The dynamics of an impulsive delay predator–prey model with stage structure and Beddington-type functional response

Nonlinear Analysis: Real World Applications 11 (2010) 3567–3576

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The dynamics of an impulsive delay predator–prey model with stage structure and Beddington-type functional responseI Yuanfu Shao a,b,∗ , Binxiang Dai a a

School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075, China

b

School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, China

article

info

Article history: Received 21 April 2009 Accepted 27 January 2010 Keywords: Attractivity Delay Impulsive effect Extinction Permanence

abstract In this paper, an impulsive delay predator–prey model with stage structure and Beddington-type functional response is established. By using the discrete dynamical system determined by the stroboscopic map, we obtain the existence and global attractivity of the predator-extinction periodic solution. By use of the theory on delay and impulsive differential equation, we study the permanence of the system. Finally, an example is given to show the effectiveness of the main results. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Recently, it has been very much in vogue to study nonlinear dynamic systems. In the 1920s, Alfred Lotka and Vito Volterra developed a simple model of interacting species that still bears their joint names. This was a nearly linear model, but the predator–prey version displayed neutrally stable cycles. From then on, the dynamic relationship between predators and their 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. It is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for IPM strategy on ecosystem, the predators are released periodically every time T , and periodic catching or spraying pesticides is also applied. Hence the predator and prey experience a change of state abruptly. Consequently, it is natural to assume that these processes act in the form of impulse. Impulsive methods have been applied in almost every field of applied sciences, which can be found in [1–5]. On the other hand, in population dynamics, a functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes. In recent years, many different functional responses are studied, such as Holling type [6–8], Watt type [9,10] and Ivlev type [11,12]. The field of research on the dynamics of impulsive predator–prey model with functional response seems to be a new increasingly interesting area, which draws many scholar’s attention [6–12]. However, in the real world, many species usually go through two or more life stages as they proceed from birth to death. Thus it is practical to introduce the stage structure into predator–prey models, see [13,14] and the references cited therein. In addition, as Kuang [15] pointed out, generally speaking, the delay differential equation exhibits much more complicated

I The paper is supported by National Natural Science Foundation of China (10971229), Foundation of Science and Technology Department of Guizhou Province (2010) and Foundation of Educational Department of Guizhou Province (20090038). ∗ Corresponding author at: School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, China. Tel.: +86 851 6702059; fax: +86 851 6702059. E-mail address: [email protected] (Y. Shao).

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.01.004

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dynamics than ordinary differential equation since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. But to our knowledge, there have been few results on impulsive delay predator–prey models with stage structure. In this paper, we are concerned with the following impulsive delay predator–prey model with stage structure and Bedding-type functional response:

  cx(t )y2 (t )   x0 (t ) = x(t )(a − bx(t )) −       α + x(t ) + β y2 (t )      cx ( t ) y ( t ) cx ( t − τ ) y ( t − τ ) 2 2  0 − d τ 2  −e λ − d1 y1 (t ) y1 (t ) = λ α + x(t ) + β y2 (t ) α + x(t − τ ) + β y2 (t − τ )     cx(t − τ )y2 (t − τ )  0 2 −d2 τ   − d y ( t ) − ry ( t ) y ( t ) = e λ  2 2 2 2   α + x(t − τ ) + β y2 (t− τ )   +   x(t +) = (1 − p)x(t ) t = nT y1 (t ) = y1 (t ) + µ, y2 (t + ) = y2 (t )

t 6= nT (1.1)

where x(t ), y1 (t ), y2 (t ) represent the densities of prey, immature predator and mature predator, respectively, a is the percapita rate of predation of the predator, b is the rate of intra-specific competition of the prey, c is the maximum numbers of the prey that can be eaten by a predator per unit of time, α is saturation constant, β scale the impact of the predator interference, λ is the conversion coefficient, d1 , d2 are the mortality rates of the immature predator and mature predator, respectively, r is the intra-specific competition rate of the mature predator, τ represents a constant time to maturity for the immature predator, p (0 ≤ p < 1) represents partial impulsive harvest to prey by catching or pesticide, µ ≥ 0 is the amount of releasing the immature predator at fixed time t = nT . The initial conditions for system (1.1) are

(φ1 (s), φ2 (s), φ3 (s)) ∈ C ([−τ , 0], R3+ ),

φi (0) > 0, i = 1, 2, 3, R3+ = {x ∈ R3 : x ≥ 0}.

(1.2)

From the biological point of view, we only consider (1.1) in the biological meaning region D = {(x(t ), y1 (t ), y2 (t )) : x(t ) ≥ 0, y1 (t ) ≥ 0, y2 (t ) ≥ 0}. In this paper, the main purpose is, by using the discrete dynamical system determined by the stroboscopic map and the theory on delay and impulsive differential equation, to obtain the existence and global attractivity of the predator-extinction periodic solution and the permanence of system (1.1). The rest of this article is organized as follows. In Section 2, we present some preliminaries. In Section 3, the existence and global attractivity of predator-extinction periodic solution of system (1.1) are studied. In Section 4, the permanence of system (1.1) is investigated. Finally, the paper concludes with a brief discussion and an example in Section 5. 2. Preliminaries Denote by f = (f1 , f2 , f3 )T the map defined by the right-hand sides of the first three equations of system (1.1). Let V : R+ × R3+ → R+ , then V is said to belong to class V0 if: (i) V is continuous in ((n − 1)T , nT ] × R3+ and for each x ∈ R3+ , lim(t ,s)→((n−1)T + ,z ) V (t , s) = V ((n − 1)T , z ) and lim(t ,s)→(nT + ,z ) V (t , s) = V (nT + , z ) exist, (ii) V is locally Lipschitzian in z. Definition 2.1. Let V ∈ V0 . Then for (t , z ) ∈ ((n − 1)T , nT ]× R3+ , the upper right derivative of V (t , z ) with respect to system (1.1) is defined as: D+ V (t , z ) = lim sup

h→0+

1 h

[V ((t + h), z + hf (t , z )) − V (t , z )].

Definition 2.2. System (1.1) is said to be permanent if there exist positive constants m and M with M > m > 0 such that each positive solution (x(t ), y1 (t ), y2 (t )) of system (1.1) satisfies m ≤ x(t ), y1 (t ), y2 (t ) ≤ M for t sufficiently large. The solution of system (1.1) is continuously differentiable on ((n − 1)T , nT ], n ∈ Z+ (the set of all non-negative integers). Obviously, the global existence and uniqueness of solutions to system (1.1) are guaranteed by the smoothness properties of f , and more details can be seen in Refs. [1,2,16]. Next, we introduce some lemmas which will be used in the proofs of the main results. Lemma 2.1 ([2]). Considering the following impulse differential inequalities:



V 0 (t ) ≤ p(t )V (t ) + q(t ), t 6= tk , V (tk+ ) ≤ dk V (tk ) + bk , t = tk

(2.1)

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where p(t ), q(t ) ∈ C (R+ , R), dk ≥ 0, and bk are constants. If the sequence {tk } satisfies 0 ≤ t0 < t1 < t2 < · · · with limk→∞ tk = ∞; and V (t ) ∈ PC 0 (R+ , R), V (t ) is left continuous at tk , k ∈ Z+ . Then Rt

Y

V (t ) ≤ V (t0 )

dk e

t0 p(s)ds

X

+

t0 < tk < t

d j bk e

tk p(s)ds

t

Z

Rt

Y

+

Y

dk e

Rt

s p(θ )dθ

q(s)ds,

t ≥ t0 .

(2.2)

t0 s
t0 < tk < t tk < tj < t

If all the directions of the inequalities in (2.1) are reversed, then (2.2) holds true for the reversed inequality. Lemma 2.2 ([14]). Considering the following equation x(t ) = ax(t − τ ) − bx(t ) − cx2 (t ) where a, b, c , τ are all positive constants, x(t ) > 0 for t ∈ [−τ , 0). Then we have (i) If a > b, then limt →∞ x(t ) = a−c b ; (ii) If a < b, then limt →∞ x(t ) = 0. Lemma 2.3. Consider the following system



x(t ) = x(t )(a − bx(t )), t 6= nT x(t + ) = (1 − p)x(t ), t = nT .

(2.3)

If

(H1 ) : p < 1 − e−aT , then system (2.3) has a positive periodic solution x∗ (t ) and for any solution of system (2.3), we have

|x(t ) − x∗ (t )| → 0 as t → ∞, where x∗ ( t ) =

a(1 − p − e−aT ) b(1 − p −

e−aT )

+

bpe−a(t −nT )

,

x∗ (0+ ) =

a(1 − p − e−aT ) b(1 − e−aT )

,

nT < t ≤ (n + 1)T .

Proof. For system (2.3), integrating the first equation over the interval (nT , (n + 1)T ], we have x(t ) =

ax(nT + ) bx(nT + ) + (a − bx(nT + ))e−a(t −nT )

.

After the successive pulse, we can obtain the following stroboscopic map of (2.3), x((n + 1)T + ) , f (x(nT + )) =

(1 − p)x(nT + ) . + (a − bx(nT + ))e−bT

(2.4)

bx(nT + )

It is easy to see that (2.4) has a unique positive fixed point x˜ = stable since

a(1−p−e−aT ) . b(1−e−aT )

According to (H1 ), we know that x˜ is locally

∂ f (x(nT + )) e−aT = < 1. + ∂ x(nT ) x(nT + )=˜x 1−p The fixed point x˜ implies that there is a corresponding positive periodic solution: x∗ ( t ) =

a(1 − p − e−aT ) b(1 − p − e−aT ) + bpe−a(t −nT )

with initial value x∗ (0+ ) =

a(1−p−e−aT ) b(1−e−aT )

for nT < t ≤ (n + 1)T .

Next, we prove the global asymptotically stable of x∗ (t ). Make the transformation x(t ) = y(1t ) , then (2.3) reduces to

 y(t ) = b − ay(t ), + y(t ) =

Let W (t , s) =

1

1−p

t 6= nT

y(t ),

t = nT .

1 −a(t −s) 0
Q

y(t ) = W (t , 0)y(0) + b

be the Cauchy matrix of (2.5), then

t

Z

W (t , s)ds 0

(2.5)

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is a solution of system (2.5). Since the solution of (2.5) is ultimately upper bounded and |y(t )−y∗ (t )| = W (t , 0)|y(0)−y∗ (0)|, where y∗ (t ) is the periodic solution of (2.5) with y∗ (0) = 1x˜ . For t ∈ (nT , (n + 1)T ], we can deduce from (H1 ) that W (t , 0) =

1

Y 0
1−p

e−



Rt

0 ads

≤

e−aT

n

1−p

→ 0.

Thus |y(t ) − y∗ (t )| → 0 as t → ∞. Therefore,

1 1 |y(t ) − y∗ (t )| − ∗ = → 0 as t → ∞. |x(t ) − x (t )| = y(t ) y (t ) y(t )y∗ (t ) ∗

The proof is completed.



Lemma 2.4. Consider the following system



u(t ) = c − du(t ), t 6= nT u(t + ) = u(t ) + µ, t = nT .

(2.6)

Then system (2.6) has a positive periodic solution u∗ (t ) and for any solution of system (2.6), we have

|u(t ) − u∗ (t )| → 0 as t → ∞, where u∗ (t ) =

c d

+

µe−d(t −nT ) 1 − e−dT

u∗ (0+ ) =

and

c d

+

µ 1 − e−dT

,

nT < t ≤ (n + 1)T .

Proof. It is easily verified that u∗ (t ) is a positive periodic solution of (2.6) with the given initial value. Suppose that u(t ) is an arbitrary solution of (2.6), then we can solve that u(t ) = (u(0+ ) − u∗ (0+ ))e−dt + u∗ (t ),

t ∈ (nT , (n + 1)T ].

Then limt →∞ |u(t ) − u (t )| = 0. This completes the proof. ∗



3. The global attractivity of the predator-extinction periodic solution First, by use of Lemmas 2.3 and 2.4, we can obtain the following result on the existence of the predator-extinction periodic solution to system (1.1). Theorem 3.1. System (1.1) has a mature predator-extinction periodic solution (x∗ (t ), y∗1 (t ), 0) for t ∈ (nT , (n + 1)T ], and for any solution (x(t ), y1 (t ), y2 (t )) of (1.1), we have x(t ) → x∗ (t ),

y1 (t ) → y∗1 (t ) as t → ∞,

where x∗ (t ) =

a(1 − p − e−aT ) b(1 − p − e−aT ) + bpe

, −a(t −nT )

y∗1 (t ) =

µe−d1 (t −nT ) , 1 − e−d1 T

nT < t ≤ (n + 1)T

with x∗ (0+ ) =

a(1 − p − e−aT ) b(1 −

e−aT )

,

y∗1 (0+ ) =

µ 1 − e−d1 T

.

Now we investigate the global attractivity of the predator-extinction periodic solution (x∗ (t ), y∗1 (t ), 0) of system (1.1). Theorem 3.2. Assume that (H1 ) holds true. Further,

(H2 )

e−d2 τ

λca(1 − p − e−aT ) b(1 − p − e−aT ) + bpe−aT

 α+

a(1 − p − e−aT ) b(1 − p − e−aT ) + bpe−aT



< d2 .

Then the mature predator-extinction periodic solution (x∗ (t ), y∗1 (t ), 0) of system (1.1) is globally attractive.

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Proof. Let (x(t ), y1 (t ), y2 (t )) be any solution of system (1.1). We obtain from the first equation and the fourth equation of (1.1) that:



x0 (t ) ≤ x(t )(a − bx(t )), t 6= nT x(t + ) = (1 − p)x(t ), t = nT .

(3.1)

Considering the following comparison equations



z10 (t ) = z1 (t )(a − bz1 (t )), t 6= nT z1 (t + ) = (1 − p)z1 (t ), t = nT

(3.2)

by using Lemma 2.3, we obtain that z1∗ (t ) =

a(1 − p − e−aT ) b(1 − p − e−aT ) + bpe−a(t −nT )

= x∗ (t ) for nT < t ≤ (n + 1)T ,

which is unique and globally attractive positive periodic solution of system (3.2). By use of comparison theorem of impulsive differential equation, there exists n1 ∈ N and an arbitrarily small positive constant ε1 such that, for all t ≥ n1 T , we have a(1 − p − e−aT )

x(t ) ≤ z1∗ (t ) + ε1 ≤

b(1 − p − e−aT ) + bpe−aT

+ ε1 , η0 .

(3.3)

Constituting (3.3) into the third equation of (1.1), for t > n1 T + τ , we have

λc η0 y2 (t − τ ) − d2 y2 (t ) − ry22 (t ). y02 (t ) ≤ e−d2 τ

(3.4)

α + η0

Considering the comparison equation

λc η0 z20 (t ) = e−d2 τ z2 (t − τ ) − d2 z2 (t ) − rz22 (t ).

(3.5)

α + η0

According to the hypotheses (H2 ), for the arbitrarily small positive constant ε1 , we have

λce−d2 τ

a(1 − p − e−aT )



b(1 − p − e−aT ) + bpe−aT

+ ε1

  α+

a(1 − p − e−aT ) b(1 − p − e−aT ) + bpe−aT

+ ε1



< d2 .

(3.6)

That is, e−d2 τ

λc η0 < d2 . α + η0

It follows from Lemma 2.2 that limt →∞ z2 (t ) = 0. Since y2 (s) = z2 (s) = φ3 (s) > 0 for all s ∈ [−τ , 0], using the comparison theorem, we can obtain that y2 (t ) → 0 as t → ∞. Without loss of generality, we assume that, for an arbitrarily small positive constant ε , 0 < y2 (t ) < ε

for all t ≥ 0.

(3.7)

In virtue of the first equation and the fourth equation of system (1.1), we have

(



x0 (t ) ≥ x(t ) a −

cε

α x(t ) = (1 − p)x(t ),

 − bx(t ) ,

t 6= nT

(3.8)

t = nT .

+

Considering the comparison system of (3.8),



(

z30 (t ) = z3 (t ) a −

cε

 − bz3 (t ) ,

α z3 (t + ) = (1 − p)z3 (t ),

t 6= nT

(3.9)

t = nT

by Lemma 2.3, we get the unique positive periodic solution of (3.9), a 1 − p − e−(a− α )T cε



z3∗ (t ) =

b 1 − p − e−(a− α )T



cε





+ bpe−(a− α )T cε

for nT < t ≤ (n + 1)T .

(3.10)

It follows from comparison theorem that, for any ε2 > 0 small enough, there exists a T1 > 0 such that, for all t > T1 , x(t ) ≥ z3∗ (t ) − ε2 . Let ε → 0, then z3∗ (t ) → x∗ (t ), and we have x(t ) ≥ x∗ (t ) − ε2 . On the other hand, we can follow from (3.1)-(3.3) that x(t ) ≤ x∗ (t ) + ε2 for t sufficiently large, which implies that x(t ) → x∗ (t ) as t → ∞.

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From the second equation and the fifth equation of system (1.1), combining (3.3) and (3.7), we have

λc η0 ε − d1 y1 (t ) α + y1 (t ) = y1 (t ) + µ.

(

y01 (t ) ≤

(3.11)

Considering the comparison system of (3.11),

λc η0 ε − d1 z4 (t ) α + z4 (t ) = z4 (t ) + µ.

(

z40 (t ) =

(3.12)

By Lemma 2.4, the unique positive periodic solution of (3.12) is z4∗ (t ) =

µe−d1 (t −nT ) λc η0 ε + d1 α 1 − e−d1 T

for t ∈ (nT , (n + 1)T ].

In view of the comparison theorem, we get that, for any ε3 > 0, there exists T2 > 0 such that y1 (t ) ≤ z4∗ (t ) + ε3 . Let ε → 0, then z4∗ (t ) → y∗1 (t ) and y1 (t ) ≤ y∗1 (t ) + ε3 .

(3.13)

Again from system (1.1), (3.3) and (3.7), we have

 

y01 (t ) ≥ −

e−d2 τ λc η0 ε

α y (t + ) = y (t ) + µ, 1 1

− d1 y1 (t ),

t 6= nT

(3.14)

t 6= nT

(3.15)

t = nT .

Considering the comparison system

 

z50 (t ) = −

e−d2 τ λc η0 ε

α z (t + ) = z (t ) + µ, 5 5

− d1 z5 (t ), t = nT .

It follows from Lemma 2.4 that, system (3.15) has a unique positive periodic solution z5∗ (t ) =

−e−d1 τ λc η0 ε µe−d1 (t −nT ) + d1 α 1 − e−d1 T

for t ∈ (nT , (n + 1)T ].

Similarly, for the sufficiently small constant ε3 > 0, there exists T3 > 0 such that y1 (t ) ≥ z5∗ (t ) − ε3 . Let ε → 0, then z5∗ (t ) → y∗1 (t ) and y1 (t ) ≥ y∗1 (t ) − ε3 .

(3.16)

It follow from (3.13) and (3.16) that y1 (t ) → y∗1 (t ) as t → ∞. This completes the proof.



4. Permanence In this section, we will consider the permanence of system (1.1). We have the following result. Theorem 4.1. Assume that −(a−

c η2

)T

α+βη2 (H3 ) 1 − p − e > 0, ξ −d2 τ (H4 ) λce − d − r η2 > 0 and 2 α+ξ +βη2 η0 η2 m0 m2 −d2 τ (H5 ) −e + α+m > 0, α+η0 +βη2 0 +β m2

where η0 , η2 , m0 , ξ and m2 are defined in (3.3), (4.3), (4.8), (4.10) and (4.15) respectively. Then system (1.1) is permanent, i.e., there exist two positive constants m and M such that m ≤ x(t ) ≤ M , m ≤ y1 (t ) ≤ M , m ≤ y2 (t ) ≤ M . Proof. Firstly, from (3.3), we have x(t ) ≤ η0 . Secondly, from system (1.1), we can obtain the following inequality y02 (t ) ≤ e−d2 τ

c η0

α + η0

y2 (t − τ ) − ry22 (t ).

(4.1)

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Consider the following comparison equation of (4.1), z60 (t ) = e−d2 τ

c η0 z6 (t − τ ) − rz62 (t ). α + η0

(4.2)

By Lemma 2.2, we have z6 (t ) ≤ e−d2 τ

c η0 r (α + η0 )

.

According to the comparison theorem of differential equation, we have y2 (t ) ≤ e−d2 τ

c η0

, η2 .

r (α + η0 )

(4.3)

Thirdly, from system (1.1), we have

λc η0 η2 − d1 y1 (t ), t 6= nT α + y1 (t ) = y1 (t ) + µ, t = nT .

(

y01 (t ) ≤

(4.4)

Using Lemma 2.4 and the comparison theorem of impulsive differential equations, we can deduce from system (4.4) that y1 (t ) ≤

λc η0 η2 µ + , η1 . d1 α 1 − e−d1 T

(4.5)

Take M = max{η0 , η1 , η2 }, then it follows from (3.3), (4.3) and (4.5) that x(t ) ≤ M ,

y1 (t ) ≤ M ,

y2 (t ) ≤ M .

Now it is in the position to prove that there exists a constant m > 0 with m < M such that x(t ) ≥ m, y1 (t ) ≥ m and y2 (t ) ≥ m. From the first equation and the fourth equation of system (1.1), we have

  

c η2



x0 (t ) ≥ x(t ) a −

 − bx(t ) ,

t 6= nT

α + βη2 x(t ) = (1 − p)x(t ), t = nT .

(4.6)

+

Considering the comparison system of (4.6),

  

c η2



z70 (t ) = z7 (t ) a −

 − bz7 (t ) ,

t 6= nT

α + βη2 z7 (t ) = (1 − p)z7 (t ), t = nT .

(4.7)

+

By using comparison theorem, there exists a ε4 > 0 small enough such that, for sufficiently large t, x(t ) ≥ z7∗ (t ) − ε4 , where z7∗ (t ) is the unique and globally stable positive periodic solution of (4.7), and



c η2

a − α+βη 2

z7∗ (t ) =



cη

1−p−e

cη

 b 1−p−e

2 )T −(a− α+βη

2 )T −(a− α+βη cη

 + bpe

2



2

2 )(t −nT ) −(a− α+βη

.

2

In view of the comparison theorem of impulsive differential equation, we derive from (H3 ) that

 x(t ) ≥

cη

2 a − α+βη 2



cη

1−p−e

2 )T −(a− α+βη

cη

 b 1−e



2

2 )T −(a− α+βη



− ε4 , m0 > 0

2

for nT < t ≤ (n + 1)T . Next, we try to find a constant m2 > 0 such that y2 (t ) ≥ m2 . Define V (t ) = y2 (t ) + λce−d2 τ

Z

t t −τ

x(θ )y2 (θ )

α + x(θ ) + β y2 (θ )

dθ.

(4.8)

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Calculating the derivative of V (t ) along the solution of system (1.1), we have V 0 (t ) =



x(t )

λce−d2 τ

α + x(t ) + β y2 (t )

 − d2 − ry2 (t ) y2 (t ).

(4.9)

By hypotheses (H4 ), we can choose two positive constants m∗2 and ε5 small enough such that

λce−d2 τ

ξ − ε5 − d2 − r η2 > 0, α + ξ − ε5 + βη2

(4.10)

where

 ξ=

cm∗

a − α2



1 − p − e−(a− cm∗

 b 1−

2 e−(a− α )T

cm∗ 2 α )T





and 0 < m∗2 (t ) <

αη2 . α + βη2

We claim that y∗2 (t ) < m∗2 cannot hold for all t ≥ t0 . Otherwise, there exists a positive constant t0 such that y2 (t ) < m∗2 for all t ≥ t0 . From system (1.1), we have

  



x (t ) ≥ x(t ) a − 0

cm∗2

α x(t ) = (1 − p)x(t ),

 − bx(t ) ,

+

t 6= nT

(4.11)

t = nT .

Then we obtain that x(t ) ≥ z8∗ (t ) − ε5 ,

(4.12)

where z8∗ (t ) is the unique solution of the comparison system of (4.11), i.e., cm∗



a − α2

z8∗ (t ) =





b 1 − p − e−(a−

1 − p − e−(a−

cm∗ 2 α )T



cm∗ 2 α )T

+ bpe−(a−



cm∗ 2

α )(t −nT )

is the unique solution of the following system:

  



z80 (t ) = z8 (t ) a −

cm∗2

α z8 (t ) = (1 − p)z8 (t ), +

 − bz8 (t ) ,

t 6= nT

(4.13)

t = nT .

It is clear that

 z8∗ (t ) ≥

cm∗ 2

a− α



cm∗

1−p−



b 1 − e−(a−

2 e−(a− α )T

cm∗ 2 α )T



 ,ξ

for nT < t ≤ (n + 1)T .

Hence x(t ) ≥ ξ − ε5 holds for nT < t ≤ (n + 1)T . Combining (4.9), we have V 0 (t ) ≥



λce−d2 τ

 ξ − ε5 − d2 − r η2 y2 (t ). α + ξ − ε5 + βη2

(4.14)

m m Let ym 2 = mint ∈[T ,T +τ ] y2 (t ), we can show that y2 (t ) ≥ y2 for t ≥ T1 . Otherwise, there exists a T2 > 0 such that y2 (t ) ≥ y2 0 for t ∈ [T1 , T1 + τ + T2 ], y2 (T1 + τ + T2 ) = ym and y ( T + τ + T ) ≤ 0. It is easily verified that 2 2 2 1

V 0 (T1 + τ + T2 ) ≥

 λce−d2 τ

 ξ − ε5 − d2 − r η2 ym 2 > 0, α + ξ − ε5 + βη2

which is a contradiction. Hence, for all t ≥ T1 , we have y2 (t ) ≥ ym 2 > 0.

Y. Shao, B. Dai / Nonlinear Analysis: Real World Applications 11 (2010) 3567–3576

3575

From (4.14) and (4.10), we have V 0 (t ) > 0, which implies V (t ) → ∞ as t → ∞. This is a contradiction to η2 η0 . Therefore, for any positive constant t0 , the inequality y2 (t ) < m∗2 cannot hold for all t ≥ t0 . V (t ) ≤ η2 + λce−dτ τ α+βη 2 +η0 There are two cases. Case 1. If y2 (t ) ≥ m∗2 holds for all t large enough, then our aim is obtained. Otherwise, we consider case 2. Case 2. If y2 (t ) is oscillatory about m∗2 . Let m2 = min{ Now we prove that y2 (t ) ≥ min



m∗2 2

, m∗2 e−(d2 +rm2 )τ ∗



m∗ 2 2

, m∗2 e−(d2 +rm2 )τ }. ∗

, m2 .

(4.15)

Obviously, there exist two positive constants t ∗ and ω such that y2 (t ∗ ) = y2 (t ∗ + ω) = m∗2 and y2 (t ) < m∗2 for t ∗ < t < t ∗ + ω, where t ∗ is sufficiently large, and the inequality (4.8) holds true for t ∗ < t < t ∗ + ω. From the continuous and bounded properties of y2 (t ), we conclude that y2 (t ) is uniformly continuous. Thus, there exists a constant T3 > 0 such that y2 (t ) >

m∗ 2 2

for all t ∗ ≤ t ≤ t ∗ + T3 . m∗

If ω ≤ T3 , then y2 (t ) > 22 , our aim is obtained. If T3 < ω ≤ τ , then we can deduce from (1.1) that for t ∗ < t ≤ t ∗ + ω, y02 (t ) ≥ −d2 y2 (t ) − ry22 (t ). According to the assumption y2 (t ∗ ) = m∗2 and y2 (t ) < m∗2 for t ∗ < t < t ∗ + ω, we have y02 (t ) ≥ (−d2 − rm∗2 )y2 (t ) for t ∗ < t ≤ t ∗ + ω ≤ t ∗ + τ . Then, we derive that y2 (t ) ≥ m∗2 e−(d2 +rm2 )τ ≥ m2 . ∗

If ω > τ , similarly we conclude that y2 (t ) ≥ m2 for t ∗ < t ≤ t ∗ + τ . Since the interval [t ∗ , t ∗ + ω] is arbitrarily chosen, we get that y2 (t ) ≥ m2 for t large enough. In view of the arguments above, the choice of m2 is independent of the positive solution of system (1.1), so y2 (t ) ≥ m2 for t sufficiently large. On the other hand, we deduce from (1.1) that

λcm0 m2 λc η0 η2 − d1 y1 (t ) + , α + η0 + βη2 α + m0 + β m2 y (t + ) = y (t ) + µ, t = nT . 1 1  

y01 (t ) ≥ −e−d2 τ

t 6= nT

(4.16)

Considering the comparison system

λc η0 η2 λcm0 m2 − d1 z 9 ( t ) + , α + η + βη α + m0 + β m2 z (t + ) = z (t ) + µ, 0 t = nT2 . 9 9  

z90 (t ) = −e−d2 τ

t 6= nT

(4.17)

Similarly, by hypotheses (H5 ) and Lemma 2.1, for ε6 > 0 small enough and t sufficiently large, we have y1 (t ) ≥ z9∗ (t ) − ε6 ≥ −e−d2 τ

λcm0 m2 µe−d1 T λc η0 η2 − ε6 , m 1 > 0 , + + d1 (α + η0 + βη2 ) d1 (α + m0 + β m2 ) 1 − e−d1 T

(4.18)

where z9∗ (t ) = −e−d2 τ

λc η 0 η 2 λcm0 m2 µe−d1 (t −nT ) + + . d1 (α + η0 + βη2 ) d1 (α + m0 + β m2 ) 1 − e−d1 T

Take m = min{m0 , m1 , m2 }, then from (4.8), (4.18) and (4.15), we have x(t ) ≥ m, y1 (t ) ≥ m and y2 (t ) ≥ m. This completes the proof.  Remark. The parameter µ has no effect on the global attractivity of the mature predator-extinction periodic solution and the permanence of system (1.1), that is, the dynamics of system (1.1) are determined by the first and third equations of (1.1). 5. Discussion and examples In this paper, we have proved that, when 1 a

ln

1 p


1 a

 ln 1 +

pλcae−d2 τ

λcaed2 τ −αd2 b (1 − p)



,

3576

Y. Shao, B. Dai / Nonlinear Analysis: Real World Applications 11 (2010) 3567–3576

i.e., the impulsive period is no larger than the threshold, or partial destruction to prey by catching or pesticide is larger than some threshold, i.e., 1 − e−aT > p > 1 − ln

λce−(aT +d2 τ ) , − d2 α b(1 − e−aT )

λce−d2 τ

the mature predator-extinction periodic solution (x∗ (t ), y∗1 (t ), 0) is globally attractive. That is, when the prey population is caught or poisoned largely, the mature natural enemy population (predator) is eradicated totally. However, due to the effects of pesticide residues on human health and on the environment, from saving resources and the principle of ecosystem balance, we hope that predator and prey can coexist. Therefore, in Section 4, we study and obtain the sufficient conditions for the permanence of (1.1). It is beneficial for human beings to keep the balance of ecosystem. Finally, we give an example to show the effectiveness of the results. Example. In system (1.1), If we take a = b = c = 1, d2 = 1/2,

α = 2, β = 1/2, d1 = 1/4, τ = 4, p = 1/2, λ = 1/4, T = 2,

then it is easy to show that conditions (H1 ), (H2 ) hold and the mature predator-extinction periodic solution of the system is globally attractive. If we let a = 2, d2 = 1/5,

b = c = 1,

τ = 10,

α = 1/2, p = 1/2,

β = 1/2, d1 = 1/4, λ = 5, T = 1.

By computation, it is easily verified that conditions (H3 )–(H5 ) hold true and the system is permanent. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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