Modelling and analysis of an impulsive SI model with Monod-Haldane functional response

Chaos, Solitons and Fractals 39 (2009) 1698–1714 www.elsevier.com/locate/chaos

Modelling and analysis of an impulsive SI model with Monod-Haldane functional response Yiping Chen, Zhijun Liu

*

Department of Mathematics, Hubei Institute for Nationalities, 445000 Enshi Hubei, PR China Accepted 18 June 2007

Abstract An impulsive SI model with Monod-Haldane functional response for pest control is proposed and investigated. First, we have proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the above system can be permanent. Then, influences of impulsive perturbation including impulse period, the time of spraying pesticide and the quantity of releasing infective pests on the above system have been studied. Moreover, numerical simulations show that the system has rich dynamical behaviors. Finally, it is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying is more effective than the classical one if the chemical control is adopted rationally. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction At present, it has become more and more popular to control plant diseases and pests by the biological means. But only when biological control is combined with conventional chemical control, can it achieve the ideal preventive effects. Biological control is the use of living organism and its metabolite to control harmful animals and plants communities or alleviate their damage (see [1–4]). Chemical control is the approach of controlling pests through the spray of pesticides which are liable to reduce the pest populations considerably and which are indispensable when there are not enough natural enemies to decrease pest populations. Though chemical control can minimize the pests, it simultaneously destroys the natural enemies of the pests so that consecutive dependence of pesticides is essential because there are hardly any other ways to control diseases and pests after the pesticides have become ineffective. Biological control is relatively safe to human beings, animals, beneficial organisms and crops, etc. Chemical control is liable to poison human beings and animals as well as crops and pose health hazards to human beings and animals because of the remaining poison in crops, and hence the use of pesticides has been more and more firmly restricted in amount. In other words, human beings have been forced to face the new challenge in the integrated pest management (IPM) program.

*

Corresponding author. Fax: +86 718 8437732. E-mail address: [email protected] (Z. Liu).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.103

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

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Over the past 20 years, the employment of insect viruses to control crop pests has become the highlights in the field of biological control methods in the world. Take tall fruit and forest trees for an example. Generally speaking, it is less effective to spray pesticides on them to control diseases and pests due to the awkward operation. It proves to be more effective to release the diseased pests, to be exact, release the infective pests which have eaten the fodder on which the suspended liquid of living microorganism or microorganism pesticides have been sprayed, in the trees with the same species of pesters. As a result, those pests of the same kind will die of infection. Such control techniques have a broad applied prospects in pester control of tea, vegetable and fruit in agriculture and forestry owing to its explicit goal, nonpollution of the environment and no harm of any natural enemies. There is extensive literature related to the applications of microbial disease to suppress pests [2,5–8], and there are a few papers on a mathematical model of the dynamical of microbial disease in pest control [3,9–14]. As we know, most of the mathematical models have so far assumed that response function serves as sigmoidal response function, while releasing infective pests (or pathogens) and spraying pesticide are co-occurring at the fixed time. In this paper, we introduce additional infective pests, which are obtained in the laboratory, into a natural SI system with spraying pesticides for pest control. According to the fact of periodic biological and chemical means for pest control and the effect of pesticide on infective pests, we construct and investigate a non-monotonic functional response SI model incorporating periodic releasing infective pests and spraying pesticides at different fixed time. We are interested in the problem that how we release the infective pests if we expect to eradicate the pest population, and how the impulsive perturbations (i.e., impulse period, the time of spraying pesticides, the quantity of releasing infective pests) affect the dynamical behaviors of the above SI system without impulsive effects. The organization of this paper is as follows. In Section 2, a basic model is proposed. Meanwhile, we give some notations, definitions and lemmas. By using Floquet theory, small amplitude perturbation method and a comparison theorem, we analyze the dynamical behaviors of such a system in Section 3. We show that there exists an asymptotically stable pest-eradication periodic solution when the period of impulsive effect is less than some threshold. The condition for the permanence of the system is also given. In Section 4, numerical simulations show that the impulsive system we consider has very complex dynamical behaviors. The complexities include period-doubling bifurcation, symmetrybreaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and non-unique dynamics, meaning that several attractors coexist. In the last section, a brief discussion is given to conclude this work. Since we are interested in the biological points, the proofs of Theorems 3.1–3.3 will be given in the Appendices.

2. Model and preliminaries The basic model we proposed is based on the following classical SI system  0 S ðtÞ ¼ bSðtÞIðtÞ; I 0 ðtÞ ¼ bSðtÞIðtÞ  wIðtÞ;

ð2:1Þ

where b > 0 is called the transmission coefficient, and x > 0 is called the death coefficient of I(t). S(t) denotes the number of susceptible insects, and I(t) denotes the number of infective insects. Now we assume that all newborns are susceptible, then the basic model considered in [15] follows a model of the epidemic under a control variable:  0 S ðtÞ ¼ bSðtÞIðtÞ; ð2:2Þ I 0 ðtÞ ¼ bSðtÞIðtÞ  wIðtÞ þ uðtÞ; where u(t) is a control variable which denotes the rate of pests infects in the laboratory. Although there are some other conditions for the above system, we consider that the variable u(t) is difficult in practice. The susceptible pests S(t) will not go to extinction with regard to human beings and some mass residing animals. Anderson and May [16] pointed out that standard incidence is more suitable than bilinear incidence. Levin et al. [17,18] have adopted an incidence form like q p bSqIp or bSN I which depends on different infective diseases and environments. Further, we suggest to model such an effect by applying a response function mx ; ð2:3Þ /ðxÞ ¼ a þ bx þ x2 called the Monod-Haldane function (see [19]). These coefficients m, a, b are the positive constants and their biological meanings can be found in [19]. Obviously, the response function (2.3) is non-monotonic. Simultaneously, based on the IPM strategy, it is more realistic to assume the releasing of the infective pests and spraying of pesticides at different fixed moment. So we develop (2.2) and consider the following Monod-Haldane functional response SI model with different fixed moment impulse:

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h i 8 9 bIðtÞSðtÞ > =  aþbSðtÞþSðtÞ S 0 ðtÞ ¼ rSðtÞ 1  SðtÞþhIðtÞ > 2 ; K > > ; > > cIðtÞSðtÞ > 0 ; > I ðtÞ ¼ 2  wIðtÞ; > aþbSðtÞþSðtÞ > <  DSðtÞ ¼ p1 SðtÞ; ; t ¼ ðn þ l  1ÞT ; > > > DIðtÞ ¼ p2 IðtÞ; > > >  > > DSðtÞ ¼ 0; > > ; t ¼ nT ; : DIðtÞ ¼ p;

t–ðn þ l  1ÞT ;

t–nT ð2:4Þ

where r > 0 is the intrinsic growth rate of pests, K > 0 is the pests capacity of environment, DS(t) = S(t+)  S(t), DI(t) = I(t+)  I(t), 0 < h < 1, p P 0 is the released amount of infective pests at t = nT, n 2 Z+, and Z+ = {1, 2, . . . }, 0 6 p1 < 1 and 0 6 p2 < 1 are the death rate of susceptible and infective pests due to spraying pesticides at t = (n + l  1)T, n 2 Z+, l 2 (0, 1), respectively, T is the period of the impulsive effect. Equations of this kind are found in almost every domain of applied sciences [22–29]. Numerous examples are given in [21]. That is, we can use a combination of biological (periodic releasing infective pests) and chemical (spraying pesticide) tactics to eradicate pests or keep the pest population below the damage level. In Sections 3 and 4, we will consider the stability of pest-eradication solution and the uniform permanence of system (2.4). Before our main results, we give some notations, definitions and lemmas which will be useful for the later sections. Let Rþ ¼ ½0; 1Þ; R2þ ¼ fx 2 R2 : x > 0g. Denote f = (f1,f2) the map defined by the right hand of system (2.4). Let V : Rþ  R2þ ! Rþ , then V is said to belong to class V0 if (1) V is continuous in ððn  1ÞT ; ðn þ l  1ÞT   R2þ and ððn þ l  1ÞT ; nT   R2þ for each x 2 R2þ , n 2 Z+. lim þ V ðt; yÞ ¼ V ðnT þ ; xÞ exist. lim þ V ðt; yÞ ¼ V ððn þ l  1ÞT þ ; xÞ and ðt;yÞ!ððnþl1ÞT ;xÞ

ðt;yÞ!ðnT ;xÞ

(2) V is locally Lipschitzian in x. Definition 2.1. If V 2 V0, then for ðt; xÞ 2 ððn  1ÞT ; ðn þ l  1ÞT   R2þ and ððn þ l  1ÞT ; nT   R2þ , the upper right derivative of V(t, x) with respect to system (2.4) is defined as 1 Dþ V ðt; xÞ ¼ limþ sup ½V ðt þ h; x þ hf ðt; xÞÞ  V ðt; xÞ: h h!0 The solution of (2.4), denoted by xðtÞ ¼ ðSðtÞ; IðtÞÞ : Rþ ! R2þ , is continuous on ((n  1)T, (n + l  1)T] and ((n + l  1)T, nT]. xððn þ l  1ÞT þ Þ ¼ limt!ðnþl1ÞT þ xðtÞ and xðnT þ Þ ¼ limþ xðtÞ exist. Obviously the global existence t!nT and uniqueness of solutions of (2.4) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-side of system (2.4) (see [20]). The following Lemma 2.1 is obvious. Lemma 2.1. If x(t) is a solution of system (2.4) with x(0+) P 0, then x(t) P 0 for all t P 0. Definition 2.2. System (2.4) is said to be permanent if there are constants m, M > 0 (independent of initial value) and a finite time T0 such that for all solutions (S(t), I(t)) with all initial values S(0+) > 0, I(0+) > 0, m 6 S(t), I(t) 6 M holds for all t P T0. Hence, T0 may depend on the initial values (S(0+), I(0+)). Now we first consider some basic properties of the following subsystem of system (2.4): 8 dIðtÞ > < dt ¼ wIðtÞ; t–ðn þ l  1ÞT ; t–nT ; DIðtÞ ¼ p2 IðtÞ; t ¼ ðn þ l  1ÞT ; > : DIðtÞ ¼ p; t ¼ nT : System (2.5) is a periodically linear system, and it is easy to see that 8 < p exp½wðtðn1ÞT  ; ðn  1ÞT < t 6 ðn þ l  1ÞT ; 1ð1p Þ expðwT Þ I  ðtÞ ¼ pð1p Þ2exp½wðtðn1ÞT  2 : ; ðn þ l  1ÞT < t 6 nT ; 1ð1p2 Þ expðwT Þ p I  ð0þ Þ ¼ I  ðnT þ Þ ¼ ; 1  ð1  p2 Þ expðwT Þ pð1  p2 Þ expðwlT Þ ðI  ðlT þ Þ ¼ ; 1  ð1  p2 Þ expðwT Þ

ð2:5Þ

ð2:6Þ

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

is a positive periodic solution of system (2.5). Since the solution of system (2.5) is 8 h i n1 > Ið0þ Þ  1ð1p2 ÞpexpðwT Þ expðwtÞ þ I  ðtÞ; > > ð1  p2 Þ > > < ðn  1ÞT < t 6 ðn þ l  1ÞT h i IðtÞ ¼ p > ð1  p Þn Ið0þ Þ  > expðwtÞ þ I  ðtÞ; > 2 1ð1p Þ expðwT Þ > 2 > : ðn þ l  1ÞT < t 6 nT

1701

ð2:7Þ

we have Lemma 2.2. System (2.5) has a positive periodic solution I*(t) and for every solution I(t) of system (2.5) we have I(t) ! I*(t) as t ! +1. The Lemma 2.2 implies that system (2.4) has a pest-eradication periodic solution (0, I*(t)).

3. Extinction and permanence Firstly, we show that all solutions of system (2.4) are uniformly ultimately bounded. Theorem 3.1. There exists a constant M > 0 such that S(t) 6 M, I(t) 6 M for each solution (S(t), I(t)) of system (2.4) with t large enough. Secondly, by using Floquet theory [21] and small amplitude perturbation method we obtain conditions for the local stability of pest-extinction periodic solution (0, I*(t)). Theorem 3.2. Let (S(t), I(t)) be any solution of system (2.4). If T <

ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1 þ ln rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1

ð3:1Þ

holds, then (0, I*(t)) is locally asymptotically stable. Now we show that system (2.4) is permanent if the condition (3.1) is reversed. Theorem 3.3. System (2.4) is permanent provided T >

ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1 þ ln rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1

ð3:2Þ

holds true. For convenience, the proofs of Theorems 3.1–3.3 are given in Appendices A–C, respectively. Remark 3.1. Because we are interested in the dynamical behavior of system (2.4) when small amplitude perturbation occurs in the parameters T, p, l, respectively, now we give some annotation of the condition (3.1) or (3.2) (1) Let g1 ðT Þ ¼ T 

ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1  ln : rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1

ð3:3Þ

1 Clearly, g1 ð0Þ ¼  1r ln 1p < 0; g1 ðT Þ ! þ1 as T ! +1, and g001 ðT Þ > 0. So g1(T) = 0 has a unique positive root, 1 denoted by Tmax. (2) Let ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1 þ ln  T: ð3:4Þ g2 ðpÞ ¼ rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1

It is obvious that the equation g2(p) = 0 exists a unique positive root for a well posed T, denoted by pmax. (3) Let ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1  T: ð3:5Þ g3 ðlÞ ¼ þ ln rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1 Clearly, the equation g3(l) = 0 has a unique positive root over (0, 1) for a well posed T, denoted by lmax.

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From Theorem 3.2 and Remark 3.1, we know that the pest-eradication periodic solution (0, I*(t)) is locally asymptotically stable when T < Tmax. In general, Tmax is called a threshold. A typical pest-eradication periodic solution of system (2.4) is shown in Fig. 1, where we observe how the variable I(t) oscillates in a stable cycle. In contrast, the pest S(t) rapidly decreases to zero and Tmax  2.1762. If the period of pulses T is larger than Tmax, the pest-eradication solution becomes unstable and undergoes a transcritical bifurcation, then the susceptible insects and infective insects can coexist on a stable limit cycle when T > Tmax and is close to Tmax (see Fig. 2), and system (2.4) can be permanent which follows from Theorem 3.3. Actually, a positive periodic solution with a small magnitude indicates that our impulsive control strategy may reduce the pest population below a certain size without eradicating it. In this case, the pests cannot reach the damage level, our impulsive control strategy is still viable. But as we continue to increase T, This limit cycle may lose its stability and system (2.4) exhibits a wide variety of dynamical behaviors. Analogically, when p P pmax or l P lmax, the pest-eradication periodic solution (0, I*(t)) is locally asymptotically stable, but reversely, system (2.4) exhibits a wide variety of dynamical behaviors too (e.g. see Figs. 3–5). In the following section, we will analyze this in detail. −39

1.2

x 10

1.4

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1

1 0.8

I(t)

s(t)

0.8 0.6

0.6 0.4 0.4

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Fig. 1. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, p = 1, S(0) = 0.5, I(0) = 0.07, T = 2.15 (*corresponding threshold: Tmax  2.1762). (a) Time-series of the susceptible insects population. (b) Time-series of the infective insects population.

3.5

x 10−3

1.4

1.2

2.5

1

2

0.8 I(t)

s(t)

3

1.5

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0 1450

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1500 t

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1520

1530

1540

1550

0

0

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1

1.5

2 s(t)

2.5

3

3.5 −3

x 10

Fig. 2. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, p = 1, S(0) = 0.5, I(0) = 0.07, T = 2.2 (*corresponding threshold: Tmax  2.1762). (a) Time-series of the susceptible insects population. (b) A T-periodic solution.

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

0.7

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1.5

0.6

0.5 1

I(t)

S(t)

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0

0

1

2

3

4

5 t

6

7

8

9

10

0

0

10

20

30

40

50

60

70

t

Fig. 3. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, p = 1.1, S(0) = 0.5, I(0) = 0.07, T = 2.15 (*corresponding threshold: pmax  0.9886). (a) Time-series of the susceptible insects population. (b) Time-series of the infective insects population.

0.7

1

1

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0.9

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I(t)

I(t)

S(t)

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0.2 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 t

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0

0.1

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0.3

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S(t)

Fig. 4. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, T = 2.15, S(0) = 0.5, I(0) = 0.07, p = 0.05 (*corresponding threshold: pmax  0.9886). (a) Time-series of the susceptible insects population. (b) Time-series of the infective insects population. (c) A 7T-periodic solution.

4. Impulsive perturbation analysis In this section, we investigate the effects of impulsive perturbation on system (2.4) by using numeric method, and we mainly investigate the influence of parameters T, p, l, respectively. For convenience,we assume keeping some parameter values of the system (2.4) as r ¼ 5; k ¼ 0:7; h ¼ 0:01; b ¼ 0:9; a ¼ 0:1; b ¼ 0:01; w ¼ 0:8:

ð3:6Þ

Before starting our discussion, we give the simulation results of the system, in which there are no impulsive perturbation, in other words, that is the unforced system. We may observe that the solution of the unforced system would tend to be a closed orbit in Fig. 6 with the rest of parameter as c = 0.9, S(0) = 0.5, I(0) = 0.07. We will point out that the solution of the unforced system would become unstable via impulsive perturbation, and clearly show the complexity of the impulsive system (2.4). Case 1. Investigation of the influence of parameter T. From (3.3) and the above parameter hypothesis of the system (2.4), we can let p1 = p2 = 0.1, p = 1, l = 0.5. By a straightforward calculation we have Tmax  2.1762. Then the influences of T may be documented by stroboscopically sampling some of the variables over a range of T values. We numerically investigate system (2.4)

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−3

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1.6

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S(t)

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0 1410

1420

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0

1510

0

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1

1.2

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1.8 −3

S(t)

x 10

Fig. 5. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, T = 2.15, p = 1, S(0) = 0.5, I(0) = 0.07, l = 0.35 (*corresponding threshold: lmax  0.3871). (a) Time-series of the susceptible insects population. (b) A T-periodic solution.

0.7

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Fig. 6. Dynamical behavior of the unforced system with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, S(0) = 0.5, I(0) = 0.07. (a) Time-series of the susceptible insects population. (b) Time-series of the infective insects population. (c) Phase portrait.

for 500 pulsing cycles at each value of T. For each T, we may plot the last 155 measures of the susceptible insects population S(t) (of cause, we can sample the infective insects population I(t)). Since we sampled at the forcing period, periodic solutions of period T appear as fixed points, periodic solutions of period 2T appear as two cycles, and so forth. When we increase T from 1.5 to 30, the resulting bifurcation diagrams (see Fig. 7) clearly show that system (2.4) has rich dynamics including periodic-doubling cascade, periodic-halving cascade, chaotic region with periodic windows, quasi-periodic oscillating, and non-unique dynamics. When T is small (T < Tmax  2.1762), the solution (0, I*(t)) is stable. But when T > Tmax  2.1762, this stability would be lost. Later, the T-period solution of system (2.4) appears, and it is still stable for 2.1762 < T < 3.049. When T P 3.049, it becomes unstable and a 2T-periodic occurs. When T > 4.63 a T-period solution occurs again. The evidence for cascade of period-doubling bifurcations leading to chaos can be seen for 9.27 < T < 10.08, 10.29 < T < 11.45, 14.19 < T < 14.29, 16.37 < T < 16.69, 18.63 < T < 18.72, respectively. The evidence for cascade of period-halving bifurcations from chaos to cycles can also be seen for 13.65 < T < 13.79, 15.39 < T < 15.54, respectively. In Fig. 8, a strange attractor coexists with a 2T-period solution when T = 10.09. A typical chaotic oscillation is captured when T = 14.5 (see Fig. 9). Case 2. Investigation of the influence of parameter p. Let p1 = p2 = 0.1, l = 0.5, T = 4, we can easily obtain pmax  1.792 from (3.4) and (3.6). Fig. 10 shows bifurcation diagrams obtained by stroboscopically sampling the susceptible insects population S(t) (of cause, we can

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

1705

Fig. 7. Bifurcation diagram of system (2.4) showing the effect of T with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.9,w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, p = 1, S(0) = 0.5, I(0) = 0.07. (a) S(t) is plotted for T over [1.5, 15]. (b) S(t) is plotted for T over [15, 30].

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I(t)

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S(t)

Fig. 8. Coexist of a strange attractor with 2T-periodic solution when T = 10.09, r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.9, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, p = 1. (a) Solution with S(0) = 0.5, I(0) = 0.07 will finally tend to a strange attractor. (b) Solution with S(0) = 1.5, I(0) = 1.07 will finally tend to a 2T-periodic solution.

sample the infective insects population I(t)) for 0.001 < p < 1.8 keeping the rest of parameter as c = 0.7, S(0) = 0.5, I(0) = 0.07. We find that by increasing the release of infective insects, system (2.4) undergoes a process of periodic-doubling cascade ! symmetry-breaking pitchfork bifurcation ! periodic-halving cascade ! quasi-periodic oscillating ! chaotic region with periodic windows ! periodic-halving cascade ! cycles. If p > pmax  1.792, then the pest will be eradicated, and pest-eradication periodic solution occurs. Specially, when p = 0.26 a strange attractor occurs, and a 6T-period solution can be seen when p = 0.295 (see Fig. 11). We also find there are occurrences of sudden changes in the types of the attractors, which is a typical feature of bifurcation diagrams. For example, a 2T-periodic solution suddenly changes to chaos when p  0.1869, a 3Tperiodic solution suddenly changes to another T-periodic solution when p  0.3379. These sudden changes can lead to non-unique attractors [30], that is, different attractors may coexist with the same T, and every attractor

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Fig. 9. A strange attractor of the system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.9, w = 0.8, p1 = 0.1, p2 = 0.1, p = 1, l = 0.5, T = 14.5, S(0) = 0.5, I(0) = 0.07. (a) Time-series of the susceptible insects population. (b) Time-series of the infective insects population. (c) Phase portrait.

Fig. 10. Bifurcation diagram of system (2.4) showing the effect of p with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, T = 4, S(0) = 0.5, I(0) = 0.07 (*corresponding threshold: pmax  1.792). (a) S(t) is plotted for T over [0.001, 1.8]. (b) S(t) is plotted for T over [0.001, 0.4].

is related to the initial values. For example, in Fig. 12, a strange attractor coexists with another 2T-period solution when p = 0.1869, and in Fig. 13, 3T-period solution coexists with a T-period solution when p = 0.3379. Case 3. Investigation of the influence of parameter l. From (3.5) and (3.6), we can let T = 2.15,p1 = p2 = 0.1, p = 1, and then we can easily obtain lmax  0.3871. But if we change T = 2.15 to T = 4, the equation g3(l) = 0 is unsolvable in range of (0, 1). We also plot bifurcation diagrams for 0.001 < p < 0.99 with c = 0.7, S(0) = 0.5, I(0) = 0.07, respectively (see Fig. 14). From the bifurcation diagram of Fig. 14a, we note that, when l P lmax  0.3871, the system will go extinct with S(t) ! 0 as t ! 1, but when l < lmax  0.3871, the solutions of system (2.4) tend to a T-period solution (see Fig. 15). In Fig. 14(b), we find that by increasing l, system (2.4) undergoes a process of T-periodic solutions ! 2T-periodic solutions. For example, Fig. 15 shows a pest-extinct solution with l = 0.45, T = 2.15 and a T-period solution with l = 0.3, T = 2.15. In Fig. 16, 2T-period solution coexists with another 2T-period solution when l = 0.85, T = 4. According to the above analysis, we can see that the solutions of the unforced system will become unstable via impulsive perturbation. System (2.4) has rich and complex dynamical behaviors. The complexities include (1) period doubling bifurcation, (2) period-halving bifurcation, (3) chaos, (4) period windows, (5) symmetry-breaking bifurcation, and (6) non-unique dynamics, meaning that several attractors coexist.

1.2

1.2

1.1

1.1

1

1

0.9

0.9

0.8

0.8 I(t)

I(t)

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.2

0.7

1707

0

0.1

0.2

0.3

S(t)

0.4

0.5

0.6

0.7

S(t)

1.2

1.1

1.1

1

1

0.9

0.9

0.8

0.8

0.7 I(t)

I(t)

Fig. 11. Phase portrait of system (2.4) when r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, T = 4, S(0) = 0.5, I(0) = 0.07. (a) Solution will finally tend to a strange attractor when p = 0.26. (b) Solution will finally tend to a 6T-periodic solution when p = 0.295.

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0

0.1

0.2

0.3

0.4 S(t)

0.5

0.6

0.7

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

S(t)

Fig. 12. Coexistence of a strange attractor with 2T-periodic when p = 0.1869, r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, T = 4. (a) Solution with S(0) = 0.5, I(0) = 0.07 will finally tend to a strange attractor. (b) Solution with S(0) = 0.5, I(0) = 1.05 will finally tend to a 2T-periodic solution.

5. Discussion In this paper, a SI impulsive model with non-monotonic functional response for pest control is proposed and investigated according to periodically releasing infective pests and spraying pesticides at different fixed times. We show that there exists an orbitally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. Further, we note that the conditions for the extinction or permanence in system (2.4) are quite different from the corresponding system without impulse. For example, the system without impulse has a positive equilibrium which is orbitally asymptotically stable (see Fig. 6), however this properties are changed via additional impulsive perturbation. In fact, choosing T, p and l as bifurcation parameters, respectively, we have obtained bifurcation diagrams (see Figs. 7, 10 and 14). Bifurcation diagrams have shown that system (2.4) can

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Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

1.2

1

1.1 0.9

1 0.8

0.9 0.7

I(t)

I(t)

0.8

0.7

0.6

0.6 0.5

0.5 0.4

0.4 0.3

0.3

0.2

0

0.1

0.2

0.3

0.4 S(t)

0.5

0.6

0.7

0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

S(t)

Fig. 13. Coexistence of 3T-periodic solution with T-periodic when p = 0.3379, r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, l = 0.5, T = 4. (a) Solution with S(0) = 0.5, I(0) = 1.05 will finally tend to a 3T-periodic solution. (b) Solution with S(0) = 0.5, I(0) = 0.07 will finally tend to a T-periodic solution.

Fig. 14. Bifurcation diagrams of system (2.4) showing the effect of l with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, p = 1, S(0) = 0.5, I(0) = 0.07. (a) S(t) is plotted for l over [0.001, 0.99] with T = 2.15. (b) S(t) is plotted for l over [0.001, 0.99] with T = 4.

take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by periodic, quasi-periodic and chaotic solutions, which implies that the presence of pulse makes the dynamic behavior more complex. Meanwhile, from Theorem 3.2 and Remark 3.1, we can easily obtain the thresholds Tmax, pmax and lmax by a straightforward calculation, hence we can choose the parameters of system (2.4) such that T < Tmax or p > pmax or l > lmax in light of the different control strategies (e.g. the pulse period is fixed, or the time of spraying pesticide is fixed, or the quantity of releasing infective pests is fixed). When T < Tmax or pPpmax or l P lmax, the pest-eradication periodic solution (0, I*(t)) is locally asymptotically stable, but reversely, the susceptible insects and infective insects can coexist on a stable limit cycle when T > Tmax or p < pmax or l < lmax and is close to Tmax (see Figs. 8, 11 and 15), and system (2.4) can be permanent which follows from Theorem 3.3. On the other hand, if p = 0, i.e. there are no periodic releasing 1 infective pests, we can obtain that T 1 ¼ 1r ln 1p is the threshold. Obviously, T1 < Tmax, i.e. impulsively releasing infec1 tive pests may lengthen the period of spraying pesticides and reduce the cost of pests control. That is, we can choose the

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

−17

2

x 10

1709

1.4

S(t)

1.2 1

1

0.8 940

960

980

1000

1020

1040

1060

1080

I(t)

0

t

0.6

1.4 1.2

I(t)

1

0.4

0.8 0.6

0.2 0.4 0.2 0

0 940

960

980

1000

1020

1040

1060

1080

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 −3

S(t)

t

x 10

Fig. 15. Dynamical behavior of system (2.4) with r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, p = 1, S(0) = 0.5, I(0) = 0.07, T = 2.15. (a) Time-series of solution with l = 0.45. (b) A T-periodic solution with l = 0.3.

1.2

1.2

1

1

0.8

0.8 I(t)

1.4

I(t)

1.4

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.1

0.2

0.3

0.4 S(t)

0.5

0.6

0.7

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

S(t)

Fig. 16. Coexistence of 2T-periodic solution with another 2T-periodic solution when l = 0.85, T = 4, r = 5, k = 0.7, h = 0.01, b = 0.9, a = 0.1, b = 0.01, c = 0.7, w = 0.8, p1 = 0.1, p2 = 0.1, p = 1. (a) Solution with S(0) = 0.5, I(0) = 0.07 will finally tend to a 2T-periodic solution. (b) Solution with S(0) = 1.5, I(0) = 1.07 will finally tend to a 2T-periodic solution.

parameters to reduce pests to tolerable levels with little cost of the releasing infective pests and minimal effect on the environment. Therefore, our impulsive strategy is more effective than the classical one if the chemical control is adopted rationally. We would like to mention here that an interesting but challenging problem associated with the studies of system (2.4) should be how to optimize the amount of periodically releasing infective pests and spraying pesticides by integrating with the above policies. We leave this for future work.

Acknowledgements The work is partially supported by the National Natural Science Foundation of China (10671001), the Youth Science Foundation of Educational Department of Hubei Province in PR China (No. Q200529001) and the Youth Group Science Foundation of Hubei Institute for Nationalities in PR China.

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Appendix A. Proof of Theorem 3.1 Define V ðtÞ ¼ V ðt; xðtÞÞ ¼ bc SðtÞ þ IðtÞ. It is obvious that V(t) 2 V0. When t ne (n + l  1)T, t 5 nT we calculate the upper right derivative of V(t) along a solution of system (2.4) and get c dSðtÞ dIðtÞ ck þ þ SðtÞ þ kIðtÞ b dt dt b   c SðtÞ þ hIðtÞ c SðtÞIðtÞ SðtÞIðtÞ ck ¼ rSðtÞ 1  þc  wIðtÞ þ SðtÞ þ kIðtÞ  b 2 2 b K b a þ bSðtÞ þ S ðtÞ b a þ bSðtÞ þ S ðtÞ cðr þ kÞ cr 2 crh ¼ SðtÞ  S ðtÞ  SðtÞIðtÞ þ ðk  wÞIðtÞ: b bK bK

Dþ V ðtÞ þ kV ðtÞ ¼

Obviously, the right hand of the above equation is bounded when 0 < k 6 w. Select a suitable k such that 0 < k 6 w. Thus we have Dþ V ðtÞ þ kV ðtÞ 6

cðr þ kÞ cr 2 SðtÞ  S ðtÞ 6 M 0 ; b bK

2

where M 0 ¼ KcðrþkÞ , that is 4br Dþ V ðtÞ þ kV ðtÞ 6 M 0 ; where t = (n + l  1)T, V((n + l  1)T+) 6 V((n + l  1)T), t = nT, V(nT+) 6 V(nT) + p, then by Lemma 2.2 in [21] we obtain M0 ekðtT Þ ekT M0 ekT V ðtÞ 6 V ð0þ Þekt þ ð1  ekt Þ þ p ! þ p kT as t ! þ1: þ p kT k 1  ekT e 1 k e 1 Therefore, V(t) is uniformly ultimately bounded, and there exists a constant M > 0 such that S(t) 6 M, S(t) 6 M for t large enough. h

Appendix B. Proof of Theorem 3.2 The locally asymptotical stability of periodic solution (0, I*(t)) may be determined by considering the behavior of small amplitude perturbation of the solution. Defining IðtÞ ¼ I  ðtÞ þ vðtÞ;

SðtÞ ¼ uðtÞ;

where v(t) is small perturbation. We have the following linearly similar system:     uðtÞ uð0Þ ¼ UðtÞ ; vðtÞ vð0Þ where U satisfies d/ ¼ dt

r

rh



þ ba I  ðtÞ K c  I ðtÞ a

! 0 /ðtÞ w

and U(0) = E, the identity matrix. Hence the fundamental solution matrix is !   R t    exp 0 r  rh þ ba I  ðsÞ ds 0 K /ðtÞ ¼ :  expðwtÞ There is no need to calculate the exact form of (*) as it is not required in the analysis that follows.The resetting impulsive condition of system (2.4) becomes      uððn þ l  1ÞT Þ 1  p1 0 uððn þ l  1ÞT þ Þ ¼ 0 1  p2 vððn þ l  1ÞT Þ vððn þ l  1ÞT þ Þ and þ uððnT Þ Þ vððnT Þþ Þ

!

 ¼

1 0 0 1



 uðnT Þ : vðnT Þ

Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

The stability of the prey-extinction   1  p1 0 1 H¼ 0 1  p2 0

1711

periodic solution (0, I*(t)) is determined by the eigenvalues of matrix H, where  0 UðT Þ: 1

Denote l1 ¼ ð1  p1 Þ exp

Z 0

T

     rh b  r þ I ðtÞ dt ; K a

l2 ¼ ð1  p2 Þ expðwT Þ:

Clearly, |l2| < 1. According to the Floquet theory [21], if |l1| < 1, i.e. Z T      rh b  ð1  p1 Þ exp r þ I ðtÞ dt < 1 K a 0 1 rha þ Kb pð1  p2 expðwlT Þ  ð1  p2 Þ expðwT ÞÞ ln > rT   1  p1 aK wð1  ð1  p2 Þ expðwT ÞÞ ðrha þ KbÞp½1  p2 expðwlT Þ  ð1  p2 Þ expðwT Þ 1 1 T < þ ln rKaw½1  ð1  p2 Þ expðwT Þ r 1  p1 holds, then (0, I*(t)) is locally asymptotically stable. The proof is completed.

h

Appendix C. Proof of Theorem 3.3 Suppose that x(t) = (S(t), I(t)) is any solution of system (2.4) with x(0) > 0. By Theorem 3.1 we have proved that there exists a constant M > 0 such that S(t) 6 M, I(t) 6 M for t large enough. We can assume S(t) 6 M, I(t) 6 M rKa and M > rhaþKb for t P 0. From system (2.4) we have dIðtÞ P wIðtÞ. Let us consider the following impulsive equation: dt 8 dzðtÞ > > > dt 6 wzðtÞ; t–ðn þ l  1ÞT ; t–nT ; > > < zðtþ Þ ¼ ð1  p ÞzðtÞ; t ¼ ðn þ l  1ÞT ; 2 ðC:1Þ > zðtþ Þ ¼ zðtÞ þ p; t ¼ nT ; > > > > : þ zð0 Þ ¼ Ið0Þ: From Lemma 2.2 and comparison theorem of impulsive equation [20], we have I(t) P z(t) and z(t) ! I*(t) as t ! +1. pð1p2 Þ expðwT Þ  e, then we have Hence, for some e > 0, we have I(t) P z(t) > I*(t)  e for all t large enough. Let m2 ¼ 1ð1p 2 Þ expðwT Þ I(t) P m2 for t large enough. Thus, we only need to find a constant m1 > 0 such that S(t) P m1 for t large enough. We will do it in the following two steps. 1. From the condition (3.2), we can choose 0 < m3 < aw , e1 > 0 small enough such that r ¼ ð1  p1 Þ exp c h i   ðrhaþKbÞpð1p2 expððwþdÞlT Þð1p2 Þ expððwþdÞT ÞÞ b >1, where d ¼ cma 3 , it is obvious that d < w. Next, e þ T  rT  rmK3 T  rh a 1 KaðwdÞð1ð1p2 Þ expððwþdÞT ÞÞ K we prove that S(t) < m3 cannot hold for all t P 0. Otherwise, 8 dIðtÞ > < dt 6 IðtÞðw þ dÞ; t–ðn þ l  1ÞT ; t–nT ; Iðtþ Þ ¼ ð1  p2 ÞIðtÞ; t ¼ ðn þ l  1ÞT ; > : þ Iðt Þ ¼ IðtÞ þ p; t ¼ nT ; then we derive I(t) 6 y(t) and y(t) ! y*(t) as t ! +1, where y(t) is the solution of 8 dyðtÞ ¼ yðtÞðw þ dÞ; t–ðn þ l  1ÞT ; t–nT ; > dt > > < yðtþ Þ ¼ ð1  p ÞyðtÞ; t ¼ ðn þ l  1ÞT ; 2 > yðtþ Þ ¼ yðtÞ þ p; t ¼ nT ; > > : yð0þ Þ ¼ Ið0þ Þ; and y  ðtÞ ¼

8 < p expððwþdÞðtðn1ÞT ÞÞ ; 1ð1p2 Þ expððwþdÞT Þ

ðn  1ÞT < t 6 ðn þ l  1ÞT ;

: pð1p2 Þ expððwþdÞðtðn1ÞT ÞÞ ; 1ð1p2 Þ expðwþdÞT Þ

ðn þ l  1ÞT < t 6 nT :

ðC:2Þ

ðC:3Þ

ðC:4Þ

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Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

Hence, there exists a T1 > 0 such that IðtÞ 6 yðtÞ < y  ðtÞ þ e1 and ( dSðtÞ dt

P SðtÞðr  Kr m3 

Sðtþ Þ ¼ ð1  p1 ÞSðtÞ;

rh K

 þ ba ðy  ðtÞ þ e1 ÞÞ;

t–ðn þ l  1ÞT ;

t ¼ ðn þ l  1ÞT ;

ðC:5Þ

for t P T1. We choose a suitable constant N 2 Z+ such that (N + l  1)T P T1. Integrating (C.5) on ((n + l  1)T, (n + l)T], n P N, we have  !   Z ðnþlÞT  r rh b  Sððn þ lÞT Þ P Sððn þ l  1ÞT Þð1  p1 Þ exp r  m3  þ ðy ðtÞ þ e1 Þ dt ¼ Sððn þ l  1ÞT Þr; K K a ðnþl1ÞT then S((N + n + l)T) P S((N + l)T)rn ! +1 as n ! +1, which is a contradiction to the bounded quantity of S(t). Therefore, there exists a t1 > 0 such that S(t1) P m3. 2. If S(t) P m3 for all t P t1, then our aim is achieved. Otherwise, let t ¼ inf t>t1 fSðtÞ < m3 g, there are also two possible cases for t*. Case (1). If t* = (n1 + l  1)T, n1 2 Z+, then we have S(t) P m3 for t 2 [t1, t*) and (1  p1)m3 6 S(t*+) = (1  p1)S(t*) < m3. Choose n2, n3 2 Z+ such that   e1 ðn2  1ÞT > ln =ðw þ dÞ; M þp ð1  p1 Þn2 rn3 expðn2 r1 T Þ > ð1  p1 Þn2 rn3 expððn2 þ 1Þr1 T Þ > 1; þ baÞM < 0. Denote T* = n2T + n3T, there must exist a t2 2 (t*, t* + T*] such that where r1 ¼ r  Kr m3  ðrh K S(t2) > m3. Otherwise, Recalling (C.3) with y(n1T+) = I(n1T+), we have  8 p > expððw þ dÞðt  n1 T ÞÞ þ y  ðtÞ; ð1  p2 Þnðn1 þ1Þ yðn1 T þ Þ  1ð1p Þ expððwþdÞT > Þ 2 > > > > < ðn  1ÞT < t 6 ðn þ l  1ÞT ; yðtÞ ¼  > nn1 p þ > expððw þ dÞðt  n1 T ÞÞ þ y  ðtÞ; Þ yðn T Þ  ð1  p > 1 2 > 1ð1p2 Þ expððwþdÞT Þ > > : ðn þ l  1ÞT < t 6 nT ; and n1 + 1 6 n 6 n1 + 1 + n2 + n3. Consequently, jyðtÞ  y  ðtÞj < ðM þ pÞ expððw þ dÞðt  n1 T ÞÞ < e1 and IðtÞ 6 yðtÞ 6 y  ðtÞ þ e1 for n1T + (n2  1)T 6 t 6 t* + T*, which implies that (C.5) holds for t* + n2T 6 t 6 t* + T*. As in step 1, we have Sðt þ T  Þ P Sðt þ n2 T Þrn3 : In view of system (2.4), we obtain ( SðtÞ   P SðtÞðr  Kr m3  rh þ ba MÞ; dt K þ

Sðt Þ ¼ ð1  p1 ÞSðtÞ;

ðC:6Þ

t–ðn þ l  1ÞT ;

t ¼ ðn þ l  1ÞT ;

ðC:7Þ

for t 2 [t*, t* + n2T]. Integrating (C.7) on [t*, t* + n2T], we have Sðt þ n2 T Þ P m3 ð1  p1 Þn2 expðn2 r1 T Þ; then we get Sðt þ T  Þ P m3 ð1  p1 Þn2 expðn2 r1 T Þrn3 > m3 ;

ðC:8Þ

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which is a contradiction. Let t ¼ inf t>t fSðtÞ > m3 g. Then for t 2 ðt ; tÞ, we obtain S(t) 6 m3 and SðtÞ ¼ m3 . For t 2 ðt ; tÞ, we have SðtÞ P m3 ð1  p1 Þn2 þn3 expððn2 þ n3 Þr1 T Þ. Denote n2 þn3 0 expððn2 þ n3 Þr1 T Þ, we have SðtÞ P m01 for t 2 ðt ; tÞ. For t > t, the same arguments m1 ¼ m3 ð1  p1 Þ can be continued since SðtÞ P m3 . Case (2). If t* 5 (n + l  1)T,n 2 Z+, then we have S(t) P m3 for t 2 [t1, t*) and S(t*) = m3. If t 2 ððn01 þ l  1ÞT ; ðn01 þ lÞT Þ; n01 2 Z þ , then there still have two possible cases for t 2 ðt ; ðn01 þ lÞT Þ. (i) If S(t) 6 m3 for all t 2 ðt ; ðn01 þ lÞT Þ, similar to case (1), we can prove that there exists a t02 2 ½ðn01 þ lÞT ; ðn01 þ lÞT þ T 0  such that Sðt02 Þ > m3 . Hence we omit it. Let ~t ¼ inf t>t fSðtÞ > m3 g. Then S(t) 6 m3 for t 2 ðt ; ~tÞ and Sð~tÞ ¼ m3 . For t 2 ðt ; ~tÞ, we have SðtÞ P m3 ð1  p1 Þn2 þn3 expððn2 þ n3 þ 1Þr1 T Þ. Let m1 ¼ m3 ð1  p1 Þn2 þn3 expððn2 þ n3 þ 1Þr1 T Þ < m01 . Then we have S(t) P m1 for t 2 ðt ; ~tÞ. For t > ~t, the same arguments can be continued since Sð~tÞ P m3 . (ii) If there exists a t 2 ðt ; ðn01 þ lÞT Þ such that S(t) > m3. Let t ¼ inf t>t fSðtÞ > m3 g. Then we have S(t) 6 m3 for t 2 (t*, ˇt) and S(tˇ) = m3. For t 2 (t*, tˇ), we have S(t) P S(t*)exp(r1(t  t*)) P m3exp(r1T) > m1. Since S(tˇ) P m3 for t > tˇ, the same arguments can be continued. Hence S(t) P m1 for all t P t1. Let X = {m1 6 x1 6 M, m2 6 x2 6 M}, by the Definition 2.2 of permanence we can find system (2.4) is permanent. This completes the proof. h

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

DeBach P, Rosen D. Biological control by natural enemies. 2nd ed. Cambridge: Cambridge University Press; 1991. Sushil, Khetan K. Microbial pest control. New York: Marcel Dekker; 2001. Mden E, Van HF, Service MW. Pest and vector control. Cambridge: Cambridge University Press; 1994. Parker FD. Management of pest population by manipulating densities of both host and parasites through periodic releases. In: Huffaker CB, editor. Biological control. New York: Plenum Press; 1971. Falcon LA. Problems associated with the use of arthropod viruses in pest control. Ann Rev Entomol 1976;21:305–24. Bailey NTJ. The mathematical theory of infectious diseases and its applications. London: Griffin; 1975. p. 413. Burges HD, Hussey NW. Microbial control of insects and mites. New York: Academic Press; 1971. p. 861. Burges HD, Hussey NW. Microbial control of insects and mites. New York: Academic Press; 1971. p. 67–95. Kor M, Sayler GS, Waltman TW. Complex dynamics in a model microbial system. Bull Math Biol 1992;54:619–48. Dwyer Greg, Elkinton Joseph S, Buonaccorsi John P. Host heterogeneity in susceptibility and disease dynamics: tests of a mathematical model. Am Nat 1997;6:685–707. Anderson RM, May RM. Regulation and stability of host–parasity population interactions, I. Regulatory processes. J Anim Ecol 1978;47:219–47. Goh BS. The potential utility of control theory to pest management. Proc Ecol Soc 1971;6:84–9. Gilbert N, Gutierrez AP, Frazer BD, Jones RE. Ecological relationships. San Francisco [CA]: W.H. Freeman and Co.; 1976. Wickwire K. Mathematical models for the control of pests and infectious diseases: a survey. Theor Popul Biol 1977;8: 182–238. Goh BS. Management and analysis of biological populations. Amsterdam: Elsevier Scientific Press Company; 1980. Anderson R, May R. Population biological of infectious diseases. Berlin, New York: Heidelberg: Springer-Verlag; 1982. Wei-min, Levin SA, Lwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS Epidemiological models. J Math Biol 1987;25:359–80. Wei-min, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 1986;23:187–240. Andrews JF. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol Bioeng 1968;10:707–23. Lakshmikantham V, Bainov D, Simeonov P. Theory of impulsive differential equations. Singapore: World Scientific; 1989. Bainov D, Simeonov P. Impulsive differential equations: periodic solutions and applications. Ptiman Monogr Surv Pure Appl Math 1993;66:54–121. Lakmeche A, Arino O. Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam Contin Discrete Impuls Syst 2000;7:265–87. Zhang SW, Tan DJ, Chen LS. Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations. Chaos, Solitons & Fractals 2006;27:980–90. Liu XN, Chen LS. Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator. Chaos, Solitons & Fractals 2003;16:311–20.

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Y. Chen, Z. Liu / Chaos, Solitons and Fractals 39 (2009) 1698–1714

[25] Zeng GZ, Chen LS, Sun LH. Complexity of an SIR epidemic dynamics model with impulsive vaccination control. Chaos, Solitons & Fractals 2005;26:495–505. [26] Hui J, Zhu DM. Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos, Solitons & Fractals 2006;29:233–51. [27] Zhang YJ, Xiu ZL, Chen LS. Dynamic complexity of a two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals 2005;26:131–9. [28] Liu ZJ, Tan RH. Impulsive harvesting and stocking in a Monod-Haldane functional response predator-prey system. Chaos, Solitons & Fractals 2007;34:454–64. [29] Liu B. The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management. Nonlinear Anal: RealWorld Appl 2005;6:227–43. [30] Kaitala V, Ylikarjula J, Heino M. Dynamic complexities in host–parasitioid interaction. J Theor Biol 1999;197:331–41.