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Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey
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Chaos, Solitons and Fractals 41 (2009) 103–112 www.elsevier.com/locate/chaos
Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey q Jianjun Jiao a,b,*, Xinzhu Meng c, Lansun Chen a b
a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, People’s Republic of China c College of Science, Shandong University of Science and Technology, Qingdao 266510, People’s Republic of China
Accepted 9 November 2007
Communicated by Prof. L. Marek-Crnjac
Abstract A predator–prey model with a stage structure for the predator, which improves the assumption that each individual predator has the same ability to capture prey, is proposed by Wang et al. [Wang W, Mulone G, Salemi F, Salone V. Permanence and stability of a stage-structured predator–prey model. J Math Anal Appl 2001;262:499–528]. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We do economic management behavior for Wang model [Wang et al., 2001] by continuous harvesting on predator and impulsive stocking on prey. Then, a delayed stage-structured Holling type II predator–prey model with impulsive stocking prey and continuous harvesting predator is established. It is also assumed that the predating products of the predator is only to increase its bearing ability. We obtain the sufficient conditions of the global attractivity of predator-extinction boundary periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Further, the numerical analysis is also inserted to illuminate the dynamics of the system. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Biological resources are renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark [18]. In recent years, the optimal management of renewable resources, which has a direct
q Supported by National Natural Science Foundation of China (10471117) and the Emphasis Subject of Guizhou College of Finance and Economics. * Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China. E-mail addresses: [email protected] (J. Jiao), [email protected] (L. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.015
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relationship to sustainable development, has been studied extensively by many authors [13]. Especially, the predator– prey models with harvesting are investigated by many papers [3,9,12,18]. There are many works concerning predator–prey system, and many good results are obtained [5–7,9,16,17,19]. The basic predator–prey model is 0 x1 ðtÞ ¼ x1 ðtÞðr ax1 bx2 ðtÞÞ; ð1:1Þ x02 ðtÞ ¼ x2 ðtÞðd þ cx1 ðtÞÞ; where x1 ðtÞ and x2 ðtÞ are the densities of the prey population and the predator population at time t, respectively, r > 0 is the intrinsic growth rate of prey, a > 0 is the coefficient of intraspecific competition, b > 0 is the per-capita rate of predation of the predator, d > 0 is the death rate of predator, c > 0 denotes the product of the per-capita rate of predation and the rate of conversing pest into predator. If rc da < 0, system (1.1) does not have any positive equilibrium point, and the only boundary equilibrium point ar ; 0 is globally asymptotically stable, which implies that the predator population will go into extinction. If the prey is stocked at a constant rate, system (1.1) can be the following differential equation: 0 x1 ðtÞ ¼ x1 ðtÞðr ax1 bx2 ðtÞÞ þ l; ð1:2Þ x02 ðtÞ ¼ x2 ðtÞðd þ cx1 ðtÞÞ: It can be easily derived that if l > dðadrcÞ , system (1.2) has a unique globally asymptotically stable positive equilibrium c2 2 d rdcad þlc . This implies that the behavior of stocking prey keeps system (1.2) permanent. While there are few papers ; bcd c [9,14] devoted to investigate the models with impulsive perturbations on prey, but their models did not consider time delay. Stage-structured models have received much attention in recent years. Mathematical analyses for single species models with stage structure were performed by many papers [6–8], Recently, Wang and Chen [13] considered predator–prey models with a stage structure for the predator to analyze the influences of a stage structure for the predator on the dynamics of predator–prey models, but these models without impulsive perturbations ignore the duration time of immature predators. And many good results of impulsive differential equations are obtained in [4,9,11]. The recent works of Wang [15] considered that the preying products of the predator is only to increase its bearing ability and a period to maturity. It can really reflect the nature of the world, For example, the large-scale fowl-run add the feedstuffs to enhance the yield of egg. But they did not take economic management behavior into account. Motivated by these ideals, we consider the following impulsive delay differential equation: 9 8 x1 ðtÞ > x01 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ b 1þcx x3 ðtÞ; > > > ðtÞ > 1 > = > > x ðtÞ x ðts Þ > 0 ws1 1 1 1 > ðtÞ ¼ kb x ðtÞ kbe x ðt s Þ wx ðtÞ; x t–ns; 1 2 > 2 1þcx1 ðtÞ 3 1þcx1 ðts1 Þ 3 > > > > > > > x ðts Þ 1 1 ; > x3 ðt s1 Þ d 3 x3 ðtÞ Ex3 ðtÞ; < x03 ðtÞ ¼ kbews1 1þcx 1 ðts1 Þ 9 ð1:3Þ Dx1 ðtÞ ¼ l; > > = > > > > Dx2 ðtÞ ¼ 0; t ¼ ns; n ¼ 1; 2 . . . ; > > > > > ; > > Dx ðtÞ ¼ 0; > > 3 : ðu1 ðfÞ; u2 ðfÞ; u3 ðfÞÞ 2 C þ ¼ Cð½s1 ; 0; R3þ Þ; ui ð0Þ > 0; i ¼ 1; 2; 3; where x1 ðtÞ denotes the density of prey at time t, x2 ðtÞ; x3 ðtÞ represent the densities of the immature individual predator and mature individual predator at time t, respectively. s1 represents a constant time to maturity, a > 0 is the intrinsic growth rate of prey, b > 0 is the coefficient of intraspecific competition, b is the capture rate of mature predator, and bx1 ðtÞ ðc > 0Þ is the Holling type II functional response. k is the rate of conversion of nutrients into the reproduction 1þcx1 ðtÞ rate of the mature predator, w is the death rate of immature predator. d 3 is the death rate of mature predator. It is also assumed that immature individuals and mature individuals are divided by age s1 and that immature individual predators do not feed on prey and do not have the ability to reproduce. This seems reasonable for a number of mammals where immature predators are raised by their parents and their reproduction rate and attacking rate for prey can be neglected. 0 < E < 1 is the effect of continuous harvesting predator, Dx1 ðtÞ ¼ x1 ðtþ Þ x1 ðtÞ, l P 0 is the stocking amount of prey at t ¼ ns; n 2 Z þ and Z þ ¼ f1; 2; . . .g, s is the period of the impulsive stocking of the prey. The purpose of this paper is to prove that system (1.3) has a predator-extinction periodic solution. Further, it is globally attractive. While due to the stocking of prey, all the mature predator population would not go into extinction for the continuous harvesting of mature predator population, that is, system (1.3) is permanent. Because the first and third equations of (1.3) do not contain x2 ðtÞ, we can simplify model (1.3) and restrict our attention to the following model:
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
8 9 x1 ðtÞ > = x01 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ b 1þcx x3 ðtÞ; > > 1 ðtÞ > t–ns; > < 0 ws1 x1 ðts1 Þ x ðt s1 Þ d 3 x3 ðtÞ Ex3 ðtÞ; ; x3 ðtÞ ¼ kbe 1þcx1 ðts1 Þ 3 > > > Dx1 ðtÞ ¼ l; > > t ¼ ns; n ¼ 1; 2 . . . ; : Dx3 ðtÞ ¼ 0;
105
ð1:4Þ
the initial conditions for (1.4) are ðu1 ðfÞ; u3 ðfÞÞ 2 C þ ¼ Cð½s1 ; 0; R2þ Þ;
ui ð0Þ > 0;
i ¼ 1; 3:
ð1:5Þ
2. Some important lemmas The solution of (1.3), denoted by xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞT , is a piecewise continuous function x : Rþ ! R3þ , xðtÞ is continuous on ðns; ðn þ 1Þs, n 2 Z þ and xðnsþ Þ ¼ limt!nsþ xðtÞ exists. Obviously the global existence and uniqueness of solutions of (1.3) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-side of system (1.3) (see [1,2]). For continuity of initial conditions, we require Z 0 u1 ðsÞ u2 ð0Þ ¼ kbews ð2:1Þ u ðsÞds; 1 þ cu1 ðsÞ 3 s1 Before we have the main results, we need to give some lemmas which will be used next. Lemma 2.1. Let ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ > 0 for s1 < t < 0. Then any solution of system (1.3) is strictly positive. Proof. Firstly, by uniqueness of solutions of system (1.3) and x01 ðtÞ ¼ 0 whenever x1 ðtÞ ¼ 0, t–ns, and x1 ðnsþ Þ ¼ x1 ðnsÞ þ l; l P 0. It is easy to see that x1 ðtÞ > 0 for all t > 0. Secondly, we show that x3 ðtÞ > 0 for all t > 0. Notice x3 ðtÞ > 0, hence if there exists t0 such that x3 ðt0 Þ ¼ 0, then t0 > 0. Assume that t0 is the first such time that x3 ðtÞ ¼ 0, that is, t0 ¼ infft > 0 : x3 ðtÞ ¼ 0g; then x1 ðt0 s1 Þ x03 ðt0 Þ ¼ kbews1 1þcx x3 ðt0 s1 Þ > 0. Hence for sufficiently small e > 0, x03 ðt0 eÞ > 0. But by the definition of 1 ðt0 s1 Þ 0 t0 , x3 ðt0 eÞ 6 0. This contradiction shows that x3 ðtÞ > 0 for all t > 0. Finally we consider the following equation: s0 ðtÞ ¼ kbews1
x1 ðt s1 Þ x3 ðt s1 Þ wsðtÞ; 1 þ cx1 ðt s1 Þ
ð2:2Þ
and comparing with (1.3), we note that if sðtÞ is the solution of (2.2) and if x1 ðtÞ can solve (1.3), then x2 ðtÞ > sðtÞ on 0 < t < s1 . Solving (2.2) gives Z t x1 ðu s1 Þ wt wðus1 Þ sðtÞ ¼ e x2 ð0Þ kbe x3 ðu s1 Þdu ; 1 þ cx1 ðu s1 Þ 0 From (2.1), one can obtain Z 0 kbews sðs1 Þ ¼ ews1
x1 ðu s1 Þ x3 ðu s1 Þdu ; 1 þ cx1 ðu s1 Þ 0 s1 R0 u1 ðsÞ By making transformation and x1 ðtÞ ¼ u1 ðtÞ, x3 ðtÞ ¼ u3 ðtÞ; t 2 ½s1 ; 0, we know that s1 kbews 1þcu u3 ðsÞds is equiv1 ðsÞ R s1 wðus1 Þ x1 ðus1 Þ x ðu s Þdu. Thus we obtain sðs Þ ¼ 0. Hence x ðtÞ > 0, Since sðtÞ is strictly decreasing, alent to 0 kbe 1 1 2 1þcx1 ðus1 Þ 3 then x2 ðtÞ > sðtÞ > 0 for t 2 ð0; s1 Þ. So x2 ðtÞ > 0 on 0 6 t 6 s1 . By induction and similar method to the proof of Theorem 1 [10], we can show that x2 ðtÞ > 0 for all t P 0. This completes the proof. h u1 ðsÞ u ðsÞds 1 þ cu1 ðsÞ 3
Z
s1
kbewðus1 Þ
Now, we show that all solutions of (1.3) are uniformly ultimately bounded. Lemma 2.2. There exists a constant M > 0 such that x1 ðtÞ 6 Mk ; x2 ðtÞ 6 M; x3 ðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ of (1.3) with all t large enough. Proof. Define V ðtÞ ¼ kx1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ. And because of w < d 3 þ E, then t–ns, we have Dþ V ðtÞ þ wV ðtÞ ¼ kðw þ aÞx1 kbx21 ðtÞ þ ðw d 3 EÞx3 ðtÞ 6 M 0 ;
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J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112 2
where M 0 ¼ kðaþwÞ , when t ¼ ns, V ðnsþ Þ ¼ V ðnsÞ þ l. By Lemma 2.2 (which can be seen in [1, p. 23]), for 4b t 2 ðns; ðn þ 1Þs we have V ðtÞ 6 V ð0Þ expðdtÞ þ
M0 l expðdðt sÞÞ l expðdsÞ M0 l expðdsÞ ð1 expðdtÞÞ þ þ þ ! t ! 1: d d 1 expðdsÞ expðdsÞ 1 expðdsÞ 1
l expðdsÞ >0 So V ðtÞ is uniformly ultimately bounded. Hence, by the definition of V ðtÞ, there exists a constant M ¼ Md0 þ expðdsÞ1 M such that x1 ðtÞ 6 k ; x2 ðtÞ 6 M; x3 ðtÞ 6 M for t large enough. The proof is complete. h
Considering the following delay equation: x0 ðtÞ ¼ a1 xðt sÞ a2 xðtÞ;
ð2:3Þ
we assume that a1 ; a2 ; s > 0; xðtÞ > 0 for s 6 t 6 0. Lemma 2.3 [8]. For system (2.3), assume that a1 < a2 . Then limt!1 xðtÞ ¼ 0: Lemma 2.4 [14]. Considering the following impulsive system: 0 v ðtÞ ¼ vðtÞða bvðtÞÞ; t–ns; vðnsþ Þ ¼ vðnsÞ þ l; t ¼ ns; n ¼ 1; 2 . . . ; where
a > 0; b > 0; l > 0. Then system t 2 ðns; ðn þ 1Þs; n 2 Z þ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 as 1Þ ðaþblÞþ ðaþblÞ þ4abl=ðe ð> abÞ. v ¼ 2b
av expðaðtnsÞÞ ; abv þbv expðaðtnsÞÞ
(2.4) which
has is
ð2:4Þ a
unique globally
positive periodic asymptotically
solution stable,
g¼ vðtÞ where
According to system (1.4), we can easily know that there exists t1 2 Z þ ; t > t1 such that x3 ðt s1 Þ ¼ 0 and x3 ðtÞ ¼ 0. Then 0 x1 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ; t–ns; ð2:5Þ Dx1 ðtÞ ¼ l; t ¼ ns; n ¼ 1; 2 . . . : ðtÞ; 0; 0Þ ¼ From (2.5) and Lemma 2.4, we know that (1.3) has a predator-extinction periodic solution ð x1g ax1 expðaðtnsÞÞ ; 0; 0 ; t 2 ðns; ðn þ 1Þs; n 2 Z or (1.4) has a predator-extinction periodic solution ð x1g ðtÞ; 0Þ ¼ þ abx þbx expðaðtnsÞÞ 1 1 ax1 expðaðtnsÞÞ ; 0 ; t 2 ðns; ðn þ 1Þs; n 2 Z þ , which is globally asymptotically stable, where abx1 þbx1 expðaðtnsÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 as ðaþblÞþ ðaþblÞ þ4abl=ðe 1Þ x1 ¼ > ab . 2b
3. Global attractivity and permanence In this section, we will obtain the sufficient condition of the global attractivity of predator-extinction periodic solution of system (1.3). Theorem 3.1. Let ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ be any solution of (1.3). If E>
a
kbax1 easws1 d3 þ bx1 eas þ cax1 eas pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
bx1
holds, where x1 ¼ attractive.
ðaþblÞ þ4abl=ðeas 1Þ ð> abÞ, 2b
ðaþblÞþ
the predator-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally
ðtÞ; 0; 0Þ of system (1.3) is equivProof. It is clear that the global attraction of predator-extinction periodic solution ð x1g g ðtÞ; 0Þ of system (1.4). So we only devote to system alent to the global attraction of pest-extinction periodic solution ð x 1 akbx easws1 (1.4). Since E > abx þbx1 eas þcax eas d 3 , we can choose e0 sufficiently small such that 1
0 ws1 @
kbe
1
1
ax1 eas abx1 þbx1 eas
þ e0
1
A < d 3 þ E; ax eas 1 þ c abx 1þbx eas þ e0 1
1
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
107
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaþblÞþ ðaþblÞ2 þ4abl=ðeas 1Þ where x1 ¼ > ab . It follows from the first equation of system (1.4) that dxdt1 ðtÞ 6 x1 ðtÞða bx1 ðtÞÞ. 2b So we consider the following comparison impulsive differential system: 8 dxðtÞ > < dt ¼ xðtÞða bxðtÞÞ; t–ns; ð3:1Þ DxðtÞ ¼ l; t ¼ ns; > : þ þ xð0 Þ ¼ x1 ð0 Þ: In view of Lemma 2.4, we obtain that the periodic solution of system (1.4) g¼ xðtÞ
ax1 expðaðt nsÞÞ ; a bx1 þ bx1 expðaðt nsÞÞ
t 2 ðns; ðn þ 1Þs; n 2 Z þ ; ð3:2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaþblÞþ ðaþblÞ2 þ4abl=ðeas 1Þ which is globally asymptotically stable, where x1 ¼ > ab . 2b ðtÞ as From Lemma 2.4 and comparison theorem of impulsive equation [2], we have x1 ðtÞ 6 xðtÞ and xðtÞ ! x1g t ! 1. Then there exists an integer k 2 > k 1 ; t > k 2 such that x1 ðtÞ 6 xðtÞ 6 x1g ðtÞ þ e0 ;
ns < t 6 ðn þ 1Þs;
n > k2 ;
that is ðtÞ þ e0 6 x1 ðtÞ < x1g
ax1 eas þ e0 , .; a bx1 þ bx1 eas
ns < t 6 ðn þ 1Þs;
n > k2 ;
and from (1.4), we get dx3 ðtÞ . 6 kbews1 x3 ðt s1 Þ ðd 3 þ EÞx3 ðtÞ; dt 1 þ c.
t > ns þ s1 ;
n > k2 :
ð3:3Þ
Consider the following comparison differential system: dyðtÞ . ¼ kbews1 yðt s1 Þ ðd 3 þ EÞyðtÞ; dt 1 þ c.
t > ns þ s1 ;
n > k2;
ð3:4Þ
. So we have kbews1 1þc. < d 3 þ E. According to Lemma 2.3, we have limt!1 yðtÞ ¼ 0. Let ðx1 ðtÞ; x3 ðtÞÞ be the solution of system (1.4) with initial conditions (1.5) and x3 ðfÞ ¼ u3 ðfÞðf 2 ½s1 ; 0Þ, yðtÞ is the solution of system (3.4) with initial conditions yðfÞ ¼ u3 ðfÞ ðf 2 ½s1 ; 0Þ. By the comparison theorem, we have
lim x3 ðtÞ < lim yðtÞ ¼ 0;
t!1
t!1
Incorporating into the positivity of x3 ðtÞ, we know that limt!1 x3 ðtÞ ¼ 0. Therefore, for any e1 > 0 (sufficiently small), there exists an integer k 3 ðk 3 s > k 2 s þ s1 Þ such that x3 ðtÞ < e1 for all t > k 3 s. For system (2.5), we have dx1 ðtÞ ð3:5Þ 6 x1 ðtÞða bx1 ðtÞÞ: dt ðtÞ; z2 ðtÞ ! x1g ðtÞ as t ! 1, while z1 ðtÞ and z2 ðtÞ are the solutions of Then we have z1 ðtÞ 6 x1 ðtÞ 6 z2 ðtÞ and z1 ðtÞ ! x1g 8 dz ðtÞ 1 > < dt ¼ z1 ðtÞ½ða kbe1 Þ bz1 ðtÞ; t–ns; ð3:6Þ z1 ðtþ Þ ¼ z1 ðtÞ þ l; t ¼ ns; > : z1 ð0þ Þ ¼ x1 ð0þ Þ; ½ða bx1 ðtÞÞ kbe1 x1 ðtÞ 6
and
8 dz ðtÞ 2 > < dt ¼ z2 ðtÞða bz2 ðtÞÞ; t–ns; z2 ðtþ Þ ¼ z2 ðtÞ þ l; t ¼ ns; > : z2 ð0þ Þ ¼ x1 ð0þ Þ;
ð3:7Þ
ðakbe1 Þz1 expððakbe1 ÞðtnsÞÞ respectively. For ns < t 6 ðn þ 1Þs, z1g ðtÞ ¼ ðakbe1 Þbz where z1 ¼ þbz1 expððakbe1 ÞðtnsÞÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðakbe1 þblÞþ ðakbe1 þblÞ2 þ4ðakbe1 Þbl=ðeðakbe1 Þs 1Þ 1 . Therefore, for any e2 > 0 there exists an integer k 4 ; n > k 4 such that > akbe 2b b g g z1 ðtÞ þ e2 < x1 ðtÞ < x1 ðtÞ e2 . Let e1 ! 0, so we have x1g ðtÞ e2 < x1 ðtÞ < x1g ðtÞ þ e2 ; for t large enough, which implies
ðtÞ as t ! 1. This completes the proof. x1 ðtÞ ! x1g
h
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J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
Remark 3.2. From Theorem 3.1, it is clear that the excess harvesting causes the extinction of predator population, and breaches the sustainable development of biological resources. The next work is to investigate the permanence of system (1.3). Before starting our theorem, we give the definition of permanence of system (1.3). Definition 3.3. System (1.3) is said to be permanent if there are constants m; M > 0 (independent of initial value) and a finite time T 0 such that for all solutions ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ with all initial values x1 ðtÞ > 0; x2 ð0þ Þ > 0; x3 ð0þ Þ > 0, m 6 x1 ðtÞ < M=k; m 6 x2 ðtÞ 6 M; m 6 x3 ðtÞ 6 M holds for all t P T 0 . Here T 0 may depend on the initial values ðx1 ð0þ Þ; x2 ð0þ Þ; ðx3 ð0þ ÞÞ. Theorem 3.4. Suppose E < d 3 þ kbews1
ðða bx3 Þ þ blÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 2b
:
Then there is a positive constant q such that each positive solution ðx1 ðtÞ; x3 ðtÞÞ of (2.5) satisfies x3 ðtÞ P q for t large enough, where x3 is determined by the equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2bðd 3 þ EÞ ¼ ðða bx Þ þ blÞ þ ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ: 3 kbews1 Proof. The first equation of (1.4) can be rewritten as Z t dx3 ðtÞ x1 ðtÞ d x1 ðuÞ ws1 ðd 3 þ EÞ x3 ðtÞ kbews1 x3 ðuÞdu: ¼ kbe dt 1 þ cx1 ðtÞ dt ts1 1 þ cx1 ðuÞ
ð3:8Þ
Let us consider any positive solution ðx1 ðtÞ; x3 ðtÞÞ of system (1.4). According to (3.8), V ðtÞ is defined as Z t x1 ðuÞ x3 ðuÞdu: V ðtÞ ¼ x3 ðtÞ þ kbews1 1 þ cx1 ðuÞ ts1 We calculate the derivative of V ðtÞ along the solution of (2.5) dV ðtÞ x1 ðtÞ ð3:9Þ ¼ kbews1 ðd 3 þ EÞ x3 ðtÞ; dt 1 þ cx1 ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðabx Þs 3 1Þ ððabx3 ÞþblÞþ ððabx3 ÞþblÞ2 þ4ðabx3 Þbl=ðe Since E < d 3 þ kbews1 ; we can easily know that there exists sufficiently 2b small e > 0 such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 ðða bx Þ þ blÞ þ ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 3 ws1 @ þ eA > d 3 þ E: kbe 2b We claim that for any t0 > 0, it is impossible that x3 ðtÞ < x3 for all t > t0 . Suppose that the claim is not valid. Then there is a t0 > 0 such that x3 ðtÞ < x3 for all t > t0 . It follows from the first equation of (1.4) that for all t > t0 dx1 ðtÞ > ½ða bx3 Þ bx1 ðtÞx1 ðtÞ: dt
ð3:10Þ
Consider the following comparison impulsive system for all t > t0 ( dvðtÞ ¼ ½ða bx3 Þ bx1 ðtÞx1 ðtÞ; t–ns; dt DvðtÞ ¼ l; t ¼ ns;
Þv expððabx3 ÞðtnsÞÞ g ¼ ðabx 3 By Lemma 2.4, we obtain vðtÞ ; ðabx Þbv þbv expððabx ÞðtnsÞÞ
ð3:11Þ
ns < t 6 ðn þ 1Þs is the unique positive periodic solution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðabx Þs 3 1Þ ððabx3 ÞþblÞþ ððabx3 ÞþblÞ2 þ4ðabx3 Þbl=ðe abx ð> b 3 Þ. By the of (3.11) which is globally asymptotically stable, where v ¼ 2b comparison theorem for impulsive differential equation [2], we know that there exists t1 ð> t0 þ s1 Þ such that the inequalg þ e holds for t P t1 , thus x1 ðtÞ P v þ e for all t P t1 . We make notation as r,v þ e for convenience. ity x1 ðtÞ P vðtÞ 3
So we have kbews1 r > d 3 þ E;
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109
then we have V 0 ðtÞ > x3 ðtÞðkbews1 r d 3 EÞ for all t > t1 . Set xm3 ¼ mint2½t1 ;t1 þs1 x3 ðtÞ; we will show that x3 ðtÞ P xm3 for all t P t1 . Suppose on the contrary, there is a T 0 > 0 such that x3 ðtÞ P xm3 for t1 6 t 6 t1 þ s1 þ T 0 ; x3 ðt1 þ s1 þ T 0 Þ ¼ xm3 and x03 ðt1 þ s1 þ T 0 Þ < 0. Hence, the first equation of system (2.5) implies that x03 ðt1 þ s1 þ T 0 Þ ¼ kbews1
x1 ðt1 þ s1 þ T 0 Þ x3 ðt1 þ s1 þ T 0 Þ ðd 3 þ EÞx3 ðt1 þ s1 þ T 0 Þ 1 þ cx1 ðt1 þ s1 þ T 0 Þ
P ðkbews1 r d 3 EÞxm3 > 0; This is a contradiction. Thus, x3 ðtÞ P xm3 for all t > t1 . As a consequence, then V0 ðtÞ > xm3 ðkbews1 r d 3 EÞ > 0 for all M . Hence, the claim is t > t1 . This implies that as t ! 1, V ðtÞ ! 1. It is a contradiction to V ðtÞ 6 M 1 þ ks1 bews1 1þcM complete. By the claim, we are left to consider two cases. First, x3 ðtÞ P x3 for all t large enough. Second, x3 ðtÞ oscillates about x3 for t large enough. Define x q ¼ min 3 ; q1 ; ð3:12Þ 2 where q1 ¼ x3 eðd 3 þEÞs1 . We hope to show that x3 ðtÞ P q for all t large enough. The conclusion is evident in the first case. For the second case, let t > 0 and n > 0 satisfy x3 ðt Þ ¼ x3 ðt þ nÞ ¼ x3 and x3 ðtÞ < x3 for all t < t < t þ n, where t is sufficiently large such that x3 ðtÞ > r for t < t < t þ n, x3 ðtÞ is uniformly continuous. The positive solutions of (2.5) are ultimately bounded and x3 ðtÞ is not affected by impulses. Hence, there is a T (0 < t < s1 and T is dependent of the choice x of t Þ such that x3 ðt Þ > 33 for t < t < t þ T . If n < T , there is nothing to prove. Let us consider the case T < n < s1 .
1.2
4e–14
1.15 1.1
3e–14
1.05 2e–14
1 0.95
1e–14 0.9 0
0.85 60
62
64
66
68
70
72
74
0
10
20
30
40
50
3e–06 2.5e–06 2e–06 1.5e–06 1e–06 5e–07 0
1.14
1.16
1.18
1.2
1.22
Fig. 1. Dynamical behavior of system (1.4) on predator-extinction periodic solution with x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; b ¼ 0:8; E ¼ 0:8; l ¼ 0:1; s ¼ 0:5; k ¼ 0:9: (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the pest ðx1 Þ population. (c) The phase portrait of prey and mature predator for global attractivity of predator-extinction periodic solution.
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Since x03 ðtÞ > ðd 3 þ EÞx3 ðtÞ and x3 ðt Þ ¼ x3 , it is clear that x3 ðtÞ P q1 for t 2 ½t ; t þ s1 . Then, proceeding exactly as the proof for the above claim, we see that x3 ðtÞ P q1 for t 2 ½t þ s1 ; t þ n. Because the kind of interval t 2 ½t ; t þ n is chosen in an arbitrary way (we only need t to be large), we concluded x3 ðtÞ P q for all large t. In the second case, in view of the above discussion, the choice of q is independent of the positive solution, and we proved that any positive solution of (2.5) satisfies x3 ðtÞ P q for all sufficiently large t. This completes the proof of the theorem. h Theorem 3.5. Suppose E < d 3 þ kbews1
ðða bx3 Þ þ blÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 2b
:
Then system (1.3) is permanent. Proof. Let ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ be any solution of system (1.3). From the first equation of system (1.3) and Theorem 3.4, we have ( dx1 ðtÞ l expðdsÞ P x1 ðtÞ½a bðMd0 þ expðdsÞ1 Þ bx1 ðtÞ; t–ns; dt ð2:13Þ Dx1 ðtÞ ¼ l; t ¼ ns; By the same argument as those in the proof of Theorem 3.1., we have that limt!1 x1 ðtÞ P p;, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðabM þblÞþ ðabMþblÞ2 þ4ðabMÞbl=ðeðabM Þs 1Þ l expðdsÞ e, M ¼ Md0 þ expðdsÞ1 and M 0 ¼ kðaþwÞ . p¼ 2b 4b In view of Theorem 3.1, the second equation of system (1.3) becomes dx3 ðtÞ M0 l expðdsÞ þ P kpbews1 wx3 ðtÞ; d dt expðdsÞ 1 M 0 l expðdsÞ d þexpðdsÞ1
kpbews1
2
e and M 0 ¼ kðaþwÞ . By Theorem 3.4 and the above It is easy to obtain limt!1 x3 ðtÞ P d, where d ¼ w 4b discussion, system (1.3) is permanent. The proof of Theorem 3.5 is complete. h
a
b 1.2
5e–24
1.1 1
4e–24
0.9
3e–24
0.8 2e–24
0.7
1e–24
0.6 0.5
0 60
62
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70
72
0
74
10
20
30
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50
c
0.5
0.4 0.3 0.2 0.1 0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Fig. 2. Dynamical behavior of system (1.4) on predator-extinction periodic solution with x1 ð0Þ ¼ 0:5; x3 ð0Þ ¼ 0:5; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 2; b ¼ 1; c ¼ 1; b ¼ 1; E ¼ 0:8; l ¼ 0:1; s ¼ 0:5; k ¼ 0:1. (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the pest ðx1 Þ population. (c) The phase portrait of prey and mature predator for global attractivity of predatorextinction periodic solution.
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
111
Remark 3.6. From Theorem 3.5, we know that the reasonable harvesting can prevent the predator population from extinction, that is, the behavior of reasonable harvesting can bring the permanence of the exploitative predator–prey system.
4. Numerical analysis and discussion In this paper, we introduce a time delay and pulse into the predator–prey system with stage structure for predator. According to the fact of biological resource management, we analyze that the predator-extinction periodic solution of system (1.3) is globally attractive. To investigate the effect of the impulsive stocking on prey and continuous harvesting on predator by numerical analysis, we assume that d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; E ¼ 0:8; s1 ¼ 0; l ¼ 0:1; b ¼ 1; k ¼ 0:9; x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1. It is obvious that the parameters satisfy the condition of Theorem 3.1. Then the predatorextinction periodic solution is global attractivity (see Fig.1), we also assume that d 3 ¼ 0:2; a ¼ 2; b ¼ 1; c ¼ 1; s1 ¼ 0; E ¼ 0:8; b ¼ 1; l ¼ 0:1; k ¼ 0:1; x1 ð0Þ ¼ 0:5; x3 ð0Þ ¼ 0:5, it is clear that the parameters satisfy the condition of Theorem 3.1, then the predator-extinction periodic solution is globally attractive (see Fig. 2), this implies that the excess harvesting may cause the extinction of predator population, and lose the merits of exploitative predator population. Finally, we assume that d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; E ¼ 0:3; l ¼ 0:5; b ¼ 1; s1 ¼ 0; k ¼ 0:9; x1 ð0Þ ¼ 0:1:1; x3 ð0Þ ¼ 1:1, it is obvious that the parameters satisfy Theorem 3.5, then system (1.4) is permanence (see Fig. 3), it illuminates that reasonable harvesting can ensure the sustainable development of biological resources. From Theorems 3.1 and 3.5, we can easily guess that there must exist a threshold l . If l < l , the predator-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally attractive. If l > l , system (1.3) is permanent, or from Theorems 3.1 and 3.5, we can also guess that there must exist a threshold E . If E > E , the pest-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally attractive. If E < E , system (1.3) is permanent. The results show that the behavior of impulsive stocking on prey plays an important role for the permanence of system (1.3), and provide tactical basis for the biological resource management. But there is an interesting problem: What is the optimal harvesting policy of system (1.3)? We will continue to study these problems in the future.
a
b
2.21
1
2.2 0.8
2.19 2.18
0.6
2.17
0.4
2.16 0.2 60
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74
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c 2.21 2.2 2.19 2.18 2.17 2.16 0.2
0.3
0.4
0.5
0.6
Fig. 3. Dynamical behavior of the permanence of system (1.4) with x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; b ¼ 2; E ¼ 0:3; s ¼ 0:5; l ¼ 0:5; k ¼ 0:95. (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the prey ðx1 Þ population. (c) Positive periodic solution of (1.4).
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References [1] Lakshmikantham V, Bainov DD, Simeonov P. Theory of impulsive differential equations. Singapore: World scientific; 1989. [2] Bainov D, Simeonov P. Impulsive differential equations: periodic solutions and applications, 66. England: Longman; 1993. [3] Song Xinyu, Chen LS. Optimal harvesting and stability for a predator–prey system with stage structure. Acta Math Appl English Series 2002;18(3):423–30. [4] Jiao Jianjun, Chen Lansun. Nonlinear incidence rate of a pest management SI model with biological and chemical control concern. Appl Math Mech (English Edition) 2007;28(4):541–51. [5] Song Xinyu, Li Yongfeng. Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals 2007;33(2):463–78. [6] Aiello WG, Freedman HI. A time-delay model of single-species growth with stage-structured. Math Biosci 1990;101:139–53. [7] Freedman HI, Gopalsamy K. Global stability in time-delayed single species dynamics. Bull Math Biol 1986;48:485–92. [8] Yang Kuang. Delay differential equation with application in population dynamics. New York: Academic Press; 1987. p. 67–70. [9] Liu Zhijun, Tan Ronghua. Impulsive harvesting and stocking in a Monod-Haldane functional response predator–prey system. Chaos, Solitons & Fractals 2007;34(2):454–64. [10] Caltagirone LE, Doutt RL. Global behavior of an SEIRS epidemic model with delays. The history of the vedalia beetle importation to California and its impact on the development of biological control. Ann Rev Entomol 1989;34:1–16. [11] Gao Shujing, Chen Lansun, Nieto Angela Torres Juan J. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 2006;24:6037–45. [12] Gakkhar Sunita, Singh Brahampal. The dynamics of a food web consisting of two preys and a harvesting predator. Chaos, Solitons & Fractals 2007;34(4):1346–56. [13] Wang W, Chen L. A predator–prey system with stage structure for predator. Comput Math Appl 1997;33(8):83–91. [14] Dong Lingzhen, Chen Lansun, Sun Lihua. Extinction and permanence of the predator–prey system with stocking of prey and harvesting of predator impulsively. Math Meth Appl Sci 2006;29:415–25. [15] Wang W, Mulone G, Salemi F, Salone V. Permanence and stability of a stage-structured predator–prey model. J Math Anal Appl 2001;262:499–528. [16] Liu Bing, Teng Zhidong, Liu Wanbo. Dynamic behaviors of the periodic Lotka Volterra competing system with impulsive perturbations. Chaos, Solitons & Fractals 2007;31(2):356–70. [17] Zhang Yujuan, Xiu Zhilong, Chen Lansun. Dynamic complexity of a two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals 2005;26(1):131–9. [18] Clark CW. Mathematical bioeconomics. New York: Wiley; 1990. [19] Wang Fengyan, Zeng Guangzhao. Chaos in a Lotka-Volterra predator–prey system with periodically impulsive ratio-harvesting the prey and time delays. Chaos, Solitons & Fractals 2007;32(4):1499–512.
Chaos, Solitons and Fractals 41 (2009) 103–112 www.elsevier.com/locate/chaos
Harvesting policy for a delayed stage-structured Holling II predator–prey model with impulsive stocking prey q Jianjun Jiao a,b,*, Xinzhu Meng c, Lansun Chen a b
a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, People’s Republic of China c College of Science, Shandong University of Science and Technology, Qingdao 266510, People’s Republic of China
Accepted 9 November 2007
Communicated by Prof. L. Marek-Crnjac
Abstract A predator–prey model with a stage structure for the predator, which improves the assumption that each individual predator has the same ability to capture prey, is proposed by Wang et al. [Wang W, Mulone G, Salemi F, Salone V. Permanence and stability of a stage-structured predator–prey model. J Math Anal Appl 2001;262:499–528]. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We do economic management behavior for Wang model [Wang et al., 2001] by continuous harvesting on predator and impulsive stocking on prey. Then, a delayed stage-structured Holling type II predator–prey model with impulsive stocking prey and continuous harvesting predator is established. It is also assumed that the predating products of the predator is only to increase its bearing ability. We obtain the sufficient conditions of the global attractivity of predator-extinction boundary periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Further, the numerical analysis is also inserted to illuminate the dynamics of the system. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Biological resources are renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark [18]. In recent years, the optimal management of renewable resources, which has a direct
q Supported by National Natural Science Foundation of China (10471117) and the Emphasis Subject of Guizhou College of Finance and Economics. * Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China. E-mail addresses: [email protected] (J. Jiao), [email protected] (L. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.015
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relationship to sustainable development, has been studied extensively by many authors [13]. Especially, the predator– prey models with harvesting are investigated by many papers [3,9,12,18]. There are many works concerning predator–prey system, and many good results are obtained [5–7,9,16,17,19]. The basic predator–prey model is 0 x1 ðtÞ ¼ x1 ðtÞðr ax1 bx2 ðtÞÞ; ð1:1Þ x02 ðtÞ ¼ x2 ðtÞðd þ cx1 ðtÞÞ; where x1 ðtÞ and x2 ðtÞ are the densities of the prey population and the predator population at time t, respectively, r > 0 is the intrinsic growth rate of prey, a > 0 is the coefficient of intraspecific competition, b > 0 is the per-capita rate of predation of the predator, d > 0 is the death rate of predator, c > 0 denotes the product of the per-capita rate of predation and the rate of conversing pest into predator. If rc da < 0, system (1.1) does not have any positive equilibrium point, and the only boundary equilibrium point ar ; 0 is globally asymptotically stable, which implies that the predator population will go into extinction. If the prey is stocked at a constant rate, system (1.1) can be the following differential equation: 0 x1 ðtÞ ¼ x1 ðtÞðr ax1 bx2 ðtÞÞ þ l; ð1:2Þ x02 ðtÞ ¼ x2 ðtÞðd þ cx1 ðtÞÞ: It can be easily derived that if l > dðadrcÞ , system (1.2) has a unique globally asymptotically stable positive equilibrium c2 2 d rdcad þlc . This implies that the behavior of stocking prey keeps system (1.2) permanent. While there are few papers ; bcd c [9,14] devoted to investigate the models with impulsive perturbations on prey, but their models did not consider time delay. Stage-structured models have received much attention in recent years. Mathematical analyses for single species models with stage structure were performed by many papers [6–8], Recently, Wang and Chen [13] considered predator–prey models with a stage structure for the predator to analyze the influences of a stage structure for the predator on the dynamics of predator–prey models, but these models without impulsive perturbations ignore the duration time of immature predators. And many good results of impulsive differential equations are obtained in [4,9,11]. The recent works of Wang [15] considered that the preying products of the predator is only to increase its bearing ability and a period to maturity. It can really reflect the nature of the world, For example, the large-scale fowl-run add the feedstuffs to enhance the yield of egg. But they did not take economic management behavior into account. Motivated by these ideals, we consider the following impulsive delay differential equation: 9 8 x1 ðtÞ > x01 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ b 1þcx x3 ðtÞ; > > > ðtÞ > 1 > = > > x ðtÞ x ðts Þ > 0 ws1 1 1 1 > ðtÞ ¼ kb x ðtÞ kbe x ðt s Þ wx ðtÞ; x t–ns; 1 2 > 2 1þcx1 ðtÞ 3 1þcx1 ðts1 Þ 3 > > > > > > > x ðts Þ 1 1 ; > x3 ðt s1 Þ d 3 x3 ðtÞ Ex3 ðtÞ; < x03 ðtÞ ¼ kbews1 1þcx 1 ðts1 Þ 9 ð1:3Þ Dx1 ðtÞ ¼ l; > > = > > > > Dx2 ðtÞ ¼ 0; t ¼ ns; n ¼ 1; 2 . . . ; > > > > > ; > > Dx ðtÞ ¼ 0; > > 3 : ðu1 ðfÞ; u2 ðfÞ; u3 ðfÞÞ 2 C þ ¼ Cð½s1 ; 0; R3þ Þ; ui ð0Þ > 0; i ¼ 1; 2; 3; where x1 ðtÞ denotes the density of prey at time t, x2 ðtÞ; x3 ðtÞ represent the densities of the immature individual predator and mature individual predator at time t, respectively. s1 represents a constant time to maturity, a > 0 is the intrinsic growth rate of prey, b > 0 is the coefficient of intraspecific competition, b is the capture rate of mature predator, and bx1 ðtÞ ðc > 0Þ is the Holling type II functional response. k is the rate of conversion of nutrients into the reproduction 1þcx1 ðtÞ rate of the mature predator, w is the death rate of immature predator. d 3 is the death rate of mature predator. It is also assumed that immature individuals and mature individuals are divided by age s1 and that immature individual predators do not feed on prey and do not have the ability to reproduce. This seems reasonable for a number of mammals where immature predators are raised by their parents and their reproduction rate and attacking rate for prey can be neglected. 0 < E < 1 is the effect of continuous harvesting predator, Dx1 ðtÞ ¼ x1 ðtþ Þ x1 ðtÞ, l P 0 is the stocking amount of prey at t ¼ ns; n 2 Z þ and Z þ ¼ f1; 2; . . .g, s is the period of the impulsive stocking of the prey. The purpose of this paper is to prove that system (1.3) has a predator-extinction periodic solution. Further, it is globally attractive. While due to the stocking of prey, all the mature predator population would not go into extinction for the continuous harvesting of mature predator population, that is, system (1.3) is permanent. Because the first and third equations of (1.3) do not contain x2 ðtÞ, we can simplify model (1.3) and restrict our attention to the following model:
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
8 9 x1 ðtÞ > = x01 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ b 1þcx x3 ðtÞ; > > 1 ðtÞ > t–ns; > < 0 ws1 x1 ðts1 Þ x ðt s1 Þ d 3 x3 ðtÞ Ex3 ðtÞ; ; x3 ðtÞ ¼ kbe 1þcx1 ðts1 Þ 3 > > > Dx1 ðtÞ ¼ l; > > t ¼ ns; n ¼ 1; 2 . . . ; : Dx3 ðtÞ ¼ 0;
105
ð1:4Þ
the initial conditions for (1.4) are ðu1 ðfÞ; u3 ðfÞÞ 2 C þ ¼ Cð½s1 ; 0; R2þ Þ;
ui ð0Þ > 0;
i ¼ 1; 3:
ð1:5Þ
2. Some important lemmas The solution of (1.3), denoted by xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞT , is a piecewise continuous function x : Rþ ! R3þ , xðtÞ is continuous on ðns; ðn þ 1Þs, n 2 Z þ and xðnsþ Þ ¼ limt!nsþ xðtÞ exists. Obviously the global existence and uniqueness of solutions of (1.3) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-side of system (1.3) (see [1,2]). For continuity of initial conditions, we require Z 0 u1 ðsÞ u2 ð0Þ ¼ kbews ð2:1Þ u ðsÞds; 1 þ cu1 ðsÞ 3 s1 Before we have the main results, we need to give some lemmas which will be used next. Lemma 2.1. Let ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ > 0 for s1 < t < 0. Then any solution of system (1.3) is strictly positive. Proof. Firstly, by uniqueness of solutions of system (1.3) and x01 ðtÞ ¼ 0 whenever x1 ðtÞ ¼ 0, t–ns, and x1 ðnsþ Þ ¼ x1 ðnsÞ þ l; l P 0. It is easy to see that x1 ðtÞ > 0 for all t > 0. Secondly, we show that x3 ðtÞ > 0 for all t > 0. Notice x3 ðtÞ > 0, hence if there exists t0 such that x3 ðt0 Þ ¼ 0, then t0 > 0. Assume that t0 is the first such time that x3 ðtÞ ¼ 0, that is, t0 ¼ infft > 0 : x3 ðtÞ ¼ 0g; then x1 ðt0 s1 Þ x03 ðt0 Þ ¼ kbews1 1þcx x3 ðt0 s1 Þ > 0. Hence for sufficiently small e > 0, x03 ðt0 eÞ > 0. But by the definition of 1 ðt0 s1 Þ 0 t0 , x3 ðt0 eÞ 6 0. This contradiction shows that x3 ðtÞ > 0 for all t > 0. Finally we consider the following equation: s0 ðtÞ ¼ kbews1
x1 ðt s1 Þ x3 ðt s1 Þ wsðtÞ; 1 þ cx1 ðt s1 Þ
ð2:2Þ
and comparing with (1.3), we note that if sðtÞ is the solution of (2.2) and if x1 ðtÞ can solve (1.3), then x2 ðtÞ > sðtÞ on 0 < t < s1 . Solving (2.2) gives Z t x1 ðu s1 Þ wt wðus1 Þ sðtÞ ¼ e x2 ð0Þ kbe x3 ðu s1 Þdu ; 1 þ cx1 ðu s1 Þ 0 From (2.1), one can obtain Z 0 kbews sðs1 Þ ¼ ews1
x1 ðu s1 Þ x3 ðu s1 Þdu ; 1 þ cx1 ðu s1 Þ 0 s1 R0 u1 ðsÞ By making transformation and x1 ðtÞ ¼ u1 ðtÞ, x3 ðtÞ ¼ u3 ðtÞ; t 2 ½s1 ; 0, we know that s1 kbews 1þcu u3 ðsÞds is equiv1 ðsÞ R s1 wðus1 Þ x1 ðus1 Þ x ðu s Þdu. Thus we obtain sðs Þ ¼ 0. Hence x ðtÞ > 0, Since sðtÞ is strictly decreasing, alent to 0 kbe 1 1 2 1þcx1 ðus1 Þ 3 then x2 ðtÞ > sðtÞ > 0 for t 2 ð0; s1 Þ. So x2 ðtÞ > 0 on 0 6 t 6 s1 . By induction and similar method to the proof of Theorem 1 [10], we can show that x2 ðtÞ > 0 for all t P 0. This completes the proof. h u1 ðsÞ u ðsÞds 1 þ cu1 ðsÞ 3
Z
s1
kbewðus1 Þ
Now, we show that all solutions of (1.3) are uniformly ultimately bounded. Lemma 2.2. There exists a constant M > 0 such that x1 ðtÞ 6 Mk ; x2 ðtÞ 6 M; x3 ðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ of (1.3) with all t large enough. Proof. Define V ðtÞ ¼ kx1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ. And because of w < d 3 þ E, then t–ns, we have Dþ V ðtÞ þ wV ðtÞ ¼ kðw þ aÞx1 kbx21 ðtÞ þ ðw d 3 EÞx3 ðtÞ 6 M 0 ;
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J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112 2
where M 0 ¼ kðaþwÞ , when t ¼ ns, V ðnsþ Þ ¼ V ðnsÞ þ l. By Lemma 2.2 (which can be seen in [1, p. 23]), for 4b t 2 ðns; ðn þ 1Þs we have V ðtÞ 6 V ð0Þ expðdtÞ þ
M0 l expðdðt sÞÞ l expðdsÞ M0 l expðdsÞ ð1 expðdtÞÞ þ þ þ ! t ! 1: d d 1 expðdsÞ expðdsÞ 1 expðdsÞ 1
l expðdsÞ >0 So V ðtÞ is uniformly ultimately bounded. Hence, by the definition of V ðtÞ, there exists a constant M ¼ Md0 þ expðdsÞ1 M such that x1 ðtÞ 6 k ; x2 ðtÞ 6 M; x3 ðtÞ 6 M for t large enough. The proof is complete. h
Considering the following delay equation: x0 ðtÞ ¼ a1 xðt sÞ a2 xðtÞ;
ð2:3Þ
we assume that a1 ; a2 ; s > 0; xðtÞ > 0 for s 6 t 6 0. Lemma 2.3 [8]. For system (2.3), assume that a1 < a2 . Then limt!1 xðtÞ ¼ 0: Lemma 2.4 [14]. Considering the following impulsive system: 0 v ðtÞ ¼ vðtÞða bvðtÞÞ; t–ns; vðnsþ Þ ¼ vðnsÞ þ l; t ¼ ns; n ¼ 1; 2 . . . ; where
a > 0; b > 0; l > 0. Then system t 2 ðns; ðn þ 1Þs; n 2 Z þ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 as 1Þ ðaþblÞþ ðaþblÞ þ4abl=ðe ð> abÞ. v ¼ 2b
av expðaðtnsÞÞ ; abv þbv expðaðtnsÞÞ
(2.4) which
has is
ð2:4Þ a
unique globally
positive periodic asymptotically
solution stable,
g¼ vðtÞ where
According to system (1.4), we can easily know that there exists t1 2 Z þ ; t > t1 such that x3 ðt s1 Þ ¼ 0 and x3 ðtÞ ¼ 0. Then 0 x1 ðtÞ ¼ x1 ðtÞða bx1 ðtÞÞ; t–ns; ð2:5Þ Dx1 ðtÞ ¼ l; t ¼ ns; n ¼ 1; 2 . . . : ðtÞ; 0; 0Þ ¼ From (2.5) and Lemma 2.4, we know that (1.3) has a predator-extinction periodic solution ð x1g ax1 expðaðtnsÞÞ ; 0; 0 ; t 2 ðns; ðn þ 1Þs; n 2 Z or (1.4) has a predator-extinction periodic solution ð x1g ðtÞ; 0Þ ¼ þ abx þbx expðaðtnsÞÞ 1 1 ax1 expðaðtnsÞÞ ; 0 ; t 2 ðns; ðn þ 1Þs; n 2 Z þ , which is globally asymptotically stable, where abx1 þbx1 expðaðtnsÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 as ðaþblÞþ ðaþblÞ þ4abl=ðe 1Þ x1 ¼ > ab . 2b
3. Global attractivity and permanence In this section, we will obtain the sufficient condition of the global attractivity of predator-extinction periodic solution of system (1.3). Theorem 3.1. Let ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ be any solution of (1.3). If E>
a
kbax1 easws1 d3 þ bx1 eas þ cax1 eas pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
bx1
holds, where x1 ¼ attractive.
ðaþblÞ þ4abl=ðeas 1Þ ð> abÞ, 2b
ðaþblÞþ
the predator-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally
ðtÞ; 0; 0Þ of system (1.3) is equivProof. It is clear that the global attraction of predator-extinction periodic solution ð x1g g ðtÞ; 0Þ of system (1.4). So we only devote to system alent to the global attraction of pest-extinction periodic solution ð x 1 akbx easws1 (1.4). Since E > abx þbx1 eas þcax eas d 3 , we can choose e0 sufficiently small such that 1
0 ws1 @
kbe
1
1
ax1 eas abx1 þbx1 eas
þ e0
1
A < d 3 þ E; ax eas 1 þ c abx 1þbx eas þ e0 1
1
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
107
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaþblÞþ ðaþblÞ2 þ4abl=ðeas 1Þ where x1 ¼ > ab . It follows from the first equation of system (1.4) that dxdt1 ðtÞ 6 x1 ðtÞða bx1 ðtÞÞ. 2b So we consider the following comparison impulsive differential system: 8 dxðtÞ > < dt ¼ xðtÞða bxðtÞÞ; t–ns; ð3:1Þ DxðtÞ ¼ l; t ¼ ns; > : þ þ xð0 Þ ¼ x1 ð0 Þ: In view of Lemma 2.4, we obtain that the periodic solution of system (1.4) g¼ xðtÞ
ax1 expðaðt nsÞÞ ; a bx1 þ bx1 expðaðt nsÞÞ
t 2 ðns; ðn þ 1Þs; n 2 Z þ ; ð3:2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaþblÞþ ðaþblÞ2 þ4abl=ðeas 1Þ which is globally asymptotically stable, where x1 ¼ > ab . 2b ðtÞ as From Lemma 2.4 and comparison theorem of impulsive equation [2], we have x1 ðtÞ 6 xðtÞ and xðtÞ ! x1g t ! 1. Then there exists an integer k 2 > k 1 ; t > k 2 such that x1 ðtÞ 6 xðtÞ 6 x1g ðtÞ þ e0 ;
ns < t 6 ðn þ 1Þs;
n > k2 ;
that is ðtÞ þ e0 6 x1 ðtÞ < x1g
ax1 eas þ e0 , .; a bx1 þ bx1 eas
ns < t 6 ðn þ 1Þs;
n > k2 ;
and from (1.4), we get dx3 ðtÞ . 6 kbews1 x3 ðt s1 Þ ðd 3 þ EÞx3 ðtÞ; dt 1 þ c.
t > ns þ s1 ;
n > k2 :
ð3:3Þ
Consider the following comparison differential system: dyðtÞ . ¼ kbews1 yðt s1 Þ ðd 3 þ EÞyðtÞ; dt 1 þ c.
t > ns þ s1 ;
n > k2;
ð3:4Þ
. So we have kbews1 1þc. < d 3 þ E. According to Lemma 2.3, we have limt!1 yðtÞ ¼ 0. Let ðx1 ðtÞ; x3 ðtÞÞ be the solution of system (1.4) with initial conditions (1.5) and x3 ðfÞ ¼ u3 ðfÞðf 2 ½s1 ; 0Þ, yðtÞ is the solution of system (3.4) with initial conditions yðfÞ ¼ u3 ðfÞ ðf 2 ½s1 ; 0Þ. By the comparison theorem, we have
lim x3 ðtÞ < lim yðtÞ ¼ 0;
t!1
t!1
Incorporating into the positivity of x3 ðtÞ, we know that limt!1 x3 ðtÞ ¼ 0. Therefore, for any e1 > 0 (sufficiently small), there exists an integer k 3 ðk 3 s > k 2 s þ s1 Þ such that x3 ðtÞ < e1 for all t > k 3 s. For system (2.5), we have dx1 ðtÞ ð3:5Þ 6 x1 ðtÞða bx1 ðtÞÞ: dt ðtÞ; z2 ðtÞ ! x1g ðtÞ as t ! 1, while z1 ðtÞ and z2 ðtÞ are the solutions of Then we have z1 ðtÞ 6 x1 ðtÞ 6 z2 ðtÞ and z1 ðtÞ ! x1g 8 dz ðtÞ 1 > < dt ¼ z1 ðtÞ½ða kbe1 Þ bz1 ðtÞ; t–ns; ð3:6Þ z1 ðtþ Þ ¼ z1 ðtÞ þ l; t ¼ ns; > : z1 ð0þ Þ ¼ x1 ð0þ Þ; ½ða bx1 ðtÞÞ kbe1 x1 ðtÞ 6
and
8 dz ðtÞ 2 > < dt ¼ z2 ðtÞða bz2 ðtÞÞ; t–ns; z2 ðtþ Þ ¼ z2 ðtÞ þ l; t ¼ ns; > : z2 ð0þ Þ ¼ x1 ð0þ Þ;
ð3:7Þ
ðakbe1 Þz1 expððakbe1 ÞðtnsÞÞ respectively. For ns < t 6 ðn þ 1Þs, z1g ðtÞ ¼ ðakbe1 Þbz where z1 ¼ þbz1 expððakbe1 ÞðtnsÞÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðakbe1 þblÞþ ðakbe1 þblÞ2 þ4ðakbe1 Þbl=ðeðakbe1 Þs 1Þ 1 . Therefore, for any e2 > 0 there exists an integer k 4 ; n > k 4 such that > akbe 2b b g g z1 ðtÞ þ e2 < x1 ðtÞ < x1 ðtÞ e2 . Let e1 ! 0, so we have x1g ðtÞ e2 < x1 ðtÞ < x1g ðtÞ þ e2 ; for t large enough, which implies
ðtÞ as t ! 1. This completes the proof. x1 ðtÞ ! x1g
h
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J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
Remark 3.2. From Theorem 3.1, it is clear that the excess harvesting causes the extinction of predator population, and breaches the sustainable development of biological resources. The next work is to investigate the permanence of system (1.3). Before starting our theorem, we give the definition of permanence of system (1.3). Definition 3.3. System (1.3) is said to be permanent if there are constants m; M > 0 (independent of initial value) and a finite time T 0 such that for all solutions ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ with all initial values x1 ðtÞ > 0; x2 ð0þ Þ > 0; x3 ð0þ Þ > 0, m 6 x1 ðtÞ < M=k; m 6 x2 ðtÞ 6 M; m 6 x3 ðtÞ 6 M holds for all t P T 0 . Here T 0 may depend on the initial values ðx1 ð0þ Þ; x2 ð0þ Þ; ðx3 ð0þ ÞÞ. Theorem 3.4. Suppose E < d 3 þ kbews1
ðða bx3 Þ þ blÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 2b
:
Then there is a positive constant q such that each positive solution ðx1 ðtÞ; x3 ðtÞÞ of (2.5) satisfies x3 ðtÞ P q for t large enough, where x3 is determined by the equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2bðd 3 þ EÞ ¼ ðða bx Þ þ blÞ þ ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ: 3 kbews1 Proof. The first equation of (1.4) can be rewritten as Z t dx3 ðtÞ x1 ðtÞ d x1 ðuÞ ws1 ðd 3 þ EÞ x3 ðtÞ kbews1 x3 ðuÞdu: ¼ kbe dt 1 þ cx1 ðtÞ dt ts1 1 þ cx1 ðuÞ
ð3:8Þ
Let us consider any positive solution ðx1 ðtÞ; x3 ðtÞÞ of system (1.4). According to (3.8), V ðtÞ is defined as Z t x1 ðuÞ x3 ðuÞdu: V ðtÞ ¼ x3 ðtÞ þ kbews1 1 þ cx1 ðuÞ ts1 We calculate the derivative of V ðtÞ along the solution of (2.5) dV ðtÞ x1 ðtÞ ð3:9Þ ¼ kbews1 ðd 3 þ EÞ x3 ðtÞ; dt 1 þ cx1 ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðabx Þs 3 1Þ ððabx3 ÞþblÞþ ððabx3 ÞþblÞ2 þ4ðabx3 Þbl=ðe Since E < d 3 þ kbews1 ; we can easily know that there exists sufficiently 2b small e > 0 such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 ðða bx Þ þ blÞ þ ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 3 ws1 @ þ eA > d 3 þ E: kbe 2b We claim that for any t0 > 0, it is impossible that x3 ðtÞ < x3 for all t > t0 . Suppose that the claim is not valid. Then there is a t0 > 0 such that x3 ðtÞ < x3 for all t > t0 . It follows from the first equation of (1.4) that for all t > t0 dx1 ðtÞ > ½ða bx3 Þ bx1 ðtÞx1 ðtÞ: dt
ð3:10Þ
Consider the following comparison impulsive system for all t > t0 ( dvðtÞ ¼ ½ða bx3 Þ bx1 ðtÞx1 ðtÞ; t–ns; dt DvðtÞ ¼ l; t ¼ ns;
Þv expððabx3 ÞðtnsÞÞ g ¼ ðabx 3 By Lemma 2.4, we obtain vðtÞ ; ðabx Þbv þbv expððabx ÞðtnsÞÞ
ð3:11Þ
ns < t 6 ðn þ 1Þs is the unique positive periodic solution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðabx Þs 3 1Þ ððabx3 ÞþblÞþ ððabx3 ÞþblÞ2 þ4ðabx3 Þbl=ðe abx ð> b 3 Þ. By the of (3.11) which is globally asymptotically stable, where v ¼ 2b comparison theorem for impulsive differential equation [2], we know that there exists t1 ð> t0 þ s1 Þ such that the inequalg þ e holds for t P t1 , thus x1 ðtÞ P v þ e for all t P t1 . We make notation as r,v þ e for convenience. ity x1 ðtÞ P vðtÞ 3
So we have kbews1 r > d 3 þ E;
3
J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
109
then we have V 0 ðtÞ > x3 ðtÞðkbews1 r d 3 EÞ for all t > t1 . Set xm3 ¼ mint2½t1 ;t1 þs1 x3 ðtÞ; we will show that x3 ðtÞ P xm3 for all t P t1 . Suppose on the contrary, there is a T 0 > 0 such that x3 ðtÞ P xm3 for t1 6 t 6 t1 þ s1 þ T 0 ; x3 ðt1 þ s1 þ T 0 Þ ¼ xm3 and x03 ðt1 þ s1 þ T 0 Þ < 0. Hence, the first equation of system (2.5) implies that x03 ðt1 þ s1 þ T 0 Þ ¼ kbews1
x1 ðt1 þ s1 þ T 0 Þ x3 ðt1 þ s1 þ T 0 Þ ðd 3 þ EÞx3 ðt1 þ s1 þ T 0 Þ 1 þ cx1 ðt1 þ s1 þ T 0 Þ
P ðkbews1 r d 3 EÞxm3 > 0; This is a contradiction. Thus, x3 ðtÞ P xm3 for all t > t1 . As a consequence, then V0 ðtÞ > xm3 ðkbews1 r d 3 EÞ > 0 for all M . Hence, the claim is t > t1 . This implies that as t ! 1, V ðtÞ ! 1. It is a contradiction to V ðtÞ 6 M 1 þ ks1 bews1 1þcM complete. By the claim, we are left to consider two cases. First, x3 ðtÞ P x3 for all t large enough. Second, x3 ðtÞ oscillates about x3 for t large enough. Define x q ¼ min 3 ; q1 ; ð3:12Þ 2 where q1 ¼ x3 eðd 3 þEÞs1 . We hope to show that x3 ðtÞ P q for all t large enough. The conclusion is evident in the first case. For the second case, let t > 0 and n > 0 satisfy x3 ðt Þ ¼ x3 ðt þ nÞ ¼ x3 and x3 ðtÞ < x3 for all t < t < t þ n, where t is sufficiently large such that x3 ðtÞ > r for t < t < t þ n, x3 ðtÞ is uniformly continuous. The positive solutions of (2.5) are ultimately bounded and x3 ðtÞ is not affected by impulses. Hence, there is a T (0 < t < s1 and T is dependent of the choice x of t Þ such that x3 ðt Þ > 33 for t < t < t þ T . If n < T , there is nothing to prove. Let us consider the case T < n < s1 .
1.2
4e–14
1.15 1.1
3e–14
1.05 2e–14
1 0.95
1e–14 0.9 0
0.85 60
62
64
66
68
70
72
74
0
10
20
30
40
50
3e–06 2.5e–06 2e–06 1.5e–06 1e–06 5e–07 0
1.14
1.16
1.18
1.2
1.22
Fig. 1. Dynamical behavior of system (1.4) on predator-extinction periodic solution with x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; b ¼ 0:8; E ¼ 0:8; l ¼ 0:1; s ¼ 0:5; k ¼ 0:9: (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the pest ðx1 Þ population. (c) The phase portrait of prey and mature predator for global attractivity of predator-extinction periodic solution.
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J. Jiao et al. / Chaos, Solitons and Fractals 41 (2009) 103–112
Since x03 ðtÞ > ðd 3 þ EÞx3 ðtÞ and x3 ðt Þ ¼ x3 , it is clear that x3 ðtÞ P q1 for t 2 ½t ; t þ s1 . Then, proceeding exactly as the proof for the above claim, we see that x3 ðtÞ P q1 for t 2 ½t þ s1 ; t þ n. Because the kind of interval t 2 ½t ; t þ n is chosen in an arbitrary way (we only need t to be large), we concluded x3 ðtÞ P q for all large t. In the second case, in view of the above discussion, the choice of q is independent of the positive solution, and we proved that any positive solution of (2.5) satisfies x3 ðtÞ P q for all sufficiently large t. This completes the proof of the theorem. h Theorem 3.5. Suppose E < d 3 þ kbews1
ðða bx3 Þ þ blÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðða bx3 Þ þ blÞ2 þ 4ða bx3 Þbl=ðeðabx3 Þs 1Þ 2b
:
Then system (1.3) is permanent. Proof. Let ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞ be any solution of system (1.3). From the first equation of system (1.3) and Theorem 3.4, we have ( dx1 ðtÞ l expðdsÞ P x1 ðtÞ½a bðMd0 þ expðdsÞ1 Þ bx1 ðtÞ; t–ns; dt ð2:13Þ Dx1 ðtÞ ¼ l; t ¼ ns; By the same argument as those in the proof of Theorem 3.1., we have that limt!1 x1 ðtÞ P p;, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðabM þblÞþ ðabMþblÞ2 þ4ðabMÞbl=ðeðabM Þs 1Þ l expðdsÞ e, M ¼ Md0 þ expðdsÞ1 and M 0 ¼ kðaþwÞ . p¼ 2b 4b In view of Theorem 3.1, the second equation of system (1.3) becomes dx3 ðtÞ M0 l expðdsÞ þ P kpbews1 wx3 ðtÞ; d dt expðdsÞ 1 M 0 l expðdsÞ d þexpðdsÞ1
kpbews1
2
e and M 0 ¼ kðaþwÞ . By Theorem 3.4 and the above It is easy to obtain limt!1 x3 ðtÞ P d, where d ¼ w 4b discussion, system (1.3) is permanent. The proof of Theorem 3.5 is complete. h
a
b 1.2
5e–24
1.1 1
4e–24
0.9
3e–24
0.8 2e–24
0.7
1e–24
0.6 0.5
0 60
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72
0
74
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c
0.5
0.4 0.3 0.2 0.1 0
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1
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1.2
Fig. 2. Dynamical behavior of system (1.4) on predator-extinction periodic solution with x1 ð0Þ ¼ 0:5; x3 ð0Þ ¼ 0:5; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 2; b ¼ 1; c ¼ 1; b ¼ 1; E ¼ 0:8; l ¼ 0:1; s ¼ 0:5; k ¼ 0:1. (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the pest ðx1 Þ population. (c) The phase portrait of prey and mature predator for global attractivity of predatorextinction periodic solution.
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Remark 3.6. From Theorem 3.5, we know that the reasonable harvesting can prevent the predator population from extinction, that is, the behavior of reasonable harvesting can bring the permanence of the exploitative predator–prey system.
4. Numerical analysis and discussion In this paper, we introduce a time delay and pulse into the predator–prey system with stage structure for predator. According to the fact of biological resource management, we analyze that the predator-extinction periodic solution of system (1.3) is globally attractive. To investigate the effect of the impulsive stocking on prey and continuous harvesting on predator by numerical analysis, we assume that d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; E ¼ 0:8; s1 ¼ 0; l ¼ 0:1; b ¼ 1; k ¼ 0:9; x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1. It is obvious that the parameters satisfy the condition of Theorem 3.1. Then the predatorextinction periodic solution is global attractivity (see Fig.1), we also assume that d 3 ¼ 0:2; a ¼ 2; b ¼ 1; c ¼ 1; s1 ¼ 0; E ¼ 0:8; b ¼ 1; l ¼ 0:1; k ¼ 0:1; x1 ð0Þ ¼ 0:5; x3 ð0Þ ¼ 0:5, it is clear that the parameters satisfy the condition of Theorem 3.1, then the predator-extinction periodic solution is globally attractive (see Fig. 2), this implies that the excess harvesting may cause the extinction of predator population, and lose the merits of exploitative predator population. Finally, we assume that d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; E ¼ 0:3; l ¼ 0:5; b ¼ 1; s1 ¼ 0; k ¼ 0:9; x1 ð0Þ ¼ 0:1:1; x3 ð0Þ ¼ 1:1, it is obvious that the parameters satisfy Theorem 3.5, then system (1.4) is permanence (see Fig. 3), it illuminates that reasonable harvesting can ensure the sustainable development of biological resources. From Theorems 3.1 and 3.5, we can easily guess that there must exist a threshold l . If l < l , the predator-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally attractive. If l > l , system (1.3) is permanent, or from Theorems 3.1 and 3.5, we can also guess that there must exist a threshold E . If E > E , the pest-extinction periodic solution ð x1g ðtÞ; 0; 0Þ of (1.3) is globally attractive. If E < E , system (1.3) is permanent. The results show that the behavior of impulsive stocking on prey plays an important role for the permanence of system (1.3), and provide tactical basis for the biological resource management. But there is an interesting problem: What is the optimal harvesting policy of system (1.3)? We will continue to study these problems in the future.
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c 2.21 2.2 2.19 2.18 2.17 2.16 0.2
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Fig. 3. Dynamical behavior of the permanence of system (1.4) with x1 ð0Þ ¼ 1:1; x3 ð0Þ ¼ 1:1; s1 ¼ 0; d 3 ¼ 0:2; a ¼ 1; b ¼ 1; c ¼ 1; b ¼ 2; E ¼ 0:3; s ¼ 0:5; l ¼ 0:5; k ¼ 0:95. (a) Time-series of the predator ðx3 Þ population. (b) Time-series of the prey ðx1 Þ population. (c) Positive periodic solution of (1.4).
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