Dynamics of a predator-prey system with pulses ... - Semantic Scholar
Applied Mathematics and Computation 204 (2008) 269–280
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Dynamics of a predator-prey system with pulses Yongfeng Li a,*, Jingan Cui a, Xinyu Song b a b
Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, PR China Department of Mathematics, Xinyang Normal University, Xinyang 464000, PR China
a r t i c l e
i n f o
Keywords: Predator-prey system Impulsive effect Permanence Extinction Stability Bifurcation
a b s t r a c t In this paper, we investigate the dynamic behaviors of a Holling II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategyperiodic releasing natural enemies and spraying pesticide at different fixed moment. By using the Floquet theory of linear periodic impulsive equation and small-amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is larger than some critical value, and meanwhile the conditions for the extinction of one of the two-prey and permanence of the remaining two species are given. Finally, we give numerical simulation, with increasing of predation rate for the super competitor and impulsive period, the system displays complicated behaviors including a sequence of direct and inverse cascades of periodic-doubling, periodic-halving, chaos and symmetry breaking bifurcation. Our results suggest a new approach in the pest control. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction It is well known, many insects are beneficial to human, but some insects are harmful to human, only these harmful insects can cause economic damage as their population reaching economic injury level. Controlling harmful insects and other arthropods has become an important issue in recent years, for instance, minimizing losses due to insect pests and insect vectors remains an essential component of the programmes in the Office of Agricultural Entomology in China. Overusing a single control tactic is discouraged to minimize damage to non-target organisms, and to preserve the quality of the environment. Overusing a single control tactic also can lead pest to produce resistance to chemical control (for example, pesticide), it will be more difficult to control pest later. Then biological and chemical control were introduced. Biological control [1–7] is the purposeful introduction one or more natural enemies 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. Natural enemies are able to play a more active role in suppressing insect pests. Usually, predators feed on not only insect pests but also other insects. There may be more than one pest species – for example, the two species of aphids predominant in small grains: the English grain aphid and the oatbird cherry aphid. Aphids’ high reproductive rate enables their populations to quickly build up to levels that can cause an economic loss. However, aphids are usually kept in check by biological control agents, such as lady beetles, parasitic wasps, and syrphid fly maggots which are often abundant in small grains. 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 releasing natural enemies, and by genetic enhancement of the enemies to increase their effectiveness at control.
* Corresponding author. E-mail address: yfl[email protected] (Y. Li). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.06.037
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Wherever possible, different pest control techniques should work together rather than against each other. In this paper, according to periodic biological and chemical control, and based on two-prey one-predator system with Holling II functional response, we suggest a simple mathematical model with pulses to describe the process of periodic releasing natural enemies and spraying pesticide (or harvesting pests) at different fixed moments. System with impulsive effects describing evolution processes are characterized by the fact that at certain moments of time they abruptly experience a change of state. Processes of such character are studied in almost every domain of applied sciences. Numerous examples are given in Bainov’s and his collaborator’s books [8,9]. Some impulsive equations have been recently introduced into population dynamics in relation to: vaccination [10,11], population ecology [12,13], and impulsive birth [14,15], chemotherapeutic [16,17]. The paper is arranged like this. A Holling II two-prey one-predator system concerning biological and chemical control is given in Section 2. In Section 3, we give some notations and lemmas. In Section 4, we consider the local stability and global asymptotic stability of the two-pest eradication periodic solution by using Floquet theory for the impulsive equation, smallamplitude perturbation skills and techniques of comparison, and in Section 5 we show that the system is permanent if the impulsive period is larger than some critical value. Moreover, we give the sufficient conditions for one of two-prey extinction and the remaining two species permanence. A brief discussion and further numerical simulation are given in the last section.
2. Model formulation Based on many experiments, Holling [18] suggested three different kinds of functional response for different kinds of species to model the phenomenon of predation, which made the standard Lotka–Volterra systems more realistic. Liu and Chen [13] investigated complex dynamics of Holling type II Lotka–Volterra predator-prey system with impulsive perturbations on the predator. Zhang and Chen [19] studied a Holling II functional response food chain model with impulsive perturbations. The model we considered in this paper is based on the following predator-prey model, where two-prey are competitive and the predator has Holling II functional response, that is
8 gzðtÞ > _ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx x ; > x ðtÞ > 1 1 > < l zðtÞ x_ 2 ðtÞ ¼ x2 ðtÞ b2 bx1 ðtÞ x2 ðtÞ 1þx2 x2 ðtÞ ; > > > > : z_ ðtÞ ¼ zðtÞ b þ dgx1 ðtÞ þ dlx2 ðtÞ ; 3 1þx1 x1 ðtÞ 1þx2 x2 ðtÞ
ð2:1Þ
where xi ðtÞ (i ¼ 1; 2) is the population size of prey (pest) species and zðtÞ is the population size of predator (natural enemies) species, bi > 0 ði ¼ 1; 2; 3Þ are intrinsic rates of increase or decrease, a > 0 and b > 0 are parameters representing competigx1 ðtÞzðtÞ lx2 ðtÞzðtÞ tive effects between two-prey, g > 0 and l > 0, 1þ x1 x1 ðtÞ and 1þx2 x2 ðtÞ are the Holling type II functional responses, d > 0 is the rate of conversing prey into predator. Model (2.1) with constant periodic releasing predator and spraying pesticide (or harvesting pests) was studied by Song and Li [20]:
9 8 gzðtÞ > x_ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx ;> > > x1 ðtÞ > > 1 > > = > > > l zðtÞ > > x_ 2 ðtÞ ¼ x2 ðtÞ b2 x2 ðtÞ bx1 ðtÞ 1þx2 x2 ðtÞ ; t 6¼ nT; > > > > > < > > dgx1 ðtÞ dlx2 ðtÞ ; z_ ðtÞ ¼ zðtÞ b3 þ 1þx1 x1 ðtÞ þ 1þx2 x2 ðtÞ ; > 9 > > > Dx ðtÞ ¼ p1 x1 ðtÞ; > > > = > 1 > > Dx2 ðtÞ ¼ p2 x2 ðtÞ; t ¼ nT; > > > > : ; DzðtÞ ¼ q;
ð2:2Þ
where Dxi ðtÞ ¼ xi ðt þ Þ xi ðtÞ; DzðtÞ ¼ zðt þ Þ zðtÞ. T is the period of the impulsive effect, pi > 0ði ¼ 1; 2Þ is the proportionality constant which represents the rate of mortality due to applying pesticide. q > 0 is the number of predator released each time. Now we will develop systems (2.1) and (2.2) by introducing a constant periodic releasing natural enemies and spraying pesticide (or harvesting pests) at different fixed moment. That is, we consider the following impulsive differential equations:
9 8 gzðtÞ > _ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx ;> x > > x ðtÞ > > 1 1 > > > = > > l zðtÞ > _ x ; ðtÞ ¼ x ðtÞ b x ðtÞ bx ðtÞ t 6¼ ðn þ l 1ÞT; t 6¼ nT; > 2 2 2 2 1 > x x ðtÞ 1þ 2 2 > > > > > > > > < z_ ðtÞ ¼ zðtÞ b þ dgx1 ðtÞ þ dlx2 ðtÞ ; ; 3 1þx1 x1 ðtÞ 1þx2 x2 ðtÞ > Dx ðtÞ ¼ p x ðtÞ; > i > i i > > t ¼ ðn þ l 1ÞT; > DzðtÞ ¼ pzðtÞ; > > > > > > Dxi ðtÞ ¼ 0; > > t ¼ nT; : DzðtÞ ¼ q;
ð2:3Þ
Y. Li et al. / Applied Mathematics and Computation 204 (2008) 269–280
271
where 0 6 l 6 1, Dxi ðtÞ ¼ xi ðtþ Þ xi ðtÞ, DzðtÞ ¼ zðt þ Þ zðtÞ, 0 6 pi < 1, 0 6 p < 1, i ¼ 1; 2, represents the fraction of pests and predator which die due to the pesticide at t ¼ ðn þ l 1ÞT, q > 0 is the number of predator released at t ¼ nT, n 2 Z þ and Z þ ¼ f1; 2; . . . ; g, T is the period of the impulsive effect. We will use a combination of biological and chemical tactics to eradicate the pest or keep the pest population below the damage level.
3. Notations and definitions Let Rþ ¼ ½0; 1Þ, R3þ ¼ fx 2 R3þ : x > 0g. X ¼ intR3þ , Z þ be the set of all non-negative integers. Denote f ¼ ðf1 ; f2 ; f3 Þ, the map defined by the right-hand side of the first three equations of system (2.3). Let V 0 ¼ fV : Rþ R3þ 7!Rþ g, then V is said to belong to class V 0 if (i) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ for each x 2 R3þ , n 2 Z þ , limðt;yÞ!ððnþl1ÞT þ ;xÞ Vðt; yÞ ¼ Vððn þ l 1ÞT þ ; xÞ and limðt;yÞ!ðnT þ ;xÞ Vðt; yÞ ¼ VðnT þ ; xÞ exist. (ii) V is locally Lipschitzian in x.
Definition 3.1. If V 2 V 0 , then for ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ , the upper right derivative of Vðt; xÞ with respect to the impulsive differential system (2.3) is defined as Dþ Vðt; xÞ ¼ limh!0þ sup 1h ½Vðt þ h; xþ hf ðt; xÞÞ Vðt; xÞ: The solution of the system (2.3) is a piecewise continuous function x : Rþ 7!R3þ ; xðtÞ is continuous in ððn 1ÞT; ðn þ l 1ÞT and ððn þ l 1ÞT; nT, ðn 2 Z þ ; 0 6 l 6 1Þ. Obviously, the smoothness properties of f guarantee the global existence and uniqueness of solution of system (2.3), for details see [8,9]. Definition 3.2. The species xi , i ¼ 1; 2 ðzÞ of (2.3) is said to be permanent if there exist positive constants m; M and T 0 such that each positive solution ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ of the system (2.3) satisfies m 6 xi ðtÞ 6 M, i ¼ 1; 2 ðm 6 zðtÞ 6 MÞ for all t > T 0 . If all species of the system are permanent, then the system is called permanent. The following lemma is obvious. Lemma 3.1. Let xðtÞ is a solution of system (2.3) with xð0þ Þ P 0, then xðtÞ P 0 for all t P 0. And xðtÞ > 0 for t P 0 if xð0þ Þ > 0. We will use a basic comparison result from Theorem 3.1 in [8]. For convenience, we state it in our notations. Lemma 3.2. Let V : Rþ R3þ ! Rþ and V 2 V 0 . Assume that
8 þ 6 ðn þ l 1ÞT; t 6¼ nT; > < D Vðt; xÞ 6 gðt; Vðt; xÞÞ; t ¼ Vðt; xðtþ ÞÞ 6 un ðVðt; xðtÞÞ; t ¼ ðn þ l 1ÞT; > : Vðt; xðtþ ÞÞ 6 wn ðVðt; xðtÞÞ; t ¼ nT;
ð3:1Þ
where g : Rþ Rþ 7!R is continuous in ððn 1ÞT; ðn þ l 1ÞTÞ and ððn þ l 1ÞT; nTÞ for x 2 Rþ ; n 2 Z þ , and limðt;yÞ!ððnþl1ÞT þ ;xÞ gðt; yÞ ¼ gððn þ l 1ÞT þ ; xÞ; limðt;yÞ!ðnT þ ;xÞ gðt; yÞ ¼ gðnT þ ; xÞ exist, functions un ; wn : Rþ ! Rþ are non-decreasing. Let rðtÞ be the maximal solution of the scalar impulsive differential equation
8 _ uðtÞ ¼ gðt; uðtÞÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < uðtþ Þ ¼ u ðuðtÞÞ; t ¼ ðn þ l 1ÞT; n þ > Þ ¼ w ðuðtÞÞ; t ¼ nT; uðt > n > : uð0þ Þ ¼ u0 ;
ð3:2Þ
existing on ½0; 1Þ. Then Vð0þ ; x0 Þ 6 u0 implies that Vðt; xðtÞÞ 6 rðtÞ, t P 0, where xðtÞ is any solution of (2.3). Similar result can be obtained when all the directions of the inequalities in the lemma are reversed and un ; wn are nonincreasing. Note that if we have some smoothness conditions of gðtÞ to guarantee the existence and uniqueness of solutions for (3.2), then rðtÞ is exactly the unique solution of (3.2). For convenience, we give some basic properties of the following system:
8 z_ ¼ b3 zðtÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < DzðtÞ ¼ pzðtÞ; t ¼ ðn þ l 1ÞT; > DzðtÞ ¼ q; t ¼ nT; > > : zð0þ Þ ¼ z0 P 0:
ð3:3Þ
Lemma 3.3. System (3.3) has a positive periodic solution ~zðtÞ and for every solution zðtÞ of (3.3) with initial value zð0þ Þ ¼ z0 P 0, we have zðtÞ ! ~zðtÞ as t ! 1.
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Proof. Clearly
~zðtÞ ¼
8 < q expðb3 ðtðn1ÞTÞÞ ; 1ð1pÞ expðb3 TÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expðb3 ðtðn1ÞTÞÞ ; 1ð1pÞ expðb3 TÞ
ðn þ l 1ÞT < t 6 nT;
is a positive periodic solution of (3.3). The solution of (3.3) with initial value zð0þ Þ ¼ z0 P 0 is
8 q > < ð1 pÞn1 z0 1ð1pÞ expðb expðb3 tÞ þ ~zðtÞ; ðn 1ÞT < t 6 ðn þ l 1ÞT; TÞ 3 zðtÞ ¼ q > : ð1 pÞn z0 1ð1pÞ expðb expðb3 tÞ þ ~zðtÞ; ðn þ l 1ÞT < t 6 nT; 3 TÞ n 2 Z þ . Hence, jzðtÞ ~zðtÞj ! 0 as t ! 1. The proof is complete. h Therefore, the system (2.3) has a two-pest eradication periodic solution
q expðb3 ðt ðn 1ÞTÞÞ ; ðn 1ÞT < t 6 ðn þ l 1ÞT; ð0; 0; ~zðtÞÞ ¼ 0; 0; 1 ð1 pÞ expðb3 TÞ qð1 pÞ expðb3 ðt ðn 1ÞTÞÞ ð0; 0; ~zðtÞÞ ¼ 0; 0; ; ðn þ l 1ÞT < t 6 nT: 1 ð1 pÞ expðb3 TÞ 4. Extinction In this section, we study the stability of the two-pest eradication periodic solution of the full system (2.3). Theorem 4.1. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of (2.3). Then ð0; 0; ~zðtÞÞ is globally asymptotically stable provided that 3 lTÞð1pÞ expðb3 TÞÞ f1 ðTÞ ¼ b1 T gqð1pbexpðb ln 3 ð1ð1pÞ expðb3 TÞÞ 3 lTÞð1pÞ expðb3 TÞÞ ln f2 ðTÞ ¼ b2 T lqð1pbexpðb 3 ð1ð1pÞ expðb3 TÞÞ
1 1p1 1 1p2
< 0; ð4:1Þ < 0:
Proof. Similar to Theorem 3.1 of Liu et al. [21], we can prove that the two-pest eradication periodic solution ð0; 0; ~zðtÞÞ is locally asymptotically stable, we omit it here. In the following, we prove the global attractivity. Noting that
x_ 1 6 x1 ðb1 x1 Þ; t 6¼ ðn þ l 1ÞT; Dx1 ¼ p1 x1 ; t ¼ ðn þ l 1ÞT:
Consider the following impulsive differential equation:
8 _ ¼ wðtÞðb1 wðtÞÞ; t 6¼ ðn þ l 1ÞT; > < wðtÞ DwðtÞ ¼ p1 wðtÞ; t ¼ ðn þ l 1ÞT; > : wð0þ Þ ¼ x1 ð0þ Þ P 0; if t 6¼ ðn þ l 1ÞT, we have x1 ðtÞ 6 wðtÞ and wðtÞ ! b1 as t ! 1; if t ¼ ðn þ l 1ÞT, we have x1 ðtþ Þ ¼ ð1 p1 Þx1 ðtÞ < x1 ðtÞ. Hence, there exists a e1 > 0 such that x1 ðtÞ < b1 þ 1 for t large enough. Without loss of generality, we may assume that x1 ðtÞ < b1 þ 1 for t > 0. Similarly, we can assume that x2 ðtÞ < b2 þ 2 for t > 0. Choose a e > 0 such that RT ~ d ¼ ð1 p1 Þ exp 0 b1 1þgxðz1ðtÞeÞ dt < 1. Note that z_ ðtÞ P b3 zðtÞ, considering the comparison system: ðb1 þe1 Þ
8 _ yðtÞ ¼ b3 yðtÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < DyðtÞ ¼ pyðtÞ; t ¼ ðn þ l 1ÞT; > DyðtÞ ¼ q; t ¼ nT; > > : yð0þ Þ ¼ z0 P 0:
ð4:2Þ
From Lemmas 3.2 and 3.3, we have zðtÞ P yðtÞ and yðtÞ ! ~zðtÞ as t ! 1. Then
zðtÞ P yðtÞ > ~zðtÞ e
ð4:3Þ
holds for all t large enough. For simplification, we may assume that (4.3) holds for all t P 0. From (2.3) we get
(
~ x_ 1 ðtÞ 6 x1 ðtÞ b1 1þgxðz1ðtÞeÞ ; ðb1 þe1 Þ
Dx1 ðtÞ ¼ p1 x1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð4:4Þ
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Integrate (4.4) on ððn þ l 1ÞT; ðn þ lÞT, which yields
x1 ððn þ lÞTÞ 6 ð1 p1 Þx1 ððn þ l 1ÞTÞ exp
Z
b1
ðnþlÞT
ðnþl1ÞT
!
gð~zðtÞ eÞ dt ¼ x1 ððn þ l 1ÞTÞd: 1 þ x1 ðb1 þ e1 Þ
Thus x1 ððn þ lÞTÞ 6 x1 ðlTÞdn and x1 ððn þ lÞTÞ ! 0 as n ! 1. Therefore, x1 ðtÞ ! 0 as n ! 1, since 0 < x1 ðtÞ 6 ð1 p1 Þx1 ððn þ l 1ÞTÞ expðb1 TÞ for ðn þ l 1ÞT < t 6 ðn þ lÞT. By the same method we can prove x2 ðtÞ ! 0 as n ! 1, so we omit it. Next, we prove that zðtÞ ! ~zðtÞ as t ! 1 if limt!1 x1 ðtÞ ¼ 0 and limt!1 x2 ðtÞ ¼ 0 hold. For 0 < e < dðgbþ3 lÞ, there exists a T 1 > 0 such that 0 < x1 ðtÞ < e and 0 < x2 ðtÞ < e hold for all t > T 1 . Without loss of generality, we may assume that 0 < x1 ðtÞ < e and 0 < x2 ðtÞ < e hold for all t > 0. Then we have b3 zðtÞ 6 z_ ðtÞ 6 zðtÞðb3 þ dge þ dleÞ. By Lemmas 3.2 and 3.3, ~1 ðtÞ as t ! 1, where yðtÞ and y1 ðtÞ are solutions of Eq. (4.2) and the we obtain yðtÞ 6 zðtÞ 6 y1 ðtÞ, yðtÞ ! ~zðtÞ and y1 ðtÞ ! y following impulsive differential equation:
8 y_1 ðtÞ ¼ y1 ðtÞðb3 þ dge þ dleÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < Dy ðtÞ ¼ py ðtÞ; t ¼ ðn þ l 1ÞT; 1 1 > Dy1 ðtÞ ¼ q; t ¼ nT; > > : y1 ð0þ Þ ¼ z0 P 0;
ð4:5Þ
respectively; and
~1 ðtÞ ¼ y
8 < q expððb3 þdgeþdleÞðtðn1ÞTÞÞ ; 1ð1pÞ expððb3 þdgeþdleÞTÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expððb3 þdgeþdleÞðtðn1ÞTÞÞ ; 1ð1pÞ expððb3 þdgeþdleÞTÞ
ðn þ l 1Þ < t 6 nT:
~1 ðtÞ þ e0 for t large enough. Let e ! 0, we get y ~1 ðtÞ ! ~zðtÞ: Hence, Therefore, there exists a e0 > 0 such that ~zðtÞ e0 < zðtÞ < y zðtÞ ! ~zðtÞ as t ! 1. This completes the proof. h 5. Permanence First, from Theorem 5.1 in [20], we know that all solutions of (2.3) are uniformly ultimately bounded. Theorem 5.1. For each positive solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ of (2.3) there exists a constant M > 0, such that xi ðtÞ 6 M(i ¼ 1; 2), and zðtÞ 6 M with t large enough. In the following, we investigate the permanence of the system (2.3). Theorem
5.2. System
(2.3)
is
permanent
gqð1p expððb3 þdlb2 ÞlTÞð1pÞ expððb3 þdlb2 ÞTÞÞ 1 ln 1p ðb3 dlb2 Þð1ð1pÞ expðb3 þdlb2 ÞTÞÞ 1
1 1p2
if > 0;
a < bb12 ,
l < dbb32 ,
b < bb21 ;
b3 g < db , 1
and
f3 ðTÞ ¼ ðb1 ab2 ÞT
3 þdgb1 ÞlTÞð1pÞ expððb3 þdgb1 ÞTÞÞ f4 ðTÞ ¼ ðb2 bb1 ÞT lqð1pðbexpððb ln 3 dgb1 Þð1ð1pÞ expððb3 þdgb1 ÞTÞÞ
> 0.
þ Proof. Suppose n o that xðtÞ is a solution of (2.3) with xð0 Þ > 0, from Theorem 5.1, we may assume that zðtÞ 6 M and b1 b2 M > max g ; l hold for t > 0. From the proof of Theorem 4.1 we can assume that x1 ðtÞ < b1 þ e1 and x2 ðtÞ < b2 þ e2 hold qð1pÞ expðb3 TÞ for t > 0. Let m ¼ 1ð1pÞ e > 0, where e > 0. According to Lemmas 3.2 and 3.3, we have zðtÞ > m for t large enough. expðb3 TÞ 1 > 0 and m 2 > 0, such that x1 ðtÞ P m 1 and x2 ðtÞ P m 2 for t large enough. We will do it in In the following, we want to find m the following two steps for convenience.
Step I: Let 0 < m1 < b3 dldðbg 2 þe2 Þ ; 0 < m2 < b3 dgdðbl1 þe1 Þ ; and dgm1 þ dlm2 < b3 . We will prove that there exist t1 ; t2 2 ð0; 1Þ, such that x1 ðt 1 Þ P m1 and x2 ðt2 Þ P m2 . Otherwise, there will be three cases: (i) There exists a t2 > 0, such that x2 ðt 2 Þ P m2 and x1 ðtÞ < m1 , for all t > 0; (ii) There exists a t1 > 0, such that x1 ðt 1 Þ P m1 and x2 ðtÞ < m2 , for all t > 0; (iii) x1 ðtÞ < m1 ; x2 ðtÞ < m2 , for all t > 0. We first consider case (i). Let e1 > 0 small enough such that
r1 ¼ ð1 p1 Þ exp ðb1 m1 aðb2 þ e2 Þ ge1 ÞT
gqð1 p expðhlTÞ ð1 pÞ expðhTÞÞ ðhÞð1 ð1 pÞ expðhTÞÞ
> 1;
where h ¼ b3 þ dgm1 þ dlðb2 þ e2 Þ. According to the above assumption, we get z_ ðtÞ 6 hzðtÞ: By Lemmas 3.2 and 3.3, we ~2 ðtÞ as t ! 1, where y2 ðtÞ is the solution of have zðtÞ 6 y2 ðtÞ and y2 ðtÞ ! y
8 y_ 2 ðtÞ ¼ hy2 ðtÞ; t 6¼ ðn þ lÞT; t 6¼ nT; > > > < Dy2 ðtÞ ¼ py2 ðtÞ; t ¼ ðn þ l 1ÞT; > > Dy2 ðtÞ ¼ q; t ¼ nT; > : y2 ð0þ Þ ¼ z0 P 0;
ð5:1Þ
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Y. Li et al. / Applied Mathematics and Computation 204 (2008) 269–280
where
~2 ðtÞ ¼ y
8 < q expðhðtðn1ÞTÞÞ ; 1ð1pÞ expðhTÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expðhðtðn1ÞTÞÞ ; 1ð1pÞ expðhTÞ
ðn þ l 1ÞT < t 6 nT:
~2 ðtÞ þ e1 . Then we have Therefore, there exists a T 1 > 0, such that zðtÞ 6 y2 ðtÞ < y
~2 ðtÞ þ e1 ÞÞ; x_ 1 ðtÞ P x1 ðtÞðb1 m1 aðb2 þ e2 Þ gðy x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
ð5:2Þ
t ¼ ðn þ l 1ÞT;
for t P T 1 . Let N 2 Z þ and ðN þ l 1ÞT P T 1 . Integrating (5.2) on ððn þ l 1ÞT; ðn þ lÞT, n P N, we have
x1 ððn þ lÞTÞ P x1 ððn þ l 1ÞTÞð1 p1 Þ exp
Z
!
ðnþlÞT
~2 ðtÞ þ e1 ÞÞdt ðb1 m1 aðb2 þ e2 Þ gðy
¼ x1 ððn þ l 1ÞTÞr1 :
ðnþl1ÞT
Then x1 ððN þ n þ lÞTÞ P x1 ðN þ lÞTÞrn1 ! 1 as n ! 1; which is a contradiction to the boundedness of x1 ðtÞ. Cases (ii) and (iii) can be analyzed by the same method as case (i), so we omit it. From the above three cases, we conclude that there exist t1 > 0, t 2 > 0 such that x1 ðt1 Þ P m1 , x2 ðt2 Þ P m2 . Step II: If x1 ðtÞ P m1 for all t P t 1 , then our aim is obtained. Otherwise, we have x1 ðtÞ < m1 for some t P t1 . Setting t ¼ inf t>t1 fx1 ðtÞ < m1 g. Then t can be divided into two cases, t is impulsive point or not impulsive point. Case (a): If t ¼ ðn1 þ l 1ÞT for n1 2 Z þ . Then x1 ðtÞ P m1 and ð1 p1 Þm1 6 x1 ðt þ Þ ¼ ð1 p1 Þx1 ðt Þ < m1 hold for t 2 ½t 1 ; t . Select n2 ; n3 2 Z þ such that
ðn2 1ÞT >
1 e1 ln ; b3 þ dgm1 þ dlðb2 þ e2 Þ Mþq
and
ð1 p1 Þn2 rn13 expðn2 d0 TÞ > ð1 p1 Þn2 rn13 expððn2 þ 1Þd0 TÞ > 1; where d0 ¼ b1 m1 aðb2 þ e2 Þ gM < 0. Set T 0 ¼ ðn2 þ n3 ÞT. We claim that there must be exist a t 2 2 ðt ; t þ T 0 such that x1 ðt 2 Þ > m1 . Otherwise, we have x1 ðtÞ < m1 for t 2 ðt ; t þ T 0 . Consider (5.2) with y2 ðn1 T þ Þ ¼ zðn1 T þ Þ, we have
8 > ~2 ðtÞ; ðn 1ÞT < t 6 ðn þ l 1ÞT; < ð1 pÞnðn1 þ1Þ y2 ðn1 T þ Þ 1ð1pÞq expðhTÞ expðhðt n1 TÞÞ þ y y2 ðtÞ ¼ > : ð1 pÞnn1 y2 ðn1 T þ Þ 1ð1pÞq expðhTÞ expðhðt n1 TÞÞ þ y ~2 ðtÞ; ðn þ l 1ÞT < t 6 nT; ~2 ðtÞj < ðM þ qÞ expðhðt n1 TÞÞ < e1 ; and zðtÞ 6 y2 ðtÞ < y ~2 ðtÞ þ e1 for t 2 ððn 1ÞT; nT; n1 þ 1 6 n 6 n1 þ n2 þ n3 . Then jy2 ðtÞ y for n1 T þ ðn2 1ÞT 6 t 6 t þ T 0 , which implies that (5.2) holds for t þ n2 T 6 t 6 t þ T 0 . As case (i), we have x1 ðt þ T 0 Þ P x1 ðt þ n2 TÞrn13 : Note that
x_ 1 ðtÞ P x1 ðtÞðb1 m1 aðb2 þ e2 Þ gMÞ; x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð5:3Þ
Integrating (5.3) on ½t ; t þ n2 T, we have x1 ðt þ n2 TÞ P m1 ð1 p1 Þn2 expðn2 d0 TÞ. Thus x1 ðt þ T 0 Þ P m1 ð1 p1 Þn2 expðn2 d0 TÞrn13 > m1 , which is a contradiction.Let ~t ¼ inf t>t fx1 ðtÞ P m1 g. Then x1 ðtÞ < m1 holds for t 2 ðt ; ~t. Since x1 ðtÞ is left continuous and x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ 6 x1 ðtÞ as t ¼ ðn þ l 1ÞT, so ~t cannot be impulsive point. Suppose t 2 ðt þ ðl 1ÞT; t þ lT; l 2 Z þ ; l 6 n2 þ n3 . Integrating (5.3) on ½t ; ~tÞ one yields
x1 ðtÞ P x1 ðtþ Þð1 p1 Þl1 expððl 1Þd0 TÞ expðd0 ðt ðt þ ðl 1ÞTÞÞÞ P m1 ð1 p1 Þl expðld0 TÞ P m1 ð1 p1 Þn2 þn3 expððn2 þ n3 Þd0 TÞ: Let m01 ¼ m1 ð1 p1 Þn2 þn3 expððn2 þ n3 Þd0 TÞ. So we have x1 ðtÞ P m01 for t 2 ðt ; ~t. For t > ~t, the same argument can be continued since x1 ð~tÞ P m1 . Case (b): If t 6¼ ðn þ l 1ÞT, n 2 Z þ . Then x1 ðtÞ P m1 for t 2 ½t 1 ; t Þ and x1 ðt Þ ¼ m1 . We can assume t 2 ððn01 þ l 1ÞT; ðn01 þ lÞTÞ; n01 2 Z þ . For t 2 ðt ; ðn01 þ lÞTÞ, there are two possible cases for x1 ðtÞ. If x1 ðtÞ 6 m1 holds for t 2 ðt ; ðn01 þ lÞTÞ. Similar to case (a), we can prove that there must exist a t02 2 ½ðn01 þ lÞT; ðn01 þ lÞT þ T 0 such that x1 ðt 02 Þ P m1 . Here, we omit it. Let t ¼ inf t>t fx1 ðtÞ > m1 g, then x1 ðtÞ ¼ m1 and 0 0 x1 ðtÞ 6 m1 hold for t 2 ðt ; tÞ. For t 2 ðt ; tÞ, we have x1 ðtÞ P m1 ð1 p1 Þl expððl þ 1Þd0 TÞ P 1 ¼ m1 ð1 p1 Þn2 þn3 expððn2 þ n3 þ 1Þd0 TÞ < m01 , then x1 ðtÞ P m 1 holds for m1 ð1 p1 Þn2 þn3 expððn2 þ n3 þ 1Þd0 TÞ. Let m 1 . Hence, x1 ðtÞ P m 1 for all t > t1 . t 2 ðt ; tÞ. For t > t, the same argument can be continued since x1 ðtÞ P m If there exists a t 2 ðt ; ðn01 þ 1ÞT such that x1 ðtÞ > m1 . Let ^t ¼ inf t>t fx1 ðtÞ > m1 g, then x1 ðtÞ 6 m1 holds for t 2 ½t ; ^tÞ and 1 : This process can be continued since x1 ð^tÞ ¼ m1 . For t 2 ðt ; ^tÞ, we have x1 ðtÞ P x1 ðt Þ expðd0 ðt t ÞÞ P m1 expðd0 TÞ > m 2 holds for all t P t 2 . The proof is complete. h x1 ð^tÞ P m1 holds for t > ^t1 . Similarly, we can prove that x2 ðtÞ P m
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Fig. 1. The trajectory fi ðTÞ has a unique positive root T ðiÞ , where a ¼ 0:6; b ¼ 0:5; b1 ¼ 3; b2 ¼ 2; b3 ¼ 3; g ¼ 0:2; l ¼ 0:7; d ¼ 0:2; w1 ¼ 1; w2 ¼ 1; p ¼ 0:4; p1 ¼ 0:4; p2 ¼ 0:3; q ¼ 0:3; l ¼ 0:5.
From the proof of Theorems 4.1 and 5.2, we can derive the following results. Corollary 5.1. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of system (2.3), then x2 and z are permanent and x1 ðtÞ ! 0 as t ! 1 provided b3 that b < bb21 ; g < db ; f1 ðTÞ < 0; f4 ðTÞ > 0. 1 Corollary 5.2. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of system (2.3), then x1 and z are permanent and x2 ðtÞ ! 0 as t ! 1 provided b3 that a < bb12 ; l < db ; f2 ðTÞ < 0; f3 ðTÞ > 0. 2 1 Remark 5.1. Since fi ð0Þ ¼ ln 1p < 0; fi ðTÞ ! þ1 as T ! þ1, and fi00 ðTÞ > 0, so fi ðTÞ ¼ 0 has a unique positive root, 1 denoted by T i ði ¼ 1; 3Þ. 1 < 0; fj ðTÞ ! þ1 as T ! þ1, and fj00 ðTÞ > 0, so fj ðTÞ ¼ 0 has a unique positive root, denoted by Since fj ð0Þ ¼ ln 1p 2 T j ðj ¼ 2; 4Þ (see Fig. 1). 6. Discussion and numerical simulation In this paper, we investigate the dynamics of a two-prey one-predator system with impulsive effect concerning biological and chemical control strategy. We show that there exists a two-pest eradication periodic solution which is globally asymptotically stable, and get the sufficient conditions for the permanence of the system (2.3). From Corollaries 5.1 and 5.2, we also get the sufficient conditions for one of two-prey extinction and the remaining two pieces permanence.
Fig. 2. Time-series of system (2.3) with x1 ; x2 eradication (T ¼ 0:1).
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Fig. 3. Time-series of system (2.3) with positive periodic solution (T ¼ 0:9).
Fig. 4. Time-series of system (2.3) with x2 eradication (T ¼ 0:19).
From Theorem 4.1, we know that the two-pest eradication periodic solution ð0; 0; ~zðtÞÞ is globally asymptotically stable if D b1 ¼ T T maxfT 3 ; T 4 g, that is to say, three species coexist if impulsive period is larger than T a ¼ 0:6; b ¼ 0:5; b1 ¼ 3; b2 ¼ 2; b3 ¼ 3; g ¼ 0:2; l ¼ 0:7; d ¼ 0:2; w1 ¼ 1; w2 ¼ 1; p1 ¼ 0:4; p2 ¼ 0:3; p ¼ 0:4; q ¼ 0:3; l ¼ 0:5, then we can calculate T 3 0:18; T 4 0:84, let T ¼ 0:9, then T > maxfT 3 ; T 4 g. Then three species coexist if impulsive period b 2 ¼ 0:84. From Corollaries 5.1 and 5.2, we know that species x1 extinct but x2 and z are permanence if is larger than T T 4 < T < T 1 ; species x2 extinct, x1 and z are permanent if T 3 < T < T 2 . Therefore, we can drive the target pest population to extinct and let the non-target pest (or harmless insect) be permanent by choosing the impulsive period, this can be seen clearly from Fig. 4. Parameters are the same as Fig. 3, then T 2 0:2; T 3 0:18, let T ¼ 0:19, then T 3 < T < T 2 . So, if one prey is target pest, another is non-target pest, we can choose the impulsive period T to obtain our aim. For example in Fig. 4, when 0:18 < T < 0:2, x2 rapidly reduces to zero, and x1 and z tend to positive periodic solution.
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Fig. 7. Spiral chaos for system (2.3) with g ¼ 7:3. (a–c) Time-series of species x1 ; x2 and z; and (d) phase portrait.
Fig. 8. Bifurcation diagrams of system (2.3) with 65 6 T 6 150 and initial value x10 ¼ 0:12; x20 ¼ 0:78; z0 ¼ 0:042.
In the following, we will investigate the influence of bifurcation diagrams on the system (2.3). Fig. 5 shows the bifurcation diagram of system (2.3) for b1 ¼ 1; b2 ¼ 1; b3 ¼ 1; a ¼ 0:99; b ¼ 1:5; d ¼ 0:5; l ¼ 1; p1 ¼ 0:05; p2 ¼ 0:05; p ¼ 0:05; l ¼ 0:05; q ¼ 0:05; w1 ¼ 0:001; w2 ¼ 0:001; T ¼ 125, and g varying from 6.8 to 8.4. Further increasing g, we can see that the dynamical behavior of system (2.3) is very complicated, including a sequence of direct and inverse cascade of periodic-dou-
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Fig. 9. Chaos of system (2.3) with T ¼ 75, (a–c) time-series of species x1 ; x2 and z with initial value x10 ¼ 0:12; x20 ¼ 0:78; z0 ¼ 0:042, and (d) phase portrait of system (2.3).
bling, chaos, and symmetry breaking bifurcation. Fig. 6 shows periodic-doubling cascade. The bifurcation diagrams show the route to chaos through cascade of periodic-doubling, which has been extensively studied by mathematicians [22,23]. A typical spiral chaos oscillation is captured when g ¼ 7:3 (see Fig. 7). Fig. 8 shows the bifurcation diagrams of system (2.3) for b1 ¼ 1; b2 ¼ 1; b3 ¼ 1; a ¼ 0:99; b ¼ 1:5; d ¼ 0:5; g ¼ 7:1; l ¼ 1; p1 ¼ 0:001; p2 ¼ 0:001; p ¼ 0:05; l ¼ 0:05; q ¼ 0:05; w1 ¼ 0:001; w2 ¼ 0:001, and T varying from 65 to 150. If we let 65 6 T 6 70, there species coexist periodically, with further increasing T, periodic-doubling cascade occurs. Further increasing T again, we can see that the dynamical behavior of system (2.3) is very complicated, including a sequence of periodichalving cascade, cycles, periodic-doubling cascade and chaos. In Fig. 8, we find that there are occurrences of sudden changes in the types of the attractors, which is a typical feature of bifurcation diagrams. For example, a T-periodic solution suddenly changes to 2T-periodic solution when T 70 and 2T-periodic solution suddenly changes to 4T-periodic solution when T 72:1 and 4T-periodic solution suddenly changes to chaos when T 73. Fig. 9 shows chaos of system (2.3) for T ¼ 75. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 10771104 and 10771179), the Henan Innovation Project for University Prominent Research Talents (No. 2005KYCX017) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] P. DeBach, D. Rosen, Biological Control by Natural Enemies, second ed., Cambridge University Press, 1991. [2] A.J. Cherry, C.J. Lomer, D. Djegui, F. Schulthess, Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in west Africa, Biocontrol 44 (1999) 301–327. [3] P. Ferron, Pest control using the fungi Beauveria and Metarhizinm, in: H.D. Burges (Ed.), Microbial Control in Pests and Plant Disease, Academic, London, 1981. [4] H.J. Freedman, Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci. 31 (1976) 207–225. [5] J. Grasman, O.A. Van Herwaarden, et al, A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control, Math. Biosci. 169 (2001) 207–216. [6] P. DeBach, Biological Control of Insect Pests and Weeds, Rheinhold, New York, 1964. [7] M.L. Luff, The potential of predators for pest control, Agri. Ecos. Environ. 10 (1983) 159–181. [8] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [9] D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, England, 1993. [10] Z. Agur, L. Cojocaru, R. Anderson, Y. Danon, Pulse mass measles vaccination across age cohorts, Proc. Natl. Acad. Sci. 90 (1993) 11698–11702. [11] B. Shulgin, L. Stone, I. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60 (1998) 1–26. [12] G. Ballinger, X. Liu, Permanence of population growth models with impulsive effects, Math. Comput. Modell. 26 (1997) 59–72.
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[13] X.N. Liu, L.S. Chen, Complex dynamics of Holling type II Lotka–Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons and Fractals 16 (2003) 311–320. [14] S.Y. Tang, L.S. Chen, Density-dependent birth rate, birth pulse and their population dynamic consequences, J. Math. Biol. 44 (2002) 185–199. [15] M.G. Roberts, R.R. Kao, The dynamics of an infectious disease in a population with birth pulse, Math. Biosci. 149 (1998) 23–36. [16] A. Lakmeche, O. Arino, Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. Cont. Discr. Impuls. Syst. 7 (2000) 265–287. [17] J.C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol. 58 (1996) 425–447. [18] C.S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Sec. Can. 45 (1965) 1–60. [19] S.W. Zhang, L.S. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos, Solitons and Fractals 24 (2005) 1269– 1278. [20] X.Y. Song, Y.F. Li, Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect, Chaos, Solitons and Fractals 33 (2007) 463– 478. [21] B. Liu, Y.J. Zhang, L.S. Chen, Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control, Chaos, Solitons and Fractals 22 (2004) 123–134. [22] A. Venkatesan, S. Parthasarathy, M. Lakshmanan, Occurrence of multiple periodic-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems, Chaos, Solitons and Fractals 18 (2003) 891–898. [23] J.P. Eckmann, Routes to chaos with special emphasis on periodic-doubling, in: G. Iooss et al. (Eds.), Chaotic Behavior of Deterministic Systems, Elsevier North Holland, Amsterdam, 1983.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of a predator-prey system with pulses Yongfeng Li a,*, Jingan Cui a, Xinyu Song b a b
Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, PR China Department of Mathematics, Xinyang Normal University, Xinyang 464000, PR China
a r t i c l e
i n f o
Keywords: Predator-prey system Impulsive effect Permanence Extinction Stability Bifurcation
a b s t r a c t In this paper, we investigate the dynamic behaviors of a Holling II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategyperiodic releasing natural enemies and spraying pesticide at different fixed moment. By using the Floquet theory of linear periodic impulsive equation and small-amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is larger than some critical value, and meanwhile the conditions for the extinction of one of the two-prey and permanence of the remaining two species are given. Finally, we give numerical simulation, with increasing of predation rate for the super competitor and impulsive period, the system displays complicated behaviors including a sequence of direct and inverse cascades of periodic-doubling, periodic-halving, chaos and symmetry breaking bifurcation. Our results suggest a new approach in the pest control. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction It is well known, many insects are beneficial to human, but some insects are harmful to human, only these harmful insects can cause economic damage as their population reaching economic injury level. Controlling harmful insects and other arthropods has become an important issue in recent years, for instance, minimizing losses due to insect pests and insect vectors remains an essential component of the programmes in the Office of Agricultural Entomology in China. Overusing a single control tactic is discouraged to minimize damage to non-target organisms, and to preserve the quality of the environment. Overusing a single control tactic also can lead pest to produce resistance to chemical control (for example, pesticide), it will be more difficult to control pest later. Then biological and chemical control were introduced. Biological control [1–7] is the purposeful introduction one or more natural enemies 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. Natural enemies are able to play a more active role in suppressing insect pests. Usually, predators feed on not only insect pests but also other insects. There may be more than one pest species – for example, the two species of aphids predominant in small grains: the English grain aphid and the oatbird cherry aphid. Aphids’ high reproductive rate enables their populations to quickly build up to levels that can cause an economic loss. However, aphids are usually kept in check by biological control agents, such as lady beetles, parasitic wasps, and syrphid fly maggots which are often abundant in small grains. 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 releasing natural enemies, and by genetic enhancement of the enemies to increase their effectiveness at control.
* Corresponding author. E-mail address: yfl[email protected] (Y. Li). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.06.037
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Wherever possible, different pest control techniques should work together rather than against each other. In this paper, according to periodic biological and chemical control, and based on two-prey one-predator system with Holling II functional response, we suggest a simple mathematical model with pulses to describe the process of periodic releasing natural enemies and spraying pesticide (or harvesting pests) at different fixed moments. System with impulsive effects describing evolution processes are characterized by the fact that at certain moments of time they abruptly experience a change of state. Processes of such character are studied in almost every domain of applied sciences. Numerous examples are given in Bainov’s and his collaborator’s books [8,9]. Some impulsive equations have been recently introduced into population dynamics in relation to: vaccination [10,11], population ecology [12,13], and impulsive birth [14,15], chemotherapeutic [16,17]. The paper is arranged like this. A Holling II two-prey one-predator system concerning biological and chemical control is given in Section 2. In Section 3, we give some notations and lemmas. In Section 4, we consider the local stability and global asymptotic stability of the two-pest eradication periodic solution by using Floquet theory for the impulsive equation, smallamplitude perturbation skills and techniques of comparison, and in Section 5 we show that the system is permanent if the impulsive period is larger than some critical value. Moreover, we give the sufficient conditions for one of two-prey extinction and the remaining two species permanence. A brief discussion and further numerical simulation are given in the last section.
2. Model formulation Based on many experiments, Holling [18] suggested three different kinds of functional response for different kinds of species to model the phenomenon of predation, which made the standard Lotka–Volterra systems more realistic. Liu and Chen [13] investigated complex dynamics of Holling type II Lotka–Volterra predator-prey system with impulsive perturbations on the predator. Zhang and Chen [19] studied a Holling II functional response food chain model with impulsive perturbations. The model we considered in this paper is based on the following predator-prey model, where two-prey are competitive and the predator has Holling II functional response, that is
8 gzðtÞ > _ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx x ; > x ðtÞ > 1 1 > < l zðtÞ x_ 2 ðtÞ ¼ x2 ðtÞ b2 bx1 ðtÞ x2 ðtÞ 1þx2 x2 ðtÞ ; > > > > : z_ ðtÞ ¼ zðtÞ b þ dgx1 ðtÞ þ dlx2 ðtÞ ; 3 1þx1 x1 ðtÞ 1þx2 x2 ðtÞ
ð2:1Þ
where xi ðtÞ (i ¼ 1; 2) is the population size of prey (pest) species and zðtÞ is the population size of predator (natural enemies) species, bi > 0 ði ¼ 1; 2; 3Þ are intrinsic rates of increase or decrease, a > 0 and b > 0 are parameters representing competigx1 ðtÞzðtÞ lx2 ðtÞzðtÞ tive effects between two-prey, g > 0 and l > 0, 1þ x1 x1 ðtÞ and 1þx2 x2 ðtÞ are the Holling type II functional responses, d > 0 is the rate of conversing prey into predator. Model (2.1) with constant periodic releasing predator and spraying pesticide (or harvesting pests) was studied by Song and Li [20]:
9 8 gzðtÞ > x_ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx ;> > > x1 ðtÞ > > 1 > > = > > > l zðtÞ > > x_ 2 ðtÞ ¼ x2 ðtÞ b2 x2 ðtÞ bx1 ðtÞ 1þx2 x2 ðtÞ ; t 6¼ nT; > > > > > < > > dgx1 ðtÞ dlx2 ðtÞ ; z_ ðtÞ ¼ zðtÞ b3 þ 1þx1 x1 ðtÞ þ 1þx2 x2 ðtÞ ; > 9 > > > Dx ðtÞ ¼ p1 x1 ðtÞ; > > > = > 1 > > Dx2 ðtÞ ¼ p2 x2 ðtÞ; t ¼ nT; > > > > : ; DzðtÞ ¼ q;
ð2:2Þ
where Dxi ðtÞ ¼ xi ðt þ Þ xi ðtÞ; DzðtÞ ¼ zðt þ Þ zðtÞ. T is the period of the impulsive effect, pi > 0ði ¼ 1; 2Þ is the proportionality constant which represents the rate of mortality due to applying pesticide. q > 0 is the number of predator released each time. Now we will develop systems (2.1) and (2.2) by introducing a constant periodic releasing natural enemies and spraying pesticide (or harvesting pests) at different fixed moment. That is, we consider the following impulsive differential equations:
9 8 gzðtÞ > _ 1 ðtÞ ¼ x1 ðtÞ b1 x1 ðtÞ ax2 ðtÞ 1þx ;> x > > x ðtÞ > > 1 1 > > > = > > l zðtÞ > _ x ; ðtÞ ¼ x ðtÞ b x ðtÞ bx ðtÞ t 6¼ ðn þ l 1ÞT; t 6¼ nT; > 2 2 2 2 1 > x x ðtÞ 1þ 2 2 > > > > > > > > < z_ ðtÞ ¼ zðtÞ b þ dgx1 ðtÞ þ dlx2 ðtÞ ; ; 3 1þx1 x1 ðtÞ 1þx2 x2 ðtÞ > Dx ðtÞ ¼ p x ðtÞ; > i > i i > > t ¼ ðn þ l 1ÞT; > DzðtÞ ¼ pzðtÞ; > > > > > > Dxi ðtÞ ¼ 0; > > t ¼ nT; : DzðtÞ ¼ q;
ð2:3Þ
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where 0 6 l 6 1, Dxi ðtÞ ¼ xi ðtþ Þ xi ðtÞ, DzðtÞ ¼ zðt þ Þ zðtÞ, 0 6 pi < 1, 0 6 p < 1, i ¼ 1; 2, represents the fraction of pests and predator which die due to the pesticide at t ¼ ðn þ l 1ÞT, q > 0 is the number of predator released at t ¼ nT, n 2 Z þ and Z þ ¼ f1; 2; . . . ; g, T is the period of the impulsive effect. We will use a combination of biological and chemical tactics to eradicate the pest or keep the pest population below the damage level.
3. Notations and definitions Let Rþ ¼ ½0; 1Þ, R3þ ¼ fx 2 R3þ : x > 0g. X ¼ intR3þ , Z þ be the set of all non-negative integers. Denote f ¼ ðf1 ; f2 ; f3 Þ, the map defined by the right-hand side of the first three equations of system (2.3). Let V 0 ¼ fV : Rþ R3þ 7!Rþ g, then V is said to belong to class V 0 if (i) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ for each x 2 R3þ , n 2 Z þ , limðt;yÞ!ððnþl1ÞT þ ;xÞ Vðt; yÞ ¼ Vððn þ l 1ÞT þ ; xÞ and limðt;yÞ!ðnT þ ;xÞ Vðt; yÞ ¼ VðnT þ ; xÞ exist. (ii) V is locally Lipschitzian in x.
Definition 3.1. If V 2 V 0 , then for ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R3þ and ððn þ l 1ÞT; nT R3þ , the upper right derivative of Vðt; xÞ with respect to the impulsive differential system (2.3) is defined as Dþ Vðt; xÞ ¼ limh!0þ sup 1h ½Vðt þ h; xþ hf ðt; xÞÞ Vðt; xÞ: The solution of the system (2.3) is a piecewise continuous function x : Rþ 7!R3þ ; xðtÞ is continuous in ððn 1ÞT; ðn þ l 1ÞT and ððn þ l 1ÞT; nT, ðn 2 Z þ ; 0 6 l 6 1Þ. Obviously, the smoothness properties of f guarantee the global existence and uniqueness of solution of system (2.3), for details see [8,9]. Definition 3.2. The species xi , i ¼ 1; 2 ðzÞ of (2.3) is said to be permanent if there exist positive constants m; M and T 0 such that each positive solution ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ of the system (2.3) satisfies m 6 xi ðtÞ 6 M, i ¼ 1; 2 ðm 6 zðtÞ 6 MÞ for all t > T 0 . If all species of the system are permanent, then the system is called permanent. The following lemma is obvious. Lemma 3.1. Let xðtÞ is a solution of system (2.3) with xð0þ Þ P 0, then xðtÞ P 0 for all t P 0. And xðtÞ > 0 for t P 0 if xð0þ Þ > 0. We will use a basic comparison result from Theorem 3.1 in [8]. For convenience, we state it in our notations. Lemma 3.2. Let V : Rþ R3þ ! Rþ and V 2 V 0 . Assume that
8 þ 6 ðn þ l 1ÞT; t 6¼ nT; > < D Vðt; xÞ 6 gðt; Vðt; xÞÞ; t ¼ Vðt; xðtþ ÞÞ 6 un ðVðt; xðtÞÞ; t ¼ ðn þ l 1ÞT; > : Vðt; xðtþ ÞÞ 6 wn ðVðt; xðtÞÞ; t ¼ nT;
ð3:1Þ
where g : Rþ Rþ 7!R is continuous in ððn 1ÞT; ðn þ l 1ÞTÞ and ððn þ l 1ÞT; nTÞ for x 2 Rþ ; n 2 Z þ , and limðt;yÞ!ððnþl1ÞT þ ;xÞ gðt; yÞ ¼ gððn þ l 1ÞT þ ; xÞ; limðt;yÞ!ðnT þ ;xÞ gðt; yÞ ¼ gðnT þ ; xÞ exist, functions un ; wn : Rþ ! Rþ are non-decreasing. Let rðtÞ be the maximal solution of the scalar impulsive differential equation
8 _ uðtÞ ¼ gðt; uðtÞÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < uðtþ Þ ¼ u ðuðtÞÞ; t ¼ ðn þ l 1ÞT; n þ > Þ ¼ w ðuðtÞÞ; t ¼ nT; uðt > n > : uð0þ Þ ¼ u0 ;
ð3:2Þ
existing on ½0; 1Þ. Then Vð0þ ; x0 Þ 6 u0 implies that Vðt; xðtÞÞ 6 rðtÞ, t P 0, where xðtÞ is any solution of (2.3). Similar result can be obtained when all the directions of the inequalities in the lemma are reversed and un ; wn are nonincreasing. Note that if we have some smoothness conditions of gðtÞ to guarantee the existence and uniqueness of solutions for (3.2), then rðtÞ is exactly the unique solution of (3.2). For convenience, we give some basic properties of the following system:
8 z_ ¼ b3 zðtÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < DzðtÞ ¼ pzðtÞ; t ¼ ðn þ l 1ÞT; > DzðtÞ ¼ q; t ¼ nT; > > : zð0þ Þ ¼ z0 P 0:
ð3:3Þ
Lemma 3.3. System (3.3) has a positive periodic solution ~zðtÞ and for every solution zðtÞ of (3.3) with initial value zð0þ Þ ¼ z0 P 0, we have zðtÞ ! ~zðtÞ as t ! 1.
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Proof. Clearly
~zðtÞ ¼
8 < q expðb3 ðtðn1ÞTÞÞ ; 1ð1pÞ expðb3 TÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expðb3 ðtðn1ÞTÞÞ ; 1ð1pÞ expðb3 TÞ
ðn þ l 1ÞT < t 6 nT;
is a positive periodic solution of (3.3). The solution of (3.3) with initial value zð0þ Þ ¼ z0 P 0 is
8 q > < ð1 pÞn1 z0 1ð1pÞ expðb expðb3 tÞ þ ~zðtÞ; ðn 1ÞT < t 6 ðn þ l 1ÞT; TÞ 3 zðtÞ ¼ q > : ð1 pÞn z0 1ð1pÞ expðb expðb3 tÞ þ ~zðtÞ; ðn þ l 1ÞT < t 6 nT; 3 TÞ n 2 Z þ . Hence, jzðtÞ ~zðtÞj ! 0 as t ! 1. The proof is complete. h Therefore, the system (2.3) has a two-pest eradication periodic solution
q expðb3 ðt ðn 1ÞTÞÞ ; ðn 1ÞT < t 6 ðn þ l 1ÞT; ð0; 0; ~zðtÞÞ ¼ 0; 0; 1 ð1 pÞ expðb3 TÞ qð1 pÞ expðb3 ðt ðn 1ÞTÞÞ ð0; 0; ~zðtÞÞ ¼ 0; 0; ; ðn þ l 1ÞT < t 6 nT: 1 ð1 pÞ expðb3 TÞ 4. Extinction In this section, we study the stability of the two-pest eradication periodic solution of the full system (2.3). Theorem 4.1. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of (2.3). Then ð0; 0; ~zðtÞÞ is globally asymptotically stable provided that 3 lTÞð1pÞ expðb3 TÞÞ f1 ðTÞ ¼ b1 T gqð1pbexpðb ln 3 ð1ð1pÞ expðb3 TÞÞ 3 lTÞð1pÞ expðb3 TÞÞ ln f2 ðTÞ ¼ b2 T lqð1pbexpðb 3 ð1ð1pÞ expðb3 TÞÞ
1 1p1 1 1p2
< 0; ð4:1Þ < 0:
Proof. Similar to Theorem 3.1 of Liu et al. [21], we can prove that the two-pest eradication periodic solution ð0; 0; ~zðtÞÞ is locally asymptotically stable, we omit it here. In the following, we prove the global attractivity. Noting that
x_ 1 6 x1 ðb1 x1 Þ; t 6¼ ðn þ l 1ÞT; Dx1 ¼ p1 x1 ; t ¼ ðn þ l 1ÞT:
Consider the following impulsive differential equation:
8 _ ¼ wðtÞðb1 wðtÞÞ; t 6¼ ðn þ l 1ÞT; > < wðtÞ DwðtÞ ¼ p1 wðtÞ; t ¼ ðn þ l 1ÞT; > : wð0þ Þ ¼ x1 ð0þ Þ P 0; if t 6¼ ðn þ l 1ÞT, we have x1 ðtÞ 6 wðtÞ and wðtÞ ! b1 as t ! 1; if t ¼ ðn þ l 1ÞT, we have x1 ðtþ Þ ¼ ð1 p1 Þx1 ðtÞ < x1 ðtÞ. Hence, there exists a e1 > 0 such that x1 ðtÞ < b1 þ 1 for t large enough. Without loss of generality, we may assume that x1 ðtÞ < b1 þ 1 for t > 0. Similarly, we can assume that x2 ðtÞ < b2 þ 2 for t > 0. Choose a e > 0 such that RT ~ d ¼ ð1 p1 Þ exp 0 b1 1þgxðz1ðtÞeÞ dt < 1. Note that z_ ðtÞ P b3 zðtÞ, considering the comparison system: ðb1 þe1 Þ
8 _ yðtÞ ¼ b3 yðtÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < DyðtÞ ¼ pyðtÞ; t ¼ ðn þ l 1ÞT; > DyðtÞ ¼ q; t ¼ nT; > > : yð0þ Þ ¼ z0 P 0:
ð4:2Þ
From Lemmas 3.2 and 3.3, we have zðtÞ P yðtÞ and yðtÞ ! ~zðtÞ as t ! 1. Then
zðtÞ P yðtÞ > ~zðtÞ e
ð4:3Þ
holds for all t large enough. For simplification, we may assume that (4.3) holds for all t P 0. From (2.3) we get
(
~ x_ 1 ðtÞ 6 x1 ðtÞ b1 1þgxðz1ðtÞeÞ ; ðb1 þe1 Þ
Dx1 ðtÞ ¼ p1 x1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð4:4Þ
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Integrate (4.4) on ððn þ l 1ÞT; ðn þ lÞT, which yields
x1 ððn þ lÞTÞ 6 ð1 p1 Þx1 ððn þ l 1ÞTÞ exp
Z
b1
ðnþlÞT
ðnþl1ÞT
!
gð~zðtÞ eÞ dt ¼ x1 ððn þ l 1ÞTÞd: 1 þ x1 ðb1 þ e1 Þ
Thus x1 ððn þ lÞTÞ 6 x1 ðlTÞdn and x1 ððn þ lÞTÞ ! 0 as n ! 1. Therefore, x1 ðtÞ ! 0 as n ! 1, since 0 < x1 ðtÞ 6 ð1 p1 Þx1 ððn þ l 1ÞTÞ expðb1 TÞ for ðn þ l 1ÞT < t 6 ðn þ lÞT. By the same method we can prove x2 ðtÞ ! 0 as n ! 1, so we omit it. Next, we prove that zðtÞ ! ~zðtÞ as t ! 1 if limt!1 x1 ðtÞ ¼ 0 and limt!1 x2 ðtÞ ¼ 0 hold. For 0 < e < dðgbþ3 lÞ, there exists a T 1 > 0 such that 0 < x1 ðtÞ < e and 0 < x2 ðtÞ < e hold for all t > T 1 . Without loss of generality, we may assume that 0 < x1 ðtÞ < e and 0 < x2 ðtÞ < e hold for all t > 0. Then we have b3 zðtÞ 6 z_ ðtÞ 6 zðtÞðb3 þ dge þ dleÞ. By Lemmas 3.2 and 3.3, ~1 ðtÞ as t ! 1, where yðtÞ and y1 ðtÞ are solutions of Eq. (4.2) and the we obtain yðtÞ 6 zðtÞ 6 y1 ðtÞ, yðtÞ ! ~zðtÞ and y1 ðtÞ ! y following impulsive differential equation:
8 y_1 ðtÞ ¼ y1 ðtÞðb3 þ dge þ dleÞ; t 6¼ ðn þ l 1ÞT; t 6¼ nT; > > > < Dy ðtÞ ¼ py ðtÞ; t ¼ ðn þ l 1ÞT; 1 1 > Dy1 ðtÞ ¼ q; t ¼ nT; > > : y1 ð0þ Þ ¼ z0 P 0;
ð4:5Þ
respectively; and
~1 ðtÞ ¼ y
8 < q expððb3 þdgeþdleÞðtðn1ÞTÞÞ ; 1ð1pÞ expððb3 þdgeþdleÞTÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expððb3 þdgeþdleÞðtðn1ÞTÞÞ ; 1ð1pÞ expððb3 þdgeþdleÞTÞ
ðn þ l 1Þ < t 6 nT:
~1 ðtÞ þ e0 for t large enough. Let e ! 0, we get y ~1 ðtÞ ! ~zðtÞ: Hence, Therefore, there exists a e0 > 0 such that ~zðtÞ e0 < zðtÞ < y zðtÞ ! ~zðtÞ as t ! 1. This completes the proof. h 5. Permanence First, from Theorem 5.1 in [20], we know that all solutions of (2.3) are uniformly ultimately bounded. Theorem 5.1. For each positive solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ of (2.3) there exists a constant M > 0, such that xi ðtÞ 6 M(i ¼ 1; 2), and zðtÞ 6 M with t large enough. In the following, we investigate the permanence of the system (2.3). Theorem
5.2. System
(2.3)
is
permanent
gqð1p expððb3 þdlb2 ÞlTÞð1pÞ expððb3 þdlb2 ÞTÞÞ 1 ln 1p ðb3 dlb2 Þð1ð1pÞ expðb3 þdlb2 ÞTÞÞ 1
1 1p2
if > 0;
a < bb12 ,
l < dbb32 ,
b < bb21 ;
b3 g < db , 1
and
f3 ðTÞ ¼ ðb1 ab2 ÞT
3 þdgb1 ÞlTÞð1pÞ expððb3 þdgb1 ÞTÞÞ f4 ðTÞ ¼ ðb2 bb1 ÞT lqð1pðbexpððb ln 3 dgb1 Þð1ð1pÞ expððb3 þdgb1 ÞTÞÞ
> 0.
þ Proof. Suppose n o that xðtÞ is a solution of (2.3) with xð0 Þ > 0, from Theorem 5.1, we may assume that zðtÞ 6 M and b1 b2 M > max g ; l hold for t > 0. From the proof of Theorem 4.1 we can assume that x1 ðtÞ < b1 þ e1 and x2 ðtÞ < b2 þ e2 hold qð1pÞ expðb3 TÞ for t > 0. Let m ¼ 1ð1pÞ e > 0, where e > 0. According to Lemmas 3.2 and 3.3, we have zðtÞ > m for t large enough. expðb3 TÞ 1 > 0 and m 2 > 0, such that x1 ðtÞ P m 1 and x2 ðtÞ P m 2 for t large enough. We will do it in In the following, we want to find m the following two steps for convenience.
Step I: Let 0 < m1 < b3 dldðbg 2 þe2 Þ ; 0 < m2 < b3 dgdðbl1 þe1 Þ ; and dgm1 þ dlm2 < b3 . We will prove that there exist t1 ; t2 2 ð0; 1Þ, such that x1 ðt 1 Þ P m1 and x2 ðt2 Þ P m2 . Otherwise, there will be three cases: (i) There exists a t2 > 0, such that x2 ðt 2 Þ P m2 and x1 ðtÞ < m1 , for all t > 0; (ii) There exists a t1 > 0, such that x1 ðt 1 Þ P m1 and x2 ðtÞ < m2 , for all t > 0; (iii) x1 ðtÞ < m1 ; x2 ðtÞ < m2 , for all t > 0. We first consider case (i). Let e1 > 0 small enough such that
r1 ¼ ð1 p1 Þ exp ðb1 m1 aðb2 þ e2 Þ ge1 ÞT
gqð1 p expðhlTÞ ð1 pÞ expðhTÞÞ ðhÞð1 ð1 pÞ expðhTÞÞ
> 1;
where h ¼ b3 þ dgm1 þ dlðb2 þ e2 Þ. According to the above assumption, we get z_ ðtÞ 6 hzðtÞ: By Lemmas 3.2 and 3.3, we ~2 ðtÞ as t ! 1, where y2 ðtÞ is the solution of have zðtÞ 6 y2 ðtÞ and y2 ðtÞ ! y
8 y_ 2 ðtÞ ¼ hy2 ðtÞ; t 6¼ ðn þ lÞT; t 6¼ nT; > > > < Dy2 ðtÞ ¼ py2 ðtÞ; t ¼ ðn þ l 1ÞT; > > Dy2 ðtÞ ¼ q; t ¼ nT; > : y2 ð0þ Þ ¼ z0 P 0;
ð5:1Þ
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where
~2 ðtÞ ¼ y
8 < q expðhðtðn1ÞTÞÞ ; 1ð1pÞ expðhTÞ
ðn 1ÞT < t 6 ðn þ l 1ÞT;
: qð1pÞ expðhðtðn1ÞTÞÞ ; 1ð1pÞ expðhTÞ
ðn þ l 1ÞT < t 6 nT:
~2 ðtÞ þ e1 . Then we have Therefore, there exists a T 1 > 0, such that zðtÞ 6 y2 ðtÞ < y
~2 ðtÞ þ e1 ÞÞ; x_ 1 ðtÞ P x1 ðtÞðb1 m1 aðb2 þ e2 Þ gðy x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
ð5:2Þ
t ¼ ðn þ l 1ÞT;
for t P T 1 . Let N 2 Z þ and ðN þ l 1ÞT P T 1 . Integrating (5.2) on ððn þ l 1ÞT; ðn þ lÞT, n P N, we have
x1 ððn þ lÞTÞ P x1 ððn þ l 1ÞTÞð1 p1 Þ exp
Z
!
ðnþlÞT
~2 ðtÞ þ e1 ÞÞdt ðb1 m1 aðb2 þ e2 Þ gðy
¼ x1 ððn þ l 1ÞTÞr1 :
ðnþl1ÞT
Then x1 ððN þ n þ lÞTÞ P x1 ðN þ lÞTÞrn1 ! 1 as n ! 1; which is a contradiction to the boundedness of x1 ðtÞ. Cases (ii) and (iii) can be analyzed by the same method as case (i), so we omit it. From the above three cases, we conclude that there exist t1 > 0, t 2 > 0 such that x1 ðt1 Þ P m1 , x2 ðt2 Þ P m2 . Step II: If x1 ðtÞ P m1 for all t P t 1 , then our aim is obtained. Otherwise, we have x1 ðtÞ < m1 for some t P t1 . Setting t ¼ inf t>t1 fx1 ðtÞ < m1 g. Then t can be divided into two cases, t is impulsive point or not impulsive point. Case (a): If t ¼ ðn1 þ l 1ÞT for n1 2 Z þ . Then x1 ðtÞ P m1 and ð1 p1 Þm1 6 x1 ðt þ Þ ¼ ð1 p1 Þx1 ðt Þ < m1 hold for t 2 ½t 1 ; t . Select n2 ; n3 2 Z þ such that
ðn2 1ÞT >
1 e1 ln ; b3 þ dgm1 þ dlðb2 þ e2 Þ Mþq
and
ð1 p1 Þn2 rn13 expðn2 d0 TÞ > ð1 p1 Þn2 rn13 expððn2 þ 1Þd0 TÞ > 1; where d0 ¼ b1 m1 aðb2 þ e2 Þ gM < 0. Set T 0 ¼ ðn2 þ n3 ÞT. We claim that there must be exist a t 2 2 ðt ; t þ T 0 such that x1 ðt 2 Þ > m1 . Otherwise, we have x1 ðtÞ < m1 for t 2 ðt ; t þ T 0 . Consider (5.2) with y2 ðn1 T þ Þ ¼ zðn1 T þ Þ, we have
8 > ~2 ðtÞ; ðn 1ÞT < t 6 ðn þ l 1ÞT; < ð1 pÞnðn1 þ1Þ y2 ðn1 T þ Þ 1ð1pÞq expðhTÞ expðhðt n1 TÞÞ þ y y2 ðtÞ ¼ > : ð1 pÞnn1 y2 ðn1 T þ Þ 1ð1pÞq expðhTÞ expðhðt n1 TÞÞ þ y ~2 ðtÞ; ðn þ l 1ÞT < t 6 nT; ~2 ðtÞj < ðM þ qÞ expðhðt n1 TÞÞ < e1 ; and zðtÞ 6 y2 ðtÞ < y ~2 ðtÞ þ e1 for t 2 ððn 1ÞT; nT; n1 þ 1 6 n 6 n1 þ n2 þ n3 . Then jy2 ðtÞ y for n1 T þ ðn2 1ÞT 6 t 6 t þ T 0 , which implies that (5.2) holds for t þ n2 T 6 t 6 t þ T 0 . As case (i), we have x1 ðt þ T 0 Þ P x1 ðt þ n2 TÞrn13 : Note that
x_ 1 ðtÞ P x1 ðtÞðb1 m1 aðb2 þ e2 Þ gMÞ; x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ;
t 6¼ ðn þ l 1ÞT;
t ¼ ðn þ l 1ÞT:
ð5:3Þ
Integrating (5.3) on ½t ; t þ n2 T, we have x1 ðt þ n2 TÞ P m1 ð1 p1 Þn2 expðn2 d0 TÞ. Thus x1 ðt þ T 0 Þ P m1 ð1 p1 Þn2 expðn2 d0 TÞrn13 > m1 , which is a contradiction.Let ~t ¼ inf t>t fx1 ðtÞ P m1 g. Then x1 ðtÞ < m1 holds for t 2 ðt ; ~t. Since x1 ðtÞ is left continuous and x1 ðt þ Þ ¼ ð1 p1 Þx1 ðtÞ 6 x1 ðtÞ as t ¼ ðn þ l 1ÞT, so ~t cannot be impulsive point. Suppose t 2 ðt þ ðl 1ÞT; t þ lT; l 2 Z þ ; l 6 n2 þ n3 . Integrating (5.3) on ½t ; ~tÞ one yields
x1 ðtÞ P x1 ðtþ Þð1 p1 Þl1 expððl 1Þd0 TÞ expðd0 ðt ðt þ ðl 1ÞTÞÞÞ P m1 ð1 p1 Þl expðld0 TÞ P m1 ð1 p1 Þn2 þn3 expððn2 þ n3 Þd0 TÞ: Let m01 ¼ m1 ð1 p1 Þn2 þn3 expððn2 þ n3 Þd0 TÞ. So we have x1 ðtÞ P m01 for t 2 ðt ; ~t. For t > ~t, the same argument can be continued since x1 ð~tÞ P m1 . Case (b): If t 6¼ ðn þ l 1ÞT, n 2 Z þ . Then x1 ðtÞ P m1 for t 2 ½t 1 ; t Þ and x1 ðt Þ ¼ m1 . We can assume t 2 ððn01 þ l 1ÞT; ðn01 þ lÞTÞ; n01 2 Z þ . For t 2 ðt ; ðn01 þ lÞTÞ, there are two possible cases for x1 ðtÞ. If x1 ðtÞ 6 m1 holds for t 2 ðt ; ðn01 þ lÞTÞ. Similar to case (a), we can prove that there must exist a t02 2 ½ðn01 þ lÞT; ðn01 þ lÞT þ T 0 such that x1 ðt 02 Þ P m1 . Here, we omit it. Let t ¼ inf t>t fx1 ðtÞ > m1 g, then x1 ðtÞ ¼ m1 and 0 0 x1 ðtÞ 6 m1 hold for t 2 ðt ; tÞ. For t 2 ðt ; tÞ, we have x1 ðtÞ P m1 ð1 p1 Þl expððl þ 1Þd0 TÞ P 1 ¼ m1 ð1 p1 Þn2 þn3 expððn2 þ n3 þ 1Þd0 TÞ < m01 , then x1 ðtÞ P m 1 holds for m1 ð1 p1 Þn2 þn3 expððn2 þ n3 þ 1Þd0 TÞ. Let m 1 . Hence, x1 ðtÞ P m 1 for all t > t1 . t 2 ðt ; tÞ. For t > t, the same argument can be continued since x1 ðtÞ P m If there exists a t 2 ðt ; ðn01 þ 1ÞT such that x1 ðtÞ > m1 . Let ^t ¼ inf t>t fx1 ðtÞ > m1 g, then x1 ðtÞ 6 m1 holds for t 2 ½t ; ^tÞ and 1 : This process can be continued since x1 ð^tÞ ¼ m1 . For t 2 ðt ; ^tÞ, we have x1 ðtÞ P x1 ðt Þ expðd0 ðt t ÞÞ P m1 expðd0 TÞ > m 2 holds for all t P t 2 . The proof is complete. h x1 ð^tÞ P m1 holds for t > ^t1 . Similarly, we can prove that x2 ðtÞ P m
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Fig. 1. The trajectory fi ðTÞ has a unique positive root T ðiÞ , where a ¼ 0:6; b ¼ 0:5; b1 ¼ 3; b2 ¼ 2; b3 ¼ 3; g ¼ 0:2; l ¼ 0:7; d ¼ 0:2; w1 ¼ 1; w2 ¼ 1; p ¼ 0:4; p1 ¼ 0:4; p2 ¼ 0:3; q ¼ 0:3; l ¼ 0:5.
From the proof of Theorems 4.1 and 5.2, we can derive the following results. Corollary 5.1. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of system (2.3), then x2 and z are permanent and x1 ðtÞ ! 0 as t ! 1 provided b3 that b < bb21 ; g < db ; f1 ðTÞ < 0; f4 ðTÞ > 0. 1 Corollary 5.2. Let ðx1 ðtÞ; x2 ðtÞ; zðtÞÞ be any solution of system (2.3), then x1 and z are permanent and x2 ðtÞ ! 0 as t ! 1 provided b3 that a < bb12 ; l < db ; f2 ðTÞ < 0; f3 ðTÞ > 0. 2 1 Remark 5.1. Since fi ð0Þ ¼ ln 1p < 0; fi ðTÞ ! þ1 as T ! þ1, and fi00 ðTÞ > 0, so fi ðTÞ ¼ 0 has a unique positive root, 1 denoted by T i ði ¼ 1; 3Þ. 1 < 0; fj ðTÞ ! þ1 as T ! þ1, and fj00 ðTÞ > 0, so fj ðTÞ ¼ 0 has a unique positive root, denoted by Since fj ð0Þ ¼ ln 1p 2 T j ðj ¼ 2; 4Þ (see Fig. 1). 6. Discussion and numerical simulation In this paper, we investigate the dynamics of a two-prey one-predator system with impulsive effect concerning biological and chemical control strategy. We show that there exists a two-pest eradication periodic solution which is globally asymptotically stable, and get the sufficient conditions for the permanence of the system (2.3). From Corollaries 5.1 and 5.2, we also get the sufficient conditions for one of two-prey extinction and the remaining two pieces permanence.
Fig. 2. Time-series of system (2.3) with x1 ; x2 eradication (T ¼ 0:1).
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Fig. 3. Time-series of system (2.3) with positive periodic solution (T ¼ 0:9).
Fig. 4. Time-series of system (2.3) with x2 eradication (T ¼ 0:19).
From Theorem 4.1, we know that the two-pest eradication periodic solution ð0; 0; ~zðtÞÞ is globally asymptotically stable if D b1 ¼ T T maxfT 3 ; T 4 g, that is to say, three species coexist if impulsive period is larger than T a ¼ 0:6; b ¼ 0:5; b1 ¼ 3; b2 ¼ 2; b3 ¼ 3; g ¼ 0:2; l ¼ 0:7; d ¼ 0:2; w1 ¼ 1; w2 ¼ 1; p1 ¼ 0:4; p2 ¼ 0:3; p ¼ 0:4; q ¼ 0:3; l ¼ 0:5, then we can calculate T 3 0:18; T 4 0:84, let T ¼ 0:9, then T > maxfT 3 ; T 4 g. Then three species coexist if impulsive period b 2 ¼ 0:84. From Corollaries 5.1 and 5.2, we know that species x1 extinct but x2 and z are permanence if is larger than T T 4 < T < T 1 ; species x2 extinct, x1 and z are permanent if T 3 < T < T 2 . Therefore, we can drive the target pest population to extinct and let the non-target pest (or harmless insect) be permanent by choosing the impulsive period, this can be seen clearly from Fig. 4. Parameters are the same as Fig. 3, then T 2 0:2; T 3 0:18, let T ¼ 0:19, then T 3 < T < T 2 . So, if one prey is target pest, another is non-target pest, we can choose the impulsive period T to obtain our aim. For example in Fig. 4, when 0:18 < T < 0:2, x2 rapidly reduces to zero, and x1 and z tend to positive periodic solution.
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Fig. 7. Spiral chaos for system (2.3) with g ¼ 7:3. (a–c) Time-series of species x1 ; x2 and z; and (d) phase portrait.
Fig. 8. Bifurcation diagrams of system (2.3) with 65 6 T 6 150 and initial value x10 ¼ 0:12; x20 ¼ 0:78; z0 ¼ 0:042.
In the following, we will investigate the influence of bifurcation diagrams on the system (2.3). Fig. 5 shows the bifurcation diagram of system (2.3) for b1 ¼ 1; b2 ¼ 1; b3 ¼ 1; a ¼ 0:99; b ¼ 1:5; d ¼ 0:5; l ¼ 1; p1 ¼ 0:05; p2 ¼ 0:05; p ¼ 0:05; l ¼ 0:05; q ¼ 0:05; w1 ¼ 0:001; w2 ¼ 0:001; T ¼ 125, and g varying from 6.8 to 8.4. Further increasing g, we can see that the dynamical behavior of system (2.3) is very complicated, including a sequence of direct and inverse cascade of periodic-dou-
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Fig. 9. Chaos of system (2.3) with T ¼ 75, (a–c) time-series of species x1 ; x2 and z with initial value x10 ¼ 0:12; x20 ¼ 0:78; z0 ¼ 0:042, and (d) phase portrait of system (2.3).
bling, chaos, and symmetry breaking bifurcation. Fig. 6 shows periodic-doubling cascade. The bifurcation diagrams show the route to chaos through cascade of periodic-doubling, which has been extensively studied by mathematicians [22,23]. A typical spiral chaos oscillation is captured when g ¼ 7:3 (see Fig. 7). Fig. 8 shows the bifurcation diagrams of system (2.3) for b1 ¼ 1; b2 ¼ 1; b3 ¼ 1; a ¼ 0:99; b ¼ 1:5; d ¼ 0:5; g ¼ 7:1; l ¼ 1; p1 ¼ 0:001; p2 ¼ 0:001; p ¼ 0:05; l ¼ 0:05; q ¼ 0:05; w1 ¼ 0:001; w2 ¼ 0:001, and T varying from 65 to 150. If we let 65 6 T 6 70, there species coexist periodically, with further increasing T, periodic-doubling cascade occurs. Further increasing T again, we can see that the dynamical behavior of system (2.3) is very complicated, including a sequence of periodichalving cascade, cycles, periodic-doubling cascade and chaos. In Fig. 8, we find that there are occurrences of sudden changes in the types of the attractors, which is a typical feature of bifurcation diagrams. For example, a T-periodic solution suddenly changes to 2T-periodic solution when T 70 and 2T-periodic solution suddenly changes to 4T-periodic solution when T 72:1 and 4T-periodic solution suddenly changes to chaos when T 73. Fig. 9 shows chaos of system (2.3) for T ¼ 75. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 10771104 and 10771179), the Henan Innovation Project for University Prominent Research Talents (No. 2005KYCX017) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] P. DeBach, D. Rosen, Biological Control by Natural Enemies, second ed., Cambridge University Press, 1991. [2] A.J. Cherry, C.J. Lomer, D. Djegui, F. Schulthess, Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in west Africa, Biocontrol 44 (1999) 301–327. [3] P. Ferron, Pest control using the fungi Beauveria and Metarhizinm, in: H.D. Burges (Ed.), Microbial Control in Pests and Plant Disease, Academic, London, 1981. [4] H.J. Freedman, Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci. 31 (1976) 207–225. [5] J. Grasman, O.A. Van Herwaarden, et al, A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control, Math. Biosci. 169 (2001) 207–216. [6] P. DeBach, Biological Control of Insect Pests and Weeds, Rheinhold, New York, 1964. [7] M.L. Luff, The potential of predators for pest control, Agri. Ecos. Environ. 10 (1983) 159–181. [8] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [9] D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, England, 1993. [10] Z. Agur, L. Cojocaru, R. Anderson, Y. Danon, Pulse mass measles vaccination across age cohorts, Proc. Natl. Acad. Sci. 90 (1993) 11698–11702. [11] B. Shulgin, L. Stone, I. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60 (1998) 1–26. [12] G. Ballinger, X. Liu, Permanence of population growth models with impulsive effects, Math. Comput. Modell. 26 (1997) 59–72.
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Y. Li et al. / Applied Mathematics and Computation 204 (2008) 269–280
[13] X.N. Liu, L.S. Chen, Complex dynamics of Holling type II Lotka–Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons and Fractals 16 (2003) 311–320. [14] S.Y. Tang, L.S. Chen, Density-dependent birth rate, birth pulse and their population dynamic consequences, J. Math. Biol. 44 (2002) 185–199. [15] M.G. Roberts, R.R. Kao, The dynamics of an infectious disease in a population with birth pulse, Math. Biosci. 149 (1998) 23–36. [16] A. Lakmeche, O. Arino, Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. Cont. Discr. Impuls. Syst. 7 (2000) 265–287. [17] J.C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol. 58 (1996) 425–447. [18] C.S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Sec. Can. 45 (1965) 1–60. [19] S.W. Zhang, L.S. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos, Solitons and Fractals 24 (2005) 1269– 1278. [20] X.Y. Song, Y.F. Li, Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect, Chaos, Solitons and Fractals 33 (2007) 463– 478. [21] B. Liu, Y.J. Zhang, L.S. Chen, Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control, Chaos, Solitons and Fractals 22 (2004) 123–134. [22] A. Venkatesan, S. Parthasarathy, M. Lakshmanan, Occurrence of multiple periodic-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems, Chaos, Solitons and Fractals 18 (2003) 891–898. [23] J.P. Eckmann, Routes to chaos with special emphasis on periodic-doubling, in: G. Iooss et al. (Eds.), Chaotic Behavior of Deterministic Systems, Elsevier North Holland, Amsterdam, 1983.
