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The dynamics of a stage-structured predator–prey system with impulsive effect and Holling mass defence
Applied Mathematical Modelling 36 (2012) 87–96
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The dynamics of a stage-structured predator–prey system with impulsive effect and Holling mass defence q Can-Yun Huang ⇑, Yan-Juan Li, Hai-Feng Huo Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 30 September 2010 Received in revised form 9 May 2011 Accepted 16 May 2011 Available online 27 May 2011 Keywords: Impulsive effect Stage structure Global attractivity Permanence
a b s t r a c t In this paper, a stage-structured Holling mass defence predator-prey model with impulsive effect is investigated. By using comparison theorem and the stroboscopic technique, sufficient conditions for the global attractivity of mature prey-extinction periodic solution and permanence of the system are obtained. Furthermore, the numerical analysis is also inserted to illuminate the feasibility of the theoretical results. Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction Biological resources is an organic part of the natural resources, and it is a kind of renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark [1]. In the recent management of renewable natural resources, biologists have been aware of the possibility that suitable harvesting and stocking could alter the genetic pattern of a resource [2] and play a key role on the permanence of ecosystems. Recent empirical evidence suggests that this effect is substantial [3–8] and that this issue deserves more attention from resource economists. Considering the exploited Biological system, human harvesting or stocking always happen in a short time or instantaneously, and the continuous action of human should be replaced with an impulsive perturbation. Consequently, impulsive differential equations provide a natural description of such systems [4]. Recently, impulsive differential equations have been extensively used as models in biology, physics, chemistry, engineering and other sciences, with particular emphasis on population dynamics [9–13]. In the real world, many species usually go through two life stages, immature and mature. In [14–19], some stage-structured models of population growth consisting of immature and mature individuals were analyzed, where the stage-structured was modeled by introduction of a constant time delay. However, as literatures [20,21] pointed out, the delay differential equation exhibits much more complicated 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 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. For the prey were simple algology cell of animal,
q
Supported by Advance Foundation of Lanzhou University (BS10200903) and the NNSF of China (10571078).
⇑ Corresponding author.
E-mail address: [email protected] (C.-Y. Huang). 0307-904X/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.038
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C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
invertebrate and vertebrate, Holling type-I, Holling type-II and Holling type-III functional response were proposed, respectively. The three functions are all monotonic in the first quadrant. It implies that, as the prey population increases, the consumption rate of prey per predator increases. However, there is experimental and observational evidence that indicates that this need not always be the case, for example, in the case of ‘‘group defense’’ in population dynamics. Group defense is a term used to describe the phenomenon whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise themselves when their numbers are large enough. An example of this phenomenon described by Holmes and Bethel [22] involves certain insect populations. For more biological information, the reader can refer to [23–25]. It is well known that large swarms of insects make individual identification difficult for their predators [26,27]. To model such an effect, Andrews [28] suggested a function
uðxÞ ¼
mx ; a þ bx þ x2
called Monod–Haldane function, and also called Holling type-IV function. The recent works of Jiang [2] and Jiao [29,30] considered that delayed predator-prey models with impulsive stocking on prey and continuous harvesting on predator, the results of which shows that the behavior of impulsive stocking prey plays an important role for the permanence of the system. In fact, harvesting on predator also should be considered as impulsive perturbation in comparison with stocking on prey. In [31], The authors proposed a stage-structured predator–prey model with disturbing pulse and time delays, and analyzed the effect of impulsive harvesting on predators. In [32], an impulsive delay predator-prey model with stage structure and Beddington-type functional response is established, in which harvesting on prey and stocking on predator are mostly stressed. To investigate the effects of impulsive stocking on immature prey and harvesting on predator, motivated by these literatures [2,29,30,32,31], we consider the following impulsive delay differential equation:
8 9 x01 ðtÞ ¼ r1 x2 ðtÞ r 1 ews x2 ðt sÞ wx1 ðtÞ; > > > > > > > = bx2 ðtÞ > 0 ws 2 > x2 ðtÞ ¼ r1 e x2 ðt sÞ 1þax ðtÞþbx2 ðtÞ x3 ðtÞ d1 x2 ðtÞ d2 x2 ðtÞ; > > t – nT; 2 2 > > > > > < x0 ðtÞ ¼ r x ðtÞ þ kbx2 ðtÞ > 2 ; x ðtÞ d x ðtÞ; 2 3 3 3 2 3 3 1þax2 ðtÞþbx2 ðtÞ 9 > > þ > ðt Þ ¼ x ðtÞ þ l ; x > > 1 1 = > > > > t ¼ nT; x2 ðt þ Þ ¼ x2 ðtÞ; > > > > ; : þ x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
ð1:1Þ
where x1(t), x2(t) represent the immature and mature prey densities respectively, x3(t) denotes the density of the predator, s represents a constant time to maturity, and r1, w, b, a, b, d1, d2, r2, k, d3 are positive constants. r1 is the birth rate of the immature prey, w(w > d1), d1 are the mortality rates of the immature prey and mature prey, b is the maximum numbers of the mature prey that can be eaten by a predator per unit of time, d2, d3 are the intra-specific competition rate of the mature prey and the predator, r2 is the intrinsic growth rate of the predator, k is the rate of conversing prey into predator, l P 0 is the stocking amount of the immature prey, p(0 6 p < 1) represents the harvest rate of the predator at t = nT, n 2 Z+ and Z+ = {1, 2, . . .}, T is the period of the impulsive effect. The initial conditions for system (1.1) are
ð/1 ; /2 ; /3 Þ 2 C ½s; 0; R3þ ;
/i ð0Þ > 0; i ¼ 1; 2; 3;
n o R3þ ¼ x 2 R3 : x P 0 :
From the biological point of view, we only consider (1.1) in the biological meaning region
D ¼ fðx1 ; x2 ; x3 Þ : x1 P 0; x2 P 0; x3 P 0g: The remainder of this paper is arranged as follows. In Section 2, we present some preliminaries. In Section 3, the sufficient conditions for the global attractivity of prey-extinction periodic solution is obtained. In Section 4, the permanence of system (1.1) is investigated. In Section 5, we give some simulations to illustrate our results. A brief conclusion follows in Section 6.
2. Preliminaries The solution of (1.1), denoted by x(t) = (x1(t), x2(t) x3(t))T, is a piecewise continuous function x : Rþ ! R3þ , x(t) is continuous on (nT, (n + 1)T], n 2 Z+ and xðnT þ Þ ¼ limt!nT þ xðtÞ exists. Obviously the global existence and uniqueness of solutions of (1.1) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-side of system (1.1) (see [33,34]). Let V: Rþ R3þ ! Rþ , then V is said to belong to class V0 if: (i) V is continuous in ððn 1ÞT; nT R3þ and for each x 2 R3þ ; lim(t,z)?((n1)T,x)V(t, z) = V((n 1)T, x) and limðt;zÞ!ðnT þ ;xÞ Vðt; zÞ ¼ VðnT þ ; xÞ exist. (ii) V is locally Lipschitzian in x.
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
89
Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ððn 1ÞT; nT R3þ , n 2 Z+. The upper right derivative of V(t, x) with respect to the impulsive differential system (1.1) is defined as
Dþ VðtÞ ¼ lim sup h!0þ
1 ½V ðt þ h; x þ hf ðt; xÞÞ Vðt; xÞ: h
Lemma 2.1 ([35,34]). Consider the following impulsive differential inequations:
x0 ðtÞ 6 ðPÞpðtÞxðtÞ þ qðtÞ; t – tk ; xðtþk Þ 6 ðPÞdk xðtk Þ þ bk ; t ¼ tk ;
where p, q 2 C(R+, R), k 2 Z+, dk P 0 and bk are constants. Assume: (A0) the sequence tk satisfies 0 6 t0 < t1 < t2 < , with limk?1tk = 1; (A1) x 2 PC0 (R+, R) and x(t) is left-continuous at tk, k 2 Z+. Then
Y
xðtÞ 6 ðPÞxðt0 Þ þ
0
X
@
t0
þ
Z
t
Y
Z
dj exp
dk exp
t tk
Z
t 0 s
Z
t
pðsÞds
t0
t 0
t k
Y
dk exp
t
!1 pðsÞds Abk
pðhÞdh qðsÞds;
t P t0 :
s
Lemma 2.2 [36]. Consider the following equation
x0 ðtÞ ¼ axðt sÞ bxðtÞ cx2 ðtÞ; where a, b, c and s are positive constants, x(t) > 0 for t 2 [s, 0]. We have (i) If a > b, then limt?+1x(t) = (a b)/c; (ii) If a < b, then limt?+1x(t) = 0. Lemma 2.3. Consider the following system
u0 ðtÞ ¼ c wuðtÞ;
t – nT;
uðtþ Þ ¼ uðtÞ þ l;
t ¼ nT:
ð2:1Þ
Then system (2.1) has a positive periodic solution u⁄(t) with period T. And for any solution u(t) of system (2.1), we have
juðtÞ u ðtÞj ! 0 as t ! 1; where
u ðtÞ ¼
c lewðtnTÞ þ ; w 1 ewT
nT < t 6 ðn þ 1ÞT
and u ð0þ Þ ¼
c l þ : w 1 ewT
Proof. Integrating the first equation of (2.1) on nT < t 6 (n + 1)T, we get
uðtÞ ¼
c c wðtnTÞ þ uðnT þ Þ e ; w w
nT < t 6 ðn þ 1ÞT:
After the successive pulse, we can obtain the following stroboscopic map of system (2.1),
uððn þ 1ÞT þ Þ ¼
c c wT þ uðnT þ Þ e þ l: w w
Eq. (2.2) has a unique positive fixed point
e¼ u
c l þ ; w 1 ewT
which implies that there is a corresponding positive periodic solution:
ð2:2Þ
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u ðtÞ ¼
c lewðtnT Þ þ ; w 1 ewT
with initial value u ð0þ Þ ¼ wc þ 1elwT for nT < t 6 (n + 1)T. Suppose that u(t) is an arbitrary solution of (2.1), using iterative technique, we have
c c c lewðtnTÞ lewt ð1 ewt Þ þ uð0þ Þewt þ lewðtTÞ þ þ lewðtnTÞ ¼ ewt þ uð0þ Þewt þ w w w 1 ewT 1 ewT ¼ ðuð0þ Þ u ð0þ ÞÞewt þ u ðtÞ; nT < t 6 ðn þ 1ÞT:
uðtÞ ¼
Then limt?1ju(t) u⁄(t)j = 0. This completes the proof.
h
Lemma 2.4 [32]. Consider the following system
x0 ðtÞ ¼ xðtÞða bxðtÞÞ; t – nT; xðt þ Þ ¼ ð1 pÞxðtÞ; t ¼ nT:
ð2:3Þ
If p < 1 eaT, then system (2.3) has a positive solution x⁄(t) and for any solution x(t) of system (2.3), we have
jxðtÞ x ðtÞj ! 0 as t ! 1; where
x ðtÞ ¼
a 1 p eaT
; aðtnTÞ bð1 p eaT Þ þ bpe
x ð0þ Þ ¼
a 1 p eaT ; bð1 eaT Þ
nT < t 6 ðn þ 1ÞT:
Lemma 2.5. There exists a constant M > 0 such that x1 ðtÞ 6 Mk, x2 ðtÞ 6 Mk ; x3(t) 6 M for each solution (x1(t), x2(t), x3(t)) of (1.1) with all t large enough, where k is positive constant defined in system (1.1) Proof. Define V(t) = kx1(t) + kx2(t) + x3(t). For t – nT, from w > d1 we have
Dþ VðtÞ þ d1 VðtÞ ¼ kðw d1 Þx1 ðtÞ þ kr 1 x2 ðtÞ kd2 x22 ðtÞ þ ðr 2 þ d1 Þx3 ðtÞ d3 x23 ðtÞ 6 kr 1 x2 ðtÞ kd2 x22 ðtÞ þ ðr 2 þ d1 Þx3 ðtÞ d3 x23 ðtÞ 6 M 0 ; kr 2
2
þd1 Þ where M 0 ¼ 4d12 þ ðr24d . 3
For t = nT, V(nT+) = kx1(nT) + kl + kx2(nT) + (1 p)x3(nT) 6 V(nT) + kl. For t 2 (nT, (n + 1)T], by Lemma 2.1 we obtain
VðtÞ 6 Vð0Þed1 t þ
Z
t
M 0 ed1 ðtsÞ ds þ
0
M0 kled1 T þ ! ; d1 ed1 T 1
X
kled1 ðtnTÞ < Vð0Þed1 t þ
0
M0 ed1 ðtTÞ kled1 T ð1 ed1 t Þ þ kl þ d1 1 ed1 T ed1 T 1
as t ! 1;
which implies that V(t) is uniformly ultimately bounded. Hence, in terms of the definition of V(t), there exists a constant
M¼
M0 kled1 T þ dT > 0; d1 e 1 1
ð2:4Þ
such that x1 ðtÞ 6 Mk, x2 ðtÞ 6 Mk, x3(t) 6 M with t large enough. The proof is completed.
h
3. Global attractivity First, by use of Lemmas 2.3 and 2.4, we can obtain the following result on the existence of the mature prey-extinction periodic solution to system (1.1). Theorem 3.1. System (1.1) has a mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ for t 2 (nT, (n + 1)T], and for any solution (x1(t), x2(t), x3(t)) of system (1.1), then
x1 ðtÞ ! x1 ðtÞ;
x3 ðtÞ ! x3 ðtÞ as t ! 1;
where
x1 ðtÞ ¼ and
lewðtnTÞ 1 ewT
; x3 ðtÞ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnTÞ
for nT < t 6 ðn þ 1ÞT;
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
x1 ð0þ Þ ¼
l
r 2 1 p er2 T : d3 ð1 er2 T Þ
x3 ð0þ Þ ¼
; 1 ewT
91
Next, we investigate the global attractivity of the prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ of system (1.1). Theorem 3.2. Assume that (H1) 1 p er2 T > 0: br 1per2 T 2 ð Þ < d2 1er2 T : (H2) ðr 1 ews d1 Þ 1 þ aM þ bM k k2 Þ 3ð Then the mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ of system (1.1) is globally attractive.
Proof. Let (x1(t), x2(t), x3(t)) be any solution of system (1.1). We can obtain from the third and sixth equation of system (1.1) that
(
x03 ðtÞ P r 2 x3 ðtÞ d3 x23 ðtÞ; þ
x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
t – nT;
ð3:1Þ
t ¼ nT:
So we consider the following auxiliary impulsive differential system:
z01 ðtÞ ¼ z1 ðtÞðr 2 d3 z1 ðtÞÞ; z1 ðt þ Þ ¼ ð1 pÞz1 ðtÞ;
t – nT;
ð3:2Þ
t ¼ nT:
Using Lemma 2.4, we obtain that
z1 ðtÞ ¼
r 2 1 p er2 T ¼ x3 ðtÞ for nT < t 6 ðn þ 1ÞT; d3 ð1 p er2 T Þ þ d3 per2 ðtnTÞ
which is unique and globally attractive positive periodic solution of system (3.1). By using comparison theorem of impulsive differential equation [34], there exist n1 2 Z+ and an arbitrarily small positive constant e0 such that
x3 ðtÞ P z1 ðtÞ P z1 ðtÞ e0 P
r 2 1 p er2 T e0 ,q; d3 ð1 er2 T Þ
ð3:3Þ
for all t P n1T. From (3.3) and Lemma 2.5, one has
0
1
bq
x02 ðtÞ 6 r1 ews x2 ðt sÞ @ þ d1 Ax2 ðtÞ d2 x22 ðtÞ; 2 1 þ aM þ bM 2 k k for t > n1T + s. Considering the auxiliary equation
0
z02 ðtÞ
¼ r1 e
ws
z2 ðt sÞ @
bq 1 þ aM þ bM k k2
2
1
þ d1 Az2 ðtÞ d2 z22 ðtÞ:
According to the hypotheses (H2), for the arbitrarily small positive constant e0, we have
r1 ews <
bq 1 þ aM þ bM k k2
2
þ d1 :
According to Lemma 2.2, we have lim?1z2(t) = 0. Since x2(s) = z2(s) = /2(s) > 0 for all s 2 [s, 0], using the comparison theorem, we can obtain that x2(t) ? 0 as t ? 1. Without loss of generality, we assume that there exists a positive constant e such that
0 < x2 ðtÞ < e;
t P 0:
ð3:4Þ
From the first, the fourth equation of (1.1) and (3.4), we have
x01 ðtÞ 6 r 1 e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – nT; t ¼ nT:
ð3:5Þ
Considering the auxiliary system of (3.5),
z03 ðtÞ ¼ r 1 e wz3 ðtÞ; þ
z3 ðt Þ ¼ z3 ðtÞ þ l;
t – nT; t ¼ nT:
By Lemma 2.3, the unique positive periodic solution of (3.6) is
ð3:6Þ
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C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
z3 ðtÞ ¼
r 1 e lewðtnTÞ þ w 1 ewT
for t 2 ðnT; ðn þ 1ÞT:
In view of the comparison theorem, for any e1 there exists T1 > 0 such that x1 ðtÞ 6 z3 ðtÞ þ e1 for t > T1. Let e ? 0, then z3 ðtÞ ! x1 ðtÞ and
x1 ðtÞ 6 x1 ðtÞ þ e1 :
ð3:7Þ
Again from system (1.1) and (3.4), we have
x01 ðtÞ P r 1 ews e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – nT;
ð3:8Þ
t ¼ nT:
Considering the auxiliary system of (3.8),
z04 ðtÞ ¼ r 1 ews e wz4 ðtÞ; z4 ðtþ Þ ¼ z4 ðtÞ þ l;
t – nT;
ð3:9Þ
t ¼ nT:
By Lemma 2.3, system (3.9) has a unique positive periodic solution
z4 ðtÞ ¼
r 1 ews e lewðtnTÞ for t 2 ðnT; ðn þ 1ÞT: þ w 1 ewT
Similarly, for the sufficiently small constants e1 > 0, there exists T2 > 0 such that x1 ðtÞ P z4 ðtÞ e1 . Let e ? 0, then z4 ðtÞ ! x1 ðtÞ and
x1 ðtÞ P x1 ðtÞ e1 :
ð3:10Þ x1 ðtÞ
It follows from (3.7) and (3.10) that x1 ðtÞ ! as t ? 1. In virtue of the third and sixth equation of system (1.1), we have
(
kbe x03 ðtÞ 6 x3 ðtÞ r 2 þ 1þa e d3 x3 ðtÞ ; þ
x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
t – nT;
ð3:11Þ
t ¼ nT:
Consider the following auxiliary system of (3.11),
(
kbe z05 ðtÞ ¼ z5 ðtÞ r 2 þ 1þa e d3 z5 ðtÞ ; þ
z5 ðt Þ ¼ ð1 pÞz5 ðtÞ;
t – nT;
ð3:12Þ
t ¼ nT:
By Lemma 2.4, we obtain the unique positive periodic solution of (3.12),
kbe kbe r 2 þ 1þa 1 p eðr2 þ1þaeÞT e z5 ðtÞ ¼ kbe kbe d3 1 p eðr2 þ1þaeÞT þ d3 peðr2 þ1þaeÞðtnTÞ
for nT < t 6 ðn þ 1ÞT:
It follows from comparison theorem that, for any e2 > 0 small enough, there exists a T3 > 0 such that x3 ðtÞ 6 z5 ðtÞ þ e2 for all t > T3. Let e ? 0, then z5 ðtÞ ! x3 ðtÞ, and we have x3 ðtÞ 6 x3 ðtÞ þ e2 . On the other hand, we can follow from 3.1, 3.2 and 3.3 that x3 ðtÞ P x3 ðtÞ e2 for t sufficiently large, which implies that x3 ðtÞ ! x3 ðtÞ as t ? 1. This completes the proof. h 4. Permanence The next work is to investigate the permanence of the system (1.1). Before starting our theorem, we give the definition of permanence. Definition 4.1. System (1.1) is said to be permanent if there exist positive constants m and M such that each positive solution (x1(t), x2(t), x3(t)) of system (1.1) satisfies m 6 x1(t), x2(t), x3(t) 6 M for t sufficiently large enough. Theorem 4.1. Assume that (H3) r1 ewsw bq d1 d2 Mk > 0; r ðm e s MÞ (H4) 1 2 w k > 0; where M, q and m2 are defined in (2.4), (4.7) and (4.10) respectively. Then system (1.1) is permanent. Proof. First, we show that there is a positive constant m2 such that each positive solution (x1(t), x2(t), x3(t)) of (1.1) satisfies x2(t) P m2 for t large enough.
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
93
The second equation of (1.1) can be rewritten as
r 1 ews
x02 ðtÞ ¼
!
bx3 ðtÞ 1 þ ax2 ðtÞ þ
2 bx2 ðtÞ
d1 d2 x2 ðtÞ x2 ðtÞ r1 ews
d dt
Z
t
x2 ðuÞdu:
ð4:1Þ
ts
Let us consider any positive solution (x1(t), x2(t), x3(t)) of system (1.1). According to (4.1), define
VðtÞ ¼ x2 ðtÞ þ r 1 ews
Z
t
x2 ðuÞdu:
ts
Calculating the derivative of V(t) along the solution of (1.1), we have ws
0
V ðtÞ ¼
r1 e
!
bx3 ðtÞ 2
1 þ ax2 ðtÞ þ bx2 ðtÞ
d1 d2 x2 ðtÞ x2 ðtÞ:
ð4:2Þ
Using Lemma 2.5, (4.2) can be written as
M x2 ðtÞ: V 0 ðtÞ > r 1 ews bx3 ðtÞ d1 d2 k
ð4:3Þ
By hypotheses (H3), we can easily know that there exists a sufficiently small e3 > 0 such that
r1 ews > bðq þ e3 Þ þ d1 þ d2
M ; k
ð4:4Þ
where
r 2 þ kbm2 1 p eðr2 þkbm2 ÞT q¼ ; d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT and m2 is determined by the following equation
r2 þ kbm2 1 p eðr2 þkbm2 ÞT 1 M ws r1 e ¼ d1 d2 : b k d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT
We claim that for any t0 > 0, it is impossible that x2 ðtÞ < m2 for all t > t0. Suppose that the claim is invalid, then there is a t0 > 0 such that x2 ðtÞ < m2 for all t > t0. It follows from the third equation of (1.1) that
x03 ðtÞ < x3 ðtÞðr 2 þ kbm2 d3 x3 ðtÞÞ;
ð4:5Þ
for all t > t0. Consider the following auxiliary impulsive system of (4.5)
z06 ðtÞ ¼ z6 ðtÞðr 2 þ kbm2 d3 z6 ðtÞÞ; z06 ðt þ Þ
¼ ð1 pÞz6 ðtÞ;
t – nT;
ð4:6Þ
t ¼ nT:
By Lemma 2.4, we have
z6 ðtÞ ¼
r 2 þ kbm2 1 p eðr2 þkbm2 ÞT ; d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞðtnTÞ
t0 < nT < t 6 ðn þ 1ÞT;
which is the unique positive periodic solution of (4.6) and globally asymptotically stable. In view of the comparison theorem for impulsive differential equation, we know that there exists t1(>t0 + s), such that
x3 ðtÞ 6 z6 ðtÞ þ e3 ; for t P t1, which implies that
1 p eðr2 þkbm2 ÞT z6 ðtÞ 6 ,q: d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT
r2 þ kbm2
ð4:7Þ
Thus
x3 ðtÞ 6 q þ e3 , r;
t P t1 :
From (4.4), we get
r1 ews > br þ d1 þ d2
M : k
ð4:8Þ
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C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
By (4.3) and (4.8), we have
M V 0 ðtÞ > r 1 ews br d1 d2 x2 ðtÞ; k
t P t1 :
ð4:9Þ
Let
xm min x2 ðtÞ: 2 ¼ t2½t 1 ;t1 þs
m We will show that x2 ðtÞ P xm 2 for all t P t1. Otherwise, there exists a T0 > 0 such that x2 ðtÞ P x2 for t1 6 t 6 t1 + s + T0, 0 x2 ðt 1 þ s þ T 0 Þ P xm and x ðt þ s þ T Þ < 0. From the second equation of system (1.1) and (4.8) we see 1 0 2 2
M m x > 0; x02 ðt1 þ s þ T 0 Þ P r 1 ews br d1 d2 k 2
which is a contradiction. Thus, x2 ðtÞ P xm 2 for all t P t1. From (4.4) and (4.9), we have
M m V 0 ðtÞ > r 1 ews br d1 d2 x > 0; k 2
t P t1 ;
which means that V(t) ? 1 as t ? 1. It is a contradiction to VðtÞ 6 Mk ð1 þ r1 sews Þ. Therefore, for any t0 > 0, the inequality x2 ðtÞ < m2 cannot hold for all t P t0. So, there exist the following two possibilities. Case 1. If x2 ðtÞ P m2 holds for all t large enough, then our aim is obtained. Case 2. Assume that x2(t) is oscillatory about m2 . Setting
m2 ¼ min
m2 ; m2 eðbMþd1 þd2 m2 Þs ; 2
ð4:10Þ
we will prove that x2(t) P m2 for all t large enough. Suppose that there exist the positive constants t⁄,n such that x2(t⁄) = x2 ðt þ nÞ ¼ m2 and x2 ðtÞ < m2 for all t⁄ < t < t⁄ + n, where t⁄ is sufficiently large, and the inequality (4.8) holds true for t⁄ < t < t⁄ + n. Since x2(t) is continuous and bounded and is not effected by impulses, we conclude that x2(t) is uniformly m
continuous. Hence, there exists a constant T (0 < T < s and T is independent of the choice of t⁄) such that x2 ðt Þ > 22 for t⁄ 6 t 6 t⁄ + T. If n 6 T, our aim is obtained. If T < n 6 s, from the second equation of (1.1) we have that for t < t < t þ n; x02 ðtÞ P bx2 ðtÞx3 ðtÞ d1 x2 ðtÞ d2 x22 ðtÞ: According to the assumption x2 ðt Þ ¼ m2 and x2 ðtÞ < m2 for t⁄ < t < t⁄ + n, we have x02 ðtÞ P ðbM þ d1 þ d2 m2 Þx2 ðtÞ for t⁄ < t 6 t⁄ + n 6 t⁄ + s. Then, we derive that ⁄ ⁄ ðbMþd1 þd2 m2 Þs . It is clear that x2(t) P m2 for t < t 6 t + n. If n P s, then we have that x2(t) P m2 for t⁄ < t < t⁄ + s. x2 ðtÞ P m2 e The same arguments can be continued, we can obtain x2(t) P m2 for t⁄ + s 6 t 6 t⁄ + n. Since the interval [t⁄, t⁄ + n] is arbitrarily chosen, we get that x2(t) P m2 for t large enough. In view of our arguments above, the choice of m2 is independent of the positive solution of (1.1) which satisfies that x2(t) P m2 for sufficiently large t. Next, from the first and the fourth equation of system (1.1), we have
(
x01 ðtÞ P r1 m2 ews Mk wx1 ðtÞ; x1 ðtþ Þ ¼ x1 ðtÞ þ l;
t – nT;
t ¼ nT:
ð4:11Þ
Considering the comparison system
(
z07 ðtÞ ¼ r 1 m2 ews Mk wz7 ðtÞ; z7 ðtþ Þ ¼ z7 ðtÞ þ l;
t – nT;
t ¼ nT:
ð4:12Þ
By using comparison theorem, there exists a e4 > 0 small enough such that x1 ðtÞ P z7 ðtÞ e4 for sufficiently large t, where z7 ðtÞ is the unique and globally stable positive periodic solution of (4.12), and
z7 ðtÞ ¼
r 1 m2 ews Mk lewðtnTÞ : þ 1 ewT w
In view of the comparison theorem of impulsive differential equation, we derive from (H4) that
r1 m2 ews Mk lewT þ x1 ðtÞ P e4 ,m1 : w 1 ewT From (3.3), let q , m3, and we have x3(t) P m3. Take m = min{m1, m2, m3}, then we have x1(t) P m, x2(t) P m, x3(t) P m. By Lemma 2.5 and the above discussion, system (1.1) is permanent. The proof is completed. h
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
95
Fig. 1. Dynamical behavior of system (1.1) with T = 10, l = 0.1, p = 0.2, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 1, a = 1, b = 0.5, d1 = 0.5, d2 = 0.2, r2 = 1, k = 3, d3 = 0.3, s = 0. (a) Time series of the immature prey population. (b) Time series of the mature prey population. (c) Time series of the predator population.
Fig. 2. Dynamical behavior of system (1.1) with T = 0.5, l = 4, p = 0.25, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 0.5, a = 1, b = 3, d1 = 0.5, d2 = 0.2, r2 = 1, k = 0.1, d3 = 0.3, s = 0. (a) Time series of the immature prey population. (b) Time series of the mature prey population. (c) Time series of the predator population.
5. Numerical simulations To investigate the impulsive effect of a stage-structured Holling mass defence predator-prey model by numerical analysis, we assume that T = 10, l = 0.1, p = 0.2, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 1, a = 1, b = 0.5, d1 = 0.5, d2 = 0.2, r2 = 1, k = 3, d3 = 0.3, s = 0. It is obvious that the parameters satisfy the condition of Theorem 3.2. Then the mature preyextinction periodic solution is global attractivity (see Fig. 1). Secondly, we assume that T = 0.5, l = 4, p = 0.25, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 0.5, a = 1, b = 3, d1 = 0.5, d2 = 0.2, r2 = 1, k = 0.1, d3 = 0.3, s = 0, it is obvious that the parameters satisfy Theorem 4.1, then system (1.1) is permanent (see Fig. 2). 6. Conclusion According to the fact of biological resource management in this paper we consider a stage-structured Holling mass defence predator-prey model with impulsive effect. We analyze that the predator-extinction periodic solution of system (1.1) is globally attractive, and obtain that the sufficient condition for the permanence of system (1.1). From Theorem 3.2, we can see that, increasing T, or reducing l or p is propitious to the global attractivity of the mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ. That is, if the immature prey is stocked too few, or the mature prey is caught excessively, then the mature prey population is extinction totally, and lose the merits of exploitative mature prey population. By Theorem 4.1, we may see that, reducing T, increasing l or p are in favor of the permanence of system (1.1). It implies that the behavior of impulsive stocking on prey plays an important role for the permanence of system (1.1), and reasonable harvesting predator can ensure the sustainable development of biological resources. From Theorems 3.2 and 4.1, we believe that there must exist a sharp threshold, but we could not get it with the current technology. Compared with other models, it is much easier to operate and manage the biological resource system. The results correspond with the reality, and provide tactical basis for the biological resource management. However, under the case of considering the manpower and the capital, i.e. the cost of control, human can take optimal strategy in the practice of biology control. So, we will continue to study the optimal harvesting policy of system (1.1) in the future. References [1] C.W. Clark, Mathematical Bioeconomics, Wiley, New York, 1990. [2] X.W. Jiang, Q. Song, M.Y. Hao, Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect, Appl. Math. Comput. 215 (2010) 4221–4229.
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[3] D.W. Coltman, P.O. Donoghue, J.T. Jorgenson, J.T. Hogg, C. Strobeck, M. Festa-Bianchet, Undesirable evolutionary consequences of trophy hunting, Nature 426 (6967) (2003) 655–658. [4] D.O. Conover, S.B. Munch, Sustaining fisheries yields over evolutionary time scales, Science 297 (5578) (2002) 94–96. [5] T. Haugen, L.A. Volestad, A century of life-history evolution in Grayling, Genetica 112-113 (2001) 475–491. [6] M. Heino, O.R. Gods, Fisheries-induced selection pressures in the context of sustainable fisheries, Bull. Mar. Sci. 70 (2002) 639–656. [7] L. Ling, W.M. Wang, Dynamics of a Ivlev-type predator–prey system with constant rate harvesting, Chaos Soliton Fract. 41 (2009) 2139–2153. [8] E.M. Olsen, M. Heino, G.R. Lilly, M.J. Morgan, J. Brattey, B. Ernande, U. Dieckmann, Maturation trends indicative of rapid evolution preceded the collapse of Northern Cod, Nature 428 (2004) 932–935. [9] J. Hui, D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Soliton Fract. 29 (2006) 233–251. [10] B. Liu, L.S. Chen, The periodic competing Lotka–Volterra model with impulsive effect, IMA J. Math. Med. Biol. 21 (2004) 129–145. [11] B. Liu, Y. Zhang, L.S. Chen, The dynamical behaviors of a Lotka–Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. RWA 6 (2005) 227–243. [12] A. D’Onofrio, Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model. 36 (2002) 473–489. [13] H. Zhang, P. Georgescu, L.S. Chen, An impulsive predator–prey system with Beddington–Deangelis functional response and time delay, Int. J. Biomath. 1 (2008) 1–17. [14] W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci. 101 (1990) 139–153. [15] F.D. Chen, M.S. You, Permanence, extinction and periodic solution of the predator-prey system with Beddington–DeAngelis functional response and stage structure for prey, Nonlinear Anal. RWA 9 (2008) 207–221. [16] H.I. Freedman, K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol. 48 (1986) 485–492. [17] S.A. Gourley, Y. Kuang, A stage structured predator–prey model and its dependence on maturation delay and death rate, J. Math. Biol. 49 (2004) 188– 200. [18] X.Y. Song, L.S. Chen, Modelling and analysis of a single species system with stage structure and harvesting, Math. Comput. Model. 36 (2002) 67–82. [19] X. Wang, W. Wang, X. Lin, Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy, Chaos Soliton Fract. 37 (2008) 706– 718. [20] Y. Kuang, Delay Differential Equation With Application in Population Dynamics, Academic Press, New York, 1993. [21] Y.K. Li, Y. Kuang, Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) 260–280. [22] J.C. Holmes, W.M. Bethel, Modification of intermediate host behavior parasites, Zool. J. Linnean Soc. 51 (1972) 123–149. [23] H.I. Freedman, G.S.K. Wolkowicz, Predator–prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol. 48 (1986) 493–508. [24] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (4) (2000) 1445–1472. [25] H. Zhu, S. Campbell, G. Wolkowicz, Bifurcation analysis of a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63 (2) (2002) 636–682. [26] J.J. Jiao, X.Z. Meng, L.S. Chen, A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators, Appl. Math. Comput. 189 (2007) 1448–1458. [27] Z.J. Liu, R.H. Tan, Impulsive harvesting and stocking in a Monod–Haldane functional response predator-prey system, Chaos Soliton Fract. 34 (2007) 454–464. [28] J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng. 10 (1968) 707– 723. [29] J.J. Jiao, X.Z. Meng, L.S. Chen, Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey, Chaos Soliton Fract. 41 (2009) 103–112. [30] J.J. Jiao, G.P. Pang, L.S. Chen, G.L. Luo, A delayed stage-structured predator–prey model with impulsive stocking on prey and continuous harvesting on predator, Appl. Math. Comput. 195 (2008) 316–325. [31] X.Y. Song, M.Y. Hao, X.Z. Meng, A stage-structured predator–prey model with disturbing pulse and time delays, Appl. Math. Model. 33 (2009) 211–223. [32] Y. Shao, B. Dai, The dynamics of an impulsive delay predator–prey model with stage structure and Beddington-type functional response, Nonlinear Anal. RWA 11 (2010) 3567–3576. [33] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, 1993. [34] V. Lakshmikantham, D.D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [35] D. Bainov, P. Simeonov, System with Impulsive Effect: Stability Theory and Applications, John Wiley and Sons, New York, 1989. [36] X.Y. Song, L.S. Chen, Optimal harvesting and stability for a two species competitive system with stage structure, Math. Biosci. 170 (2001) 173–186.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The dynamics of a stage-structured predator–prey system with impulsive effect and Holling mass defence q Can-Yun Huang ⇑, Yan-Juan Li, Hai-Feng Huo Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 30 September 2010 Received in revised form 9 May 2011 Accepted 16 May 2011 Available online 27 May 2011 Keywords: Impulsive effect Stage structure Global attractivity Permanence
a b s t r a c t In this paper, a stage-structured Holling mass defence predator-prey model with impulsive effect is investigated. By using comparison theorem and the stroboscopic technique, sufficient conditions for the global attractivity of mature prey-extinction periodic solution and permanence of the system are obtained. Furthermore, the numerical analysis is also inserted to illuminate the feasibility of the theoretical results. Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved.
1. Introduction Biological resources is an organic part of the natural resources, and it is a kind of renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark [1]. In the recent management of renewable natural resources, biologists have been aware of the possibility that suitable harvesting and stocking could alter the genetic pattern of a resource [2] and play a key role on the permanence of ecosystems. Recent empirical evidence suggests that this effect is substantial [3–8] and that this issue deserves more attention from resource economists. Considering the exploited Biological system, human harvesting or stocking always happen in a short time or instantaneously, and the continuous action of human should be replaced with an impulsive perturbation. Consequently, impulsive differential equations provide a natural description of such systems [4]. Recently, impulsive differential equations have been extensively used as models in biology, physics, chemistry, engineering and other sciences, with particular emphasis on population dynamics [9–13]. In the real world, many species usually go through two life stages, immature and mature. In [14–19], some stage-structured models of population growth consisting of immature and mature individuals were analyzed, where the stage-structured was modeled by introduction of a constant time delay. However, as literatures [20,21] pointed out, the delay differential equation exhibits much more complicated 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 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. For the prey were simple algology cell of animal,
q
Supported by Advance Foundation of Lanzhou University (BS10200903) and the NNSF of China (10571078).
⇑ Corresponding author.
E-mail address: [email protected] (C.-Y. Huang). 0307-904X/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.038
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invertebrate and vertebrate, Holling type-I, Holling type-II and Holling type-III functional response were proposed, respectively. The three functions are all monotonic in the first quadrant. It implies that, as the prey population increases, the consumption rate of prey per predator increases. However, there is experimental and observational evidence that indicates that this need not always be the case, for example, in the case of ‘‘group defense’’ in population dynamics. Group defense is a term used to describe the phenomenon whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise themselves when their numbers are large enough. An example of this phenomenon described by Holmes and Bethel [22] involves certain insect populations. For more biological information, the reader can refer to [23–25]. It is well known that large swarms of insects make individual identification difficult for their predators [26,27]. To model such an effect, Andrews [28] suggested a function
uðxÞ ¼
mx ; a þ bx þ x2
called Monod–Haldane function, and also called Holling type-IV function. The recent works of Jiang [2] and Jiao [29,30] considered that delayed predator-prey models with impulsive stocking on prey and continuous harvesting on predator, the results of which shows that the behavior of impulsive stocking prey plays an important role for the permanence of the system. In fact, harvesting on predator also should be considered as impulsive perturbation in comparison with stocking on prey. In [31], The authors proposed a stage-structured predator–prey model with disturbing pulse and time delays, and analyzed the effect of impulsive harvesting on predators. In [32], an impulsive delay predator-prey model with stage structure and Beddington-type functional response is established, in which harvesting on prey and stocking on predator are mostly stressed. To investigate the effects of impulsive stocking on immature prey and harvesting on predator, motivated by these literatures [2,29,30,32,31], we consider the following impulsive delay differential equation:
8 9 x01 ðtÞ ¼ r1 x2 ðtÞ r 1 ews x2 ðt sÞ wx1 ðtÞ; > > > > > > > = bx2 ðtÞ > 0 ws 2 > x2 ðtÞ ¼ r1 e x2 ðt sÞ 1þax ðtÞþbx2 ðtÞ x3 ðtÞ d1 x2 ðtÞ d2 x2 ðtÞ; > > t – nT; 2 2 > > > > > < x0 ðtÞ ¼ r x ðtÞ þ kbx2 ðtÞ > 2 ; x ðtÞ d x ðtÞ; 2 3 3 3 2 3 3 1þax2 ðtÞþbx2 ðtÞ 9 > > þ > ðt Þ ¼ x ðtÞ þ l ; x > > 1 1 = > > > > t ¼ nT; x2 ðt þ Þ ¼ x2 ðtÞ; > > > > ; : þ x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
ð1:1Þ
where x1(t), x2(t) represent the immature and mature prey densities respectively, x3(t) denotes the density of the predator, s represents a constant time to maturity, and r1, w, b, a, b, d1, d2, r2, k, d3 are positive constants. r1 is the birth rate of the immature prey, w(w > d1), d1 are the mortality rates of the immature prey and mature prey, b is the maximum numbers of the mature prey that can be eaten by a predator per unit of time, d2, d3 are the intra-specific competition rate of the mature prey and the predator, r2 is the intrinsic growth rate of the predator, k is the rate of conversing prey into predator, l P 0 is the stocking amount of the immature prey, p(0 6 p < 1) represents the harvest rate of the predator at t = nT, n 2 Z+ and Z+ = {1, 2, . . .}, T is the period of the impulsive effect. The initial conditions for system (1.1) are
ð/1 ; /2 ; /3 Þ 2 C ½s; 0; R3þ ;
/i ð0Þ > 0; i ¼ 1; 2; 3;
n o R3þ ¼ x 2 R3 : x P 0 :
From the biological point of view, we only consider (1.1) in the biological meaning region
D ¼ fðx1 ; x2 ; x3 Þ : x1 P 0; x2 P 0; x3 P 0g: The remainder of this paper is arranged as follows. In Section 2, we present some preliminaries. In Section 3, the sufficient conditions for the global attractivity of prey-extinction periodic solution is obtained. In Section 4, the permanence of system (1.1) is investigated. In Section 5, we give some simulations to illustrate our results. A brief conclusion follows in Section 6.
2. Preliminaries The solution of (1.1), denoted by x(t) = (x1(t), x2(t) x3(t))T, is a piecewise continuous function x : Rþ ! R3þ , x(t) is continuous on (nT, (n + 1)T], n 2 Z+ and xðnT þ Þ ¼ limt!nT þ xðtÞ exists. Obviously the global existence and uniqueness of solutions of (1.1) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-side of system (1.1) (see [33,34]). Let V: Rþ R3þ ! Rþ , then V is said to belong to class V0 if: (i) V is continuous in ððn 1ÞT; nT R3þ and for each x 2 R3þ ; lim(t,z)?((n1)T,x)V(t, z) = V((n 1)T, x) and limðt;zÞ!ðnT þ ;xÞ Vðt; zÞ ¼ VðnT þ ; xÞ exist. (ii) V is locally Lipschitzian in x.
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89
Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ððn 1ÞT; nT R3þ , n 2 Z+. The upper right derivative of V(t, x) with respect to the impulsive differential system (1.1) is defined as
Dþ VðtÞ ¼ lim sup h!0þ
1 ½V ðt þ h; x þ hf ðt; xÞÞ Vðt; xÞ: h
Lemma 2.1 ([35,34]). Consider the following impulsive differential inequations:
x0 ðtÞ 6 ðPÞpðtÞxðtÞ þ qðtÞ; t – tk ; xðtþk Þ 6 ðPÞdk xðtk Þ þ bk ; t ¼ tk ;
where p, q 2 C(R+, R), k 2 Z+, dk P 0 and bk are constants. Assume: (A0) the sequence tk satisfies 0 6 t0 < t1 < t2 < , with limk?1tk = 1; (A1) x 2 PC0 (R+, R) and x(t) is left-continuous at tk, k 2 Z+. Then
Y
xðtÞ 6 ðPÞxðt0 Þ þ
0
X
@
t0
þ
Z
t
Y
Z
dj exp
dk exp
t tk
Z
t 0 s
Z
t
pðsÞds
t0
t 0
t k
Y
dk exp
t
!1 pðsÞds Abk
pðhÞdh qðsÞds;
t P t0 :
s
Lemma 2.2 [36]. Consider the following equation
x0 ðtÞ ¼ axðt sÞ bxðtÞ cx2 ðtÞ; where a, b, c and s are positive constants, x(t) > 0 for t 2 [s, 0]. We have (i) If a > b, then limt?+1x(t) = (a b)/c; (ii) If a < b, then limt?+1x(t) = 0. Lemma 2.3. Consider the following system
u0 ðtÞ ¼ c wuðtÞ;
t – nT;
uðtþ Þ ¼ uðtÞ þ l;
t ¼ nT:
ð2:1Þ
Then system (2.1) has a positive periodic solution u⁄(t) with period T. And for any solution u(t) of system (2.1), we have
juðtÞ u ðtÞj ! 0 as t ! 1; where
u ðtÞ ¼
c lewðtnTÞ þ ; w 1 ewT
nT < t 6 ðn þ 1ÞT
and u ð0þ Þ ¼
c l þ : w 1 ewT
Proof. Integrating the first equation of (2.1) on nT < t 6 (n + 1)T, we get
uðtÞ ¼
c c wðtnTÞ þ uðnT þ Þ e ; w w
nT < t 6 ðn þ 1ÞT:
After the successive pulse, we can obtain the following stroboscopic map of system (2.1),
uððn þ 1ÞT þ Þ ¼
c c wT þ uðnT þ Þ e þ l: w w
Eq. (2.2) has a unique positive fixed point
e¼ u
c l þ ; w 1 ewT
which implies that there is a corresponding positive periodic solution:
ð2:2Þ
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u ðtÞ ¼
c lewðtnT Þ þ ; w 1 ewT
with initial value u ð0þ Þ ¼ wc þ 1elwT for nT < t 6 (n + 1)T. Suppose that u(t) is an arbitrary solution of (2.1), using iterative technique, we have
c c c lewðtnTÞ lewt ð1 ewt Þ þ uð0þ Þewt þ lewðtTÞ þ þ lewðtnTÞ ¼ ewt þ uð0þ Þewt þ w w w 1 ewT 1 ewT ¼ ðuð0þ Þ u ð0þ ÞÞewt þ u ðtÞ; nT < t 6 ðn þ 1ÞT:
uðtÞ ¼
Then limt?1ju(t) u⁄(t)j = 0. This completes the proof.
h
Lemma 2.4 [32]. Consider the following system
x0 ðtÞ ¼ xðtÞða bxðtÞÞ; t – nT; xðt þ Þ ¼ ð1 pÞxðtÞ; t ¼ nT:
ð2:3Þ
If p < 1 eaT, then system (2.3) has a positive solution x⁄(t) and for any solution x(t) of system (2.3), we have
jxðtÞ x ðtÞj ! 0 as t ! 1; where
x ðtÞ ¼
a 1 p eaT
; aðtnTÞ bð1 p eaT Þ þ bpe
x ð0þ Þ ¼
a 1 p eaT ; bð1 eaT Þ
nT < t 6 ðn þ 1ÞT:
Lemma 2.5. There exists a constant M > 0 such that x1 ðtÞ 6 Mk, x2 ðtÞ 6 Mk ; x3(t) 6 M for each solution (x1(t), x2(t), x3(t)) of (1.1) with all t large enough, where k is positive constant defined in system (1.1) Proof. Define V(t) = kx1(t) + kx2(t) + x3(t). For t – nT, from w > d1 we have
Dþ VðtÞ þ d1 VðtÞ ¼ kðw d1 Þx1 ðtÞ þ kr 1 x2 ðtÞ kd2 x22 ðtÞ þ ðr 2 þ d1 Þx3 ðtÞ d3 x23 ðtÞ 6 kr 1 x2 ðtÞ kd2 x22 ðtÞ þ ðr 2 þ d1 Þx3 ðtÞ d3 x23 ðtÞ 6 M 0 ; kr 2
2
þd1 Þ where M 0 ¼ 4d12 þ ðr24d . 3
For t = nT, V(nT+) = kx1(nT) + kl + kx2(nT) + (1 p)x3(nT) 6 V(nT) + kl. For t 2 (nT, (n + 1)T], by Lemma 2.1 we obtain
VðtÞ 6 Vð0Þed1 t þ
Z
t
M 0 ed1 ðtsÞ ds þ
0
M0 kled1 T þ ! ; d1 ed1 T 1
X
kled1 ðtnTÞ < Vð0Þed1 t þ
0
M0 ed1 ðtTÞ kled1 T ð1 ed1 t Þ þ kl þ d1 1 ed1 T ed1 T 1
as t ! 1;
which implies that V(t) is uniformly ultimately bounded. Hence, in terms of the definition of V(t), there exists a constant
M¼
M0 kled1 T þ dT > 0; d1 e 1 1
ð2:4Þ
such that x1 ðtÞ 6 Mk, x2 ðtÞ 6 Mk, x3(t) 6 M with t large enough. The proof is completed.
h
3. Global attractivity First, by use of Lemmas 2.3 and 2.4, we can obtain the following result on the existence of the mature prey-extinction periodic solution to system (1.1). Theorem 3.1. System (1.1) has a mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ for t 2 (nT, (n + 1)T], and for any solution (x1(t), x2(t), x3(t)) of system (1.1), then
x1 ðtÞ ! x1 ðtÞ;
x3 ðtÞ ! x3 ðtÞ as t ! 1;
where
x1 ðtÞ ¼ and
lewðtnTÞ 1 ewT
; x3 ðtÞ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnTÞ
for nT < t 6 ðn þ 1ÞT;
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
x1 ð0þ Þ ¼
l
r 2 1 p er2 T : d3 ð1 er2 T Þ
x3 ð0þ Þ ¼
; 1 ewT
91
Next, we investigate the global attractivity of the prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ of system (1.1). Theorem 3.2. Assume that (H1) 1 p er2 T > 0: br 1per2 T 2 ð Þ < d2 1er2 T : (H2) ðr 1 ews d1 Þ 1 þ aM þ bM k k2 Þ 3ð Then the mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ of system (1.1) is globally attractive.
Proof. Let (x1(t), x2(t), x3(t)) be any solution of system (1.1). We can obtain from the third and sixth equation of system (1.1) that
(
x03 ðtÞ P r 2 x3 ðtÞ d3 x23 ðtÞ; þ
x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
t – nT;
ð3:1Þ
t ¼ nT:
So we consider the following auxiliary impulsive differential system:
z01 ðtÞ ¼ z1 ðtÞðr 2 d3 z1 ðtÞÞ; z1 ðt þ Þ ¼ ð1 pÞz1 ðtÞ;
t – nT;
ð3:2Þ
t ¼ nT:
Using Lemma 2.4, we obtain that
z1 ðtÞ ¼
r 2 1 p er2 T ¼ x3 ðtÞ for nT < t 6 ðn þ 1ÞT; d3 ð1 p er2 T Þ þ d3 per2 ðtnTÞ
which is unique and globally attractive positive periodic solution of system (3.1). By using comparison theorem of impulsive differential equation [34], there exist n1 2 Z+ and an arbitrarily small positive constant e0 such that
x3 ðtÞ P z1 ðtÞ P z1 ðtÞ e0 P
r 2 1 p er2 T e0 ,q; d3 ð1 er2 T Þ
ð3:3Þ
for all t P n1T. From (3.3) and Lemma 2.5, one has
0
1
bq
x02 ðtÞ 6 r1 ews x2 ðt sÞ @ þ d1 Ax2 ðtÞ d2 x22 ðtÞ; 2 1 þ aM þ bM 2 k k for t > n1T + s. Considering the auxiliary equation
0
z02 ðtÞ
¼ r1 e
ws
z2 ðt sÞ @
bq 1 þ aM þ bM k k2
2
1
þ d1 Az2 ðtÞ d2 z22 ðtÞ:
According to the hypotheses (H2), for the arbitrarily small positive constant e0, we have
r1 ews <
bq 1 þ aM þ bM k k2
2
þ d1 :
According to Lemma 2.2, we have lim?1z2(t) = 0. Since x2(s) = z2(s) = /2(s) > 0 for all s 2 [s, 0], using the comparison theorem, we can obtain that x2(t) ? 0 as t ? 1. Without loss of generality, we assume that there exists a positive constant e such that
0 < x2 ðtÞ < e;
t P 0:
ð3:4Þ
From the first, the fourth equation of (1.1) and (3.4), we have
x01 ðtÞ 6 r 1 e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – nT; t ¼ nT:
ð3:5Þ
Considering the auxiliary system of (3.5),
z03 ðtÞ ¼ r 1 e wz3 ðtÞ; þ
z3 ðt Þ ¼ z3 ðtÞ þ l;
t – nT; t ¼ nT:
By Lemma 2.3, the unique positive periodic solution of (3.6) is
ð3:6Þ
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C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
z3 ðtÞ ¼
r 1 e lewðtnTÞ þ w 1 ewT
for t 2 ðnT; ðn þ 1ÞT:
In view of the comparison theorem, for any e1 there exists T1 > 0 such that x1 ðtÞ 6 z3 ðtÞ þ e1 for t > T1. Let e ? 0, then z3 ðtÞ ! x1 ðtÞ and
x1 ðtÞ 6 x1 ðtÞ þ e1 :
ð3:7Þ
Again from system (1.1) and (3.4), we have
x01 ðtÞ P r 1 ews e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – nT;
ð3:8Þ
t ¼ nT:
Considering the auxiliary system of (3.8),
z04 ðtÞ ¼ r 1 ews e wz4 ðtÞ; z4 ðtþ Þ ¼ z4 ðtÞ þ l;
t – nT;
ð3:9Þ
t ¼ nT:
By Lemma 2.3, system (3.9) has a unique positive periodic solution
z4 ðtÞ ¼
r 1 ews e lewðtnTÞ for t 2 ðnT; ðn þ 1ÞT: þ w 1 ewT
Similarly, for the sufficiently small constants e1 > 0, there exists T2 > 0 such that x1 ðtÞ P z4 ðtÞ e1 . Let e ? 0, then z4 ðtÞ ! x1 ðtÞ and
x1 ðtÞ P x1 ðtÞ e1 :
ð3:10Þ x1 ðtÞ
It follows from (3.7) and (3.10) that x1 ðtÞ ! as t ? 1. In virtue of the third and sixth equation of system (1.1), we have
(
kbe x03 ðtÞ 6 x3 ðtÞ r 2 þ 1þa e d3 x3 ðtÞ ; þ
x3 ðt Þ ¼ ð1 pÞx3 ðtÞ;
t – nT;
ð3:11Þ
t ¼ nT:
Consider the following auxiliary system of (3.11),
(
kbe z05 ðtÞ ¼ z5 ðtÞ r 2 þ 1þa e d3 z5 ðtÞ ; þ
z5 ðt Þ ¼ ð1 pÞz5 ðtÞ;
t – nT;
ð3:12Þ
t ¼ nT:
By Lemma 2.4, we obtain the unique positive periodic solution of (3.12),
kbe kbe r 2 þ 1þa 1 p eðr2 þ1þaeÞT e z5 ðtÞ ¼ kbe kbe d3 1 p eðr2 þ1þaeÞT þ d3 peðr2 þ1þaeÞðtnTÞ
for nT < t 6 ðn þ 1ÞT:
It follows from comparison theorem that, for any e2 > 0 small enough, there exists a T3 > 0 such that x3 ðtÞ 6 z5 ðtÞ þ e2 for all t > T3. Let e ? 0, then z5 ðtÞ ! x3 ðtÞ, and we have x3 ðtÞ 6 x3 ðtÞ þ e2 . On the other hand, we can follow from 3.1, 3.2 and 3.3 that x3 ðtÞ P x3 ðtÞ e2 for t sufficiently large, which implies that x3 ðtÞ ! x3 ðtÞ as t ? 1. This completes the proof. h 4. Permanence The next work is to investigate the permanence of the system (1.1). Before starting our theorem, we give the definition of permanence. Definition 4.1. System (1.1) is said to be permanent if there exist positive constants m and M such that each positive solution (x1(t), x2(t), x3(t)) of system (1.1) satisfies m 6 x1(t), x2(t), x3(t) 6 M for t sufficiently large enough. Theorem 4.1. Assume that (H3) r1 ewsw bq d1 d2 Mk > 0; r ðm e s MÞ (H4) 1 2 w k > 0; where M, q and m2 are defined in (2.4), (4.7) and (4.10) respectively. Then system (1.1) is permanent. Proof. First, we show that there is a positive constant m2 such that each positive solution (x1(t), x2(t), x3(t)) of (1.1) satisfies x2(t) P m2 for t large enough.
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
93
The second equation of (1.1) can be rewritten as
r 1 ews
x02 ðtÞ ¼
!
bx3 ðtÞ 1 þ ax2 ðtÞ þ
2 bx2 ðtÞ
d1 d2 x2 ðtÞ x2 ðtÞ r1 ews
d dt
Z
t
x2 ðuÞdu:
ð4:1Þ
ts
Let us consider any positive solution (x1(t), x2(t), x3(t)) of system (1.1). According to (4.1), define
VðtÞ ¼ x2 ðtÞ þ r 1 ews
Z
t
x2 ðuÞdu:
ts
Calculating the derivative of V(t) along the solution of (1.1), we have ws
0
V ðtÞ ¼
r1 e
!
bx3 ðtÞ 2
1 þ ax2 ðtÞ þ bx2 ðtÞ
d1 d2 x2 ðtÞ x2 ðtÞ:
ð4:2Þ
Using Lemma 2.5, (4.2) can be written as
M x2 ðtÞ: V 0 ðtÞ > r 1 ews bx3 ðtÞ d1 d2 k
ð4:3Þ
By hypotheses (H3), we can easily know that there exists a sufficiently small e3 > 0 such that
r1 ews > bðq þ e3 Þ þ d1 þ d2
M ; k
ð4:4Þ
where
r 2 þ kbm2 1 p eðr2 þkbm2 ÞT q¼ ; d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT and m2 is determined by the following equation
r2 þ kbm2 1 p eðr2 þkbm2 ÞT 1 M ws r1 e ¼ d1 d2 : b k d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT
We claim that for any t0 > 0, it is impossible that x2 ðtÞ < m2 for all t > t0. Suppose that the claim is invalid, then there is a t0 > 0 such that x2 ðtÞ < m2 for all t > t0. It follows from the third equation of (1.1) that
x03 ðtÞ < x3 ðtÞðr 2 þ kbm2 d3 x3 ðtÞÞ;
ð4:5Þ
for all t > t0. Consider the following auxiliary impulsive system of (4.5)
z06 ðtÞ ¼ z6 ðtÞðr 2 þ kbm2 d3 z6 ðtÞÞ; z06 ðt þ Þ
¼ ð1 pÞz6 ðtÞ;
t – nT;
ð4:6Þ
t ¼ nT:
By Lemma 2.4, we have
z6 ðtÞ ¼
r 2 þ kbm2 1 p eðr2 þkbm2 ÞT ; d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞðtnTÞ
t0 < nT < t 6 ðn þ 1ÞT;
which is the unique positive periodic solution of (4.6) and globally asymptotically stable. In view of the comparison theorem for impulsive differential equation, we know that there exists t1(>t0 + s), such that
x3 ðtÞ 6 z6 ðtÞ þ e3 ; for t P t1, which implies that
1 p eðr2 þkbm2 ÞT z6 ðtÞ 6 ,q: d3 1 p eðr2 þkbm2 ÞT þ d3 peðr2 þkbm2 ÞT
r2 þ kbm2
ð4:7Þ
Thus
x3 ðtÞ 6 q þ e3 , r;
t P t1 :
From (4.4), we get
r1 ews > br þ d1 þ d2
M : k
ð4:8Þ
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C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
By (4.3) and (4.8), we have
M V 0 ðtÞ > r 1 ews br d1 d2 x2 ðtÞ; k
t P t1 :
ð4:9Þ
Let
xm min x2 ðtÞ: 2 ¼ t2½t 1 ;t1 þs
m We will show that x2 ðtÞ P xm 2 for all t P t1. Otherwise, there exists a T0 > 0 such that x2 ðtÞ P x2 for t1 6 t 6 t1 + s + T0, 0 x2 ðt 1 þ s þ T 0 Þ P xm and x ðt þ s þ T Þ < 0. From the second equation of system (1.1) and (4.8) we see 1 0 2 2
M m x > 0; x02 ðt1 þ s þ T 0 Þ P r 1 ews br d1 d2 k 2
which is a contradiction. Thus, x2 ðtÞ P xm 2 for all t P t1. From (4.4) and (4.9), we have
M m V 0 ðtÞ > r 1 ews br d1 d2 x > 0; k 2
t P t1 ;
which means that V(t) ? 1 as t ? 1. It is a contradiction to VðtÞ 6 Mk ð1 þ r1 sews Þ. Therefore, for any t0 > 0, the inequality x2 ðtÞ < m2 cannot hold for all t P t0. So, there exist the following two possibilities. Case 1. If x2 ðtÞ P m2 holds for all t large enough, then our aim is obtained. Case 2. Assume that x2(t) is oscillatory about m2 . Setting
m2 ¼ min
m2 ; m2 eðbMþd1 þd2 m2 Þs ; 2
ð4:10Þ
we will prove that x2(t) P m2 for all t large enough. Suppose that there exist the positive constants t⁄,n such that x2(t⁄) = x2 ðt þ nÞ ¼ m2 and x2 ðtÞ < m2 for all t⁄ < t < t⁄ + n, where t⁄ is sufficiently large, and the inequality (4.8) holds true for t⁄ < t < t⁄ + n. Since x2(t) is continuous and bounded and is not effected by impulses, we conclude that x2(t) is uniformly m
continuous. Hence, there exists a constant T (0 < T < s and T is independent of the choice of t⁄) such that x2 ðt Þ > 22 for t⁄ 6 t 6 t⁄ + T. If n 6 T, our aim is obtained. If T < n 6 s, from the second equation of (1.1) we have that for t < t < t þ n; x02 ðtÞ P bx2 ðtÞx3 ðtÞ d1 x2 ðtÞ d2 x22 ðtÞ: According to the assumption x2 ðt Þ ¼ m2 and x2 ðtÞ < m2 for t⁄ < t < t⁄ + n, we have x02 ðtÞ P ðbM þ d1 þ d2 m2 Þx2 ðtÞ for t⁄ < t 6 t⁄ + n 6 t⁄ + s. Then, we derive that ⁄ ⁄ ðbMþd1 þd2 m2 Þs . It is clear that x2(t) P m2 for t < t 6 t + n. If n P s, then we have that x2(t) P m2 for t⁄ < t < t⁄ + s. x2 ðtÞ P m2 e The same arguments can be continued, we can obtain x2(t) P m2 for t⁄ + s 6 t 6 t⁄ + n. Since the interval [t⁄, t⁄ + n] is arbitrarily chosen, we get that x2(t) P m2 for t large enough. In view of our arguments above, the choice of m2 is independent of the positive solution of (1.1) which satisfies that x2(t) P m2 for sufficiently large t. Next, from the first and the fourth equation of system (1.1), we have
(
x01 ðtÞ P r1 m2 ews Mk wx1 ðtÞ; x1 ðtþ Þ ¼ x1 ðtÞ þ l;
t – nT;
t ¼ nT:
ð4:11Þ
Considering the comparison system
(
z07 ðtÞ ¼ r 1 m2 ews Mk wz7 ðtÞ; z7 ðtþ Þ ¼ z7 ðtÞ þ l;
t – nT;
t ¼ nT:
ð4:12Þ
By using comparison theorem, there exists a e4 > 0 small enough such that x1 ðtÞ P z7 ðtÞ e4 for sufficiently large t, where z7 ðtÞ is the unique and globally stable positive periodic solution of (4.12), and
z7 ðtÞ ¼
r 1 m2 ews Mk lewðtnTÞ : þ 1 ewT w
In view of the comparison theorem of impulsive differential equation, we derive from (H4) that
r1 m2 ews Mk lewT þ x1 ðtÞ P e4 ,m1 : w 1 ewT From (3.3), let q , m3, and we have x3(t) P m3. Take m = min{m1, m2, m3}, then we have x1(t) P m, x2(t) P m, x3(t) P m. By Lemma 2.5 and the above discussion, system (1.1) is permanent. The proof is completed. h
C.-Y. Huang et al. / Applied Mathematical Modelling 36 (2012) 87–96
95
Fig. 1. Dynamical behavior of system (1.1) with T = 10, l = 0.1, p = 0.2, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 1, a = 1, b = 0.5, d1 = 0.5, d2 = 0.2, r2 = 1, k = 3, d3 = 0.3, s = 0. (a) Time series of the immature prey population. (b) Time series of the mature prey population. (c) Time series of the predator population.
Fig. 2. Dynamical behavior of system (1.1) with T = 0.5, l = 4, p = 0.25, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 0.5, a = 1, b = 3, d1 = 0.5, d2 = 0.2, r2 = 1, k = 0.1, d3 = 0.3, s = 0. (a) Time series of the immature prey population. (b) Time series of the mature prey population. (c) Time series of the predator population.
5. Numerical simulations To investigate the impulsive effect of a stage-structured Holling mass defence predator-prey model by numerical analysis, we assume that T = 10, l = 0.1, p = 0.2, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 1, a = 1, b = 0.5, d1 = 0.5, d2 = 0.2, r2 = 1, k = 3, d3 = 0.3, s = 0. It is obvious that the parameters satisfy the condition of Theorem 3.2. Then the mature preyextinction periodic solution is global attractivity (see Fig. 1). Secondly, we assume that T = 0.5, l = 4, p = 0.25, x1(0) = 0.5, x2(0) = 0.5, x3(0) = 0.5, r1 = 1, w = 1, b = 0.5, a = 1, b = 3, d1 = 0.5, d2 = 0.2, r2 = 1, k = 0.1, d3 = 0.3, s = 0, it is obvious that the parameters satisfy Theorem 4.1, then system (1.1) is permanent (see Fig. 2). 6. Conclusion According to the fact of biological resource management in this paper we consider a stage-structured Holling mass defence predator-prey model with impulsive effect. We analyze that the predator-extinction periodic solution of system (1.1) is globally attractive, and obtain that the sufficient condition for the permanence of system (1.1). From Theorem 3.2, we can see that, increasing T, or reducing l or p is propitious to the global attractivity of the mature prey-extinction periodic solution ðx1 ðtÞ; 0; x3 ðtÞÞ. That is, if the immature prey is stocked too few, or the mature prey is caught excessively, then the mature prey population is extinction totally, and lose the merits of exploitative mature prey population. By Theorem 4.1, we may see that, reducing T, increasing l or p are in favor of the permanence of system (1.1). It implies that the behavior of impulsive stocking on prey plays an important role for the permanence of system (1.1), and reasonable harvesting predator can ensure the sustainable development of biological resources. From Theorems 3.2 and 4.1, we believe that there must exist a sharp threshold, but we could not get it with the current technology. Compared with other models, it is much easier to operate and manage the biological resource system. The results correspond with the reality, and provide tactical basis for the biological resource management. However, under the case of considering the manpower and the capital, i.e. the cost of control, human can take optimal strategy in the practice of biology control. So, we will continue to study the optimal harvesting policy of system (1.1) in the future. References [1] C.W. Clark, Mathematical Bioeconomics, Wiley, New York, 1990. [2] X.W. Jiang, Q. Song, M.Y. Hao, Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect, Appl. Math. Comput. 215 (2010) 4221–4229.
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[3] D.W. Coltman, P.O. Donoghue, J.T. Jorgenson, J.T. Hogg, C. Strobeck, M. Festa-Bianchet, Undesirable evolutionary consequences of trophy hunting, Nature 426 (6967) (2003) 655–658. [4] D.O. Conover, S.B. Munch, Sustaining fisheries yields over evolutionary time scales, Science 297 (5578) (2002) 94–96. [5] T. Haugen, L.A. Volestad, A century of life-history evolution in Grayling, Genetica 112-113 (2001) 475–491. [6] M. Heino, O.R. Gods, Fisheries-induced selection pressures in the context of sustainable fisheries, Bull. Mar. Sci. 70 (2002) 639–656. [7] L. Ling, W.M. Wang, Dynamics of a Ivlev-type predator–prey system with constant rate harvesting, Chaos Soliton Fract. 41 (2009) 2139–2153. [8] E.M. Olsen, M. Heino, G.R. Lilly, M.J. Morgan, J. Brattey, B. Ernande, U. Dieckmann, Maturation trends indicative of rapid evolution preceded the collapse of Northern Cod, Nature 428 (2004) 932–935. [9] J. Hui, D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Soliton Fract. 29 (2006) 233–251. [10] B. Liu, L.S. Chen, The periodic competing Lotka–Volterra model with impulsive effect, IMA J. Math. Med. Biol. 21 (2004) 129–145. [11] B. Liu, Y. Zhang, L.S. Chen, The dynamical behaviors of a Lotka–Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. RWA 6 (2005) 227–243. [12] A. D’Onofrio, Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model. 36 (2002) 473–489. [13] H. Zhang, P. Georgescu, L.S. Chen, An impulsive predator–prey system with Beddington–Deangelis functional response and time delay, Int. J. Biomath. 1 (2008) 1–17. [14] W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci. 101 (1990) 139–153. [15] F.D. Chen, M.S. You, Permanence, extinction and periodic solution of the predator-prey system with Beddington–DeAngelis functional response and stage structure for prey, Nonlinear Anal. RWA 9 (2008) 207–221. [16] H.I. Freedman, K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol. 48 (1986) 485–492. [17] S.A. Gourley, Y. Kuang, A stage structured predator–prey model and its dependence on maturation delay and death rate, J. Math. Biol. 49 (2004) 188– 200. [18] X.Y. Song, L.S. Chen, Modelling and analysis of a single species system with stage structure and harvesting, Math. Comput. Model. 36 (2002) 67–82. [19] X. Wang, W. Wang, X. Lin, Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy, Chaos Soliton Fract. 37 (2008) 706– 718. [20] Y. Kuang, Delay Differential Equation With Application in Population Dynamics, Academic Press, New York, 1993. [21] Y.K. Li, Y. Kuang, Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) 260–280. [22] J.C. Holmes, W.M. Bethel, Modification of intermediate host behavior parasites, Zool. J. Linnean Soc. 51 (1972) 123–149. [23] H.I. Freedman, G.S.K. Wolkowicz, Predator–prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol. 48 (1986) 493–508. [24] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (4) (2000) 1445–1472. [25] H. Zhu, S. Campbell, G. Wolkowicz, Bifurcation analysis of a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63 (2) (2002) 636–682. [26] J.J. Jiao, X.Z. Meng, L.S. Chen, A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators, Appl. Math. Comput. 189 (2007) 1448–1458. [27] Z.J. Liu, R.H. Tan, Impulsive harvesting and stocking in a Monod–Haldane functional response predator-prey system, Chaos Soliton Fract. 34 (2007) 454–464. [28] J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng. 10 (1968) 707– 723. [29] J.J. Jiao, X.Z. Meng, L.S. Chen, Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey, Chaos Soliton Fract. 41 (2009) 103–112. [30] J.J. Jiao, G.P. Pang, L.S. Chen, G.L. Luo, A delayed stage-structured predator–prey model with impulsive stocking on prey and continuous harvesting on predator, Appl. Math. Comput. 195 (2008) 316–325. [31] X.Y. Song, M.Y. Hao, X.Z. Meng, A stage-structured predator–prey model with disturbing pulse and time delays, Appl. Math. Model. 33 (2009) 211–223. [32] Y. Shao, B. Dai, The dynamics of an impulsive delay predator–prey model with stage structure and Beddington-type functional response, Nonlinear Anal. RWA 11 (2010) 3567–3576. [33] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, 1993. [34] V. Lakshmikantham, D.D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [35] D. Bainov, P. Simeonov, System with Impulsive Effect: Stability Theory and Applications, John Wiley and Sons, New York, 1989. [36] X.Y. Song, L.S. Chen, Optimal harvesting and stability for a two species competitive system with stage structure, Math. Biosci. 170 (2001) 173–186.