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Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control
ARTICLE IN PRESS
Journal of Theoretical Biology 235 (2005) 495–503 www.elsevier.com/locate/yjtbi
Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control$ Yongzhen Peia,b, Lansun Chena,, Qingrui Zhangc, Changguo Lib a
Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116023, PR China b Department of Basic Science of Institute of Military Traffic, Tianjin 300161, PR China c Department of Chemical Engineering, Qingdao University of Science and Technology, Qingdao 266042, PR China Received 25 October 2004; received in revised form 6 February 2005; accepted 7 February 2005 Available online 27 April 2005 Communicated by Karl Sigmund
Abstract In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one. r 2005 Elsevier Ltd. All rights reserved. Keywords: Impulsive biological control; Holling II functional response; Permanence; Extinction
1. Introduction Based on experiments, Holling (1965) suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka–Volterra system more realistic. Letting X and Y be the densities of a prey and its predator, respectively, a model of prey–predator system with Holling functional response is given by the equations X 0 ¼ XgðX Þ Y FðX Þ, Y 0 ¼ dY þ eY FðX Þ,
$ This work is supported by National Natural Science Foundation of China (10171106). Corresponding author. E-mail addresses: [email protected] (Y. Pei), [email protected] (L. Chen).
0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2005.02.003
where gðX Þ is the per capita rate of change of prey density in absence of the predator, d, e are positive parameters and FðX Þ is a Holling functional response of a predator which takes following concrete forms (1) FðX Þ is a Holling I functional response function, i.e. ( cX if X pX 0 ; FðX Þ ¼ cX 0 if X 4X 0 ; where c is a positive constant characterizing the threshold of prey concentration above which the predation rate is constant and under which the predation rate is similar to the Lotka–Volterra one. (2) If we take into account the time a predator uses in handling the prey it has captured, Holling find the predator has a Holling II functional response. That is FðX Þ ¼
aX , 1 þ wX
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where a; w are positive constants. Clearly, FðX Þ is a strictly increasing function. (3) If FðX Þ is Holling III functional response function, that is FðX Þ ¼
aX 2 . b2 þ X 2
By the definition, It is an S-type function. But some experiments and observations indicate that the non-monotonic response occurs at a level: when the nutrient concentrations reaches a high level an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, Andrews (1968) suggested a function: FðX Þ ¼
mX , a þ bX þ X 2
called the Monod–Haldane function, or Holling type-IV function. Sugie and Howell (1980) proposed a simplified Monod-Haldane function of the form FðX Þ ¼
mX , a þ X2
which describes the phenomenon of group defense whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise themselves when their numbers are large enough. Based on the predator–prey system with Holling functional response, many authors have suggested impulsive differential equation (Bainov and Simeonov, 1993; Samoilenko and Perestyuk, 1995; Lakshmikantham et al., 1989) to model the process of periodic releasing natural enemies and spraying pesticides at different fixed time which is called integrated pest management. Biological control (DeBach, 1964; DeBach and Rosen, 1991; Freedman, 1976; Grasman et al., 2001) which a component of an integrated pest management strategy is the purposeful introduction and establishment of one or more natural enemies from region of origin of an exotic pest, specifically for the purpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economic damage. Virtually all insect and mite pests have some natural enemies. One approach to biological control is augmentation, which is manipulation of existing natural enemies to increase their effectiveness. This can be achieved by mass production and periodic release of natural enemies of the pest, and by genetic enhancement of the enemies to increase their effectiveness at control. The pioneering project of biological control began in 1888 when the new legendary predator, the vedalia beetle was imported from Australia and established in California, where it rapidly suppressed populations of cottony cushion scale that had been
decimating the developing citrus industry (Caltagirone and Doutt, 1989). Another important method for pest control is chemical control. Pesticides are useful because they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss. However, pesticide pollution is not only recognized as a major health hazard to human beings and to natural enemies but also is resited by pests. And the latter results in the use of high rates and more toxic materials to combat pests. Beneficial insects are often susceptible to chemical insecticides applied for the target pest. One of the side effects of high rates of pesticide use is that natural enemies and other small animals that might otherwise feed on pests are killed and pests population burst out once again after beneficial insects being killed. So in this paper, we only apply impulsive biological control to beat the pest in view of green agriculture and environment protection. As mentioned above, impulsive equations are found in almost every domain of applied science and have been studied in many investigations, among which are: impulsive birth (Roberts and Kao, 1998; Tang and Chen, 2002), impulsive vaccination (Shulgin et al., 1998; Donofrio, 1997), chemotherapeutic treatment of disease (Jin et al., 2004; Panetta, 1996) and population ecology (Lakmeche and Arino, 2000; Ballinger and Liu, 1997) They generally describe phenomena that are subject to steep or instantaneous changes which make the differential system more intractable. According to Hassell (1976), Holling type II functional response is the most common type of functional response among arthropod predators. Liu and Chen (2003) has investigated the predator–prey system with periodic constant impulsive immigration of predator: x01 ðtÞ ¼ axðtÞ bx21 ðtÞ
axðtÞ1 x2 ðtÞ ; 1 þ ox1 ðtÞ
tant,
kaxðtÞ1 x2 ðtÞ cx2 ðtÞ; tant, 1 þ ox1 ðtÞ x1 ðntþ Þ ¼ x1 ðntÞ; x2 ðntþ Þ ¼ x2 ðntÞ þ p.
x02 ðtÞ ¼
ð1:1Þ
The system (1.1) describes a one-prey one-predator model with Holing II type. Wherever possible, multinatural enemies control techniques should work together rather than against one. In this paper, the biological control technique of multi-predators is adopted to kill pest. So if one wishes to eradicate the pest population or keep the pest population below the damage level, how do we release the natural enemies? One of the main purposes of this paper is to construct a mathematical model according to the fact of periodic biological control for pest control and to investigate the dynamics of such system. In Section 2, we analyse the dynamics of one-prey multi-predator model with Holling II functional response. And based on this
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model, we suggest an impulsive differential equation to model the process of periodic releasing natural enemies. In Section 3, we give some notations and lemmas. By using the Floquent theory of impulsive, small amplitude perturbation skills and techniques of comparison, we consider the local stability of pest-eradication periodic solution and the permanence of the system in Section 4. A brief discussion of the main results is given in the last section. In the following, the main results of system (1.1) are summarized in the lemmas which has been studied completely in Ballinger and Liu (1997) Lemma 1.1. ð0; x2 ðtÞÞ is exponentially stable if p4atc=a: Where x2 ðtÞ ¼ pecðtntÞ =ð1 ect Þ: Lemma 1.2. System (1.1) is permanent if poatc=a:
2. Model formulation Firstly, Let us discuss the following system with Holling II functional response: x0 ðtÞ ¼ axðtÞ bx2 ðtÞ
m X ai xðtÞyi ðtÞ , 1 þ oi xðtÞ k¼1
ki ai xðtÞyi ðtÞ y0i ðtÞ ¼ ci yi ðtÞ; 1 þ oi xðtÞ
i ¼ 1; 2; . . . ; m,
ð2:1Þ
where xðtÞ; yi ðtÞ are the densities of the prey (pest) and the predator i at time t, respectively, a being the intrinsic growth rate of prey, b being the rate of intraspecific competition or density dependence, and ci being the death rate of predator i, and ki being the rate of conversing prey into predator i. Remark 2.1. The equilibrium ð0; 0; . . . ; 0Þ of system (2.1) is unstable. Remark 2.2. The equilibrium ð0; y1þ ; . . . ; ymþ Þ of system (2.1) does not exist. Where yiþ 40: Theorem 2.1. The necessary conditions of all predators that survive are ci 0o o1; i ¼ 1; 2; . . . ; m. (2.2) ki ai wi ci In this paper, with the idea of impulsive biological control, we will study the following system: 9 Pm ai xðtÞyi ðtÞ > 2 0 > x ðtÞ ¼ axðtÞ bx ðtÞ k¼1 > 1 þ oi xðtÞ = tanT, ki ai xðtÞyi ðtÞ > > ci yi ðtÞ y0i ðtÞ ¼ > ; 1 þ oi xðtÞ xðnT þ Þ ¼ xðnTÞ; yi ðnT þ Þ ¼ yi ðnTÞ þ pi , i ¼ 1; . . . ; m,
497
ðxð0þ Þ; y1 ð0þ Þ; . . . ; ym ð0þ ÞÞT ¼ ðx0 ; y01 ; . . . ; y0m ÞT 9X 0 ,
ð2:3Þ
where pi is the release amount of predator i at time t ¼ nT; n 2 N and N ¼ f1; 2; . . .g; and T is the period of the impulsive effect. For convenience, we denote the solution of Eq. (2.3) by X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ:
3. Preliminaries Firstly, we give some notations, definitions and lemmas. Let Rþ ¼ ½0; 1 ; Rðmþ1Þ ¼ fX 2 Rðmþ1Þ jX X0g: Deþ note by f ¼ ðf 1 ; f 2 ; . . . ; f mþ1 Þ the map defined by the right hand of system (2.3), and N the set of all nonnegative integers. Let V : Rþ Rðmþ1Þ !Rþ ; then V is þ said to belong to class V 0 if ðmþ1Þ and for (i) V is continuous in ðnT; ðn þ 1ÞT Rþ ðmþ1Þ each X 2 Rþ ; n 2 N; limðt;yÞ!ðnT þ ;X Þ V ðt; yÞ ¼ V ðnT þ ; X Þ exists. (ii) V is locally Lipschitzian in X :
Definition 3.1. Let V 2 V 0 ; for ðt; X Þ 2 ðnT; ðn þ 1ÞT ðmþ1Þ Rþ ; the upper right derivative of V ðt; X Þ with respect to the impulsive differential system (2.3) is defined as 1 Dþ V ðt; X Þ ¼ limþ sup ½V ðtþh; X þhf ðt; X ÞÞV ðt; X Þ . h h!0 The solution of the system (2.3) X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞ : Rþ !Rþ Rðmþ1Þ ; is continuþ ously differential on ðnT; ðn þ 1ÞT ; n 2 N and X ðnT þ Þ ¼ limt!nT þ X ðtÞ exists. The global existence and uniqueness of the solution of system (2.3) is guaranteed by smoothness of f (Lakshmikantham et al., 1989). The following lemma is obvious. Lemma 3.1. Suppose X ðtÞ is a solution of Eq. (2.3) with X ð0þ ÞX0; then X ðtÞX0 for all tX0: Definition 3.2. System (2.3) is said to be permanent if there exist two positive constants m; M and T 0 40; such that each positive solution ðxðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) satisfies mpxðtÞpM; mpyi ðtÞpM for all t4T 0 ; i ¼ 1; 2; . . . ; m: We will use the following important comparison theorem on impulsive differential equation (Lakshmikantham et al., 1989). Lemma 3.2. Let V 2 V 0 : Assume that ( þ D V ðt; X Þpgðt; V ðt; X ÞÞ; tanT; V ðt; X ðtþ ÞÞpjn ðV ðt; X ÞÞ; t ¼ nT;
(3.1)
where g : Rþ Rþ ! R is continuous in ðnT; ðn þ 1ÞT Rþ and for u 2 Rþ ; n 2 N; limðt;yÞ!ðnT þ ;uÞ :
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V ðt; yÞ ¼ V ðnT þ ; uÞ exists, jn : Rþ ! Rþ is nondecreasing. Let rðtÞ be the maximal solution of the scalar impulsive differential equation
Theorem 4.1. Let ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ be any solution of Eq. (2.3), then ð0; y1 ðtÞ; Pm. . .ai;pyi m ðtÞÞ is asymptotically stable provided that To i¼1 aci :
u0 ðtÞ ¼ gðt; uðtÞÞ;
We show that all solutions of Eq. (2.3) are uniformly ultimately bounded.
þ
uðt Þ ¼ jn ðuðtÞÞ; uð0þ Þ ¼ u0 ,
tanT, t ¼ nT, ð3:2Þ þ
exists on ½0; 1 : Then V ð0 ; X 0 Þpu0 implies that V ðt; X ðtÞÞprðtÞ for tX0; where X ðtÞ is any solution of Eq. (2.3). Similar results can be obtained when all the directions of the inequalities in Eq. (3.1) are reversed. Note that if one has some smoothness conditions of g to guarantee the existence and uniqueness of solutions for Eq. (3.2), so rðtÞ is exactly the unique solution of Eq. (3.2). For convenience, we give some basic properties of the following system: y0i ðtÞ ¼ ci yi ðtÞ;
tanT,
þ
yi ðt Þ ¼ yi ðtÞ þ pi ; yi ð0þ Þ ¼ y0i .
t ¼ nT,
Theorem 4.2. There exist a constant M40; such that xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m for each solution X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) with all t large enough. In the following, let us investigate the permanence of system (2.3). P ai pi Theorem 4.3. System (2.3) is permanent if T4 m i¼1 aci : For the sake of the biological points, the proofs of Theorems 2.1, 4.1, 4.2 and 4.3 are given in Appendixes. The results of Theorems 4.1 and 4.3 are further confirmed by numerical simulations in Figs. 1–8. Let a ¼ 8; b ¼ 5; e1 ¼ 1; e2 ¼ 1; d 1 ¼ 0:3; d 2 ¼ 0:1; p1 p2 m1 ¼ 0:95; m2 ¼ 0:8; p1 ¼ 2; p2 ¼ 1; then ad þ ad ¼ 1 2
ð3:3Þ
Clearly, system (3.3) has a positive periodic solution yi ðtÞ ¼
pi eci ðtnTÞ , 1 eci T
t 2 ðnT; ðn þ 1ÞT ; n 2 N; yi ð0þ Þ ¼
pi . 1 eci T
Since the solution of the system (3.3) is pi yi ðtÞ ¼ yi ð0þ Þ eci t þ yi ðtÞ, 1 eci T t 2 ðnT; ðn þ 1ÞT ; yi ð0þ ÞX0. We have Lemma 3.3. For every positive periodic solution yi ðtÞ of system (3.3) and every solution yi ðtÞ ði ¼ 1; 2; . . . ; mÞ of system (2.3) with y0þ X0; it follows that yi ðtÞ ! yi ðtÞ as t ! 1: Therefore, the complete expression for the pesteradication periodic solution of the system (2.3) is obtained
p1 ec1 ðtnTÞ pm ecm ðtnTÞ ð0; y1 ðtÞ; . . . ; ym ðtÞÞ ¼ 0; ;...; 1 ec1 T 1 ecm T
Fig. 1. Phase portraits of system (2.3).
for t 2 ðnT; ðn þ 1ÞT : In the following, we investigate the extinction and permanence of system (2.3).
4. Extinction and permanence Firstly, we study the stability of prey-eradication periodic solution.
Fig. 2. Time series of population x1 :
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Fig. 3. Time series of population x2 :
Fig. 6. Time series of population x1 :
Fig. 7. Time series of population x2 : Fig. 4. Time series of population x3 :
Fig. 5. Phase portraits of system (2.3).
Fig. 8. Time series of population x3 :
499
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2:0833: Choose T ¼ 2; by Theorem 4.1, ð0; y1 ðtÞ; . . . ; ym ðtÞÞ is asymptotically stable (see Figs. 1–4). Choose T ¼ 3; by Theorem 4.3, system (2.3) is permanent (see Figs. 5–8). 5. Conclusions
Clearly, the integrand of above equality is negative definite, so yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ: ci Assume that ki ai w 41; ði ¼ 1; . . . ; mÞ holds, then i ci ( ) ci m Z t ðk a w c ÞðxðxÞ X i i i i ki ai wi ci Þ yi ðtÞ ¼ yi ð0Þ exp dx , 1 þ wi xðxÞ i¼1 0 i ¼ 1; . . . ; m.
In this paper, the dynamic behaviors of Holling II one-prey multi-predator with impulsive effect concerning periodic biological control for pest at different fixed moment is investigated. We have shown that there exist an asymptotically stable pest-eradication periodic solution if the impulsive periodic is less than a threshold. When the stability of pest-eradication periodic solution disappears, the system (2.3) is permanent, which in line with reality from a biological point of view. In the following, let us compare the results of system (2.3) with systems (1.1) and (2.1). System (2.1) is a model with classical biological control technique. From Remarks 2.1 and 2.2, it can be shown that the pest eradication equilibrium ð0; y1þ ; . . . ; ymþ Þ does not exist, and the equilibrium ð0; 0; . . . ; 0Þ is unstable, which means that one cannot eradicate pest steadily. However, from the impulsive control system (2.3), the pest can be eradicated provided Pm ai pithat impulsive period T is less than a threshold i¼1 aci ; where pi is the releasing amount of natural enemy which can be controlled in advance. The result reflects the ability of human’ reconstructed nature. On the other hand, in system (1.1), the threshold of pest-eradication is ap ac : Clearly, the two thresholds indicate the advantage of multi-predator control compared with one-predator control. In addition, the condition (2.2) of system (2.1) must be satisfied if predators survive. Such condition is relevant to wi and ki which represents the rate of conversing prey into predator, while the permanent condition of system (2.3) is not relevant to the two parameters. Appendix A. Proof of Theorem 2.1 Condition (2.2) implies that ki ai wi ci 40: Assuming that ki ai wi ci o0; we have ( ) m Z t X ðki ai wi ci ÞxðxÞ ci dx , yi ðtÞ ¼ yi ð0Þ exp 1 þ wi xðxÞ i¼1 0 i ¼ 1; . . . ; m.
Since on the half space xXab; x0 ðtÞo0; xðtÞ is strictly monotone decreasing. So for t large enough, the integrand of above equality is negative. Hence yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ: Thus xðtÞ ! ab as t ! 1: In fact, by the Lemma (Markus, 1956), we can prove the conclusion. Let 8 < x0 ðtÞ ¼ xðtÞða bxðtÞÞ Pm ai xðtÞyi ðtÞ ; k¼1 1 þ oi xðtÞ A: : xð0Þ ¼ x0 40; ( A1 :
Since yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ; by the Lemma (Markus, 1956) xðtÞ ! a=b as t ! 1: This completes the proof. &
Appendix B. Proof of Theorem 4.1 The local stability of periodic solution ð0; y1 ðtÞ; . . . ; ym ðtÞÞ may be determined by considering the behavior of small amplitude perturbations of the solution. Define xðtÞ ¼ uðtÞ; yi ðtÞ ¼ yi ðtÞ þ vi ðtÞ;
ði ¼ 1; 2; . . . ; mÞ.
where vi ðtÞ; ði ¼ 1; 2; . . . ; mÞ are small perturbations. Eq. (2.3) can be expanded in a Taylor series after neglecting higher order terms, the linearized equations read: 8 ) P u0 ðtÞ ¼ ða m ai yi ðtÞÞuðtÞ > i¼1 > < 0otoT; v0i ðtÞ ¼ ai yi ðtÞuðtÞ ci vi ðtÞ (A.1) > > : uðnT þ Þ ¼ uðTÞ; v ðnT þ Þ ¼ vðTÞ: i
Let FðtÞ be fundamental matrix of Eq. (A.1), then FðtÞ must satisfy
P a m 0 0 i¼1 ai yi ðtÞ B k1 a1 y ðtÞ c1 0 B 1 dFðtÞ B B a y ðtÞ c 0 k 2 2 2 2 ¼B B dt . . . B .. .. .. @ 0 0 0 0
x0 ðtÞ ¼ xðtÞða bxðtÞÞ; xð0Þ ¼ x0 40:
1 0 0 C C C 0 0 CFðtÞ9AFðtÞ C .. C .. . C . A km am ym ðtÞ cm 0 0
(A.2)
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and Fð0Þ ¼ I the identity matrix. The linearization of Eq. (2.3) of the forth becomes 1 0 0 10 uðnTÞ 1 uðnT þ Þ 1 0 0 0 C C B B v1 ðnTÞ C B v1 ðnT þ Þ C B 0 1 0 0C CB C B C B B C B C ¼ B .. .. .. C. B . B .. .. C @ . . . .. C C B B . . A A A @ @ þ 0 0 0 1 vm ðnTÞ vm ðnT Þ Thus, the 0 1 B0 B M¼B B .. @.
monodromy matrix of Eq. (A.1) is 1 0 0 0 1 0 0C C .. C .. .. CFðTÞ. . . . A 0 0 0 1
From ð
RT
Eq.
(A.2),
A dtÞ
it
follows
501
According to Lemma 3.2, we have Pm 1
l1 ðl0 tÞ k¼1 k pi þ þ ðl TÞ i e V ðtÞp V ð0 Þ l0 e 0 1 l1 ð1 enl0 T Þeðl0 TÞ el0 ðtnTÞ þ . l0 where t 2 ðnT; ðn þ 1ÞT : Hence, ! m X l1 1 eðl0 TÞ . lim V ðtÞp þ pi ðl TÞ t!1 k l0 e 0 1 k¼1 i Therefore V ðt; xÞ is ultimately bounded. Hence there exists a constant M40; such that xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m for each solution X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) with all t large enough. The proof is completed. &
that
FðTÞ ¼
¯ A
Fð0Þe 0 9Fð0Þe : Let m; m1 ; . . . ; mm be eigenvalues ¯ then of matrix A; ! Z T m m X X ai pi m¼ a ai yi ðtÞ dt ¼ aT , ci 0 i¼1 i¼1 mi ¼ ci To0; i ¼ 1; . . . ; m.
Appendix D. Proof of Theorem 4.3 Suppose X ðtÞ is a solution of Eq. (2.3) with X 0 40: From Theorem 4.2, we may assume xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m and M4ab; tX0: Let ci T
Therefore, all eigenvalues of M, namely, e ; e ; . . . ; e have absolute values less than one if and only if P ai pi To m i¼1 aci : According to Floquent theory of impulsive differential equation, the pest eradication solution ð0; y1 ðtÞ; . . . ; ym ðtÞÞ is locally stable. This completes the proof. &
pi e mi ¼ 1e ci T i 40; i 40; ði ¼ 1; . . . ; mÞ: According to Lemmas 3.2 and 3.3, it follows that yi ðtÞ4mi ði ¼ 1; . . . ; mÞ; for all t large enough. In the following, we want to find mmþ1 40 such that xðtÞXmmþ1 for all t large enough. We will do it in the following two steps for convenience. P ai pi 0 Step 1: Since T4 m i¼1 aci ; we can select mmþ1 40; 0 0 i 40 ði ¼ 1; . . . ; mÞ small enough such that mmþ1 oab and
Appendix C. Proof of Theorem 4.2
di ¼
m
m1
mm
ki ai m0mþ1 oci ; 1 þ wi m0mþ1
Define V ðt; X Þ as s ¼ aT bm0mþ1 T
m X 1 V ðt; X Þ ¼ xðtÞ þ y ðtÞ. k i k¼1 i
It is obvious that V 2 V 0 ; we calculate the upper right derivative of V ðt; X Þ along a solution of system (2.3) and get the following impulsive differential equation 8 þ D V ðt; X Þ þ lV ðt; X Þ > > > > > < ¼ ða þ lÞx bx2 þ Pm l ci y ; tanT; i k¼1 ki > > Pm 1 > > þ > : V ðt; X ðt ÞÞ ¼ V ðt; X Þ þ k¼1 k pi ; t ¼ nT: i
0olo minfc1 ; . . . ; cm g; then lci k¼1 ki yi is bounded. Select two l0 and l1 such that
Let Pm
ða þ lÞx bx2 þ positive constants
8 þ < D V ðt; X Þp l0 V ðt; X Þ þ l1 ; tanT; Pm 1 p ; t ¼ nT: : V ðt; X ðtþ ÞÞ ¼ V ðt; X Þ þ k¼1 ki i
ði ¼ 1; . . . ; mÞ, m m X X ai pi 0i ai T40. c d i i i¼1 i¼1
We will prove that there exists t1 2 ð0; 1Þ such that xðt1 ÞXm0mþ1 : Otherwise, according to above assumption, it is easy to see y0i ðtÞpyi ðtÞðci þ di Þ;
tanT; i ¼ 1; . . . ; m.
By Lemmas 3.2 and 3.3, we get yi ðtÞpY i ðtÞ and Y i ðtÞ ! Y¯ i ðtÞ; where ðci þdi ÞðtnTÞ
pe ; t 2 ðnT; ðn þ 1ÞT Y¯ i ðtÞ ¼ i 1 eðci þdi ÞT and Y i ðtÞ is the solution of the following equation 8 0 > < Y i ðtÞpðci þ di ÞY i ðtÞ; tanT; Y i ðtþ Þ ¼ Y i ðtÞ þ pi ; t ¼ nT; (A.3) > : Y ð0þ Þ ¼ Y : i
0i
Therefore there exists a T i 40 such that yi ðtÞpY i ðtÞpY¯ i ðtÞ þ 0i ,
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0
x ðtÞXxðtÞ a
bm0mþ1
m X
! ai ðY¯ i ðtÞ þ
0i Þ
.
(A.4)
i¼1
Let N i 2 N and N i TXT i : Integrating Eq. (A.4) on ðnT; ðn þ 1ÞT ; nX maxfN 1 ; . . . ; N m g; we have R ðnþ1ÞT Pm ðabm0mþ1 a ðY¯ i ðtÞþ0i ÞÞ i¼1 i xððn þ 1ÞTÞXxðnTÞe nT ¼ xðnTÞes .
Then xððn þ kÞTÞXxðnTÞeks ! 1;
as k ! 1,
which is a contradiction to the boundedness of xðtÞ: Step 2: If xðtÞXm0mþ1 for t4t1 ; the result is apparent. Otherwise, if xðtÞom0mþ1 for some t4t1 ; setting t ¼ inf tXt1 fxðtÞom0mþ1 g; we have xðtÞXm0mþ1 for ½t; t Þ: It is easy to see xðt Þ ¼ m0mþ1 ; since xðtÞ is continuous. Suppose t 2 ½n1 T; ðn1 þ 1ÞTÞ; n1 2 N: Select n2i ; n3 2 N such that
0i n2i 4T 2i ¼ ln ðci þ di Þ, M þ pi
which is a contradiction. Let ¯t ¼ inf tXt fxðtÞXm0mþ1 g; then xð¯tÞ ¼ m0mþ1 and Eq. (A.5) holds for t 2 ½t ; ¯tÞ: Integrating Eq. (A.5) on ½t ; ¯t yields
xðtÞXxðt Þeðs1 ðtt ÞÞ Xm0mþ1 eðs1 ðn2 þ1þn3 ÞTÞ . Letting m0mþ1 eðs1 ðn2 þ1þn3 ÞTÞ ¼ mmþ1 ; for t4t¯; the same argument can be continued since xð¯tÞXm0mþ1 for tXt1 : Hence xðtÞXmmþ1 for all tXt1 : Case (ii). There exists a t0 2 ðt ; ðn1 þ 1ÞT such that xðt0 ÞXm0mþ1 : Let t ¼ inf t4t fxðtÞXm0mþ1 g; then xðtÞom0mþ1 for t 2 ½t ; tÞ and xðtÞ ¼ m0mþ1 : For t 2 ½t ; tÞ Eq. (A.5) holds.Integrating Eq. (A.5) on t 2 ½t ; tÞ; we have
xðtÞXxðt Þeðs1 ðtt ÞÞ Xm0mþ1 eðs1 TÞ 4mmþ1 . This process can be continued since xðtÞXm0mþ1 ; and we have xðtÞXmmþ1 for t4t1 : Thus on both cases we get xðtÞXmmþ1 for t4t1 : This completes the proof. &
References
eððn2 þ1Þs1 TÞ eðn3 sÞ 41,
P where s1 ¼ a bm0mþ1 M m i¼1 ai o0; n2 ¼ maxfn21 ; . . . ; n2m g; T¯ ¼ n2 T þ n3 T: We claim that there must exist ¯ such that xðt0 ÞXm0mþ1 : a t0 2 ððn1 þ 1ÞT; ðn1 þ 1ÞT þ T 0 ¯ Otherwise, xðtÞommþ1 ; t 2 ððn1 þ 1ÞT; ðn1 þ 1ÞT þ T : Considering Eq. (A.3) with Y i ððn1 þ1Þ T þ Þ ¼ yi ððn1 þ 1ÞT þ Þ; we have pi Y i ðtÞ ¼ Y i ððn1 þ 1ÞT þ Þ 1 eðci þdi ÞT ððci þdi Þðtðn1 þ1ÞTÞÞ e þ Y¯ i ðtÞ for t 2 ðnT; ðnþ1ÞT ; n1 þ 1pnon1 þ 1 þ n2 þ n3 : Then jY i ðtÞ Y¯ i ðtÞjoðM þ pi Þeððci di Þn2i TÞ o0i , yi ðtÞpY i ðtÞpY¯ i ðtÞ þ 0i ¯ which implies Eq. for ðn1 þ 1 þ n2 Þptpðn1 þ 1ÞT þ T; ¯ Similar as (A.4) holds on ½ðn1 þ 1 þ n2 Þ; ðn1 þ 1ÞT þ T : in Step 1, we have xððn1 þ 1 þ n2 þ n3 ÞTÞXxððn1 þ 1 þ n2 ÞTÞeðn3 sÞ , There are two possible cases for t 2 ðt ; ðn1 þ 1ÞT : Case (i): If xðtÞom0mþ1 for t 2 ðt ; ðn1 þ 1ÞT ; then xðtÞom0mþ1 for all t 2 ðt ; ðn1 þ 1 þ n2 ÞT : System (2.3) gives ! m X 0 0 x ðtÞXxðtÞ a bmmþ1 ai M ¼ s1 xðtÞ. (A.5) i¼1
Integrating Eq. (A.5) on ½t ; ðn1 þ 1 þ n2 ÞT yields xððn1 þ 1 þ n2 ÞTÞXm0mþ1 eðs1 ðn2 þ1ÞTÞ . Then xððn1 þ 1 þ n2 þ n3 ÞTÞXm0mþ1 eðs1 ðn2 þ1ÞTÞ eðn3 sÞ 4m0mþ1 .
Andrews, J.F., 1968. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723. Bainov, D., Simeonov, P., 1993. Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, New York. Ballinger, G., Liu, X., 1997. Permanence of population growth models with impulsive effects. Math. Comput. Modell. 26, 59–72. Caltagirone, L.E., Doutt, R.L., 1989. The history of the vedalia beetle importation to California and its impact on the development of biological control. Ann. Rev. Entomol. 34, 1–16. DeBach, P., 1964. Biological Control of Insect Pests and Weeds. Rheinhold, New York. DeBach, P., Rosen, D., 1991. Biological Control by Natural Enemies, second ed. Cambridge University Press, Cambridge. Donofrio, A., 1997. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Comput. Modell. 26, 59–72. Freedman, H.J., 1976. Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci. 31, 207–225. Grasman, J., Van Herwaarden, O.A., et al., 2001. A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169, 207–216. Hassell, M.P., 1976. The Dynamics of Competition and Predation. Edward Arnold, London, p. 68. Holling, C.S., 1965. The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Ent. Sec. Can. 45, 1–60. Jin, Z., Zhien, M., Han, 2004. The existence of periodic solutions of the n-species lotka-Volterra competition system with impulsive. Chaos Solitions Fract. 22, 181–188. Lakmeche, A., Arino, O., 2000. Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn. Contin. Discrete Impulsive Syst. 7, 165–187. Lakshmikantham, V., Bainov, D., Simeonov, P., 1989. Theory of Impulsive Differential Equations. World Scientific, Singapore. Liu, X., Chen, L., 2003. Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator. Chaos Solitons Fract. 6, 311–320.
ARTICLE IN PRESS Y. Pei et al. / Journal of Theoretical Biology 235 (2005) 495–503 Markus, L., 1956. Asymptotically autonomous differential system. Contributions to the Theory of Nonlinear Oscillation, vol. 3. Princeton University Press, Princeton, pp. 7–29. Panetta, J.C., 1996. A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425–447. Roberts, M.G., Kao, R.R., 1998. The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149, 23–36.
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Samoilenko, A.M., Perestyuk, N.A., 1995. Impulsive Differential Equations. World Scientific, Singapore. Shulgin, B., Stone, L., Agur, Z., 1998. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1–26. Sugie, J., Howell, J.A., 1980. Kinetics of phenol oxidation by washed cell. Biotechnol. Bioeng. 23, 2039–2049. Tang, S.Y., Chen, L.S., 2002. Density-dependent birth rate, birth pulse and their population dynamic consequences. J. Math. Biol. 44, 185–199.
Journal of Theoretical Biology 235 (2005) 495–503 www.elsevier.com/locate/yjtbi
Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control$ Yongzhen Peia,b, Lansun Chena,, Qingrui Zhangc, Changguo Lib a
Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116023, PR China b Department of Basic Science of Institute of Military Traffic, Tianjin 300161, PR China c Department of Chemical Engineering, Qingdao University of Science and Technology, Qingdao 266042, PR China Received 25 October 2004; received in revised form 6 February 2005; accepted 7 February 2005 Available online 27 April 2005 Communicated by Karl Sigmund
Abstract In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one. r 2005 Elsevier Ltd. All rights reserved. Keywords: Impulsive biological control; Holling II functional response; Permanence; Extinction
1. Introduction Based on experiments, Holling (1965) suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka–Volterra system more realistic. Letting X and Y be the densities of a prey and its predator, respectively, a model of prey–predator system with Holling functional response is given by the equations X 0 ¼ XgðX Þ Y FðX Þ, Y 0 ¼ dY þ eY FðX Þ,
$ This work is supported by National Natural Science Foundation of China (10171106). Corresponding author. E-mail addresses: [email protected] (Y. Pei), [email protected] (L. Chen).
0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2005.02.003
where gðX Þ is the per capita rate of change of prey density in absence of the predator, d, e are positive parameters and FðX Þ is a Holling functional response of a predator which takes following concrete forms (1) FðX Þ is a Holling I functional response function, i.e. ( cX if X pX 0 ; FðX Þ ¼ cX 0 if X 4X 0 ; where c is a positive constant characterizing the threshold of prey concentration above which the predation rate is constant and under which the predation rate is similar to the Lotka–Volterra one. (2) If we take into account the time a predator uses in handling the prey it has captured, Holling find the predator has a Holling II functional response. That is FðX Þ ¼
aX , 1 þ wX
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where a; w are positive constants. Clearly, FðX Þ is a strictly increasing function. (3) If FðX Þ is Holling III functional response function, that is FðX Þ ¼
aX 2 . b2 þ X 2
By the definition, It is an S-type function. But some experiments and observations indicate that the non-monotonic response occurs at a level: when the nutrient concentrations reaches a high level an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, Andrews (1968) suggested a function: FðX Þ ¼
mX , a þ bX þ X 2
called the Monod–Haldane function, or Holling type-IV function. Sugie and Howell (1980) proposed a simplified Monod-Haldane function of the form FðX Þ ¼
mX , a þ X2
which describes the phenomenon of group defense whereby predation is decreased, or even prevented altogether, due to the increased ability of the prey to better defend or disguise themselves when their numbers are large enough. Based on the predator–prey system with Holling functional response, many authors have suggested impulsive differential equation (Bainov and Simeonov, 1993; Samoilenko and Perestyuk, 1995; Lakshmikantham et al., 1989) to model the process of periodic releasing natural enemies and spraying pesticides at different fixed time which is called integrated pest management. Biological control (DeBach, 1964; DeBach and Rosen, 1991; Freedman, 1976; Grasman et al., 2001) which a component of an integrated pest management strategy is the purposeful introduction and establishment of one or more natural enemies from region of origin of an exotic pest, specifically for the purpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economic damage. Virtually all insect and mite pests have some natural enemies. One approach to biological control is augmentation, which is manipulation of existing natural enemies to increase their effectiveness. This can be achieved by mass production and periodic release of natural enemies of the pest, and by genetic enhancement of the enemies to increase their effectiveness at control. The pioneering project of biological control began in 1888 when the new legendary predator, the vedalia beetle was imported from Australia and established in California, where it rapidly suppressed populations of cottony cushion scale that had been
decimating the developing citrus industry (Caltagirone and Doutt, 1989). Another important method for pest control is chemical control. Pesticides are useful because they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss. However, pesticide pollution is not only recognized as a major health hazard to human beings and to natural enemies but also is resited by pests. And the latter results in the use of high rates and more toxic materials to combat pests. Beneficial insects are often susceptible to chemical insecticides applied for the target pest. One of the side effects of high rates of pesticide use is that natural enemies and other small animals that might otherwise feed on pests are killed and pests population burst out once again after beneficial insects being killed. So in this paper, we only apply impulsive biological control to beat the pest in view of green agriculture and environment protection. As mentioned above, impulsive equations are found in almost every domain of applied science and have been studied in many investigations, among which are: impulsive birth (Roberts and Kao, 1998; Tang and Chen, 2002), impulsive vaccination (Shulgin et al., 1998; Donofrio, 1997), chemotherapeutic treatment of disease (Jin et al., 2004; Panetta, 1996) and population ecology (Lakmeche and Arino, 2000; Ballinger and Liu, 1997) They generally describe phenomena that are subject to steep or instantaneous changes which make the differential system more intractable. According to Hassell (1976), Holling type II functional response is the most common type of functional response among arthropod predators. Liu and Chen (2003) has investigated the predator–prey system with periodic constant impulsive immigration of predator: x01 ðtÞ ¼ axðtÞ bx21 ðtÞ
axðtÞ1 x2 ðtÞ ; 1 þ ox1 ðtÞ
tant,
kaxðtÞ1 x2 ðtÞ cx2 ðtÞ; tant, 1 þ ox1 ðtÞ x1 ðntþ Þ ¼ x1 ðntÞ; x2 ðntþ Þ ¼ x2 ðntÞ þ p.
x02 ðtÞ ¼
ð1:1Þ
The system (1.1) describes a one-prey one-predator model with Holing II type. Wherever possible, multinatural enemies control techniques should work together rather than against one. In this paper, the biological control technique of multi-predators is adopted to kill pest. So if one wishes to eradicate the pest population or keep the pest population below the damage level, how do we release the natural enemies? One of the main purposes of this paper is to construct a mathematical model according to the fact of periodic biological control for pest control and to investigate the dynamics of such system. In Section 2, we analyse the dynamics of one-prey multi-predator model with Holling II functional response. And based on this
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model, we suggest an impulsive differential equation to model the process of periodic releasing natural enemies. In Section 3, we give some notations and lemmas. By using the Floquent theory of impulsive, small amplitude perturbation skills and techniques of comparison, we consider the local stability of pest-eradication periodic solution and the permanence of the system in Section 4. A brief discussion of the main results is given in the last section. In the following, the main results of system (1.1) are summarized in the lemmas which has been studied completely in Ballinger and Liu (1997) Lemma 1.1. ð0; x2 ðtÞÞ is exponentially stable if p4atc=a: Where x2 ðtÞ ¼ pecðtntÞ =ð1 ect Þ: Lemma 1.2. System (1.1) is permanent if poatc=a:
2. Model formulation Firstly, Let us discuss the following system with Holling II functional response: x0 ðtÞ ¼ axðtÞ bx2 ðtÞ
m X ai xðtÞyi ðtÞ , 1 þ oi xðtÞ k¼1
ki ai xðtÞyi ðtÞ y0i ðtÞ ¼ ci yi ðtÞ; 1 þ oi xðtÞ
i ¼ 1; 2; . . . ; m,
ð2:1Þ
where xðtÞ; yi ðtÞ are the densities of the prey (pest) and the predator i at time t, respectively, a being the intrinsic growth rate of prey, b being the rate of intraspecific competition or density dependence, and ci being the death rate of predator i, and ki being the rate of conversing prey into predator i. Remark 2.1. The equilibrium ð0; 0; . . . ; 0Þ of system (2.1) is unstable. Remark 2.2. The equilibrium ð0; y1þ ; . . . ; ymþ Þ of system (2.1) does not exist. Where yiþ 40: Theorem 2.1. The necessary conditions of all predators that survive are ci 0o o1; i ¼ 1; 2; . . . ; m. (2.2) ki ai wi ci In this paper, with the idea of impulsive biological control, we will study the following system: 9 Pm ai xðtÞyi ðtÞ > 2 0 > x ðtÞ ¼ axðtÞ bx ðtÞ k¼1 > 1 þ oi xðtÞ = tanT, ki ai xðtÞyi ðtÞ > > ci yi ðtÞ y0i ðtÞ ¼ > ; 1 þ oi xðtÞ xðnT þ Þ ¼ xðnTÞ; yi ðnT þ Þ ¼ yi ðnTÞ þ pi , i ¼ 1; . . . ; m,
497
ðxð0þ Þ; y1 ð0þ Þ; . . . ; ym ð0þ ÞÞT ¼ ðx0 ; y01 ; . . . ; y0m ÞT 9X 0 ,
ð2:3Þ
where pi is the release amount of predator i at time t ¼ nT; n 2 N and N ¼ f1; 2; . . .g; and T is the period of the impulsive effect. For convenience, we denote the solution of Eq. (2.3) by X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ:
3. Preliminaries Firstly, we give some notations, definitions and lemmas. Let Rþ ¼ ½0; 1 ; Rðmþ1Þ ¼ fX 2 Rðmþ1Þ jX X0g: Deþ note by f ¼ ðf 1 ; f 2 ; . . . ; f mþ1 Þ the map defined by the right hand of system (2.3), and N the set of all nonnegative integers. Let V : Rþ Rðmþ1Þ !Rþ ; then V is þ said to belong to class V 0 if ðmþ1Þ and for (i) V is continuous in ðnT; ðn þ 1ÞT Rþ ðmþ1Þ each X 2 Rþ ; n 2 N; limðt;yÞ!ðnT þ ;X Þ V ðt; yÞ ¼ V ðnT þ ; X Þ exists. (ii) V is locally Lipschitzian in X :
Definition 3.1. Let V 2 V 0 ; for ðt; X Þ 2 ðnT; ðn þ 1ÞT ðmþ1Þ Rþ ; the upper right derivative of V ðt; X Þ with respect to the impulsive differential system (2.3) is defined as 1 Dþ V ðt; X Þ ¼ limþ sup ½V ðtþh; X þhf ðt; X ÞÞV ðt; X Þ . h h!0 The solution of the system (2.3) X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞ : Rþ !Rþ Rðmþ1Þ ; is continuþ ously differential on ðnT; ðn þ 1ÞT ; n 2 N and X ðnT þ Þ ¼ limt!nT þ X ðtÞ exists. The global existence and uniqueness of the solution of system (2.3) is guaranteed by smoothness of f (Lakshmikantham et al., 1989). The following lemma is obvious. Lemma 3.1. Suppose X ðtÞ is a solution of Eq. (2.3) with X ð0þ ÞX0; then X ðtÞX0 for all tX0: Definition 3.2. System (2.3) is said to be permanent if there exist two positive constants m; M and T 0 40; such that each positive solution ðxðtÞ; y1 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) satisfies mpxðtÞpM; mpyi ðtÞpM for all t4T 0 ; i ¼ 1; 2; . . . ; m: We will use the following important comparison theorem on impulsive differential equation (Lakshmikantham et al., 1989). Lemma 3.2. Let V 2 V 0 : Assume that ( þ D V ðt; X Þpgðt; V ðt; X ÞÞ; tanT; V ðt; X ðtþ ÞÞpjn ðV ðt; X ÞÞ; t ¼ nT;
(3.1)
where g : Rþ Rþ ! R is continuous in ðnT; ðn þ 1ÞT Rþ and for u 2 Rþ ; n 2 N; limðt;yÞ!ðnT þ ;uÞ :
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V ðt; yÞ ¼ V ðnT þ ; uÞ exists, jn : Rþ ! Rþ is nondecreasing. Let rðtÞ be the maximal solution of the scalar impulsive differential equation
Theorem 4.1. Let ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ be any solution of Eq. (2.3), then ð0; y1 ðtÞ; Pm. . .ai;pyi m ðtÞÞ is asymptotically stable provided that To i¼1 aci :
u0 ðtÞ ¼ gðt; uðtÞÞ;
We show that all solutions of Eq. (2.3) are uniformly ultimately bounded.
þ
uðt Þ ¼ jn ðuðtÞÞ; uð0þ Þ ¼ u0 ,
tanT, t ¼ nT, ð3:2Þ þ
exists on ½0; 1 : Then V ð0 ; X 0 Þpu0 implies that V ðt; X ðtÞÞprðtÞ for tX0; where X ðtÞ is any solution of Eq. (2.3). Similar results can be obtained when all the directions of the inequalities in Eq. (3.1) are reversed. Note that if one has some smoothness conditions of g to guarantee the existence and uniqueness of solutions for Eq. (3.2), so rðtÞ is exactly the unique solution of Eq. (3.2). For convenience, we give some basic properties of the following system: y0i ðtÞ ¼ ci yi ðtÞ;
tanT,
þ
yi ðt Þ ¼ yi ðtÞ þ pi ; yi ð0þ Þ ¼ y0i .
t ¼ nT,
Theorem 4.2. There exist a constant M40; such that xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m for each solution X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) with all t large enough. In the following, let us investigate the permanence of system (2.3). P ai pi Theorem 4.3. System (2.3) is permanent if T4 m i¼1 aci : For the sake of the biological points, the proofs of Theorems 2.1, 4.1, 4.2 and 4.3 are given in Appendixes. The results of Theorems 4.1 and 4.3 are further confirmed by numerical simulations in Figs. 1–8. Let a ¼ 8; b ¼ 5; e1 ¼ 1; e2 ¼ 1; d 1 ¼ 0:3; d 2 ¼ 0:1; p1 p2 m1 ¼ 0:95; m2 ¼ 0:8; p1 ¼ 2; p2 ¼ 1; then ad þ ad ¼ 1 2
ð3:3Þ
Clearly, system (3.3) has a positive periodic solution yi ðtÞ ¼
pi eci ðtnTÞ , 1 eci T
t 2 ðnT; ðn þ 1ÞT ; n 2 N; yi ð0þ Þ ¼
pi . 1 eci T
Since the solution of the system (3.3) is pi yi ðtÞ ¼ yi ð0þ Þ eci t þ yi ðtÞ, 1 eci T t 2 ðnT; ðn þ 1ÞT ; yi ð0þ ÞX0. We have Lemma 3.3. For every positive periodic solution yi ðtÞ of system (3.3) and every solution yi ðtÞ ði ¼ 1; 2; . . . ; mÞ of system (2.3) with y0þ X0; it follows that yi ðtÞ ! yi ðtÞ as t ! 1: Therefore, the complete expression for the pesteradication periodic solution of the system (2.3) is obtained
p1 ec1 ðtnTÞ pm ecm ðtnTÞ ð0; y1 ðtÞ; . . . ; ym ðtÞÞ ¼ 0; ;...; 1 ec1 T 1 ecm T
Fig. 1. Phase portraits of system (2.3).
for t 2 ðnT; ðn þ 1ÞT : In the following, we investigate the extinction and permanence of system (2.3).
4. Extinction and permanence Firstly, we study the stability of prey-eradication periodic solution.
Fig. 2. Time series of population x1 :
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Fig. 3. Time series of population x2 :
Fig. 6. Time series of population x1 :
Fig. 7. Time series of population x2 : Fig. 4. Time series of population x3 :
Fig. 5. Phase portraits of system (2.3).
Fig. 8. Time series of population x3 :
499
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2:0833: Choose T ¼ 2; by Theorem 4.1, ð0; y1 ðtÞ; . . . ; ym ðtÞÞ is asymptotically stable (see Figs. 1–4). Choose T ¼ 3; by Theorem 4.3, system (2.3) is permanent (see Figs. 5–8). 5. Conclusions
Clearly, the integrand of above equality is negative definite, so yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ: ci Assume that ki ai w 41; ði ¼ 1; . . . ; mÞ holds, then i ci ( ) ci m Z t ðk a w c ÞðxðxÞ X i i i i ki ai wi ci Þ yi ðtÞ ¼ yi ð0Þ exp dx , 1 þ wi xðxÞ i¼1 0 i ¼ 1; . . . ; m.
In this paper, the dynamic behaviors of Holling II one-prey multi-predator with impulsive effect concerning periodic biological control for pest at different fixed moment is investigated. We have shown that there exist an asymptotically stable pest-eradication periodic solution if the impulsive periodic is less than a threshold. When the stability of pest-eradication periodic solution disappears, the system (2.3) is permanent, which in line with reality from a biological point of view. In the following, let us compare the results of system (2.3) with systems (1.1) and (2.1). System (2.1) is a model with classical biological control technique. From Remarks 2.1 and 2.2, it can be shown that the pest eradication equilibrium ð0; y1þ ; . . . ; ymþ Þ does not exist, and the equilibrium ð0; 0; . . . ; 0Þ is unstable, which means that one cannot eradicate pest steadily. However, from the impulsive control system (2.3), the pest can be eradicated provided Pm ai pithat impulsive period T is less than a threshold i¼1 aci ; where pi is the releasing amount of natural enemy which can be controlled in advance. The result reflects the ability of human’ reconstructed nature. On the other hand, in system (1.1), the threshold of pest-eradication is ap ac : Clearly, the two thresholds indicate the advantage of multi-predator control compared with one-predator control. In addition, the condition (2.2) of system (2.1) must be satisfied if predators survive. Such condition is relevant to wi and ki which represents the rate of conversing prey into predator, while the permanent condition of system (2.3) is not relevant to the two parameters. Appendix A. Proof of Theorem 2.1 Condition (2.2) implies that ki ai wi ci 40: Assuming that ki ai wi ci o0; we have ( ) m Z t X ðki ai wi ci ÞxðxÞ ci dx , yi ðtÞ ¼ yi ð0Þ exp 1 þ wi xðxÞ i¼1 0 i ¼ 1; . . . ; m.
Since on the half space xXab; x0 ðtÞo0; xðtÞ is strictly monotone decreasing. So for t large enough, the integrand of above equality is negative. Hence yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ: Thus xðtÞ ! ab as t ! 1: In fact, by the Lemma (Markus, 1956), we can prove the conclusion. Let 8 < x0 ðtÞ ¼ xðtÞða bxðtÞÞ Pm ai xðtÞyi ðtÞ ; k¼1 1 þ oi xðtÞ A: : xð0Þ ¼ x0 40; ( A1 :
Since yi ðtÞ ! 0 as t ! 1; ði ¼ 1; . . . ; mÞ; by the Lemma (Markus, 1956) xðtÞ ! a=b as t ! 1: This completes the proof. &
Appendix B. Proof of Theorem 4.1 The local stability of periodic solution ð0; y1 ðtÞ; . . . ; ym ðtÞÞ may be determined by considering the behavior of small amplitude perturbations of the solution. Define xðtÞ ¼ uðtÞ; yi ðtÞ ¼ yi ðtÞ þ vi ðtÞ;
ði ¼ 1; 2; . . . ; mÞ.
where vi ðtÞ; ði ¼ 1; 2; . . . ; mÞ are small perturbations. Eq. (2.3) can be expanded in a Taylor series after neglecting higher order terms, the linearized equations read: 8 ) P u0 ðtÞ ¼ ða m ai yi ðtÞÞuðtÞ > i¼1 > < 0otoT; v0i ðtÞ ¼ ai yi ðtÞuðtÞ ci vi ðtÞ (A.1) > > : uðnT þ Þ ¼ uðTÞ; v ðnT þ Þ ¼ vðTÞ: i
Let FðtÞ be fundamental matrix of Eq. (A.1), then FðtÞ must satisfy
P a m 0 0 i¼1 ai yi ðtÞ B k1 a1 y ðtÞ c1 0 B 1 dFðtÞ B B a y ðtÞ c 0 k 2 2 2 2 ¼B B dt . . . B .. .. .. @ 0 0 0 0
x0 ðtÞ ¼ xðtÞða bxðtÞÞ; xð0Þ ¼ x0 40:
1 0 0 C C C 0 0 CFðtÞ9AFðtÞ C .. C .. . C . A km am ym ðtÞ cm 0 0
(A.2)
ARTICLE IN PRESS Y. Pei et al. / Journal of Theoretical Biology 235 (2005) 495–503
and Fð0Þ ¼ I the identity matrix. The linearization of Eq. (2.3) of the forth becomes 1 0 0 10 uðnTÞ 1 uðnT þ Þ 1 0 0 0 C C B B v1 ðnTÞ C B v1 ðnT þ Þ C B 0 1 0 0C CB C B C B B C B C ¼ B .. .. .. C. B . B .. .. C @ . . . .. C C B B . . A A A @ @ þ 0 0 0 1 vm ðnTÞ vm ðnT Þ Thus, the 0 1 B0 B M¼B B .. @.
monodromy matrix of Eq. (A.1) is 1 0 0 0 1 0 0C C .. C .. .. CFðTÞ. . . . A 0 0 0 1
From ð
RT
Eq.
(A.2),
A dtÞ
it
follows
501
According to Lemma 3.2, we have Pm 1
l1 ðl0 tÞ k¼1 k pi þ þ ðl TÞ i e V ðtÞp V ð0 Þ l0 e 0 1 l1 ð1 enl0 T Þeðl0 TÞ el0 ðtnTÞ þ . l0 where t 2 ðnT; ðn þ 1ÞT : Hence, ! m X l1 1 eðl0 TÞ . lim V ðtÞp þ pi ðl TÞ t!1 k l0 e 0 1 k¼1 i Therefore V ðt; xÞ is ultimately bounded. Hence there exists a constant M40; such that xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m for each solution X ðtÞ ¼ ðxðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; ym ðtÞÞ of system (2.3) with all t large enough. The proof is completed. &
that
FðTÞ ¼
¯ A
Fð0Þe 0 9Fð0Þe : Let m; m1 ; . . . ; mm be eigenvalues ¯ then of matrix A; ! Z T m m X X ai pi m¼ a ai yi ðtÞ dt ¼ aT , ci 0 i¼1 i¼1 mi ¼ ci To0; i ¼ 1; . . . ; m.
Appendix D. Proof of Theorem 4.3 Suppose X ðtÞ is a solution of Eq. (2.3) with X 0 40: From Theorem 4.2, we may assume xðtÞpM; yi ðtÞpM; i ¼ 1; . . . ; m and M4ab; tX0: Let ci T
Therefore, all eigenvalues of M, namely, e ; e ; . . . ; e have absolute values less than one if and only if P ai pi To m i¼1 aci : According to Floquent theory of impulsive differential equation, the pest eradication solution ð0; y1 ðtÞ; . . . ; ym ðtÞÞ is locally stable. This completes the proof. &
pi e mi ¼ 1e ci T i 40; i 40; ði ¼ 1; . . . ; mÞ: According to Lemmas 3.2 and 3.3, it follows that yi ðtÞ4mi ði ¼ 1; . . . ; mÞ; for all t large enough. In the following, we want to find mmþ1 40 such that xðtÞXmmþ1 for all t large enough. We will do it in the following two steps for convenience. P ai pi 0 Step 1: Since T4 m i¼1 aci ; we can select mmþ1 40; 0 0 i 40 ði ¼ 1; . . . ; mÞ small enough such that mmþ1 oab and
Appendix C. Proof of Theorem 4.2
di ¼
m
m1
mm
ki ai m0mþ1 oci ; 1 þ wi m0mþ1
Define V ðt; X Þ as s ¼ aT bm0mþ1 T
m X 1 V ðt; X Þ ¼ xðtÞ þ y ðtÞ. k i k¼1 i
It is obvious that V 2 V 0 ; we calculate the upper right derivative of V ðt; X Þ along a solution of system (2.3) and get the following impulsive differential equation 8 þ D V ðt; X Þ þ lV ðt; X Þ > > > > > < ¼ ða þ lÞx bx2 þ Pm l ci y ; tanT; i k¼1 ki > > Pm 1 > > þ > : V ðt; X ðt ÞÞ ¼ V ðt; X Þ þ k¼1 k pi ; t ¼ nT: i
0olo minfc1 ; . . . ; cm g; then lci k¼1 ki yi is bounded. Select two l0 and l1 such that
Let Pm
ða þ lÞx bx2 þ positive constants
8 þ < D V ðt; X Þp l0 V ðt; X Þ þ l1 ; tanT; Pm 1 p ; t ¼ nT: : V ðt; X ðtþ ÞÞ ¼ V ðt; X Þ þ k¼1 ki i
ði ¼ 1; . . . ; mÞ, m m X X ai pi 0i ai T40. c d i i i¼1 i¼1
We will prove that there exists t1 2 ð0; 1Þ such that xðt1 ÞXm0mþ1 : Otherwise, according to above assumption, it is easy to see y0i ðtÞpyi ðtÞðci þ di Þ;
tanT; i ¼ 1; . . . ; m.
By Lemmas 3.2 and 3.3, we get yi ðtÞpY i ðtÞ and Y i ðtÞ ! Y¯ i ðtÞ; where ðci þdi ÞðtnTÞ
pe ; t 2 ðnT; ðn þ 1ÞT Y¯ i ðtÞ ¼ i 1 eðci þdi ÞT and Y i ðtÞ is the solution of the following equation 8 0 > < Y i ðtÞpðci þ di ÞY i ðtÞ; tanT; Y i ðtþ Þ ¼ Y i ðtÞ þ pi ; t ¼ nT; (A.3) > : Y ð0þ Þ ¼ Y : i
0i
Therefore there exists a T i 40 such that yi ðtÞpY i ðtÞpY¯ i ðtÞ þ 0i ,
ARTICLE IN PRESS Y. Pei et al. / Journal of Theoretical Biology 235 (2005) 495–503
502
0
x ðtÞXxðtÞ a
bm0mþ1
m X
! ai ðY¯ i ðtÞ þ
0i Þ
.
(A.4)
i¼1
Let N i 2 N and N i TXT i : Integrating Eq. (A.4) on ðnT; ðn þ 1ÞT ; nX maxfN 1 ; . . . ; N m g; we have R ðnþ1ÞT Pm ðabm0mþ1 a ðY¯ i ðtÞþ0i ÞÞ i¼1 i xððn þ 1ÞTÞXxðnTÞe nT ¼ xðnTÞes .
Then xððn þ kÞTÞXxðnTÞeks ! 1;
as k ! 1,
which is a contradiction to the boundedness of xðtÞ: Step 2: If xðtÞXm0mþ1 for t4t1 ; the result is apparent. Otherwise, if xðtÞom0mþ1 for some t4t1 ; setting t ¼ inf tXt1 fxðtÞom0mþ1 g; we have xðtÞXm0mþ1 for ½t; t Þ: It is easy to see xðt Þ ¼ m0mþ1 ; since xðtÞ is continuous. Suppose t 2 ½n1 T; ðn1 þ 1ÞTÞ; n1 2 N: Select n2i ; n3 2 N such that
0i n2i 4T 2i ¼ ln ðci þ di Þ, M þ pi
which is a contradiction. Let ¯t ¼ inf tXt fxðtÞXm0mþ1 g; then xð¯tÞ ¼ m0mþ1 and Eq. (A.5) holds for t 2 ½t ; ¯tÞ: Integrating Eq. (A.5) on ½t ; ¯t yields
xðtÞXxðt Þeðs1 ðtt ÞÞ Xm0mþ1 eðs1 ðn2 þ1þn3 ÞTÞ . Letting m0mþ1 eðs1 ðn2 þ1þn3 ÞTÞ ¼ mmþ1 ; for t4t¯; the same argument can be continued since xð¯tÞXm0mþ1 for tXt1 : Hence xðtÞXmmþ1 for all tXt1 : Case (ii). There exists a t0 2 ðt ; ðn1 þ 1ÞT such that xðt0 ÞXm0mþ1 : Let t ¼ inf t4t fxðtÞXm0mþ1 g; then xðtÞom0mþ1 for t 2 ½t ; tÞ and xðtÞ ¼ m0mþ1 : For t 2 ½t ; tÞ Eq. (A.5) holds.Integrating Eq. (A.5) on t 2 ½t ; tÞ; we have
xðtÞXxðt Þeðs1 ðtt ÞÞ Xm0mþ1 eðs1 TÞ 4mmþ1 . This process can be continued since xðtÞXm0mþ1 ; and we have xðtÞXmmþ1 for t4t1 : Thus on both cases we get xðtÞXmmþ1 for t4t1 : This completes the proof. &
References
eððn2 þ1Þs1 TÞ eðn3 sÞ 41,
P where s1 ¼ a bm0mþ1 M m i¼1 ai o0; n2 ¼ maxfn21 ; . . . ; n2m g; T¯ ¼ n2 T þ n3 T: We claim that there must exist ¯ such that xðt0 ÞXm0mþ1 : a t0 2 ððn1 þ 1ÞT; ðn1 þ 1ÞT þ T 0 ¯ Otherwise, xðtÞommþ1 ; t 2 ððn1 þ 1ÞT; ðn1 þ 1ÞT þ T : Considering Eq. (A.3) with Y i ððn1 þ1Þ T þ Þ ¼ yi ððn1 þ 1ÞT þ Þ; we have pi Y i ðtÞ ¼ Y i ððn1 þ 1ÞT þ Þ 1 eðci þdi ÞT ððci þdi Þðtðn1 þ1ÞTÞÞ e þ Y¯ i ðtÞ for t 2 ðnT; ðnþ1ÞT ; n1 þ 1pnon1 þ 1 þ n2 þ n3 : Then jY i ðtÞ Y¯ i ðtÞjoðM þ pi Þeððci di Þn2i TÞ o0i , yi ðtÞpY i ðtÞpY¯ i ðtÞ þ 0i ¯ which implies Eq. for ðn1 þ 1 þ n2 Þptpðn1 þ 1ÞT þ T; ¯ Similar as (A.4) holds on ½ðn1 þ 1 þ n2 Þ; ðn1 þ 1ÞT þ T : in Step 1, we have xððn1 þ 1 þ n2 þ n3 ÞTÞXxððn1 þ 1 þ n2 ÞTÞeðn3 sÞ , There are two possible cases for t 2 ðt ; ðn1 þ 1ÞT : Case (i): If xðtÞom0mþ1 for t 2 ðt ; ðn1 þ 1ÞT ; then xðtÞom0mþ1 for all t 2 ðt ; ðn1 þ 1 þ n2 ÞT : System (2.3) gives ! m X 0 0 x ðtÞXxðtÞ a bmmþ1 ai M ¼ s1 xðtÞ. (A.5) i¼1
Integrating Eq. (A.5) on ½t ; ðn1 þ 1 þ n2 ÞT yields xððn1 þ 1 þ n2 ÞTÞXm0mþ1 eðs1 ðn2 þ1ÞTÞ . Then xððn1 þ 1 þ n2 þ n3 ÞTÞXm0mþ1 eðs1 ðn2 þ1ÞTÞ eðn3 sÞ 4m0mþ1 .
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