Positive periodic solutions for a class of neutral delay Gause-type predator–prey system

Positive periodic solutions for a class of neutral delay Gause-type predator–prey system

Nonlinear Analysis 71 (2009) 4438–4447 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Po...

570KB Sizes 0 Downloads 29 Views

Nonlinear Analysis 71 (2009) 4438–4447

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Positive periodic solutions for a class of neutral delay Gause-type predator–prey systemI Guirong Liu ∗ , Weiping Yan, Jurang Yan School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China

article

abstract

info

Article history: Received 25 November 2008 Accepted 3 March 2009

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 (t ) = x(t )[r (t ) − a(t )x(t − σ1 ) − ρ x0 (t − σ2 )] − φ(t , x(t ))y(t − τ1 (t )), y0 (t ) = y(t )[−d(t ) + h(t , x(t − τ2 (t )))].



MSC: 34K13 34C25 92D25

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. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Predator–prey system Periodic solution Neutral Coincidence degree

1. Introduction Modelling of ecological systems has received a great deal of attention from theoretical ecologists in the last few decades. Much focus has been on mathematical models of these systems, since they have substantially contributed to the understanding of the dynamics of systems by forging strong links between models and available data. Mathematical analysis of such models frequently presents interesting and challenging aspects of dynamical systems theory. There is a large body of literature on population dynamics in ecological modelling, particularly in predator–prey systems [1–22]. A well-known model of such systems is the predator–prey model of Gause-type with a monotonic-bounded functional response [5]. The dynamics of this model is described by the following differential equations: x(t )

 

x (t ) = rx(t ) 1 −



y (t ) = y(t )(−D + µφ(x(t ))),

0

0



K



− y(t )φ(x(t )),

(1.1)

where x and y are the prey and the predator population size, 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 µ and D are the conversion rate of prey to predator and predator death rate, respectively. The function φ is called the functional response of predator to prey, and describes the change in the rate of exploitation of prey by a predator as a result of a change in the prey density. Indeed, the growth of the predator is enhanced in the presence of the prey by an amount proportional to the number of prey. Thus, this functional

I This work was supported by the Mathematical Tianyuan Foundation of China (No. 10826080).



Corresponding author. E-mail addresses: [email protected] (G. Liu), [email protected] (W. Yan), [email protected] (J. Yan).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.002

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

4439

response can be interpreted as the proportion of prey that is eaten by the predator. A more detailed biological description of this model can be found in [5,14]. Most of the published studies have considered the model (1.1) with functional responses which satisfy the following hypotheses [6,13,16]:

(A1 ) (A2 ) (A3 ) (A4 )

φ(0) = 0; φ 0 (x) > 0, for x ≥ 0; φ 00 (x) < 0, for x ≥ 0; limx→+∞ φ(x) = c < ∞.

These studies have provided a detailed dynamics analysis of (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 [12], the model (1.1) is extended to incorporate a seasonal functional response φ(t , x) as: x(t )

 

x0 (t ) = rx(t ) 1 −



y (t ) = y(t )(−D + µφ(t , x(t ))),



0

K



− y(t )φ(t , x(t )),

(1.2)

where φ(t , x) satisfies the following conditions:

(A∗1 ) (A∗2 ) (A∗3 ) (A∗4 )

φ(t , 0) = 0, for t ∈ R; φx (t , x) > 0, for x > 0, t ∈ R; φxx (t , x) < 0, for x > 0, t ∈ R; limx→+∞ φ(t , x) = c < ∞.

Under the assumptions (A∗1 )–(A∗4 ), the functional response φ covers many examples appearing in the literature [8,11]. Recently, Ding and Jiang [3] studied the existence of positive periodic solutions for the following delayed Gause-type predator–prey systems:

 

x0 (t ) = x(t )f

 Z t,

−γ0

x(t + θ )dµ(θ )



− g (t , x(t ))y(t − τ (t )),

−γ



(1.3)

y0 (t ) = y(t )[−d(t ) + h(t , x(t − σ (t )))].

In 1991, Kuang [10] studied the local stability and oscillation of the following neutral delay Gause-type predator–prey system x(t − τ ) + ρ x0 (t − τ )

 

x0 (t ) = rx(t ) 1 −



y (t ) = y(t )[−α + β p(x(t − σ ))].





K

0

− y(t )p(x(t )),

(1.4)

As pointed out by Freedman and Wu [23] and Kuang [9], 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 of the following more general neutral delay Gause-type predator–prey system:



x0 (t ) = x(t ) r (t ) − a(t )x(t − σ1 ) − ρ x0 (t − σ2 ) − φ(t , x(t ))y(t − τ1 (t )), y0 (t ) = y(t )[−d(t ) + h(t , x(t − τ2 (t )))].





(1.5)

For convenience, we will use the notations:

|f |0 = max {|f (t )|}, t ∈[0,ω]

f¯ =

1

ω

ω

Z 0

f (t )dt ,

fˆ =

1

ω

Z

ω

|f (t )|dt

0

and Q (v) =

1

ω

ω

Z

q(t , v)dt ,

0

where f (t ) is a continuous ω-periodic function, q(t , v) is a continuous function and ω-periodic with respect to t. In this paper, we always make the following assumptions for system (1.5).

(H1 ) ρ R ω> 0, σ1 , σ2 are R ω three constants. τ1 (t ), τ2 (t ), r (t ), a(t ) and d(t ) are continuous ω-periodic functions. In addition, r ( t ) dt > 0 , d(t )dt > 0 and a(t ) > 0 for any t ∈ [0, ω]; 0 0 (H2 ) φ(t , v) and h(t , v) are continuous functions and ω-periodic with respect to t. φv (t , v) and hv (t , v) are also continuous functions.

4440

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

(H3 ) φ(t , 0) = 0, φv (t , v) > 0 and φvv (t , v) < 0 for t ∈ R, v ≥ 0. (H4 ) Rh(t , 0) = 0, hv (t , v)R> 0 for t ∈ R, v ≥ 0. ω ω (H5 ) 0 d(t )dt < supv≥0 0 h(t , v)dt. n o (H6 ) ρ eB < 1 , where B = ln A + ρ A + (ˆr + r¯ )ω and A = maxt ∈[0,ω] a2(rt¯) . Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [24], to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5). As corollaries, some applications are listed. 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 : Dom L ⊂ 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 Ker L = codim Im L < +∞ 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 . The mapping N is said to be L-compact on Ω , if Ω is an open bounded subset of X , QN (Ω ) is bounded and KP (I −Q )N :Ω → 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 [24, p. 40]). Let Ω ⊂ X be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on Ω . Suppose (i) for each λ ∈ (0, 1), x ∈ ∂ Ω ∩ Dom L, Lx 6= λNx; (ii) for each x ∈ ∂ Ω ∩ Ker L, QNx 6= 0; (iii) deg{JQN , Ω ∩ Ker L, 0} 6= 0. Then Lx = Nx has at least one solution in Ω ∩ Dom L. By (H4 ), we have H 0 (v) =

1

ω

ω

Z

hv (t , v)dt > 0.

0

Hence, H (v) is strictly increasing on [0, +∞). By this, (H1 ), (H4 ) and (H5 ), one can easily see that equation H (v) = d¯ has a unique 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

(H7 ) a¯ eD < r¯ , where D = | ln v0 | +

1 1−ρ eB

(ˆr + r¯ )ω.

Then system (1.5) has at least one ω-periodic solution with strictly positive components. Proof. Consider the following system: u1 0 (t ) = r (t ) − a(t )eu1 (t −σ1 ) − ρ eu1 (t −σ2 ) u1 0 (t − σ2 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) ,  u2 0 (t ) = −d(t ) + h t , eu1 (t −τ2 (t )) ,





(2.1)

where all functions are defined as ones in system (1.5). It is easy to see that if system (2.1) has one ω-periodic solution ∗ ∗ (u∗1 (t ), u∗2 (t ))T , then (x∗ (t ), y∗ (t ))T = (eu1 (t ) , eu2 (t ) )T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.1) has one ω-periodic solution. Take X = u = (u1 (t ), u2 (t ))T ∈ C 1 (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2





and Z = u = (u1 (t ), u2 (t ))T ∈ C (R, R2 ) : ui (t + ω) = ui (t ), t ∈ R, i = 1, 2





and denote

|u|∞ = max {|u1 (t )| + |u2 (t )|}, t ∈[0,ω]

kuk = |u|∞ + |u0 |∞ .

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

4441

Then X and Z are Banach spaces when they are endowed with the norms k · k and | · |∞ , respectively. Let L : X → Z and N : X → Z be L(u1 (t ), u2 (t ))T = (u1 0 (t ), u2 0 (t ))T and r (t ) − a(t )eu1 (t −σ1 ) − ρ eu1 (t −σ2 ) u1 0 (t − σ2 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) u (t )  . N 1 = u2 (t ) −d(t ) + h t , eu1 (t −τ2 (t ))











With these notations system (2.1) can be written in the form Lu = Nu,

u ∈ X.



Obviously, Ker L = R2 , Im L = (u1 (t ), u2 (t ))T ∈ Z : 0 ui (t )dt = 0, i = 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



u1 (t ) u¯ = ¯1 , u2 (t ) u2



u1 (t ) u¯ = ¯1 , u2 u2 (t )



 P



 



u1 ( t ) ∈X u2 ( t )



and

 Q

 



u1 ( t ) ∈ Z. u2 ( t )



Then P and Q are continuous projectors such that Im P = Ker L,

Ker Q = Im L = Im (I − Q ).

Furthermore, the generalized inverse (to L) KP : Im L → Ker P ∩ Dom L has the form KP (u) =

t

Z

u(s)ds − 0

ω

Z

1

ω

0

Z

t

u(s)dsdt .

0

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

1 Z

ω

r (t ) − a(t )eu1 (t −σ1 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt



 Z0 (QN )u =  ω ω  1 −d(t ) + h t , eu1 (t −τ2 (t )) dt ω 0





  

and t

Z

u1 (s−σ1 )

r (s) − a(s)e − φ s, e  Z0 (KP (I − Q )N )u =   t  −d(s) + h s, eu1 (s−τ2 (s)) ds



u1 (s)



e

u2 (s−τ1 (s))−u1 (s)



ds − ρ e



u1 (t −σ2 )

−e

u1 (−σ2 )

    

0



1

Z

ω

Z

t

u1 (s−σ1 )

u1 (s)

u2 (s−τ1 (s))−u1 (s)

1

r (s) − a(s)e − φ s, e e dsdt − ρ ω ω Z0 ω Z0 t − 1   u1 (s−τ2 (s)) −d(s) + h s, e dsdt ω 0 0  Z ω    u (s−τ (s))−u (s)  t 1 u1 (s−σ1 ) u1 (s) 2 1 1 − r ( s ) − a ( s ) e − φ s , e e ds  ω 2  .   Z0 ω −  t    1 u1 (s−τ2 (s)) − −d(s) + h s, e ds ω 2 0







ω

Z



e

0

u1 (t −σ2 )

−e

u1 (−σ2 )

 

dt 

 

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 QN (Ω ) is bounded, KP (I − Q )N (Ω ) is compact for any open bounded set Ω ⊂ X . Hence N is L-compact on Ω for any open bounded set Ω ⊂ X . In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω ⊂ X .

4442

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

Corresponding to the operator equation Lu = λNu, λ ∈ (0, 1), we have u1 0 (t ) = λ r (t ) − a(t )eu1 (t −σ1 ) − ρ eu1 (t −σ2 ) u1 0 (t − σ2 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) ,

(





u2 0 (t ) = λ −d(t ) + h t , eu1 (t −τ2 (t ))







.

(2.2)

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

Z

r (t ) − a(t )eu1 (t −σ1 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt = 0







(2.3)

0

and ω

Z

  −d(t ) + h t , eu1 (t −τ2 (t )) dt = 0.

0

That is ω

Z

a(t )eu1 (t −σ1 ) + φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt = r¯ ω







(2.4)

0

and ω

Z

h t , eu1 (t −τ2 (t )) dt = d¯ ω.



(2.5)

0

By (2.2) and (2.4), (H1 ) and (H3 ), we have ω

Z 0

Z ω d    u1 (t −σ2 ) r (t ) − a(t )eu1 (t −σ1 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt u ( t ) + λρ e dt = λ 1 dt 0 Z ω Z ω    |r (t )| dt + ≤ a(t )eu1 (t −σ1 ) + φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt 0

0

= (ˆr + r¯ )ω.

(2.6)

In addition, (2.4) implies that ω

Z

a(t )eu1 (t −σ1 ) dt ≤ r¯ ω.

(2.7)

0

In view of (2.7) and the periodicity of u1 (t ), we find r¯ ω ≥

ω

Z

a(t )e

u1 (t −σ1 )

Z

σ2 −σ1

0

ω

Z =

ω+σ2 −σ1

dt =

a(s − σ2 + σ1 )eu1 (s−σ2 ) ds

a(t − σ2 + σ1 )eu1 (t −σ2 ) dt

(2.8)

0

and r¯ ω ≥

ω

Z

a(t )eu1 (t −σ1 ) dt =

Z

a(s + σ1 )eu1 (s) ds

−σ1

0

ω

Z

ω−σ1

=

a(t + σ1 )eu1 (t ) dt .

(2.9)

0

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

Z

a(t + σ1 )eu1 (t ) + a(t − σ2 + σ1 )eu1 (t −σ2 ) dt ≤ 2r¯ ω.





0

According to the mean value theorem of differential calculus, we see that there exists ξ ∈ [0, ω] such that a(ξ + σ1 )eu1 (ξ ) + a(ξ − σ2 + σ1 )eu1 (ξ −σ2 ) ≤ 2r¯ . This, together with (H6 ), yields u1 (ξ ) ≤ ln

2r¯ a(ξ + σ1 )

≤ ln A

(2.10)

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

4443

and eu1 (ξ −σ2 ) ≤

2r¯ a(ξ + σ1 − σ2 )

≤ A.

(2.11)

For any t ∈ [0, ω], one can know from (2.6), (2.10) and (2.11) that u1 (t ) + λρ e

u1 (t −σ2 )

≤ u1 (ξ ) + λρ e

u1 (ξ −σ2 )

ω

Z + 0

d   u1 (t −σ2 ) dt u1 (t ) + λρ e dt

≤ ln A + ρ A + (ˆr + r¯ )ω = B. As λρ eu1 (t −σ2 ) > 0, one can find that u1 (t ) ≤ B,

t ∈ [0, ω].

(2.12)

Since (u1 (t ), u2 (t )) ∈ X , there exist ξi , ηi ∈ [0, ω] (i = 1, 2) such that T

ui (ξi ) = min {ui (t )},

ui (ηi ) = max {ui (t )}.

t ∈[0,ω]

(2.13)

t ∈[0,ω]

Set

α1 = min {φv (t , eB )},

α2 = max {φv (t , 0)}.

t ∈[0,ω]

t ∈[0,ω]

(2.14)

In view of (H2 ) and (H3 ), it is easy to see 0 < α1 < α2 . From (2.3) and (2.12)–(2.14), (H2 ) and (H3 ), one can find that

ω(¯r + a¯ e ) ≥ B

ω

Z

Z0 ω = Z0 ω = Z0 ω ≥

r (t ) − a(t )eu1 (t −σ1 ) dt





 φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt    φ t , eu1 (t ) − φ(t , 0) eu2 (t −τ1 (t ))−u1 (t ) dt φv (t , eB )eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt

0

≥ ωα1 eu2 (ξ2 ) , which implies u2 (ξ2 ) ≤ ln



r¯ + a¯ eB



α1

.

(2.15)

In view of (2.2) and (2.5) and (H4 ), we have ω

Z 0

|u2 0 (t )|dt = λ Z ≤

Z

ω

 −d(t ) + h t , eu1 (t −τ2 (t )) dt 0 Z ω ω  |d(t )| dt + h t , eu1 (t −τ2 (t )) dt

0

0

= (dˆ + d¯ )ω.

(2.16)

From (2.15) and (2.16), one can find that, for any t ∈ [0, ω], u2 (t ) ≤ u2 (ξ2 ) +

ω

Z

|u2 0 (t )|dt ≤ ln

0



r¯ + a¯ eB

α1



+ (dˆ + d¯ )ω =: β1 .

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

Z 0

Z

ω

 r (t ) − a(t )eu1 (t −σ1 ) − ρ eu1 (t −σ2 ) u1 0 (t − σ2 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt 0 Z ω ω    |r (t )| dt + a(t )eu1 (t −σ1 ) + φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt 0 0 Z ω u (t −σ ) 0 1 2 e +ρ u1 (t − σ2 ) dt

|u1 0 (t )|dt = λ Z ≤

0

(2.17)

4444

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

≤ (ˆr + r¯ )ω + ρ eB = (ˆr + r¯ )ω + ρ e

B

ω

Z

Z0 ω

0 u1 (t − σ2 ) dt 0 u1 (t ) dt .

0

This, together with (H6 ), implies that ω

Z

1

|u1 0 (t )|dt ≤

1 − ρ eB

0

(ˆr + r¯ )ω.

(2.18)

From (2.5) and (2.13) and the monotonicity of h(t , v) with respect to v , we have H e

u1 (η1 )





ω

Z

1

ω

h t , eu1 (t −τ2 (t )) dt = d¯



0

and H e

u1 (ξ1 )





1

ω

Z

ω

h t , eu1 (t −τ2 (t )) dt = d¯ .



0

By these and the monotonicity of H, we obtain eu1 (η1 ) ≥ v0 ,

eu1 (ξ1 ) ≤ v0 .

That is, u1 (η1 ) ≥ ln v0 ,

u1 (ξ1 ) ≤ ln v0 .

Hence, there exists ζ ∈ [0, ω] such that u1 (ζ ) = ln v0 . It follows from (2.18) that for any t ∈ [0, ω],

|u1 (t )| ≤ |u1 (ζ )| +

ω

Z

|u1 0 (t )|dt ≤ | ln v0 | +

0

1 1 − ρ eB

(ˆr + r¯ )ω = D.

(2.19)

It follows from (2.14) and (H3 ) that, for any t ∈ [0, ω], v > 0,

φ(t , v) = φ(t , v) − φ(t , 0) ≤ α2 v.

(2.20)

In view of (2.3), (2.13), (2.19) and (2.20), we have

ω r¯ − a¯ e 

D



ω

Z



≤ Z0 ω



 φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt

= Z0 ω

α2 eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) dt

≤ 0

= α2

r (t ) − a(t )eu1 (t −σ1 ) dt

ω

Z

eu2 (t −τ1 (t )) dt

0

≤ α2 ωeu2 (η2 ) . Further, from (H7 ), we obtain u2 (η2 ) ≥ ln r¯ − a¯ eD − ln α2 .



This, together with (2.16), leads to u2 (t ) ≥ u2 (η2 ) −

ω

Z

 |u2 0 (t )|dt ≥ ln r¯ − a¯ eD − ln α2 − (dˆ + d¯ )ω =: β2 ,

t ∈ [0, ω].

(2.21)

0

In view of (2.17) and (2.21), we have

|u2 |0 ≤ max{|β1 |, |β2 |} =: β3 . In addition, it follows from (2.2), (2.12), (2.20) and (2.22) that for any t ∈ [0, ω],

   |u1 0 (t )| = λ r (t ) − a(t )eu1 (t −σ1 ) − ρ eu1 (t −σ2 ) u1 0 (t − σ2 ) − φ t , eu1 (t ) eu2 (t −τ1 (t ))−u1 (t ) ≤ |r |0 + |a|0 eB + ρ eB |u1 0 |0 + α2 eβ3 .

(2.22)

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

4445

This, together with (H6 ), implies that

|u1 0 |0 ≤

1 1−ρ

eB

  |r |0 + |a|0 eB + α2 eβ3 =: β4 .

(2.23)

From (2.2) and (2.12), we have

|u2 0 (t )| ≤ |d|0 + max {h(t , eB )} =: β5 ,

t ∈ [0, ω].

t ∈[0,ω]

(2.24)

In view of (2.19) and (2.22)–(2.24), one can find that

kuk = |u|∞ + |u0 |∞ ≤ β3 + β4 + β5 + D. From (H7 ), we have r¯ > a¯ eD ≥ a¯ e| ln v0 | ≥ a¯ eln v0 = a¯ v0 , which implies that the algebraic equations



r¯ − a¯ eu1 − Φ eu1 eu2 −u1 = 0, −d¯ + H eu1 = 0



(2.25)

have a unique solution (u∗1 , u∗2 )T ∈ R2 . Set β = β3 + β4 + β5 + D + β0 , where β0 is taken sufficiently large such that the unique solution of (2.25) satisfies (u∗1 , u∗2 )T = |u∗1 | + |u∗2 | < β0 . Clearly, β is independent of λ. We now take

 Ω = (u1 (t ), u2 (t ))T ∈ X : (u1 (t ), u2 (t ))T < β . This satisfies condition (i) in Lemma 2.1. When (u1 (t ), u2 (t ))T ∈ ∂ Ω ∩ Ker L = ∂ Ω ∩ R2 , (u1 (t ), u2 (t ))T is a constant vector in R2 with |u1 | + |u2 | = β . Thus, we have

  QN

u1 u2

 =

r¯ − a¯ eu1 − Φ eu1 eu2 −u1 −d¯ + H eu1





  6=

0 . 0

This proves that condition (ii) in Lemma 2.1 is satisfied. Taking J = I : Im Q → Ker L, (u1 , u2 )T → (u1 , u2 )T , a direct calculation shows that

n 













deg{JQN , Ω ∩ Ker L, 0} = sign Φ eu1 H 0 eu1 eu2

o

= 1 6= 0.

By now we have proved that Ω satisfies all the requirements in Lemma 2.1. Hence, (2.1) has at least one ω-periodic solution. Accordingly, system (1.5) has at least one ω-periodic solution with strictly positive components. The proof of Theorem 2.1 is complete.  Remark 2.1. From Theorem 2.1, we can see that the deviating arguments σ1 , σ2 , τ1 (t ), τ2 (t ) have no effect on the existence of positive periodic solution of system (1.5). Remark 2.2. It is easy to see that (H5 ) is also the necessary condition for the existence of positive ω-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 ρ = 0. Consequently, we can obtain the following corollary. Corollary 2.1. In addition to (H1 )–(H5 ), suppose further that

(H8 ) a¯ eQ < r¯ , where Q = | ln v0 | + (ˆr + r¯ )ω. Then the following delayed Gause-type predator–prey system



x0 (t ) = x(t )[r (t ) − a(t )x(t − σ1 )] − φ(t , x(t ))y(t − τ1 (t )), y0 (t ) = y(t )[−d(t ) + h(t , x(t − τ2 (t )))]

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



x0 (t ) = x(t )[r (t ) − a(t )x(t − σ1 ) − ρ x0 (t − σ2 )] − φ(t , x(t ))y(t − τ3 (t , x(t ), y(t ))), y0 (t ) = y(t )[−d(t ) + h(t , x(t − τ4 (t , x(t ), y(t ))))],

where τi (t , x, y) (i = 3, 4) are continuous functions and ω-periodic functions with respect to t.

(2.26)

4446

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

Theorem 2.2. Suppose that (H1 )–(H7 ) hold. Then system (2.26) has at least one ω-periodic solution with strictly positive components. Proof. The proof is similar to the proof of Theorem 2.1 and hence is omitted here.



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:

   b(t )y(t − τ1 (t )) 0 0   , x ( t ) = x ( t ) r ( t ) − a ( t ) x ( t − σ ) − ρ x ( t − σ ) − 1 2  1 + mx(t )    c (t )x(t − τ2 (t ))  y0 (t ) = y(t ) −d(t ) + , 1 + mx(t − τ2 (t ))

(3.1)

which can be obtained by letting

φ(t , v) =

b(t )v 1 + mv

,

h(t , v) =

c (t )v 1 + mv

in system (1.5), where m > 0, ρ > 0, σ1 and σ2 are constants. In addition, r , d, τ1 , τ2 ∈ C (R, R) and a, b, c ∈ C (R, R+ ) are ω-periodic functions and r¯ > 0, d¯ > 0. Using Theorem 2.1, we have the following result. Theorem 3.1. In addition to (H6 ), suppose further that md¯ < c¯

and a¯ eD < r¯ , ¯ 1 where D = ln c¯ −dmd¯ + 1−ρ (ˆr + r¯ )ω. Then system (3.1) has at least one ω-periodic solution with strictly positive components. eB Example 3.2. Consider the following neutral delay predator–prey system with Ivlev-type functional response: x0 (t ) = x(t ) r (t ) − a(t )x(t − σ1 ) − ρ x0 (t − σ2 ) − b(t ) 1 − e−mx(t ) y(t − τ1 (t )),

(







y0 (t ) = y(t ) −d(t ) + c (t ) 1 − e−mx(t −τ2



 (t ))

,

(3.2)

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

 φ(t , v) = b(t ) 1 − e−mv ,

h(t , v) = c (t ) 1 − e−mv ,



where functions r , a, b, c , d, τi (i = 1, 2) and constants σ1 , σ2 , m, ρ are defined as in system (3.1). Using Theorem 2.1, we have the following result. Theorem 3.2. In addition to (H6 ), suppose further that c¯ > d¯

and a¯ eD < r¯ ,

1 where D = | ln v0 | + 1−ρ (ˆr + r¯ )ω and v0 = eB positive components.

1 m

ln c¯ −c¯ d¯ . Then system (3.2) has at least one ω-periodic solution with strictly

In order to illustrate the feasibility of our results, we give the following example. Example 3.3. In system (3.1), let m = 1, ρ = 3 − sin(20π t ), c (t ) = 2 + sin(20π t ), d(t ) = 1 + rˆ = r¯ = 3,

a¯ =

1 2

,

c¯ = 2,

, r (t ) = 3 + 2 sin(20π t ), a(t ) = 12 − 14 cos(20π t ), b(t ) = (20π t ). Further, a straightforward calculation shows that

1 100 1 cos 2

d¯ = 1,

A = 24,

and B = ln A + ρ A + (ˆr + r¯ )ω =

21 25

+ ln 24 = 4.0181.

Hence,

ρ eB =

1 100

e4.0181 = 0.5560 < 1.

ω=

1 10

G. Liu et al. / Nonlinear Analysis 71 (2009) 4438–4447

¯ In addition, D = ln c¯ −dmd¯ + a¯ eD =

1 2

1 1−ρ eB

4447

(ˆr + r¯ )ω = 1.3514. Further,

e1.3514 = 1.9314 < r¯ .

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

1 -periodic solution with strictly 10

References [1] A. Ardito, P. Ricciardi, Lyapunov functions for a generalized Gause-type model, J. Math. Biol. 33 (1995) 816–823. [2] J. Cui, Y. Takeuchi,Permanence, extinction and periodic solution of predator–prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl. 317 (2006) 464–474. [3] X. Ding, J. Jiang, Positive periodic solutions in delayed Gause-type predator–prey systems, J. Math. Anal. Appl. 339 (2008) 1220–1230. [4] X. Ding, J. Jiang, Multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with non-monotonic numerical responses, Math. Comput. Modelling 47 (2008) 1323–1331. [5] H.I. Freedman, Deterministic mathematical models in population ecology, in: HIFR Conf., 2nd ed., Edmonton, 1980. [6] M. Hesaaraki, S.M. Moghadas, Existence of limit cycles for predator–prey systems with a class of functional responses, Ecol. Modell 142 (2001) 1–9. [7] X. Hu, G. Liu, J. Yan, Existence of multiple positive periodic solutions of delayed predator–prey models with functional responses, Comput. Math. Appl. 52 (2006) 1453–1462. [8] A.A. King, W.M. Schaffer, The rainbow bridge: Hamiltonian limits and resonance in predator–prey dynamics, J. Math. Biol. 39 (1999) 439–469. [9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [10] Y. Kuang, On neutral delay logistic Gause-type predator–prey systems, Dyn. Stab. Syst. 6 (1991) 173–189. [11] X. Liu, L. Chen, Complex dynamics of Holling-type II Lotka-Volterra predator–prey system with impulsive perturbations in the predator, Chaos Solitons Fractals 16 (2003) 311–320. [12] S.M. Moghadas, M.E. Alexander, Dynamics of a generalized Gause-type predator–prey model with a seasonal functional response, Chaos Solitons Fractals 23 (2005) 55–65. [13] S.M. Moghadas, M.E. Alexander, B.D. Corbett, A non-standard numerical scheme for a generalized Gause-type predator–prey model, Physica D 188 (2004) 134–151. [14] J.D. Murray, Mathematical biology, Springer-Verlag, Berlin Heidelberg, New York, 1989. [15] A. Sikder, A.B. Roy, Persistence of a generalized Gause-type two prey-two predator pair linked by competition Mathematical Biosciences 122 (1994) 1–23. [16] J. Sugie, R. Kohno, R. Miyazaki, On a predator–prey system of Holling type, Proc. Amer. Math. Soc. 125 (1997) 2041–2050. [17] R. Xu, M.A.J. Chaplain, Persistence and global stability in a delayed Gause-type predator–prey system without dominating instantaneous negative feedbacks, J. Math. Anal. Appl. 265 (2002) 148–162. [18] R. Xu, Z. Ma, Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure, Chaos Solitons Fractals 38 (2008) 669–684. [19] S.J. Yang, B. Shi, Periodic solution for a three-stage-structured predator–prey system with time delay, J. Math. Anal. Appl. 341 (2008) 287–294. [20] Z. Zhang, Z. Hou, L. Wang, Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking, Nonlinear Anal. 68 (2008) 2608–2622. [21] C.J. Zhao, On a periodic predator–prey system with time delays, J. Math. Anal. Appl. 331 (2007) 978–985. [22] T. Zhao, Y. Kuang, H.L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator–prey systems, Nonlinear Anal. 28 (1997) 1373–1394. [23] H.I. Freedman, J. Wu, Periodic solutions of single species models with periodic delay, SIAM J. Math. Anal. Appl. 23 (1992) 689–701. [24] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.