Existence of positive periodic solutions for neutral delay Gause-type predator–prey system

Existence of positive periodic solutions for neutral delay Gause-type predator–prey system

Applied Mathematical Modelling 35 (2011) 5741–5750 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.el...

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Applied Mathematical Modelling 35 (2011) 5741–5750

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Existence of positive periodic solutions for neutral delay Gause-type predator–prey system 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

Article history: Received 7 June 2010 Received in revised form 28 April 2011 Accepted 8 May 2011 Available online 18 May 2011

a b s t r a c t By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions for neutral delay Gausetype predator–prey system.



x0 ðtÞ ¼ xðtÞ½rðtÞ  aðtÞxðt  rðtÞÞ  bðtÞx0 ðt  rðtÞÞ  /ðt; xðtÞÞyðt  s1 ðtÞÞ; y0 ðtÞ ¼ yðtÞ½dðtÞ þ hðt; xðt  s2 ðtÞÞÞ:

Keywords: Predator–prey system Periodic solution Neutral Coincidence degree

In addition, our results are applicable to neutral delay predator–prey systems with different types of functional responses such as Holling-type II and Ivlev-type. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction There has been a surge of interest in developing and analyzing models of interacting species in ecosystems. A much studied category of such models is that of two-species interaction, the so called predator–prey models [1–13], which has been critically important for analyzing the dynamics of complex ecological systems such as food chains. A well-known model of such systems is the predator–prey model of Gause-type, given by the following system of differential equations [14–17]:

(

  x0 ðtÞ ¼ rxðtÞ 1  xðtÞ  yðtÞ/ðxðtÞÞ; K y0 ðtÞ ¼ yðtÞðD þ lwðxðtÞÞÞ;

ð1:1Þ

where x and y are the prey and the predator densities, respectively. The parameter r is the prey’s intrinsic growth rate which describes the exponential growth of the prey population at low densities. The carrying capacity K represents the prey biomass at equilibrium in the absence of predators. The parameters l and D are the conversion rate of prey to predator and predator death rate, respectively. The functional response of predator to prey is denoted by / and defined as the change in the rate of predation by a single predator per unit time as a function of prey density. The growth of the predator depends on the presence of prey and is proportional to the number of predators. Thus, the numerical response w is defined as the change in reproduction rate with changing prey density. For the derivation of equations in (1.1) and more biological clarification, the reader may consult [14,15]. Most of the published studies have considered the model (1.1) with different types of functional responses such as sigmoidal, Holling-types II and III, and Ivlev-type. However, they neglect the effect of seasonality on the dynamics of the model. In [18], the model (1.1) is extended to incorporate a seasonal functional response /(t, x) as: 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 (No. 2009021001-1, No. 2010021001-1). ⇑ Corresponding author. E-mail addresses: [email protected] (G. Liu), [email protected] (J. Yan).

0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.006

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

(

  x0 ðtÞ ¼ rxðtÞ 1  xðtÞ  yðtÞ/ðt; xðtÞÞ; K

ð1:2Þ

y0 ðtÞ ¼ yðtÞðD þ l/ðt; xðtÞÞÞ; where /(t, x) satisfies the following conditions: (A1) (A2) (A3) (A4)

/(t, 0) = 0, for t 2 R; /x(t, x) > 0, for x > 0, t 2 R; /xx(t, x) < 0, for x > 0, t 2 R; limx?+1/(t, x) = c < 1.

Under the assumptions (A1)–(A4), the functional response / covers many examples appearing in the literature [19,20]. Recently, Ding and Jiang [21] studied the existence of positive periodic solutions for the following delayed Gause-type predator–prey systems:

(

 R  c x0 ðtÞ ¼ xðtÞf t; c 0 xðt þ hÞdlðhÞ  gðt; xðtÞÞyðt  sðtÞÞ; y0 ðtÞ ¼ yðtÞ½dðtÞ þ hðt; xðt  rðtÞÞÞ:

ð1:3Þ

In 1991, Kuang [22] 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:

In this paper, motivated by the above work, we shall consider the following more general neutral delay Gause-type predator–prey system:



x0 ðtÞ ¼ xðtÞ½rðtÞ  aðtÞxðt  rðtÞÞ  bðtÞx0 ðt  rðtÞÞ  /ðt; xðtÞÞyðt  s1 ðtÞÞ; y0 ðtÞ ¼ yðtÞ½dðtÞ þ hðt; xðt  s2 ðtÞÞÞ:

ð1:5Þ

As pointed out by Freedman and Wu [23] and Kuang [24], 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. To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.5). In addition, Gaines and Mawhin’s continuation theorem based on coincidence degree theory [25] is an effective tool in finding solutions to ordinary differential equation, delay differential equation and boundary value problem. Recently, by using the continuation theorem, many results concerned with the global existence of positive periodic solutions of biological models are obtained. However, only a few papers have been published on the existence of periodic solutions of neutral delay population models. The reason for it lies in the following two aspects. The first is that the criterion of L-compact of nonlinear operator N on the set X is difficult to establish; the second is that a priori bound of solutions is not easy to estimate. In this paper, our aim is, by using some analytical techniques to get over this difficulty, and employing Gaines and Mawhin’s continuation theorem to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5). 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

and

Kðv Þ ¼

1

x

Z

x

kðt; v Þdt;

0

where f(t) is a continuous x-periodic function, k(t, v) is a continuous function and x-periodic with respect to t. In this paper, we always make the following assumptions for system (1.5). (H1) r(t), a(t), b(t), d(t), s1(t), s2(t) and r(t) are continuous x-periodic functions. /(t, v) and h(t, v) are continuous functions  > 0 and a(t) > 0 and x-periodic with respect to t. /v(t, v) and hv(t, v) are also continuous functions. In addition, r > 0; d for any t 2 [0, x]; (H2) b 2 C1(R, [0, 1)), r 2 C2(R, R), r0 (t) < 1 and c(t) > 0, where

cðtÞ ¼ aðtÞ  g 0 ðtÞ;

gðtÞ ¼

bðtÞ ; 1  r0 ðtÞ

t 2 R:

(H3) /(t, 0) = 0, /v(t, v) > 0 and /vv(t, v) < 0 for t 2 R,v P 0.

G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

5743

(H4) h(t, 0) = 0, hv(t, v) > 0 for t 2 R, v P 0. Rx Rx (H5) 0 dðtÞdt < supv P0 0 hðt; v Þdt. (H6) eBmax{jbj0, jgj0} < 1, where

     1  r0 ðtÞ 2r þ jgj0 max þ ð^r þ r Þx: B ¼ ln 2r max t2½0;x t2½0;x cðtÞ cðtÞ

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 dim KerL = codim Im L < +1 and Im L 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 Im P = Ker L, Ker Q = Im L = Im (I  Q). It follows that the restriction LP of L to Dom L \ Ker P : (I  P)X ? Im L is invertible. Denote the inverse of LP by KP. Let X be an open bounded subset of X. Denote the closure of X by X. The mapping N is said to be L-compact on X, if QNðXÞ is bounded and K P ðI  Q ÞN : X ! X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q ? Ker L. Lemma 2.1 (Continuation Theorem [25, p.40]). Let X  X be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on X. Suppose (i) for each k 2 (0, 1), x 2 oX \ Dom L, Lx – kNx; (ii) for each x 2 oX \ Ker L, QNx – 0; (iii) deg{JQN, X \ Ker L, 0} – 0. Then Lx = Nx has at least one solution in X \ DomL. By (H4), we have

H 0 ðv Þ ¼

1

Z

x

0

x

hv ðt; v Þdt > 0;

for

v > 0:

 has a unique Hence, H(v) is strictly increasing on [0, +1). By this, (H1), (H4) and (H5), one can easily see that equation Hðv Þ ¼ d positive solution. Let v0 > 0 be its solution. We are now in a position to state and prove our main result. Theorem 2.1. In addition to (H1)–(H6), suppose further that x (H7) ceD < r, where D ¼ j ln v 0 j þ 1jgj

0e

B

 0 B jg j0 e þ ð^r þ r Þ .

Then system (1.5) has at least one x-periodic solution with strictly positive components. Proof. Consider the following system:

(

u01 ðtÞ ¼ rðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  / t; eu1 ðtÞ eu2 ðts1 ðtÞÞu1 ðtÞ ;

u02 ðtÞ ¼ dðtÞ þ h t; eu1 ðts2 ðtÞÞ ;

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

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

Z ¼ fu ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 CðR; R2 Þ : ui ðt þ xÞ ¼ ui ðtÞ; t 2 R; i ¼ 1; 2g 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 kk and jj1, respectively. Let L : X ? Z and N : X ? Z be

Lðu1 ðtÞ; u2 ðtÞÞT ¼ ðu01 ðtÞ; u02 ðtÞÞT

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

and



u1 ðtÞ

N

"

 ¼

u2 ðtÞ

rðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dðtÞ þ hðt; eu1 ðts2 ðtÞÞ Þ

# :

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

Lu ¼ Nu;

u 2 X:

n o Rx Obviously, Ker L ¼ R2 ; Im L ¼ ðu1 ðtÞ; u2 ðtÞÞT 2 Z : 0 ui ðtÞdt ¼ 0; i ¼ 1; 2 is closed in Z, and dim Ker L = codim Im L = 2. Therefore L is a Fredholm mapping of index zero. Now define two projectors P : X ? X and Q : Z ? Z as

P



 1 u ; 2 u



   1 u u1 ðtÞ ¼ ;  u2 ðtÞ u2



u1 ðtÞ



 ¼

u2 ðtÞ

u1 ðtÞ



2X

u2 ðtÞ

and

Q



 u1 ðtÞ 2 Z: u2 ðtÞ

Then P and Q are continuous projectors such that

Im P ¼ Ker L;

Ker Q ¼ Im L ¼ Im ðI  Q Þ:

Furthermore, the generalized inverse (to L) KP : Im L ? Ker P \ Dom L has the form

K P ðuÞ ¼

Z

t

uðsÞds 

0

1

x

Z 0

x

Z

t

uðsÞdsdt:

0

Then QN : X ? Z and KP(I  Q)N : X ? X read

" ðQNÞu ¼

1

x 1

x

# rðtÞ  cðtÞeu1 ðtrðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt Rx  dðtÞ þ hðt; eu1 ðts2 ðtÞÞ Þ dt 0 Rx  0

and

2Rt 

3 rðsÞ  cðsÞeu1 ðsrðsÞÞ  /ðs; eu1 ðsÞ Þeu2 ðss1 ðsÞÞu1 ðsÞ ds 7 gðtÞeu1 ðtrðtÞÞ þ gð0Þeu1 ðrð0ÞÞ 5 Rt  u1 ðss2 ðsÞÞ dðsÞ þ hðs; e Þ ds 2 0R x R t  3

1 rðsÞ  cðsÞeu1 ðsrðsÞÞ  / s; eu1 ðsÞ eu2 ðss1 ðsÞÞu1 ðsÞ dsdt 6 x 0 1 R0 x  7 u1 ðtrðtÞÞ 7 6  gð0Þeu1 ðrð0ÞÞ dt 4  x 0 gðtÞe 5 

R R x t 1 dðsÞ þ h s; eu1 ðss2 ðsÞÞ dsdt 0 x 0 "

Rx  # t 1 u1 ðsrðsÞÞ  /ðs; eu1 ðsÞ Þeu2 ðss1 ðsÞÞu1 ðsÞ ds x  2 0 rðsÞ  cðsÞe :  t 1 R x  u1 ðss2 ðsÞÞ Þ ds x  2 0 dðsÞ þ hðs; e

6 ðK P ðI  Q ÞNÞu ¼ 4

0

It is obvious that QN and KP(I  Q)N 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 = kNu, k 2 (0, 1), we have

(

 u01 ðtÞ ¼ k rðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ ; u02 ðtÞ ¼ k½dðtÞ þ hðt; eu1 ðts2 ðtÞÞ Þ:

ð2:2Þ

Suppose that (u1(t), u2(t))T 2 X is a solution of (2.2) for a certain k 2 (0, 1). Integrating (2.2) over the interval [0, x] leads to

Z 0

x

rðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt ¼ 0

ð2:3Þ

and

Z 0

x

dðtÞ þ h t; eu1 ðts2 ðtÞÞ dt ¼ 0:

ð2:4Þ

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

Note that

Z

x

0

Z x

0 bðtÞ u1 ðtrðtÞÞ 0 dt ¼ gðtÞ eu1 ðtrðtÞÞ dt e 0 ðtÞ 1  r 0 0 Z x Z x  u1 ðtrðtÞÞ x 0 u1 ðtrðtÞÞ ¼ gðtÞe  g ðtÞe dt ¼  g 0 ðtÞeu1 ðtrðtÞÞ dt; 0

bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞdt ¼

Z

x

0

0

which, together with (2.3), yields

Z

x

cðtÞeu1 ðtrðtÞÞ þ /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt ¼ r x:

ð2:5Þ

0

From (2.4), we have

Z

x

 x: hðt; eu1 ðts2 ðtÞÞ Þdt ¼ d

ð2:6Þ

0

From (2.2) and (2.5), (H2) and (H3), one can find

Z 0

Z x  d u1 ðtÞ þ kgðtÞeu1 ðtrðtÞÞ dt ¼ k rðtÞ  cðtÞeu1 ðtrðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt dt 0 Z x Z x  jrðtÞjdt þ cðtÞeu1 ðtrðtÞÞ þ /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt ¼ ð^r þ r Þx: 6

x

0

ð2:7Þ

0

Let t = w(p) be the inverse function of p = t  r(t). It is easy to see that c(w(p)) and r0 (w(p)) are all x-periodic functions. Further, it follows from (2.5) and (H2) that

rx P

Z

x

cðtÞeu1 ðtrðtÞÞ dt ¼

Z

xrðxÞ

cðwðpÞÞeu1 ðpÞ

rð0Þ

0

1 dp ¼ 1  r0 ðwðpÞÞ

Z 0

x

cðwðpÞÞ eu1 ðpÞ dp ¼ 1  r0 ðwðpÞÞ

Z 0

x

cðwðtÞÞ eu1 ðtÞ dt; 1  r0 ðwðtÞÞ

which yields that

2r x P

Z

x

0

 cðwðtÞÞ u1 ðtÞ u1 ðtrðtÞÞ dt: þ cðtÞe e 1  r0 ðwðtÞÞ

According to the mean value theorem of differential calculus, we see that there exists n 2 [0, x] such that

cðwðnÞÞ eu1 ðnÞ þ cðnÞeu1 ðnrðnÞÞ 6 2r : 1  r0 ðwðnÞÞ This, together with (H2), yields

   1  r0 ðtÞ u1 ðnÞ 6 ln 2r max t2½0;x cðtÞ

and

eu1 ðnrðnÞÞ 6 max

t2½0;x



 2r ; cðtÞ

which, together with (2.7) and (H6), imply that, for any t 2 [0, x],

Z x d  u1 ðtrðtÞÞ u1 ðtÞ þ kgðtÞeu1 ðtrðtÞÞ 6 u1 ðnÞ þ kgðnÞeu1 ðnrðnÞÞ þ dt u1 ðtÞ þ kgðtÞe dt 0      0 1  r ðtÞ 2r 6 ln 2r max þ jgj0 max þ ð^r þ r Þx ¼ B: t2½0;x t2½0;x cðtÞ cðtÞ

As kgðtÞeu1 ðtrðtÞÞ P 0, one can find that

u1 ðtÞ 6 B;

t 2 ½0; x:

ð2:8Þ

Since (u1(t), u2(t))T 2 X, there exist ni, gi 2 [0, x] (i = 1, 2) such that

ui ðni Þ ¼ min fui ðtÞg; t2½0;x

ui ðgi Þ ¼ max fui ðtÞg: t2½0;x

ð2:9Þ

Set

a1 ¼ min f/v ðt; eB Þg; a2 ¼ max f/v ðt; 0Þg: t2½0;x

t2½0;x

ð2:10Þ

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

In view of (H1) and (H3), it is easy to see 0 < a1 < a2. From (2.5) and (2.8)–(2.10), (H1) and (H3), one can find that

xðr þ ceB Þ P

Z

x

½rðtÞ  cðtÞeu1 ðtrðtÞÞ dt ¼

Z

0

Z

x

/ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt ¼

0

x

Z

x

½/ðt; eu1 ðtÞ Þ  /ðt; 0Þeu2 ðts1 ðtÞÞu1 ðtÞ dt

0

/v ðt; eB Þeu1 ðtÞ eu2 ðts1 ðtÞÞu1 ðtÞ dt P xa1 eu2 ðn2 Þ ;

P 0

which implies

u2 ðn2 Þ 6 ln

 r þ ceB :

ð2:11Þ

a1

In view of (2.2) and (2.6) and (H4), we have

Z

Z

x

ju02 ðtÞjdt ¼ k

0

x

j  dðtÞ þ hðt; eu1 ðts2 ðtÞÞ Þjdt 6

0

Z

x

jdðtÞjdt þ

0

Z

x

^ þ dÞ  x: hðt; eu1 ðts2 ðtÞÞ Þdt ¼ ðd

ð2:12Þ

0

From (2.11) and (2.12), one can find that, for any t 2 [0, x],

Z

u2 ðtÞ 6 u2 ðn2 Þ þ

 r þ ceB

x

ju02 ðtÞjdt 6 ln

0

a1

^ þ dÞ  x ¼: b : þ ðd 1

ð2:13Þ

In view of (2.2), (2.5) and (2.8), we obtain

Z

Z

x

ju01 ðtÞjdt ¼ k

0

x

0

Z

jrðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  /ðt; eu1 ðtÞ tÞeu2 ðts1 ðtÞÞu1 ðtÞ jdt

x

jrðtÞjdt þ

6 0

Z

x

cðtÞeu1 ðtrðtÞÞ þ /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt þ

Z

0

Z

x

jg 0 ðtÞeu1 ðtrðtÞÞ jdt

0

x

jbðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞjdt Z x jbðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞjdt: 6 ð^r þ rÞx þ jg 0 j0 eB x þ þ

0

0

In addition,

Z 0

x

Z xrðxÞ bðwðpÞÞ 1 u1 ðpÞ 0 dp u ðpÞ dp ¼ e 1 1  r0 ðwðpÞÞ 1  r0 ðwðpÞÞ rð0Þ rð0Þ Z xrðxÞ Z xrðxÞ Z x ¼ jgðwðpÞÞeu1 ðpÞ u01 ðpÞjdp 6 jgj0 eB u01 ðpÞjdp ¼ jgj0 eB ju01 ðpÞjdp

jbðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞjdt ¼

Z

xrðxÞ

jbðwðpÞÞeu1 ðpÞ u01 ðpÞj

rð0Þ

¼ jgj0 e

B

Z

rð0Þ

0

x

0

ju01 ðtÞjdt;

which implies that

Z 0

x

u0 ðtÞ dt 6 ð^r þ r Þx þ jg 0 j eB x þ jgj eB 0 0 1

Z

x

0

u0 ðtÞ dt: 1

From (H6), we obtain

Z 0

x

u0 ðtÞ dt 6 1

 1 ð^r þ rÞx þ jg 0 j0 eB x : 1  jgj0 eB

From (2.6) and (2.9) and the monotonicity of h(t, v) with respect to v, we have

Hðeu1 ðg1 Þ Þ P

1

Z

x

x

 hðt; eu1 ðts2 ðtÞÞ Þdt ¼ d

0

and

Hðeu1 ðn1 Þ Þ 6

1

Z

x

x

 hðt; eu1 ðts2 ðtÞÞ Þdt ¼ d:

0

By these and the monotonicity of H, we obtain

eu1 ðg1 Þ P v 0 ;

eu1 ðn1 Þ 6 v 0 :

That is,

u1 ðg1 Þ P ln v 0 ;

u1 ðn1 Þ 6 ln v 0 :

ð2:14Þ

G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

5747

Hence, there exists f 2 [0, x] such that u1(f) = ln v0. It follows from (2.14) and (H7) that for any t 2 [0, x],

ju1 ðtÞj 6 ju1 ðfÞj þ

Z

x

0

ju01 ðtÞjdt 6 j ln v 0 j þ

 0 B 1 jg j0 e x þ ð^r þ r Þx ¼ D: 1  jgj0 eB

It follows from (2.10) and (H3) that, for any t 2 [0, x],

ð2:15Þ

v > 0,

/ðt; v Þ ¼ /ðt; v Þ  /ðt; 0Þ 6 a2 v :

ð2:16Þ

In view of (2.5), (2.9), (2.15) and (2.16), we have



Z



x r  ceD 6

x

0

¼ a2

rðtÞ  cðtÞeu1 ðtrðtÞÞ dt ¼

Z

x

/ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ dt 6

0

Z

x

Z

x

a2 eu1 ðtÞ eu2 ðts1 ðtÞÞu1 ðtÞ dt

0

eu2 ðts1 ðtÞÞ dt 6 a2 xeu2 ðg2 Þ :

0

Further, from (H7), we obtain

u2 ðg2 Þ P lnðr  ceD Þ  ln a2 : This, together with (2.12), leads to

u2 ðtÞ P u2 ðg2 Þ 

Z

x 0

^ þ dÞ  x ¼: b ; ju02 ðtÞjdt P ln r  ceD  ln a2  ðd 2

t 2 ½0; x:

ð2:17Þ

In view of (2.13) and (2.17), we have

ju2 j0 6 maxfjb1 j; jb2 jg ¼: b3 :

ð2:18Þ

In addition, it follows from (2.2), (2.8), (2.16) and (2.18) that for any t 2 [0, x],

 ju01 ðtÞj ¼ k rðtÞ  aðtÞeu1 ðtrðtÞÞ  bðtÞeu1 ðtrðtÞÞ u01 ðt  rðtÞÞ  /ðt; eu1 ðtÞ Þeu2 ðts1 ðtÞÞu1 ðtÞ 6 jrj0 þ jaj0 eB þ jbj0 eB ju01 j0 þ a2 eb3 : This, together with (H6), implies that

ju01 j0 6

 1 jrj0 þ jaj0 eB þ a2 eb3 ¼: b4 : 1  jbj0 eB

ð2:19Þ

From (2.2) and (2.8), we have

ju02 ðtÞj 6 jdj0 þ max fhðt; eB Þg ¼: b5 ;

t 2 ½0; x:

t2½0;x

ð2:20Þ

In view of (2.15) and (2.18)–(2.20), one can find that

kuk ¼ juj1 þ ju0 j1 6 b3 þ b4 þ b5 þ D: From (H7), we have

r > ceD P cej ln v 0 j P celn v 0 ¼ cv 0 ; which implies that the algebraic equations



r  ceu1  Uðeu1 Þeu2 u1 ¼ 0;  þ Hðeu1 Þ ¼ 0 d

ð2:21Þ

2 have a unique solution ðu1; u2 ÞT 2 R . Set b = b3 + b4 + b5 + D + b0, where b0 is taken sufficiently large such that the unique    T solution of (2.21) satisfies ðu1 ; u2 Þ  ¼ ju1 j þ ju2 j < b0 . Clearly, b is independent of k.

We now take

X ¼ fðu1 ðtÞ; u2 ðtÞÞT 2 X : kðu1 ðtÞ; u2 ðtÞÞT k < bg: This satisfies condition (i) in Lemma 2.1. When (u1(t), u2(t))T 2 oX \ Ker L = oX \ R2, (u1(t) ,u2(t))T is a constant vector in R2 with ju1j + ju2j = b. Thus, we have

QN



u1 u2



 ¼

r  ceu1  Uðeu1 Þeu2 u1  þ Hðeu1 Þ d

 –

  0 : 0

This proves that condition (ii) in Lemma 2.1 is satisfied.

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

Taking J = I : Im Q ? Ker L, (u1, u2)T ? (u1, u2)T, a direct calculation shows that

    degfJQN; X \ KerL; 0g ¼ sign Uðeu1 ÞH0 ðeu1 Þeu2 ¼ 1 – 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. It is easy to see that (H5) is also the necessary condition for the existence of positive x-periodic solutions of system (1.5). Remark 2.2. The time delays s1 and s2 have no influence on the existence of positive periodic solutions to system (1.5). Remark 2.3. From the proof of Theorem 2.1, we see that Theorem 2.1 is also valid if b(t)  0 for t 2 R. Consequently, we can obtain the following corollary. Corollary 2.1. In addition to (H1)–(H5), suppose further that eW < r, where W ¼ j ln v 0 j þ ð^r þ r Þx. (H8) a Then the following delay Gause-type predator–prey system



x0 ðtÞ ¼ xðtÞ½rðtÞ  aðtÞxðt  rðtÞÞ  /ðt; xðtÞÞyðt  s1 ðtÞÞ; y0 ðtÞ ¼ yðtÞ½dðtÞ þ hðt; xðt  s2 ðtÞÞÞ

has at least one x-periodic solution with strictly positive components. Next consider the following neutral Gause-type predator–prey system with state dependent delays



x0 ðtÞ ¼ xðtÞ½rðtÞ  aðtÞxðt  rðtÞÞ  bðtÞx0 ðt  rðtÞÞ  /ðt; xðtÞÞyðt  s3 ðt; xðtÞ; yðtÞÞÞ; y0 ðtÞ ¼ yðtÞ½dðtÞ þ hðt; xðt  s4 ðt; xðtÞ; yðtÞÞÞÞ;

ð2:22Þ

where si(t, x, y) (i = 3, 4) are continuous functions and x-periodic functions with respect to t. Theorem 2.2. Suppose that (H1)–(H7) hold. Then system (2.22) 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

3. Applications In this section, we shall give some applications of the above results. Example 3.1. Consider the following neutral delay predator–prey system with Holling-II functional response:

8 h i s1 ðtÞÞ > < x0 ðtÞ ¼ xðtÞ rðtÞ  aðtÞxðt  rðtÞÞ  bðtÞx0 ðt  rðtÞÞ  eðtÞyðt ; 1þmxðtÞ h i f ðtÞxðt s ðtÞÞ > : y0 ðtÞ ¼ yðtÞ dðtÞ þ 1þmxðts2 ðtÞÞ ; 2

ð3:1Þ

which can be obtained by letting

/ðt; v Þ ¼

eðtÞv ; 1 þ mv

hðt; v Þ ¼

f ðtÞv 1 þ mv

in system (1.5), where m > 0 is a constant. In addition, e, f 2 C(R, (0, 1)) are x-periodic functions. Using Theorem 2.1, we have the following result. Theorem 3.1. In addition to (H1), (H2) and (H6), suppose further that

 < f and ceD < r; md

 0 B  x d where D ¼ ln f m r þ rÞ . Then system (3.1) has at least one x-periodic solution with strictly positive  þ 1jgj eB jg j0 e þ ð^ d 0 components.

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

Example 3.2. Consider the following neutral delay predator–prey system with Ivlev-type functional response:

(

x0 ðtÞ ¼ xðtÞ½rðtÞ  aðtÞxðt  rðtÞÞ  bðtÞx0 ðt  rðtÞÞ  eðtÞð1  emxðtÞ Þyðt  s1 ðtÞÞ;  y0 ðtÞ ¼ yðtÞ dðtÞ þ f ðtÞð1  emxðts2 ðtÞÞ Þ ;

ð3:2Þ

which is a special case of system (1.5) by letting

/ðt; v Þ ¼ eðtÞð1  emv Þ;

hðt; v Þ ¼ f ðtÞð1  emv Þ;

where m > 0 is a constant. In addition, e, f 2 C(R, (0, 1)) are x-periodic functions. Using Theorem 2.1, we have the following result. Theorem 3.2. In addition to (H1), (H2) and (H6), suppose further that

 < f and ceD < r ; d 

f x 1 0 B ^  where D ¼ j ln v 0 j þ 1jgj B ðjg j0 e þ r þ r Þ and v 0 ¼ m ln  . Then system (3.2) has at least one x-periodic solution with strictly f d 0e positive components. In order to illustrate the feasibility of our results, we give the following example.

 1 1  12 cosð20ptÞ ; Example 3.3. In system (3.1), let m ¼ 1; rðtÞ ¼ 3 þ 2 sinð20ptÞ; aðtÞ ¼ 12  14 cosð20ptÞ; bðtÞ ¼ 100 1 1 dðtÞ ¼ 1 þ 2 cosð20ptÞ; eðtÞ ¼ 3  sinð20ptÞ; f ðtÞ ¼ 2 þ sinð20ptÞ; rðtÞ ¼ 40p sinð20ptÞ. In addition, s1(t) and s2(t) are arbitrary continuous x-periodic functions. A straightforward calculation shows that

^r ¼ r ¼ 3;

¼ a

1 ; 2

f ¼ 2;

 ¼ 1; d

jbj0 ¼

3 ; 200



1 10

and

gðtÞ ¼

1 ; 100

cðtÞ ¼ aðtÞ ¼

1 ; 100

jg 0 j0 ¼ 0;

1 1  cosð20ptÞ: 2 4

Further,

jgj0 ¼

c ¼

1 ; 2

B ¼ ln 12 þ 0:84;

D ¼ 0:8310:

 < f . In addition, Hence, md

eB maxfjbj0 ; jgj0 g ¼ 12  e0:84 

3 ¼ 0:4169 < 1 200

and

ceD ¼

1  e0:8310 ¼ 1:1478 < r : 2

Consequently, all the conditions in Theorem 3.1 hold. Therefore, system Theorem 3.1 has at least one with strictly positive components.

1 -periodic 10

solution

4. Discussion In this paper, we have discussed the combined effects of periodicity of the ecological and environmental parameters and time delays due to the gestations on the dynamics of a neutral delay Gause-type predator–prey system. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to a neutral delay Gause-type predator–prey system. By Theorem 2.1, we see that the time delays s1 and s2 have no influence on the existence of positive periodic solutions to system (1.5). However, the time delay r plays the important role in determining the existence of positive periodic solutions of (1.5). In addition, we have not provided criteria for the uniqueness and stability of positive periodic solutions to system (1.5). So far, many investigators have successfully studied the existence of positive periodic solutions for population models by using coincidence degree theory and topological degree theory. But they cannot obtain the conditions of stability for population models discussed by them. This is mainly because only the boundedness of positive periodic solutions can be obtained in the study of the existence part of positive periodic solutions, but, according to the usual methods, the boundedness of a positive solution must be obtained in the study of the stability part. In addition, some investigators considered the existence and stability of positive periodic solutions for periodic population models. They first established the existence of positive periodic solutions, then established the stability of positive periodic solutions by using a Lyapunov functional. However,

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G. Liu, J. Yan / Applied Mathematical Modelling 35 (2011) 5741–5750

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