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Permanence of a stage-structured predator–prey system with impulsive stocking prey and harvesting predator
Applied Mathematics and Computation 235 (2014) 32–42
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Permanence of a stage-structured predator–prey system with impulsive stocking prey and harvesting predator q Xinhui Wang, Canyun Huang ⇑ 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
Keywords: Permanence Impulses Stage-structure Global attractivity
a b s t r a c t In this paper we propose a stage-structured predator prey system with impulsive stocking immature prey and harvesting predator at different moment. We analyze the global attractivity of the mature prey-extinction periodic solution, and obtain sufficient conditions for the permanence of the system. Numerical simulations are also inserted to verify the feasibility of the theoretical results. Moreover, the obtained results show that impulsive stocking immature prey or harvesting predator may play a key role on the permanence of the system and provide tactical basis for the biological resources management. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Since biological resources can reproduce themselves in a natural way, it is regarded as the most important part of the natural resources. In recent years, how to get maximal profit and protect the sustainable development of biological system have already drawn a great deal of attention of biologists and resources economists [1]. Recent literatures [2–4] revealed that harvesting and stocking prey or predator can influence ecological system, but overusing these methods sometimes can destroy the ecological balance of renewable resources. To protect the permanence of biological resources, it will be a dominant theme to seek for the suitable threshold of harvesting and stocking in mathematical ecology. Compared with the growth process of biological resources, manual harvesting or stocking always happen in a short time, which should be described by impulsive perturbation instead of continuous effect. Consequently, impulsive differential equations provide a more realistic description of such exploited system [12,9–11,5,8,6,7]. Recently, impulsive differential equations have been extensively used as models in biology, physics, chemistry, engineering and other sciences, with particular emphasis on population dynamics [13–17]. In [12], Wang proposed a system with impulsive immigration of predator. The effect of harvesting on prey and predator with constant efforts has been discussed by Negi [11]. In [18], Shao and Li discussed an impulsive predator–prey system with stage structure and generalized functional response, to model the diffusion by impulses. Sufficient conditions are established for the existence of a predator-extinction positive periodic solution and the permanence of the system. Numerical simulation shows that impulses and functional response affect the dynamics of the system. In the natural world, many species usually go through two distinct life stages from birth to death, immature and mature. The immature takes s units of time to mature and it is well-known that time delay s is an important factor of mathematical models in ecology. The predator–prey models with stage-structure have been investigated by many researchers [19–24]. Time delay and impulse are incorporated into predator–prey models, which greatly enriches biologic background. But the system become so complicated that it causes us greatly difficulties to study the model. The investigation of impulsive delay q
Supported by Natural Science Foundation of Gansu Province (1107RJZA164).
⇑ Corresponding author.
E-mail address: [email protected] (C. Huang). http://dx.doi.org/10.1016/j.amc.2014.02.092 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
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differential equations is inchoate, and focus on theoretical analysis [26,25,27]. In [9] Liu studied a stage-structured model with impulsive perturbations on predator. The recent works of Jiang [8] and Jiao [6,7] considered some delayed predator– prey models with impulsive stocking on prey and continuous harvesting on predator, in which the influences of impulses and stage-structure are analyzed. The dynamical relationships between predator and prey can be represented by the functional response, which refers to the change in the density of prey attached per unit time per predator as the prey density change. When the prey are simple algology cell of animal, invertebrate and vertebrate, Holling type-I, Holling type-II and Holling type-III functional response are proposed, respectively. Luo [10] considered a stage-structured predator–prey Leslie–Gower Holling II model with disturbing pulse. In [5], the authors analyzed a stage-structured predator–prey model with Holling mass defence, and discussed the effect of impulsive stocking prey and harvesting predator at same fixed time. In 1975, Beddington [29] and DeAngelis et al. [30] introduced an additional functional response
uðxÞ ¼
mx : a þ x þ by
This functional response has an extra term in the denominator, which models the mutual interference between predators and avoids the ‘‘low densities problem’’ of the ratio-dependent type functional response. In [31], an impulsive delayed 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. In fact, stocking prey and harvesting predator usually be at different moment in the real world. The paper [32] established a Lotka–Volterra model with birth pulse and impulsive catching or poisoning for the prey at different moment, and obtained sufficient conditions of the global attractivity of predator-extinction periodic solution and the permanence of the system. Motivated by these literatures [6–8,22,28,31–34], we consider the following impulsive stage-structured differential system
8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > :
x01 ðtÞ ¼ r 1 x2 ðt Þ r 1 ews x2 ðt sÞ cx1 ðt Þ; Þyðt Þ x02 ðtÞ ¼ r 1 ews x2 ðt sÞ aþxcx22ððttÞþby d1 x2 ðtÞ d2 x22 ðt Þ; ðt Þ 2 ðt Þ y0 ðt Þ ¼ r2 yðt Þ þ aþxkcx yðt Þ d3 y2 ðt Þ; 2 ðt Þþbyðt Þ 9 þ x1 ðt Þ ¼ x1 ðt Þ þ l; > = t ¼ ðn þ l 1ÞT; x2 ðtþ Þ ¼ x2 ðt Þ; > ; yðtþ Þ ¼ yðtÞ; 9 x1 ðtþ Þ ¼ x1 ðt Þ; > = t ¼ nT; x2 ðtþ Þ ¼ x2 ðt Þ; > ; yðtþ Þ ¼ ð1 pÞyðt Þ;
9 > > = > > ;
t – ðn þ l 1ÞT; nT;
ð1:1Þ
where x1 ðtÞ; x2 ðt Þ represent the immature and mature prey densities respectively, yðtÞ denotes the density of the predator, s represents a constant time to maturity, and r1 ; w; c; a; b; d1 ; d2 ; r 2 ; k; d3 are positive constants. r1 is the birth rate of the immature prey, wðw > d1 Þ; d1 denotes the mortality rate of the mature prey, c 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, r 2 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 at t ¼ ðn þ l 1ÞT; l is a constant (0 < l < 1), pð0 6 p < 1Þ represents the harvesting rate on the predator at t ¼ nT; n 2 Z þ and Z þ ¼ f1; 2; . . .g; T is the seasonal period of the biological resources. 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, we only consider system (1.1) in the following region
D ¼ fðx1 ; x2 ; yÞ : x1 P 0; x2 P 0; y P 0g: The reminder of this paper is arranged as follows. In Section 2, some fundamental notations, definitions and lemmas are firstly given. In Section 3, we consider the global attractivity of the mature prey-extinction periodic solution. In Section 4, we investigate the sufficient conditions for the permanence of system (1.1). Numerical simulations are presented to verify the feasibility of theoretical results in last section. 2. Notations and preliminaries Let Rþ ¼ ½0; þ1Þ; R3þ ¼ fx 2 R3 ; x P 0g. The solution of (1.1), denoted by xðt Þ ¼ ðx1 ðt Þ; x2 ðtÞ; yðtÞÞT , is a piecewise continu ous function x : Rþ ! R3þ ; xðt Þ is continuous on ððn 1ÞT; ðn þ l 1ÞT and ððn þ l 1ÞT; nT; x ðn l þ 1ÞT þ þ ¼ limt!ðnlþ1ÞT þ xðt Þ and x nT ¼ limt!nT þ xðt Þ exist. Obviously, the global existence and uniqueness of solution of (1.1) is guaranteed by the smoothness properties of f, which denotes the right-side of the first three equations of system (1.1) (see [35,36]).
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Let V : Rþ R3þ ! Rþ , then V is said to belong to class V 0 if: (i) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ . For each x 2 R3þ ; limðt;zÞ!ðKT ;xÞ V ðt; zÞ ¼ V ðKT; xÞ and limðt;zÞ!ðKT þ ;xÞ V ðt; zÞ ¼ V KT þ ; x exist, where K ¼ n; n þ l 1. (ii) V is locally Lipschitzian in x. Definition 2.1. For V 2 V 0 ; ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R3þ or ððn þ l 1ÞT; nT R3þ , the upper right derivative of V ðt; xÞ with respect to system (1.1) is defined as
1 Dþ V ðtÞ ¼ lim sup ½V ðt þ h; x þ hf ðt; xÞÞ V ðt; xÞ: h h!0þ Lemma 2.1. [37,36]. Consider the following impulsive differential inequations
(
u0 ðt Þ 6 ðPÞpðt Þuðt Þ þ qðt Þ; t – tk ; u tþk 6 ðPÞdk uðtk Þ þ bk ; t ¼ tk ;
where p; q 2 C ðRþ ; RÞ; k 2 Z þ ; dk P 0 and bk are constants. Assume: (A1) the sequence tk satisfies 0 6 t0 < t 1 < t 2 < , with limk!1 tk ¼ 1; (A2) u 2 PC 0 ðRþ ; RÞ, where PC 0 ðRþ ; RÞ denote the set of functions u : Rþ ! R, and u is continuously differential in ðtk1 ; tk and left-continuous at tk ; k 2 Z þ . Then
uðtÞ 6 ðPÞuðt0 Þ
Y
dk exp
Z
t
t0
t 0
0 !1 Z t Z X Y @ pðsÞds þ dj exp pðsÞds Abk þ t 0
t k
tk
t
Y
dk exp
Z
t0 s
t
pðhÞdh qðsÞds; t P t0 :
s
Lemma 2.2 [38]. Consider the following equation
u0 ðt Þ ¼ auðt sÞ buðt Þ cu2 ðt Þ; where a; b; c and s are positive constants, uðt Þ > 0 for t 2 ½s; 0. We have (i) If a > b, then limt!þ1 uðtÞ ¼ ða bÞ=c; (ii) If a < b, then limt!þ1 uðtÞ ¼ 0. Lemma 2.3. Consider the following system
u0 ðt Þ ¼ c wuðt Þ; t – nT; uðtþ Þ ¼ uðt Þ þ l; t ¼ nT:
ð2:1Þ
Then system (2.1) has a positive periodic solution u ðt Þ. For any solution uðtÞ of system (2.1), we have
juðt Þ u ðtÞj ! 0 as t ! 1; where
u ðt Þ ¼
c l u 0þ ¼ þ ; w 1 ewT
c lewðtnT Þ þ ; w 1 ewT
nT < t 6 ðn þ 1ÞT:
Lemma 2.4 [5]. Consider the following system
u0 ðt Þ ¼ uðtÞða buðtÞÞ; uðtþ Þ ¼ ð1 pÞuðt Þ;
t – nT;
ð2:2Þ
t ¼ nT:
If p < 1 eaT , then system (2.2) has a positive solution u ðt Þ. For any solution uðtÞ of system (2.2), we have
juðt Þ u ðtÞj ! 0 as t ! 1; where
u ðt Þ ¼
a 1 p eaT bð1 p eaT Þ þ bpe
; aðtnT Þ
a 1 p eaT u 0þ ¼ ; bð1 eaT Þ
nT < t 6 ðn þ 1ÞT:
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Lemma 2.5. There exists a constant M > 0 such that x1 ðtÞ 6 Mk ; x2 ðtÞ 6 Mk ; yðt Þ 6 M for each solution ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ of (1.1) with all t large enough, where k is the positive constant defined in system (1.1). Proof. Define V ðt Þ ¼ kx1 ðt Þ þ kx2 ðtÞ þ yðtÞ. For t – ðn þ l 1ÞT; nT, calculating the upper right derivative of V ðtÞ along the solution of system (1.1), we have
Dþ V ðt Þ þ d1 V ðtÞ ¼ kðw d1 Þx1 ðt Þ þ kr 1 x2 ðt Þ kd2 x22 ðtÞ þ ðr 2 þ d1 ÞyðtÞ d3 y2 ðtÞ 6 kr1 x2 ðt Þ kd2 x22 ðtÞ þ ðr 2 þ d1 ÞyðtÞ d3 y2 ðt Þ 6 M0 ; r2
2
1 þr 2 Þ where M 0 ¼ kðr1 d þ d23 . 4d2
For t ¼ nT; V ðtþ Þ ¼ kx1 ðt Þ þ kx2 ðt Þ þ ð1 pÞyðtÞ 6 V ðtÞ. For t ¼ ðn þ l 1ÞT; V ðt þ Þ ¼ kx1 ðtÞ þ kx2 ðtÞ þ yðt Þ þ l. By Lemma 2.1 we obtain
V ðt Þ 6 V ð0Þer2 t þ
Z
X
t
M 0 er2 ðtsÞ ds þ
0
kler2 ðtnT Þ < V ð0Þer2 t þ
0
M0 k le ! þ rT ; r2 e2 1
M0 er2 ðtT Þ kler2 T 1 er2 t þ kl þ rT r T r2 1e2 e2 1
r2 T
as t ! 1;
which implies that V ðt Þ is uniformly ultimately bounded. Hence, in terms of the definition of V ðt Þ, there exists a constant
M¼
M0 kler2 T þ > 0; r 2 er 2 T 1
ð2:3Þ
such that x1 ðt Þ 6 Mk ; x2 ðtÞ 6 Mk ; yðtÞ 6 M with t large enough. The proof is completed.
h
3. Global attractivity From Lemmas 2.3 and 2.4, we can obtain the following result on the existence of the mature prey-extinction periodic solution of system (1.1). Theorem 3.1. System (1.1) has a mature prey-extinction periodic solution ðx1 ðt Þ; 0; y ðtÞÞ ðn þ l 1ÞT [ ððn þ l 1ÞT; nT and for any solution ðx1 ðtÞ; x2 ðtÞ; yðt ÞÞ of system (1.1), then
x1 ðtÞ ! x1 ðt Þ;
for
t 2 ððn 1ÞT;
yðt Þ ! y ðtÞ as t ! 1;
where
x1 ðtÞ ¼
lecðtðnþl1ÞT Þ 1 ecT
;
ðn þ 1 1ÞT < t 6 ðn þ 1ÞT;
and
y ðt Þ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnT Þ
nT < t 6 ðn þ 1ÞT:
Next, we investigate the global attractivity of the mature prey-extinction periodic solution x1 ðt Þ; 0; y ðt Þ of system (1.1). Theorem 3.2. Assume that (H1) 1 p er2 T > 0; cr2 ð1per2 T Þ (H2) ðr 1 ews d1 Þ a þ Mk þ bM < d 1er2 T . Þ 3ð Then the mature prey-extinction periodic solution x1 ðtÞ; 0; y ðt Þ of system (1.1) is globally attractive. Proof. Let ðx1 ðtÞ; x2 ðt Þ; yðt ÞÞ be any solution of system (1.1). From the third and ninth equation of system (1.1) we obtain
(
y0 ðtÞ P r2 yðt Þ d3 y2 ðt Þ; yðtþ Þ ¼ ð1 pÞyðtÞ;
t – nT;
t ¼ nT:
ð3:1Þ
Consider the following auxiliary impulsive differential system
z01 ðt Þ ¼ z1 ðt Þðr 2 d3 z1 ðtÞÞ; þ
z1 ðt Þ ¼ ð1 pÞz1 ðtÞ;
t – nT;
t ¼ nT:
ð3:2Þ
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Using Lemma 2.4, we obtain
z1 ðtÞ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnT Þ
nT < t 6 ðn þ 1ÞT;
which is unique and globally attractive positive periodic solution of system (3.1). By using comparison theorem of impulsive differential equation [36], there exist a n1 2 Z þ and an arbitrarily small positive constant e0 such that
yðtÞ 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 n1 T. For t > n1 T þ s, from (3.3) and Lemma 2.5 we have
! cq x ðtÞ d2 x22 ðt Þ: þ d a þ Mk þ bM 1 2
x02 ðtÞ 6 r1 ews x2 ðt sÞ
ð3:4Þ
According to the hypotheses ðH2 Þ, we have
r1 ews <
cq þd : a þ Mk þ bM 1
ð3:5Þ
Consider the auxiliary impulsive differential equation of (3.4)
z02 ðtÞ
¼ r1 e
ws
! cq z2 ðt sÞ þ d z ðtÞ d2 z22 ðt Þ; t > n1 T þ s: a þ Mk þ bM 1 2
From 2.2 and (3.5), we have limt!1 z2 ðt Þ ¼ 0. Since x2 ðsÞ ¼ z2 ðsÞ ¼ /2 ðsÞ > 0 for all s 2 ½s; 0, we can obtain that x2 ðtÞ ! 0 as t ! 1. Without loss of generality, we assume that there exists a positive constant e such that
0 < x2 ðt Þ < e;
t P 0:
ð3:6Þ
From the first, the fourth equation of (1.1) and (3.6), we have
x01 ðt Þ 6 r 1 e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – ðn þ l 1ÞT; t ¼ ðn þ l 1ÞT:
ð3:7Þ
Consider the auxiliary system of (3.7)
z03 ðtÞ ¼ r 1 e wz3 ðt Þ; z3 ðtþ Þ ¼ z3 ðtÞ þ l;
t – ðn þ l 1ÞT; t ¼ ðn þ l 1ÞT:
ð3:8Þ
By Lemma 2.3, the unique positive periodic solution of system (3.8) is
z3 ðtÞ ¼
r 1 e lewðtðnþl1ÞÞ þ ; w 1 ewT
t 2 ððn þ l 1ÞT; ðn þ lÞT :
In view of the comparison theorem, for any z3 ðt Þ ! x1 ðt Þ and
e1 , there exists a n2 such that x1 ðtÞ 6 z3 ðtÞ þ e1 for t > n2 T. Let e ! 0, then
x1 ðtÞ 6 x1 ðtÞ þ e1 :
ð3:9Þ
Again from system (1.1) and (3.6), we have
x01 ðt Þ P r 1 ews e wx1 ðtÞ; t – ðn þ l 1ÞT; x1 ðt þ Þ ¼ x1 ðtÞ þ l; t ¼ ðn þ l 1ÞT:
ð3:10Þ
Consider the auxiliary system of system (3.10)
z04 ðtÞ ¼ r 1 ews e wz4 ðtÞ; z4 ðtþ Þ ¼ z4 ðtÞ þ l;
t – ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð3:11Þ
By Lemma 2.3, system (3.11) has a unique positive periodic solution
z4 ðtÞ ¼
r 1 ews e lewðtðnþl1ÞÞ þ ; w 1 ewT
Similarly, for the constant and
x1 ðtÞ P x1 ðtÞ e1 :
t 2 ððn þ l 1ÞT; ðn þ lÞT :
e1 > 0, there exists a n3 > n2 such that x1 ðtÞ P z4 ðtÞ e1 for t > n3 T. Let e ! 0, then z4 ðtÞ ! x1 ðtÞ ð3:12Þ
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
37
It follows from (3.9) and (3.12) that x1 ðtÞ ! x1 ðtÞ as t ! 1. In virtue of the third and ninth equation of system (1.1), we have
(
y0 ðtÞ 6 yðtÞ r 2 þ akcþee d3 yðt Þ ; þ
yðt Þ ¼ ð1 pÞyðtÞ;
t – nT;
ð3:13Þ
t ¼ nT:
Consider the following auxiliary system of system (3.13)
(
z05 ðtÞ ¼ z5 ðt Þ r 2 þ akcþee d3 z5 ðtÞ ; þ
z5 ðt Þ ¼ ð1 pÞz5 ðt Þ;
t – nT;
ð3:14Þ
t ¼ nT:
By Lemma 2.4, we obtain the unique positive periodic solution of system (3.14)
kce r 2 þ akcþee 1 p eðr2 þaþeÞT z5 ðtÞ ¼ ; kce kce d3 1 p eðr2 þaþeÞT þ d3 peðr2 þaþeÞðtnT Þ
nT < t 6 ðn þ 1ÞT:
It follows from comparison theorem that, for any sufficiently small constant e2 > 0, there exists a n4 > 0 such that yðtÞ 6 z5 ðtÞ þ e2 for all t > n4 T. Let e ! 0, then z5 ðtÞ ! y ðt Þ, and we have yðt Þ 6 y ðt Þ þ e2 . On the other hand, we can follow from (3.1), (3.2) and (3.3) that yðtÞ P y ðt Þ e2 for t sufficiently large, which implies that yðt Þ ! y ðtÞ as t ! 1. This completes the proof. h
4. Permanence Next work is to investigate the permanence of 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 ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (1.1) satisfies m 6 x1 ðtÞ; x2 ðtÞ; yðtÞ 6 M for t sufficiently large enough. Theorem 4.1. Assume that cq (H3) r1 ews aþbq d1 d2 Mk > 0;
(H4)
r 1 ðm2 ews M kÞ w
> 0,
where M; q and m2 are defined in (2.3), (4.5) and (4.11) respectively. Then system (1.1) is permanent. Proof. Firstly, we show that there is a positive constant m2 such that each positive solution ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ of system (1.1) satisfies x2 ðtÞ P m2 for t large enough. The second equation of (1.1) can be rewritten as
x02 ðtÞ ¼ r 1 ews
Z t cyðt Þ d x2 ðuÞdu: d1 d2 x2 ðt Þ x2 ðt Þ r 1 ews dt ts a þ x2 ðtÞ þ byðtÞ
ð4:1Þ
Let ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ be an solution of system (1.1), define
V ðt Þ ¼ x2 ðt Þ þ r 1 ews
Z
t
x2 ðuÞdu:
ts
Calculating the derivative of V ðt Þ along the solution of system (1.1), we have
V 0 ðtÞ ¼ r 1 ews
cyðt Þ d1 d2 x2 ðtÞ x2 ðt Þ: a þ x2 ðtÞ þ byðtÞ
ð4:2Þ
By 2.5, (4.2) can be written as
V 0 ðtÞ > r 1 ews
cyðt Þ M x2 ðtÞ: d1 d2 k a þ byðtÞ
From hypotheses ðH3 Þ, we can easily know that there exists a sufficiently small
r1 ews >
cðq þ e3 Þ M þ d þ d2 ; k a þ bðq þ e3 Þ 1
ð4:3Þ
e3 > 0 such that ð4:4Þ
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
where
1 p eðr2 þkcm2 ÞT q¼ d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞT
r2 þ kcm2
and m2 is determined by the following equation
ð4:5Þ
1 p eðr2 þkcm2 ÞT ¼ : c b r 1 ews d1 d2 Mk d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞT
a r1 ews d1 d2 Mk
r2 þ kcm2
We claim that for any t 0 > 0, it is impossible that x2 ðt Þ < m2 for all t > t 0 . Suppose that the claim is invalid, then there is a t0 > 0 such that x2 ðt Þ < m2 for all t > t0 . It follows from the third equation of (1.1) that
y0 ðt Þ < yðt Þ r 2 þ kcm2 d3 yðt Þ ;
t > t0 :
ð4:6Þ
Consider the following auxiliary impulsive system
(
z06 ðtÞ ¼ z6 ðtÞ r 2 þ kcm2 d3 z6 ðt Þ ; z06 ðtþ Þ ¼ ð1 pÞz6 ðt Þ;
t – nT;
ð4:7Þ
t ¼ nT:
By Lemma 2.4, we have
1 p eðr2 þkcm2 ÞT ¼ ; d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞðtnT Þ
z6 ðtÞ
r2 þ kcm2
t 0 < nT < t 6 ðn þ 1ÞT;
which is the unique positive periodic solution of (4.7) and globally asymptotically stable. In view of the comparison theorem, we know that there exists a t1 ð> t 0 þ sÞ, such that
yðtÞ 6 z6 ðtÞ þ e3 ;
t P t1 :
It follows from (4.5) that
z6 ðtÞ 6 q;
t P t1 :
ð4:8Þ
Thus
yðtÞ 6 q þ e3 , r;
t P t1 :
ð4:9Þ
From (4.4), we get
r1 ews >
cr M þ d þ d2 : k a þ br 1
By (4.3) and (4.9), we have
V 0 ðt Þ > r 1 ews
cr M x2 ðt Þ; d1 d2 k a þ br
t P t1 :
ð4:10Þ
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 t 1 . Otherwise, there exists a d > 0 such that x2 ðt Þ P x2 for t 1 6 t 6 t 1 þ s d, and 0 x2 ðt 1 þ s dÞ P xm ; x ð t þ s d Þ < 0. From the second equation of system (1.1) and (4.9), we have 1 2 2
x02 ðt1 þ s dÞ P r 1 ews
cr M m x > 0; d1 d2 k 2 a þ br
which is a contradiction. Thus, x2 ðt Þ P xm 2 for all t P t 1 . From (4.4) and (4.10), we obtain
V 0 ðt Þ > r 1 ews
cr M m x > 0; d1 d2 k 2 a þ br
t P t1 ;
which means that V ðt Þ ! 1 as t ! 1. It is a contradiction to V ðtÞ 6 Mk ð1 þ r 1 sews Þ. Therefore, for any t 0 > 0, the inequality x2 ðt Þ < m2 cannot hold for all t P t0 . So there exist the following two possibilities. Case 1. If x2 ðt Þ P m2 holds for all t large enough, then our aim is obtained. Case 2. Assume that x2 ðt Þ is oscillatory about m2 .
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
39
Setting
m2 ¼ min
cM m2 ; m2 eðaþbMþd1 þd2 m2 Þs : 2
ð4:11Þ
We will prove that x2 ðt Þ P m2 for all t large enough. Suppose that there exist two positive constants t ; n such that x2 ðt Þ ¼ x2 ðt þ nÞ ¼ m2 and x2 ðtÞ < m2 for all t < t < t þ n, where t is sufficiently large. And the inequality (4.9) holds true for t < t < t þ n. Since x2 ðtÞ is not effected by impulses, it is continuous. Furthermore x2 ðt Þ is uniformly continuous and bounded on ½t ; t þ n. Hence, there exists a constant k (0 < k < s and k is independent of the choice of t ) such that x2 ðt Þ >
m2 for t 2 cx2 ðt Þyðt Þ aþbyðtÞ
6 t 6 t þ k. If n 6 k, our aim is met. If k < n 6 s, from the second equation of (1.1) we have that
d1 x2 ðt Þ d2 x22 ðt Þ for t < t < t þ n. According to the assumption x2 ðt Þ ¼ m2 and x2 ðtÞ < m2 for cM þ d1 þ d2 m2 x2 ðt Þ for t < t 6 t þ n 6 t þ s. Then, we derive that x2 ðtÞ P t < t < t þ n, we have x02 ðtÞ P aþbM x02 ðt Þ
P
cM m2 eðaþbMþd1 þd2 m2 Þs and x2 ðt Þ P m2 for t < t 6 t þ n. If n P s, it is obvious that x2 ðt Þ P m2 for t < t < t þ s. The same arguments can be continued, we can obtain x2 ðtÞ P m2 for t þ s 6 t 6 t þ n. Since the interval ½t ; t þ n is arbitrarily chosen, we get that x2 ðt Þ P m2 for sufficiently large t. Next, from the first and the fourth equation of system (1.1), we have
(
x01 ðtÞ P r 1 m2 ews Mk wx1 ðt Þ; þ
x1 ðt Þ ¼ x1 ðtÞ þ l;
t – ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð4:12Þ
Consider the following comparison system
(
z07 ðtÞ ¼ r 1 m2 ews Mk wz7 ðt Þ; t – ðn þ l 1ÞT; z7 ðtþ Þ ¼ z7 ðt Þ þ l; t ¼ ðn þ l 1ÞT:
ð4:13Þ
By Lemma 2.3 and (4.13), there exists a small positive e4 such that x1 ðtÞ P z7 ðtÞ e4 for sufficiently large t, where z7 ðt Þ is the unique and globally stable positive periodic solution of (4.13), and
z7 ðtÞ ¼
r 1 m2 ews Mk lewðtðnþl1ÞT Þ þ : 1 ewT w
In view of Lemma 2.2 and ðH4 Þ, we have
x1 ð t Þ P
r 1 m2 ews Mk lewT þ e4 , m1 : w 1 ewT
From (3.3), let q , m3 , and we have yðt Þ P m3 . Let m ¼ min fm1 ; m2 ; m3 g, then we have x1 ðt Þ P m; x2 ðtÞ P m; yðt Þ P m. By Lemma 2.5 and the above discussion, system (1.1) is permanent. The proof is completed. h
5. Numerical simulations and conclusions In this paper, we have investigated the dynamic behaviors of a stage-structured predator–prey model concerning impulsive control strategy for biological management. We analyzed the global attractivity of the mature prey-extinction periodic solution of the system (1.1), and obtained sufficient conditions for the permanence of the system (1.1). To illustrate the theoretical results, we employ Matlab software to give some numerical simulations in this section. We firstly assume that T ¼ 2; l ¼ 0:3; l ¼ 15; p ¼ 0:8; x1 ð0Þ ¼ 20; x2 ð0Þ ¼ 20; x3 ð0Þ ¼ 10; r 1 ¼ 3; w ¼ 1; c ¼ 1; a ¼ 3; b ¼ 1; d1 ¼ 0:1; d2 ¼ 0:05; r 2 ¼ 3; k ¼ 1; d3 ¼ 0:1; s ¼ 0:3. It is obvious that the parameters satisfy the condition of Theorem 3.2, then the mature prey-extinction periodic solution is global attractivity, as it is shown in Fig. 1. Especially, we find that the mature prey decrease to zero rapidly. From Theorem 3.2, we conclude that increasing T and reducing p are in favor of the global attractivity of the mature prey-extinction solution x1 ðtÞ; 0; y ðt Þ of system (1.1). This means that if the period is prolonged, or the harvesting rate on the predator is reduced to some extent, the mature prey will be extinct totally. To avoid the extinction of mature prey, one should diminish the interval between two harvesting on predator or expand the harvesting rate. We also assume that T ¼ 2; l ¼ 0:5; l ¼ 7; p ¼ 0:5; x1 ð0Þ ¼ 10; x2 ð0Þ ¼ 20; x3 ð0Þ ¼ 15; r1 ¼ 1; w ¼ 0:8; c ¼ 0:2; a ¼ 0:5; b ¼ 0:2; d1 ¼ 0:1; d2 ¼ 0:1; r 2 ¼ 2; k ¼ 3; d3 ¼ 0:1; s ¼ 0:5. By calculations, the parameters fulfill the conditions of Theorem 4.1, so system (1.1) is permanent, which means the prey and predator can coexist (see Fig. 2). By Theorem 4.1, we may see that such activity as reducing T, increasing l or p are in favor of the permanence of system (1.1). That is to say, the behavior of impulsive stocking on prey or reasonable harvesting predator will play an important role on the permanence of system (1.1), which can ensure the sustainable development of biological resources. Since time delay s is contained in the condition of both Theorems 3.2 and 4.1, it is very important to consider the juvenile period of the stocking immature prey. Compared with other models, the results correspond with reality in the management
40
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Fig. 1. (a) Time series of the immature prey population x1 ðtÞ for periodic oscillation, (b) Time series of the mature prey population x2 ðtÞ for extinction, (c) Time series of the predator population yðt Þ for periodic oscillation.
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
41
Fig. 2. (a) Time series of the immature prey population x1 ðtÞ for permanence, (b) Time series of the mature prey population x2 ðtÞ for permanence, (c) Time series of the predator population yðtÞ for permanence.
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
of the biological resources system. However, how can we optimize the sharp threshold, such as the length of period and the amount of harvesting predator or stocking immature prey, is another interesting problem. We will continue to study these problems in the future. References [1] C.W. Clark, Mathematical Bioeconomics, Wiley, New York, 1990. [2] 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. [3] T. Haugen, L.A. Volestad, A century of life-history evolution in Grayling, Genetica 112–113 (2001) 475–491. [4] M. Heino, O.R. Gods, Fisheries-induced selection pressures in the context of sustainable fisheries, Bull. Mar. Sci. 70 (2002) 639–656. [5] C.Y. Huang, Y.J. Li, The dynamics of a stage-structured predator–prey system with impulsive effect and Holling mass defence, Appl. Math. Modell. 36 (2012) 87–96. [6] 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. [7] 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. [8] 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. [9] K.Y. Liu, L.S. Chen, An Ivlev’s functional response predator–prey model with time delay and impulsive perturbations on predators, J. Dalian Univ. Technol. 48 (6) (2008) 926–931. [10] G.H. Luo, S.L. Sun, A stage-structured predator–prey Leslie–Gower Holling II model with disturbing pulse, J. Beihua Univ. (Nat. Sci.) 11 (2010) 24–31. [11] K. Negi, S. Gakkhar, Dynamics in a Beddington–DeAngelis prey–predator system with impulsive harvesting, Ecol. Modell. 206 (2007) 421–430. [12] H.L. Wang, W.M. Wang, The dynamical complexity of a Ivlev-type predator–prey system with impulsive effect, Chaos Soliton Fract. 38 (2008) 1168– 1176. [13] 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. [14] J. Hui, D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Soliton Fract. 29 (2006) 233–251. [15] B. Liu, L.S. Chen, The periodic competing Lotka–Volterra model with impulsive effect, IMA J. Math. Med. Biol. 21 (2004) 129–145. [16] 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. [17] 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. [18] Y.F. Shao, Y. Li, Dynamical analysis of a stage structured predator–prey system with impulsive diffusion and generic functional response, Appl. Math. Comput. 220 (2013) 472–481. [19] A. Gourley, Y. Kuang, A stage structured predator–prey model and its dependence on through-stage delay and death rate, J. Math. Biol. 49 (2) (2004) 188–200. [20] A. Hastings, Age-dependent predation is not a simple process. I. Continuous time models, J. Theor. Popul. Biol. 23 (3) (1983) 347–362. [21] 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 (2) (2007) 1448–1458. [22] 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. [23] 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. [24] 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. [25] X. Liu, G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal. TMA 53 (2003) 1041–1062. [26] B. Leonid, B. Elena, Linearized oscillation theory for a nonlinear delay impulsive equation, J. Comput. Appl. Math. 161 (2003) 477–495. [27] J. Yan, Stability for impulsive delay differential equations, Nonlinear Anal. TMA 63 (2005) 66–80. [28] H.D. Chen, F. Wang, Stage-structured predator–prey model with delay and impulsive perturbations on predator, J. Biomath. 24 (3) (2009) 489–495. [29] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975) 331–340. [30] D.L. DeAngelis, R.A. Goldstein, R.V. ONeill, A model for trophic interaction, Ecology 56 (1975) 881–892. [31] 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. [32] Z.Y. Xiang, D. Long, X.Y. Song, A delayed Lotka–Volterra model with birth pulse and impulsive effect at different moment on the prey, Appl. Math. Comput. 219 (2013) 10263–10270. [33] Y. Duan, B. Liu, Q. Zhang, Dynamics of a pest control model with time delay and age structured, J. Anshan Normal Univ. 11 (6) (2009–2012) 9–11. [34] X.Z. Meng, J.J. Jiao, L.S. Chen, The dynamics of an age structured predator–prey model with disturbing pulse and time delays, Nonliear Anal. RWA 9 (2) (2008) 547–561. [35] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, Longman, England, 1993. [36] V. Lakshmikantham, D.D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [37] D. Bainov, P. Simeonov, System with Impulsive Effect: Stability Theory and Applications, John Wiley and Sons, New York, 1989. [38] 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 Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Permanence of a stage-structured predator–prey system with impulsive stocking prey and harvesting predator q Xinhui Wang, Canyun Huang ⇑ 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
Keywords: Permanence Impulses Stage-structure Global attractivity
a b s t r a c t In this paper we propose a stage-structured predator prey system with impulsive stocking immature prey and harvesting predator at different moment. We analyze the global attractivity of the mature prey-extinction periodic solution, and obtain sufficient conditions for the permanence of the system. Numerical simulations are also inserted to verify the feasibility of the theoretical results. Moreover, the obtained results show that impulsive stocking immature prey or harvesting predator may play a key role on the permanence of the system and provide tactical basis for the biological resources management. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Since biological resources can reproduce themselves in a natural way, it is regarded as the most important part of the natural resources. In recent years, how to get maximal profit and protect the sustainable development of biological system have already drawn a great deal of attention of biologists and resources economists [1]. Recent literatures [2–4] revealed that harvesting and stocking prey or predator can influence ecological system, but overusing these methods sometimes can destroy the ecological balance of renewable resources. To protect the permanence of biological resources, it will be a dominant theme to seek for the suitable threshold of harvesting and stocking in mathematical ecology. Compared with the growth process of biological resources, manual harvesting or stocking always happen in a short time, which should be described by impulsive perturbation instead of continuous effect. Consequently, impulsive differential equations provide a more realistic description of such exploited system [12,9–11,5,8,6,7]. Recently, impulsive differential equations have been extensively used as models in biology, physics, chemistry, engineering and other sciences, with particular emphasis on population dynamics [13–17]. In [12], Wang proposed a system with impulsive immigration of predator. The effect of harvesting on prey and predator with constant efforts has been discussed by Negi [11]. In [18], Shao and Li discussed an impulsive predator–prey system with stage structure and generalized functional response, to model the diffusion by impulses. Sufficient conditions are established for the existence of a predator-extinction positive periodic solution and the permanence of the system. Numerical simulation shows that impulses and functional response affect the dynamics of the system. In the natural world, many species usually go through two distinct life stages from birth to death, immature and mature. The immature takes s units of time to mature and it is well-known that time delay s is an important factor of mathematical models in ecology. The predator–prey models with stage-structure have been investigated by many researchers [19–24]. Time delay and impulse are incorporated into predator–prey models, which greatly enriches biologic background. But the system become so complicated that it causes us greatly difficulties to study the model. The investigation of impulsive delay q
Supported by Natural Science Foundation of Gansu Province (1107RJZA164).
⇑ Corresponding author.
E-mail address: [email protected] (C. Huang). http://dx.doi.org/10.1016/j.amc.2014.02.092 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
33
differential equations is inchoate, and focus on theoretical analysis [26,25,27]. In [9] Liu studied a stage-structured model with impulsive perturbations on predator. The recent works of Jiang [8] and Jiao [6,7] considered some delayed predator– prey models with impulsive stocking on prey and continuous harvesting on predator, in which the influences of impulses and stage-structure are analyzed. The dynamical relationships between predator and prey can be represented by the functional response, which refers to the change in the density of prey attached per unit time per predator as the prey density change. When the prey are simple algology cell of animal, invertebrate and vertebrate, Holling type-I, Holling type-II and Holling type-III functional response are proposed, respectively. Luo [10] considered a stage-structured predator–prey Leslie–Gower Holling II model with disturbing pulse. In [5], the authors analyzed a stage-structured predator–prey model with Holling mass defence, and discussed the effect of impulsive stocking prey and harvesting predator at same fixed time. In 1975, Beddington [29] and DeAngelis et al. [30] introduced an additional functional response
uðxÞ ¼
mx : a þ x þ by
This functional response has an extra term in the denominator, which models the mutual interference between predators and avoids the ‘‘low densities problem’’ of the ratio-dependent type functional response. In [31], an impulsive delayed 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. In fact, stocking prey and harvesting predator usually be at different moment in the real world. The paper [32] established a Lotka–Volterra model with birth pulse and impulsive catching or poisoning for the prey at different moment, and obtained sufficient conditions of the global attractivity of predator-extinction periodic solution and the permanence of the system. Motivated by these literatures [6–8,22,28,31–34], we consider the following impulsive stage-structured differential system
8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > :
x01 ðtÞ ¼ r 1 x2 ðt Þ r 1 ews x2 ðt sÞ cx1 ðt Þ; Þyðt Þ x02 ðtÞ ¼ r 1 ews x2 ðt sÞ aþxcx22ððttÞþby d1 x2 ðtÞ d2 x22 ðt Þ; ðt Þ 2 ðt Þ y0 ðt Þ ¼ r2 yðt Þ þ aþxkcx yðt Þ d3 y2 ðt Þ; 2 ðt Þþbyðt Þ 9 þ x1 ðt Þ ¼ x1 ðt Þ þ l; > = t ¼ ðn þ l 1ÞT; x2 ðtþ Þ ¼ x2 ðt Þ; > ; yðtþ Þ ¼ yðtÞ; 9 x1 ðtþ Þ ¼ x1 ðt Þ; > = t ¼ nT; x2 ðtþ Þ ¼ x2 ðt Þ; > ; yðtþ Þ ¼ ð1 pÞyðt Þ;
9 > > = > > ;
t – ðn þ l 1ÞT; nT;
ð1:1Þ
where x1 ðtÞ; x2 ðt Þ represent the immature and mature prey densities respectively, yðtÞ denotes the density of the predator, s represents a constant time to maturity, and r1 ; w; c; a; b; d1 ; d2 ; r 2 ; k; d3 are positive constants. r1 is the birth rate of the immature prey, wðw > d1 Þ; d1 denotes the mortality rate of the mature prey, c 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, r 2 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 at t ¼ ðn þ l 1ÞT; l is a constant (0 < l < 1), pð0 6 p < 1Þ represents the harvesting rate on the predator at t ¼ nT; n 2 Z þ and Z þ ¼ f1; 2; . . .g; T is the seasonal period of the biological resources. 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, we only consider system (1.1) in the following region
D ¼ fðx1 ; x2 ; yÞ : x1 P 0; x2 P 0; y P 0g: The reminder of this paper is arranged as follows. In Section 2, some fundamental notations, definitions and lemmas are firstly given. In Section 3, we consider the global attractivity of the mature prey-extinction periodic solution. In Section 4, we investigate the sufficient conditions for the permanence of system (1.1). Numerical simulations are presented to verify the feasibility of theoretical results in last section. 2. Notations and preliminaries Let Rþ ¼ ½0; þ1Þ; R3þ ¼ fx 2 R3 ; x P 0g. The solution of (1.1), denoted by xðt Þ ¼ ðx1 ðt Þ; x2 ðtÞ; yðtÞÞT , is a piecewise continu ous function x : Rþ ! R3þ ; xðt Þ is continuous on ððn 1ÞT; ðn þ l 1ÞT and ððn þ l 1ÞT; nT; x ðn l þ 1ÞT þ þ ¼ limt!ðnlþ1ÞT þ xðt Þ and x nT ¼ limt!nT þ xðt Þ exist. Obviously, the global existence and uniqueness of solution of (1.1) is guaranteed by the smoothness properties of f, which denotes the right-side of the first three equations of system (1.1) (see [35,36]).
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Let V : Rþ R3þ ! Rþ , then V is said to belong to class V 0 if: (i) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ . For each x 2 R3þ ; limðt;zÞ!ðKT ;xÞ V ðt; zÞ ¼ V ðKT; xÞ and limðt;zÞ!ðKT þ ;xÞ V ðt; zÞ ¼ V KT þ ; x exist, where K ¼ n; n þ l 1. (ii) V is locally Lipschitzian in x. Definition 2.1. For V 2 V 0 ; ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R3þ or ððn þ l 1ÞT; nT R3þ , the upper right derivative of V ðt; xÞ with respect to system (1.1) is defined as
1 Dþ V ðtÞ ¼ lim sup ½V ðt þ h; x þ hf ðt; xÞÞ V ðt; xÞ: h h!0þ Lemma 2.1. [37,36]. Consider the following impulsive differential inequations
(
u0 ðt Þ 6 ðPÞpðt Þuðt Þ þ qðt Þ; t – tk ; u tþk 6 ðPÞdk uðtk Þ þ bk ; t ¼ tk ;
where p; q 2 C ðRþ ; RÞ; k 2 Z þ ; dk P 0 and bk are constants. Assume: (A1) the sequence tk satisfies 0 6 t0 < t 1 < t 2 < , with limk!1 tk ¼ 1; (A2) u 2 PC 0 ðRþ ; RÞ, where PC 0 ðRþ ; RÞ denote the set of functions u : Rþ ! R, and u is continuously differential in ðtk1 ; tk and left-continuous at tk ; k 2 Z þ . Then
uðtÞ 6 ðPÞuðt0 Þ
Y
dk exp
Z
t
t0
t 0
0 !1 Z t Z X Y @ pðsÞds þ dj exp pðsÞds Abk þ t 0
t k
tk
t
Y
dk exp
Z
t0 s
t
pðhÞdh qðsÞds; t P t0 :
s
Lemma 2.2 [38]. Consider the following equation
u0 ðt Þ ¼ auðt sÞ buðt Þ cu2 ðt Þ; where a; b; c and s are positive constants, uðt Þ > 0 for t 2 ½s; 0. We have (i) If a > b, then limt!þ1 uðtÞ ¼ ða bÞ=c; (ii) If a < b, then limt!þ1 uðtÞ ¼ 0. Lemma 2.3. Consider the following system
u0 ðt Þ ¼ c wuðt Þ; t – nT; uðtþ Þ ¼ uðt Þ þ l; t ¼ nT:
ð2:1Þ
Then system (2.1) has a positive periodic solution u ðt Þ. For any solution uðtÞ of system (2.1), we have
juðt Þ u ðtÞj ! 0 as t ! 1; where
u ðt Þ ¼
c l u 0þ ¼ þ ; w 1 ewT
c lewðtnT Þ þ ; w 1 ewT
nT < t 6 ðn þ 1ÞT:
Lemma 2.4 [5]. Consider the following system
u0 ðt Þ ¼ uðtÞða buðtÞÞ; uðtþ Þ ¼ ð1 pÞuðt Þ;
t – nT;
ð2:2Þ
t ¼ nT:
If p < 1 eaT , then system (2.2) has a positive solution u ðt Þ. For any solution uðtÞ of system (2.2), we have
juðt Þ u ðtÞj ! 0 as t ! 1; where
u ðt Þ ¼
a 1 p eaT bð1 p eaT Þ þ bpe
; aðtnT Þ
a 1 p eaT u 0þ ¼ ; bð1 eaT Þ
nT < t 6 ðn þ 1ÞT:
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Lemma 2.5. There exists a constant M > 0 such that x1 ðtÞ 6 Mk ; x2 ðtÞ 6 Mk ; yðt Þ 6 M for each solution ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ of (1.1) with all t large enough, where k is the positive constant defined in system (1.1). Proof. Define V ðt Þ ¼ kx1 ðt Þ þ kx2 ðtÞ þ yðtÞ. For t – ðn þ l 1ÞT; nT, calculating the upper right derivative of V ðtÞ along the solution of system (1.1), we have
Dþ V ðt Þ þ d1 V ðtÞ ¼ kðw d1 Þx1 ðt Þ þ kr 1 x2 ðt Þ kd2 x22 ðtÞ þ ðr 2 þ d1 ÞyðtÞ d3 y2 ðtÞ 6 kr1 x2 ðt Þ kd2 x22 ðtÞ þ ðr 2 þ d1 ÞyðtÞ d3 y2 ðt Þ 6 M0 ; r2
2
1 þr 2 Þ where M 0 ¼ kðr1 d þ d23 . 4d2
For t ¼ nT; V ðtþ Þ ¼ kx1 ðt Þ þ kx2 ðt Þ þ ð1 pÞyðtÞ 6 V ðtÞ. For t ¼ ðn þ l 1ÞT; V ðt þ Þ ¼ kx1 ðtÞ þ kx2 ðtÞ þ yðt Þ þ l. By Lemma 2.1 we obtain
V ðt Þ 6 V ð0Þer2 t þ
Z
X
t
M 0 er2 ðtsÞ ds þ
0
kler2 ðtnT Þ < V ð0Þer2 t þ
0
M0 k le ! þ rT ; r2 e2 1
M0 er2 ðtT Þ kler2 T 1 er2 t þ kl þ rT r T r2 1e2 e2 1
r2 T
as t ! 1;
which implies that V ðt Þ is uniformly ultimately bounded. Hence, in terms of the definition of V ðt Þ, there exists a constant
M¼
M0 kler2 T þ > 0; r 2 er 2 T 1
ð2:3Þ
such that x1 ðt Þ 6 Mk ; x2 ðtÞ 6 Mk ; yðtÞ 6 M with t large enough. The proof is completed.
h
3. Global attractivity From Lemmas 2.3 and 2.4, we can obtain the following result on the existence of the mature prey-extinction periodic solution of system (1.1). Theorem 3.1. System (1.1) has a mature prey-extinction periodic solution ðx1 ðt Þ; 0; y ðtÞÞ ðn þ l 1ÞT [ ððn þ l 1ÞT; nT and for any solution ðx1 ðtÞ; x2 ðtÞ; yðt ÞÞ of system (1.1), then
x1 ðtÞ ! x1 ðt Þ;
for
t 2 ððn 1ÞT;
yðt Þ ! y ðtÞ as t ! 1;
where
x1 ðtÞ ¼
lecðtðnþl1ÞT Þ 1 ecT
;
ðn þ 1 1ÞT < t 6 ðn þ 1ÞT;
and
y ðt Þ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnT Þ
nT < t 6 ðn þ 1ÞT:
Next, we investigate the global attractivity of the mature prey-extinction periodic solution x1 ðt Þ; 0; y ðt Þ of system (1.1). Theorem 3.2. Assume that (H1) 1 p er2 T > 0; cr2 ð1per2 T Þ (H2) ðr 1 ews d1 Þ a þ Mk þ bM < d 1er2 T . Þ 3ð Then the mature prey-extinction periodic solution x1 ðtÞ; 0; y ðt Þ of system (1.1) is globally attractive. Proof. Let ðx1 ðtÞ; x2 ðt Þ; yðt ÞÞ be any solution of system (1.1). From the third and ninth equation of system (1.1) we obtain
(
y0 ðtÞ P r2 yðt Þ d3 y2 ðt Þ; yðtþ Þ ¼ ð1 pÞyðtÞ;
t – nT;
t ¼ nT:
ð3:1Þ
Consider the following auxiliary impulsive differential system
z01 ðt Þ ¼ z1 ðt Þðr 2 d3 z1 ðtÞÞ; þ
z1 ðt Þ ¼ ð1 pÞz1 ðtÞ;
t – nT;
t ¼ nT:
ð3:2Þ
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X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Using Lemma 2.4, we obtain
z1 ðtÞ ¼
r 2 1 p er2 T ; d3 ð1 p er2 T Þ þ d3 per2 ðtnT Þ
nT < t 6 ðn þ 1ÞT;
which is unique and globally attractive positive periodic solution of system (3.1). By using comparison theorem of impulsive differential equation [36], there exist a n1 2 Z þ and an arbitrarily small positive constant e0 such that
yðtÞ 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 n1 T. For t > n1 T þ s, from (3.3) and Lemma 2.5 we have
! cq x ðtÞ d2 x22 ðt Þ: þ d a þ Mk þ bM 1 2
x02 ðtÞ 6 r1 ews x2 ðt sÞ
ð3:4Þ
According to the hypotheses ðH2 Þ, we have
r1 ews <
cq þd : a þ Mk þ bM 1
ð3:5Þ
Consider the auxiliary impulsive differential equation of (3.4)
z02 ðtÞ
¼ r1 e
ws
! cq z2 ðt sÞ þ d z ðtÞ d2 z22 ðt Þ; t > n1 T þ s: a þ Mk þ bM 1 2
From 2.2 and (3.5), we have limt!1 z2 ðt Þ ¼ 0. Since x2 ðsÞ ¼ z2 ðsÞ ¼ /2 ðsÞ > 0 for all s 2 ½s; 0, we can obtain that x2 ðtÞ ! 0 as t ! 1. Without loss of generality, we assume that there exists a positive constant e such that
0 < x2 ðt Þ < e;
t P 0:
ð3:6Þ
From the first, the fourth equation of (1.1) and (3.6), we have
x01 ðt Þ 6 r 1 e wx1 ðtÞ; x1 ðt þ Þ ¼ x1 ðtÞ þ l;
t – ðn þ l 1ÞT; t ¼ ðn þ l 1ÞT:
ð3:7Þ
Consider the auxiliary system of (3.7)
z03 ðtÞ ¼ r 1 e wz3 ðt Þ; z3 ðtþ Þ ¼ z3 ðtÞ þ l;
t – ðn þ l 1ÞT; t ¼ ðn þ l 1ÞT:
ð3:8Þ
By Lemma 2.3, the unique positive periodic solution of system (3.8) is
z3 ðtÞ ¼
r 1 e lewðtðnþl1ÞÞ þ ; w 1 ewT
t 2 ððn þ l 1ÞT; ðn þ lÞT :
In view of the comparison theorem, for any z3 ðt Þ ! x1 ðt Þ and
e1 , there exists a n2 such that x1 ðtÞ 6 z3 ðtÞ þ e1 for t > n2 T. Let e ! 0, then
x1 ðtÞ 6 x1 ðtÞ þ e1 :
ð3:9Þ
Again from system (1.1) and (3.6), we have
x01 ðt Þ P r 1 ews e wx1 ðtÞ; t – ðn þ l 1ÞT; x1 ðt þ Þ ¼ x1 ðtÞ þ l; t ¼ ðn þ l 1ÞT:
ð3:10Þ
Consider the auxiliary system of system (3.10)
z04 ðtÞ ¼ r 1 ews e wz4 ðtÞ; z4 ðtþ Þ ¼ z4 ðtÞ þ l;
t – ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð3:11Þ
By Lemma 2.3, system (3.11) has a unique positive periodic solution
z4 ðtÞ ¼
r 1 ews e lewðtðnþl1ÞÞ þ ; w 1 ewT
Similarly, for the constant and
x1 ðtÞ P x1 ðtÞ e1 :
t 2 ððn þ l 1ÞT; ðn þ lÞT :
e1 > 0, there exists a n3 > n2 such that x1 ðtÞ P z4 ðtÞ e1 for t > n3 T. Let e ! 0, then z4 ðtÞ ! x1 ðtÞ ð3:12Þ
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
37
It follows from (3.9) and (3.12) that x1 ðtÞ ! x1 ðtÞ as t ! 1. In virtue of the third and ninth equation of system (1.1), we have
(
y0 ðtÞ 6 yðtÞ r 2 þ akcþee d3 yðt Þ ; þ
yðt Þ ¼ ð1 pÞyðtÞ;
t – nT;
ð3:13Þ
t ¼ nT:
Consider the following auxiliary system of system (3.13)
(
z05 ðtÞ ¼ z5 ðt Þ r 2 þ akcþee d3 z5 ðtÞ ; þ
z5 ðt Þ ¼ ð1 pÞz5 ðt Þ;
t – nT;
ð3:14Þ
t ¼ nT:
By Lemma 2.4, we obtain the unique positive periodic solution of system (3.14)
kce r 2 þ akcþee 1 p eðr2 þaþeÞT z5 ðtÞ ¼ ; kce kce d3 1 p eðr2 þaþeÞT þ d3 peðr2 þaþeÞðtnT Þ
nT < t 6 ðn þ 1ÞT:
It follows from comparison theorem that, for any sufficiently small constant e2 > 0, there exists a n4 > 0 such that yðtÞ 6 z5 ðtÞ þ e2 for all t > n4 T. Let e ! 0, then z5 ðtÞ ! y ðt Þ, and we have yðt Þ 6 y ðt Þ þ e2 . On the other hand, we can follow from (3.1), (3.2) and (3.3) that yðtÞ P y ðt Þ e2 for t sufficiently large, which implies that yðt Þ ! y ðtÞ as t ! 1. This completes the proof. h
4. Permanence Next work is to investigate the permanence of 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 ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (1.1) satisfies m 6 x1 ðtÞ; x2 ðtÞ; yðtÞ 6 M for t sufficiently large enough. Theorem 4.1. Assume that cq (H3) r1 ews aþbq d1 d2 Mk > 0;
(H4)
r 1 ðm2 ews M kÞ w
> 0,
where M; q and m2 are defined in (2.3), (4.5) and (4.11) respectively. Then system (1.1) is permanent. Proof. Firstly, we show that there is a positive constant m2 such that each positive solution ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ of system (1.1) satisfies x2 ðtÞ P m2 for t large enough. The second equation of (1.1) can be rewritten as
x02 ðtÞ ¼ r 1 ews
Z t cyðt Þ d x2 ðuÞdu: d1 d2 x2 ðt Þ x2 ðt Þ r 1 ews dt ts a þ x2 ðtÞ þ byðtÞ
ð4:1Þ
Let ðx1 ðt Þ; x2 ðtÞ; yðtÞÞ be an solution of system (1.1), define
V ðt Þ ¼ x2 ðt Þ þ r 1 ews
Z
t
x2 ðuÞdu:
ts
Calculating the derivative of V ðt Þ along the solution of system (1.1), we have
V 0 ðtÞ ¼ r 1 ews
cyðt Þ d1 d2 x2 ðtÞ x2 ðt Þ: a þ x2 ðtÞ þ byðtÞ
ð4:2Þ
By 2.5, (4.2) can be written as
V 0 ðtÞ > r 1 ews
cyðt Þ M x2 ðtÞ: d1 d2 k a þ byðtÞ
From hypotheses ðH3 Þ, we can easily know that there exists a sufficiently small
r1 ews >
cðq þ e3 Þ M þ d þ d2 ; k a þ bðq þ e3 Þ 1
ð4:3Þ
e3 > 0 such that ð4:4Þ
38
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
where
1 p eðr2 þkcm2 ÞT q¼ d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞT
r2 þ kcm2
and m2 is determined by the following equation
ð4:5Þ
1 p eðr2 þkcm2 ÞT ¼ : c b r 1 ews d1 d2 Mk d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞT
a r1 ews d1 d2 Mk
r2 þ kcm2
We claim that for any t 0 > 0, it is impossible that x2 ðt Þ < m2 for all t > t 0 . Suppose that the claim is invalid, then there is a t0 > 0 such that x2 ðt Þ < m2 for all t > t0 . It follows from the third equation of (1.1) that
y0 ðt Þ < yðt Þ r 2 þ kcm2 d3 yðt Þ ;
t > t0 :
ð4:6Þ
Consider the following auxiliary impulsive system
(
z06 ðtÞ ¼ z6 ðtÞ r 2 þ kcm2 d3 z6 ðt Þ ; z06 ðtþ Þ ¼ ð1 pÞz6 ðt Þ;
t – nT;
ð4:7Þ
t ¼ nT:
By Lemma 2.4, we have
1 p eðr2 þkcm2 ÞT ¼ ; d3 1 p eðr2 þkcm2 ÞT þ d3 peðr2 þkcm2 ÞðtnT Þ
z6 ðtÞ
r2 þ kcm2
t 0 < nT < t 6 ðn þ 1ÞT;
which is the unique positive periodic solution of (4.7) and globally asymptotically stable. In view of the comparison theorem, we know that there exists a t1 ð> t 0 þ sÞ, such that
yðtÞ 6 z6 ðtÞ þ e3 ;
t P t1 :
It follows from (4.5) that
z6 ðtÞ 6 q;
t P t1 :
ð4:8Þ
Thus
yðtÞ 6 q þ e3 , r;
t P t1 :
ð4:9Þ
From (4.4), we get
r1 ews >
cr M þ d þ d2 : k a þ br 1
By (4.3) and (4.9), we have
V 0 ðt Þ > r 1 ews
cr M x2 ðt Þ; d1 d2 k a þ br
t P t1 :
ð4:10Þ
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 t 1 . Otherwise, there exists a d > 0 such that x2 ðt Þ P x2 for t 1 6 t 6 t 1 þ s d, and 0 x2 ðt 1 þ s dÞ P xm ; x ð t þ s d Þ < 0. From the second equation of system (1.1) and (4.9), we have 1 2 2
x02 ðt1 þ s dÞ P r 1 ews
cr M m x > 0; d1 d2 k 2 a þ br
which is a contradiction. Thus, x2 ðt Þ P xm 2 for all t P t 1 . From (4.4) and (4.10), we obtain
V 0 ðt Þ > r 1 ews
cr M m x > 0; d1 d2 k 2 a þ br
t P t1 ;
which means that V ðt Þ ! 1 as t ! 1. It is a contradiction to V ðtÞ 6 Mk ð1 þ r 1 sews Þ. Therefore, for any t 0 > 0, the inequality x2 ðt Þ < m2 cannot hold for all t P t0 . So there exist the following two possibilities. Case 1. If x2 ðt Þ P m2 holds for all t large enough, then our aim is obtained. Case 2. Assume that x2 ðt Þ is oscillatory about m2 .
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
39
Setting
m2 ¼ min
cM m2 ; m2 eðaþbMþd1 þd2 m2 Þs : 2
ð4:11Þ
We will prove that x2 ðt Þ P m2 for all t large enough. Suppose that there exist two positive constants t ; n such that x2 ðt Þ ¼ x2 ðt þ nÞ ¼ m2 and x2 ðtÞ < m2 for all t < t < t þ n, where t is sufficiently large. And the inequality (4.9) holds true for t < t < t þ n. Since x2 ðtÞ is not effected by impulses, it is continuous. Furthermore x2 ðt Þ is uniformly continuous and bounded on ½t ; t þ n. Hence, there exists a constant k (0 < k < s and k is independent of the choice of t ) such that x2 ðt Þ >
m2 for t 2 cx2 ðt Þyðt Þ aþbyðtÞ
6 t 6 t þ k. If n 6 k, our aim is met. If k < n 6 s, from the second equation of (1.1) we have that
d1 x2 ðt Þ d2 x22 ðt Þ for t < t < t þ n. According to the assumption x2 ðt Þ ¼ m2 and x2 ðtÞ < m2 for cM þ d1 þ d2 m2 x2 ðt Þ for t < t 6 t þ n 6 t þ s. Then, we derive that x2 ðtÞ P t < t < t þ n, we have x02 ðtÞ P aþbM x02 ðt Þ
P
cM m2 eðaþbMþd1 þd2 m2 Þs and x2 ðt Þ P m2 for t < t 6 t þ n. If n P s, it is obvious that x2 ðt Þ P m2 for t < t < t þ s. The same arguments can be continued, we can obtain x2 ðtÞ P m2 for t þ s 6 t 6 t þ n. Since the interval ½t ; t þ n is arbitrarily chosen, we get that x2 ðt Þ P m2 for sufficiently large t. Next, from the first and the fourth equation of system (1.1), we have
(
x01 ðtÞ P r 1 m2 ews Mk wx1 ðt Þ; þ
x1 ðt Þ ¼ x1 ðtÞ þ l;
t – ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð4:12Þ
Consider the following comparison system
(
z07 ðtÞ ¼ r 1 m2 ews Mk wz7 ðt Þ; t – ðn þ l 1ÞT; z7 ðtþ Þ ¼ z7 ðt Þ þ l; t ¼ ðn þ l 1ÞT:
ð4:13Þ
By Lemma 2.3 and (4.13), there exists a small positive e4 such that x1 ðtÞ P z7 ðtÞ e4 for sufficiently large t, where z7 ðt Þ is the unique and globally stable positive periodic solution of (4.13), and
z7 ðtÞ ¼
r 1 m2 ews Mk lewðtðnþl1ÞT Þ þ : 1 ewT w
In view of Lemma 2.2 and ðH4 Þ, we have
x1 ð t Þ P
r 1 m2 ews Mk lewT þ e4 , m1 : w 1 ewT
From (3.3), let q , m3 , and we have yðt Þ P m3 . Let m ¼ min fm1 ; m2 ; m3 g, then we have x1 ðt Þ P m; x2 ðtÞ P m; yðt Þ P m. By Lemma 2.5 and the above discussion, system (1.1) is permanent. The proof is completed. h
5. Numerical simulations and conclusions In this paper, we have investigated the dynamic behaviors of a stage-structured predator–prey model concerning impulsive control strategy for biological management. We analyzed the global attractivity of the mature prey-extinction periodic solution of the system (1.1), and obtained sufficient conditions for the permanence of the system (1.1). To illustrate the theoretical results, we employ Matlab software to give some numerical simulations in this section. We firstly assume that T ¼ 2; l ¼ 0:3; l ¼ 15; p ¼ 0:8; x1 ð0Þ ¼ 20; x2 ð0Þ ¼ 20; x3 ð0Þ ¼ 10; r 1 ¼ 3; w ¼ 1; c ¼ 1; a ¼ 3; b ¼ 1; d1 ¼ 0:1; d2 ¼ 0:05; r 2 ¼ 3; k ¼ 1; d3 ¼ 0:1; s ¼ 0:3. It is obvious that the parameters satisfy the condition of Theorem 3.2, then the mature prey-extinction periodic solution is global attractivity, as it is shown in Fig. 1. Especially, we find that the mature prey decrease to zero rapidly. From Theorem 3.2, we conclude that increasing T and reducing p are in favor of the global attractivity of the mature prey-extinction solution x1 ðtÞ; 0; y ðt Þ of system (1.1). This means that if the period is prolonged, or the harvesting rate on the predator is reduced to some extent, the mature prey will be extinct totally. To avoid the extinction of mature prey, one should diminish the interval between two harvesting on predator or expand the harvesting rate. We also assume that T ¼ 2; l ¼ 0:5; l ¼ 7; p ¼ 0:5; x1 ð0Þ ¼ 10; x2 ð0Þ ¼ 20; x3 ð0Þ ¼ 15; r1 ¼ 1; w ¼ 0:8; c ¼ 0:2; a ¼ 0:5; b ¼ 0:2; d1 ¼ 0:1; d2 ¼ 0:1; r 2 ¼ 2; k ¼ 3; d3 ¼ 0:1; s ¼ 0:5. By calculations, the parameters fulfill the conditions of Theorem 4.1, so system (1.1) is permanent, which means the prey and predator can coexist (see Fig. 2). By Theorem 4.1, we may see that such activity as reducing T, increasing l or p are in favor of the permanence of system (1.1). That is to say, the behavior of impulsive stocking on prey or reasonable harvesting predator will play an important role on the permanence of system (1.1), which can ensure the sustainable development of biological resources. Since time delay s is contained in the condition of both Theorems 3.2 and 4.1, it is very important to consider the juvenile period of the stocking immature prey. Compared with other models, the results correspond with reality in the management
40
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
Fig. 1. (a) Time series of the immature prey population x1 ðtÞ for periodic oscillation, (b) Time series of the mature prey population x2 ðtÞ for extinction, (c) Time series of the predator population yðt Þ for periodic oscillation.
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
41
Fig. 2. (a) Time series of the immature prey population x1 ðtÞ for permanence, (b) Time series of the mature prey population x2 ðtÞ for permanence, (c) Time series of the predator population yðtÞ for permanence.
42
X. Wang, C. Huang / Applied Mathematics and Computation 235 (2014) 32–42
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