2.2), (2.4) and ðH1 Þ, we have

Rx d u1 ðtÞ þ kqeu1 ðtr2 Þ dt 0 dt

R x Rx u1 ðtÞ u2 ðtÞ e 6 0 jrðtÞjdt rðtÞ aðtÞeu1 ðtr1 Þ mcðtÞe 2 e2u2 ðtÞ þe2u1 ðtÞ dt 0 i Rx h u1 ðtÞ u2 ðtÞ e dt ¼ ð^r þ rÞx: þ 0 aðtÞeu1 ðtr1 Þ þ mcðtÞe 2 e2u2 ðtÞ þe2u1 ðtÞ

ð2:5Þ

¼k

ð2:6Þ

G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

4345

In addition, (2.4) implies that

Z

x

aðtÞeu1 ðtr1 Þ dt 6 r x:

ð2:7Þ

0

In view of (2.7) and the periodicity of u1 ðtÞ, we ﬁnd

rx P

Z

x

aðtÞeu1 ðtr1 Þ dt ¼

Z

xþr2 r1

aðs r2 þ r1 Þeu1 ðsr2 Þ ds ¼

r2 r1

0

Z

x

aðt r2 þ r1 Þeu1 ðtr2 Þ dt

ð2:8Þ

0

and

rx P

Z

x

aðtÞeu1 ðtr1 Þ dt ¼

Z

xr1

aðs þ r1 Þeu1 ðsÞ ds ¼

Z

r1

0

x

aðt þ r1 Þeu1 ðtÞ dt:

ð2:9Þ

0

From (2.8) and (2.9), it is easy to see that

Z

x

aðt þ r1 Þeu1 ðtÞ þ aðt r2 þ r1 Þeu1 ðtr2 Þ dt 6 2r x:

0

According to the mean value theorem of differential calculus, we see that there exists n 2 ½0; x such that

aðn þ r1 Þeu1 ðnÞ þ aðn r2 þ r1 Þeu1 ðnr2 Þ 6 2r: This, together with ðH2 Þ, yields

u1 ðnÞ 6 ln

2r 6 ln A aðn þ r1 Þ

ð2:10Þ

eu1 ðnr2 Þ 6

2r 6 A: aðn þ r1 r2 Þ

ð2:11Þ

and

For any t 2 ½0; x, one can know from (2.6), (2.10), (2.11) and ðH2 Þ that

u1 ðtÞ þ kqeu1 ðtr2 Þ 6 u1 ðnÞ þ kqeu1 ðnr2 Þ þ

Z

x

0 u1 ðtr2 Þ

As kqe

d u1 ðtÞ þ kqeu1 ðtr2 Þ dt 6 ln A þ qA þ ð^r þ r Þx ¼ B: dt

> 0, one can ﬁnd that

u1 ðtÞ 6 B;

t 2 ½0; x:

ð2:12Þ

In view of (2.2), (2.4) and (2.12), we obtain

Z

u1 ðtÞ u2 ðtÞ e rðtÞ aðtÞeu1 ðtr1 Þ qeu1 ðtr2 Þ u0 ðt r2 Þ cðtÞe dt 1 m2 e2u2 ðtÞ þ e2u1 ðtÞ 0 Z x Z x Z x u ðtr Þ 0 cðtÞeu1 ðtÞ eu2 ðtÞ e 1 2 u ðt r2 Þ dt 6 dt þ jrðtÞjdt þ aðtÞeu1 ðtr1 Þ þ 2 2u ðtÞ q 1 2u1 ðtÞ 2 m e þ e 0 0 0 Z x Z x 0 0 B B u1 ðt r2 Þ dt ¼ ð^r þ r Þx þ qe u1 ðtÞ dt: 6 ð^r þ rÞx þ qe

x

ju01 ðtÞjdt ¼ k

0

x

Z

0

0

This, together with ðH2 Þ, implies that

Z 0

x

ju01 ðtÞjdt 6

1 ð^r þ r Þx: 1 qeB

ð2:13Þ

Since ðu1 ðtÞ; u2 ðtÞÞT 2 X, there exist ni ; gi 2 ½0; x ði ¼ 1; 2Þ such that

ui ðgi Þ ¼ max fui ðtÞg:

ui ðni Þ ¼ min fui ðtÞg; t2½0;x

ð2:14Þ

t2½0;x 2

From (2.3), (2.14) and inequality a2 þ b P 2ab, we have

xðr aeu1 ðg1 Þ Þ 6

Z

x

rðtÞ aðtÞeu1 ðtr1 Þ dt ¼

0

Z 0

x

cðtÞeu1 ðtÞ eu2 ðtÞ dt 6 m2 e2u2 ðtÞ þ e2u1 ðtÞ

Z 0

x

cðtÞeu1 ðtÞ eu2 ðtÞ 1 cx; dt ¼ 2m 2meu2 ðtÞ eu1 ðtÞ

which, together with ðH3 Þ, implies that

u1 ðg1 Þ P ln

2mr c : 2ma

ð2:15Þ

From (2.13) and (2.15), one can ﬁnd that, for any t 2 ½0; x,

u1 ðtÞ P u1 ðg1 Þ

Z

x 0

ju01 ðtÞjdt P ln

2mr c 1 ð^r þ rÞx ¼: b1 : 2ma 1 qeB

ð2:16Þ

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G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

It follows from (2.12) and (2.16) that

ju1 j0 6 maxfjBj; jb1 jg ¼: b2 :

ð2:17Þ

According to (2.5) and Lemma 2.2, we know that there exists f 2 ½0; x such that

d e2u1 ðfsðfÞÞ ¼ : m2 e2u2 ðfsðfÞÞ þ e2u1 ðfsðfÞÞ h Set f sðfÞ ¼ d þ kx, where d 2 ½0; x and k is an integer. Then

d e2u1 ðdÞ ¼ : 2u ðdÞ þe 1 h

ð2:18Þ

m2 e2u2 ðdÞ

In view of (2.14), (2.17), (2.18) and the monotonicity of the function

x2 , x2 þl

one can ﬁnd that

d e2u1 ðdÞ e2b2 6 2 2u ðn Þ 6 2 2u ðn Þ 2u ðdÞ 2 2 1 2 2 þ e2b2 þe m e h m e and

d e2u1 ðdÞ e2b2 P 2 2u ðg Þ P 2 2u ðg Þ : 2 2 þ e2u1 ðdÞ 2 2 þ e2b2 m e m e h Further, it follows from ðH4 Þ that

! d h 1 u2 ðn2 Þ 6 b2 þ ln 2 m2 d

ð2:19Þ

! d h 1 u2 ðg2 Þ P b2 þ ln : 2 m2 d

ð2:20Þ

and

In addition, it follows from (2.2), (2.5) and ðH1 Þ that, for any t 2 ½0; x,

Z 0

Z hðtÞe2u1 ðtsðtÞÞ dt 6 dðtÞ þ 2 2u ðt s ðtÞÞ 2u ðt s ðtÞÞ m e 2 þe 1

x

Z u0 ðtÞ dt ¼ k 2

x

0

x

jdðtÞjdt þ

Z

0

0

x

hðtÞe2u1 ðtsðtÞÞ ^ þ dÞ x; dt ¼ ðd þ e2u1 ðtsðtÞÞ

m2 e2u2 ðtsðtÞÞ

which, together with (2.19) and (2.20), implies that for any t 2 ½0; x,

u2 ðtÞ 6 u2 ðn2 Þ þ

Z

x

ju02 ðtÞjdt

0

! d h 1 ^ 6 b2 þ ln þ ðd þ dÞx ¼: b3 2 m2 d

and

u2 ðtÞ P u2 ðg2 Þ

Z

x

0

ju02 ðtÞjdt

! d h 1 ^ P b2 þ ln ðd þ dÞx ¼: b4 : 2 m2 d

Hence,

ju2 j0 6 maxfjb3 j; jb4 jg ¼: b5 :

ð2:21Þ

From (2.2), (2.12) and (2.21), one can ﬁnd that for any t 2 ½0; x,

ju01 ðtÞj ¼ k rðtÞ aðtÞeu1 ðtr1 Þ qeu1 ðtr2 Þ u01 ðt r2 Þ

cðtÞeu1 ðtÞ eu2 ðtÞ 1 6 jrj0 þ jaj0 eB þ qeB ju01 j0 þ jcj0 2m m2 e2u2 ðtÞ þ e2u1 ðtÞ

and

ju02 ðtÞj ¼ k dðtÞ þ

hðtÞe2u1 ðtsðtÞÞ 6 jdðtÞj þ jhðtÞj 6 jdj þ jhj : 0 0 2u ðt s ðtÞÞ þe 1

m2 e2u2 ðtsðtÞÞ

These, together with ðH2 Þ, yield

ju01 j0 6

1 1 B ¼: b6 : jrj þ jaj e þ jcj 0 0 0 1 qeB 2m

ð2:22Þ

and

ju02 j0 6 jdj0 þ jhj0 ¼: b7 :

ð2:23Þ

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G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

From (2.17), (2.21)–(2.23), we have

kuk ¼ juj1 þ ju0 j1 6 b2 þ b5 þ b6 þ b7 : In addition, it follows from ðH3 Þ and ðH4 Þ that the system of algebraic equations

8 u u eu1 c 2 e2u12e 2 2u1 ¼ 0; < r a m e þe : d þh

e2u1 m2 e2u2 þe2u1

ð2:24Þ

¼0

have a unique solution ðu1 ; u2 ÞT 2 R2 . Moreover, a single calculation yields

! lcd r h ; u2 ¼ u1 þ ln l; h a

u1 ¼ ln where l ¼

qﬃﬃﬃﬃﬃﬃﬃ d T h . Set b ¼ b2 þ b5 þ b6 þ b7 þ b0 , where b0 is taken sufﬁciently large such that the unique solution ðu1 ; u2 Þ of m2 d

(2.24) satisﬁes

T ðu1 ; u2 Þ ¼ u1 þ u2 < b0 : Clearly, b is independent of k. We now take

n

o

X ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 X : ðu1 ðtÞ; u2 ðtÞÞT < b : This satisﬁes condition (i) in Lemma 2.1. When ðu1 ðtÞ; u2 ðtÞÞT 2 @ X \ KerL ¼ @ X \ R2 ; ðu1 ðtÞ; u2 ðtÞÞT is a constant vector in R2 with ju1 j þ ju2 j ¼ b. Thus, we have

" QN

u1 u2

#

2 6 ¼4

u

u

eu1 c 2 e2u12e 2 2u1 r a m e þe e2u1 þh d m2 e2u2 þe2u1

3 7 5–

" # 0 : 0

This proves that condition (ii) in Lemma 2.1 is satisﬁed. Taking J ¼ I : ImQ ! KerL; ðu1 ; u2 ÞT ! ðu1 ; u2 ÞT , a direct calculation shows that

2 6 degfJQN; X \ KerL; 0g ¼ sgn det 6 4

eu1 ceu1 eu2 a 2m2 h

2u 2u 2 e 1 2u 2u 2 2 2 ðm e þe 1 Þ

m2 e

2u 2u 1e 2 2u 2u ðm2 e 2 þe 1 Þ2

e

ceu1 eu2

2u 2u 2 e 1 2u 2u 2 2 2 ðm e þe 1 Þ

2m2 h

2u 2u 1e 2 2u 2u ðm2 e 2 þe 1 Þ2

m2 e

e

3

( 7 7 ¼ sgn 2m2 a h 5

)

e3u1 e2u2 ðm2 e2u2 þ e2u1 Þ2

– 0:

By now we have proved that X satisﬁes all the requirements in Lemma 2.1. Hence, (2.1) has at least one x-periodic solution. Accordingly, system (1.5) has at least one x-periodic solution with strictly positive components. The proof of Theorem 2.1 is complete. h Remark 2.1. From Theorem 2.1, we can see that the deviating arguments positive periodic solution of system (1.5).

r1 ; r2 ; sðtÞ have no effect on the existence of

Remark 2.2. It is easy to see that ðH4 Þ is also the necessary condition for the existence of positive x-periodic solutions of system (1.5). Remark 2.3. From the proof of Theorem 2.1, we see that Theorem 2.1 is also valid for both advance type and mixed type if

q ¼ 0. Consequently, we can obtain the following corollary. Corollary 2.1. Assume that ðH1 Þ; ðH3 Þ and ðH4 Þ hold. Then the following delayed ratio-dependent predator–prey model with Holling type III functional response

8 cðtÞx2 ðtÞ > < x0 ðtÞ ¼ xðtÞ½rðtÞ aðtÞxðt r1 Þ m2 y2 ðtÞþx2 ðtÞ yðtÞ; h i > : y0 ðtÞ ¼ yðtÞ dðtÞ þ 2 2 hðtÞx2 ðts2ðtÞÞ m y ðtsðtÞÞþx ðtsðtÞÞ has at least one x-periodic solution with strictly positive components.

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G. Liu, J. Yan / Applied Mathematics and Computation 218 (2011) 4341–4348

Next consider the following neutral ratio-dependent predator–prey model with state dependent delays

8 2 ðtÞ < x0 ðtÞ ¼ xðtÞ½rðtÞ aðtÞxðt r1 Þ qx0 ðt r2 Þ 2 cðtÞx yðtÞ; m y2 ðtÞþx2 ðtÞ h i 2 ðtdðt;xðtÞ;yðtÞÞÞ hðtÞx : y0 ðtÞ ¼ yðtÞ dðtÞ þ 2 2 ; m y ðtdðt;xðtÞ;yðtÞÞÞþx2 ðtdðt;xðtÞ;yðtÞÞÞ

ð2:25Þ

where dðt; x; yÞ is continuous functions and x-periodic functions with respect to t. Theorem 2.2. Suppose that ðH1 Þ–ðH4 Þ hold. Then system (2.25) has at least one x -periodic solution with strictly positive components. Proof. The proof is similar to the proof of Theorem 2.1 and hence is omitted here. h In order to illustrate the feasibility of our results, we give the following example. Example 2.1. In system (1.5), let m ¼ 1; q ¼ 14 ; rðtÞ ¼ 3 cosð12ptÞ; aðtÞ ¼ 7 þ sinð12ptÞ; cðtÞ ¼ 2 þ sinð12ptÞ; dðtÞ ¼ 3 sinð12ptÞ; hðtÞ ¼ 4 þ cosð12ptÞ. Further, a straightforward calculation shows that

^r ¼ r ¼ 3;

c ¼ 2;

¼ 3; d

¼ 4; h

1 6

x¼ ; A¼1

and

B ¼ ln A þ qA þ ð^r þ r Þx ¼

1 þ 1 ¼ 1:25: 4

< h. In addition, Hence, c < 2mr and d

1 4

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