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Staged-structured Lotka–Volterra predator–prey models for pest management
Applied Mathematics and Computation 203 (2008) 258–265
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Staged-structured Lotka–Volterra predator–prey models for pest management Ruiqing Shi a,b,*, Lansun Chen a a b
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, People’s Republic of China
a r t i c l e
i n f o
Keywords: Stage structured Biological control Impulsive Pest-eradication equilibrium Pest-eradication periodic solution
a b s t r a c t In this paper, two predator–prey models with stage structure are constructed and investigated. In the first model, continuous biological control is taken. The existence and local stability of two equilibriums are studied. By the Liapunov stability theorem, we obtain the condition for the global asymptotical stability of the trivial equilibrium (i.e., pest-eradication equilibrium). In the second model, impulsive biological control is taken. By use of the Floquet’s theorem, small-amplitude perturbation method and comparison techniques, we get the condition which guarantees the global asymptotical stability of the pest-eradication periodic solution. The sufficient condition for the permanence of the impulsive system is also obtained. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction According to reports of Food and Agriculture Organization of the United Nations, the warfare between human and pests (such as locust, Aphis, cotton bollworm, etc.) has sustained for thousands of years. With the development of society and the progress of science and technology, man has adopted some advanced and modern weapons such as chemical pesticides, biological pesticides, remote sensing and measuring, computers, atomic energy and so on. At last, some brilliant achievements have been obtained. However, the warfare is not over, and will continue. A great deal of and a large variety of pesticides were used to control pests. Pesticides are useful because they can quickly kill a significant portion of a pest population and sometimes provide the only feasible method for preventing economic loss. However, pesticide pollution is also recognized as a major health hazard to human beings and beneficial insects. In the natural world, many pest individuals have a life story that takes them through two stages, immature and mature. The natural enemy (predator) only catch the mature pests (prey), for the immature pests are protected by their eggshells. There are some literatures about single-species models with stage structure [1,3,5,12]; while other models are about competitive systems or predator–prey systems with stage structure [2,4,7,8,14]. There are still some other models concerning stage structure [3,6,9–11,13,17,18]. Liu and Chen [19] developed the Holling II Lotka–Volterra predator–prey (natural enemy–pest) system by periodic constant impulsive immigration of natural enemy. They gave the conditions for extinction of the pest and permanence of the system and studied the influences on the inherent oscillation caused by the impulsive perturbations. But the paper did not consider the stage structure. Wang and Chen [14] studied a predator–prey system with stage structure for the predator, but they did not consider biological control. In this paper, we consider the Lotka–Volterra predator–prey models with stage structure for the prey (pest). For the purpose of pest management, we shall release natural
* Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China. E-mail addresses: [email protected] (R. Shi), [email protected] (L. Chen). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.032
R. Shi, L. Chen / Applied Mathematics and Computation 203 (2008) 258–265
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enemy (predator), then more of the pests (prey) will be captured. The natural enemy may be cultivated in the laboratory, or migrate from other areas. We can release the natural enemy continuously, or impulsively. The main purpose of this paper is to construct two realistic models of biological control systems for pest management, to investigate their dynamics and to compare the results obtained for the ordinary differential model, corresponding to the continuous biological control, with those obtained for the impulsive differential model, corresponding to the impulsive biological control. The present paper is organized as follows. In Section 2, the main biological assumptions are formulated, and two models are constructed. In Section 3, by using the qualitative theory of ordinary differential equations, we investigate the behavior of the ordinary system which models the process of continuous release of natural enemy. In Section 4, we study the impulsive system which models the process of periodic release of natural enemy at fixed moments. By use of the Floquet’s theory for impulsive differential equations, small-amplitude perturbation methods and comparison techniques, we get the global asymptotic stability of the pest-eradication periodic solution and the condition for the permanence of the system. Finally, a brief discussion is provided in the last section. 2. Models formulation The basic stage-structured Lotka–Volterra predator–prey model is 8 0 > < x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > : 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ;
ð1Þ
where x1 ðtÞ and x2 ðtÞ denote the densities of immature and mature prey (pest) population, respectively, yðtÞ denotes the density of the predator (natural enemy) population. All coefficients are positive constants. The model is derived with the following assumptions. ðH1 Þ At any time t P 0 birth into the immature prey population is proportional to the existing mature prey population, with proportionality constant r. ðH2 Þ The immature prey population has the natural death rate l, and the predator has a natural death rate d. The death rate of the mature prey population has a logistic nature, that is, proportional to the square of the population with proportionality constant b. ðH3 Þ The immature prey population enter into mature population with proportionality constant d. ðH4 Þ Only the mature preys are preyed by the predator, and the immature preys are protected by their eggshells, where a is the per-capita rate of predation of the predator, and 0 < k < 1 is the rate of conversing prey into predator.In this paper, we construct the following two models, and the parameters are of the same with those in model (1), with the assumption follows. ðH5 Þ The predator population are released with constant rate of p > 0 in model (2), or with constant number p > 0 periodically with period s in model (3), respectively: 8 0 > < x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; ð2Þ x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > : 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ þ p: And 8 0 9 x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; > > > = > > > > x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > > > > ; > < y0 ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ; > Dx1 ðtÞ ¼ 0; 9 > > > = > > > Dx ðtÞ ¼ 0; > t ¼ ns; n ¼ 1; 2; 3; . . . > 2 > > > ; : DyðtÞ ¼ p;
t 6¼ ns; n ¼ 1; 2; 3; . . . ;
3. Qualitative analysis for system (2) Denote p0 ¼
drd : aðl þ dÞ
By simple calculation, we have Lemma 3.1. (1) System (2) always has a trivial equilibrium (pest-eradication equilibrium) E0 ¼ ðx01 ; x02 ; y0 Þ ¼ ð0; 0; pdÞ.
ð3Þ
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(2) If p < p0 , then system (2) has a unique positive equilibrium (natural enemy–pest coexistence equilibrium) E1 ¼ ðx1 ; x2 ; y Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr
ay
r x2 , x2 ¼ lþd b where x1 ¼ lþd
, and y ¼
ðkadr lþd dbÞþ
2
2 ðkadr lþd dbÞ þ4ka pb 2
2ka
.
Theorem 3.2. If p > p0 , then the trivial equilibrium E0 is locally asymptotically stable. If p < p0 , then E0 is unstable, and the positive equilibrium E1 is locally asymptotically stable. Proof. The characteristic equation at E0 is ðk þ dÞ½k2 þ ðl þ d þ ay0 Þk þ aðl þ dÞy0 dr ¼ 0: If p > p0 holds, then all of the eigenvalues of the above characteristic equation have negative real part. Thus, E0 is locally asymptotically stable. If p < p0 holds, then one of the eigenvalues of the above characteristic equation has positive real part. Thus, E0 is unstable. By Lemma 3.1, we know that the positive equilibrium E1 exists when p < p0 . The characteristic equation at E1 is k3 þ a1 k2 þ a2 k þ a3 ¼ 0; where dr p þ bx2 þ > 0; a1 ¼ ðl þ dÞ þ lþd y p dr p 2 þ bx2 a2 ¼ ðl þ dÞ bx2 þ þ þ ka x2 y > 0; y lþd y p 2 a3 ¼ ðl þ dÞ bx2 þ ka x2 y > 0 y and a1 a2 a3 ¼
dr p p dr p dr p 2 8 p þ bx2 þ ðl þ dÞ bx2 þ þ þ bx þ Þ þ ka x y þ ðbx 2 2 2 lþd y y l þ d y lþd y y p dr p þ ðl þ dÞ ðl þ dÞ bx2 þ þ > 0: y l þ d y
By the Hurwitz criterion, we get that all of the three eigenvalues of the above characteristic equation have negative real part. Thus, E1 is locally asymptotically stable, whenever it exists. This completes the proof. h Theorem 3.3. The trivial equilibrium E0 is globally asymptotically stable if p P p0 . Proof. Define the Liapunov function as follows: yðtÞ VðtÞ ¼ c1 x1 ðtÞ þ c2 x2 ðtÞ þ c3 yðtÞ y0 y0 ln 0 ; y where ci are positive constants to be determined (i ¼ 1; 2; 3), then dVðtÞ y0 ½kax2 ðtÞyðtÞ dyðtÞ þ p ¼ c1 ½rx2 ðtÞ ðl þ dÞx1 ðtÞ þ c2 ½dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ þ c3 1 dt ð2Þ yðtÞ c3 d y0 ðyðtÞ y0 Þ2 þ c1 rx2 ðtÞ ½c1 ðl þ dÞ c2 dx1 ðtÞ ax2 ðtÞyðtÞ c2 c3 k 1 c2 bx22 ðtÞ: ¼ yðtÞ yðtÞ If we set c2 ¼ kc3 ¼ l þ d, and c1 ðl þ dÞ ¼ c2 d, then dVðtÞ c3 d apðl þ dÞ 0 2 2 ðyðtÞ y x2 ðtÞ: ¼ Þ c bx ðtÞ þ dr 2 2 dt ð2Þ yðtÞ d < 0. Therefore E0 is globally asymptotically stable. The proof is complete. If p P p0 , then we easily get dVðtÞ dt
h
ð2Þ
Remark 1. From the above theorems, we can see that if the coefficients satisfy the condition p < p0 , then, the prey (pests) will persist, and they will do harm to the crops. Under this condition, we can release predator (natural enemy) with rate p P p0 , and by the result of Theorem 3.3, we know that the prey (pests) will be controlled. 4. Qualitative analysis for system (3) Definition 4.1. System (3) is said to be permanent if there are constants m; M > 0 (independent of initial value) and a finite time T 0 such that for all solutions ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ with all initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0,
R. Shi, L. Chen / Applied Mathematics and Computation 203 (2008) 258–265
m 6 x1 ðtÞ þ x2 ðtÞ 6 M; m 6 yðtÞ 6 M x2 ð0þ Þ; yð0þ ÞÞ.
hold for all t P T 0 . Here T 0
261
may depend on the initial values ðx1 ð0þ Þ;
Lemma 4.2 [15,16]. Suppose ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ is a solution of system (3) with initial values x1 ð0þ Þ P 0; x2 ð0þ Þ P 0; yð0þ Þ P 0, then x1 ðtÞ P 0; x2 ðtÞ P 0; yðtÞ P 0, for all t P 0. And further x1 ðtÞ > 0; x2 ðtÞ > 0; yðtÞ > 0, if x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. Lemma 4.3 [15]. Let the function m 2 PC 0 ½Rþ ; R satisfies the inequalities ( m0 ðtÞ 6 pðtÞmðtÞ þ qðtÞ; t P t0 ; t 6¼ tk ; k ¼ 1; 2; . . . ; mðt þ k Þ 6 dk mðt k Þ þ bk ;
t ¼ tk ;
where p; q 2 PC½Rþ ; R and dk P 0; bk are constants, then 0 1 Rt Rt Z Y Y X pðsÞds pðsÞds t t0 @ Abk þ dk e þ dj e k mðtÞ 6 mðt0 Þ t 0
t 0
t k
t
Y
Rt pðrÞdr dk e s qðsÞds
t 0 s
for all t P t 0 . If x1 ðtÞ ¼ x2 ðtÞ ¼ 0, for all t P 0, then we get the subsystem of system (3) as follows: 0 y ðtÞ ¼ dyðtÞ; t¼ 6 ns; n ¼ 1; 2; . . . ; yðnsþ Þ ¼ yðnsÞ þ p;
t ¼ ns; n ¼ 1; 2; . . .
ð4Þ
dðtnsÞ
~ð0þ Þ ¼ 1epds is a positive periodic solution of system (4). The ~ðtÞ ¼ pe1eds ; t 2 ðns; ðn þ 1Þs; n 2 Z þ ¼ f1; 2; 3; . . .g; y Obviously y þ p dt ~ solution of system (4) is yðtÞ ¼ ðyð0 Þ 1eds Þe þ yðtÞ, t 2 ðns; ðn þ 1Þs; n 2 Z þ . Therefore, the complete expression for the dðtnsÞ ~ðtÞÞ ¼ ð0; 0; pe1eds Þ; t 2 ðns; ðn þ 1Þs; n 2 Z þ . And we pest-eradication periodic solution of system (3) is obtained as ð0; 0; y derive ~ðtÞ as t ! 1. Lemma 4.4 [16]. For every solution of system (4) with initial condition yð0þ Þ > 0, it follows that yðtÞ ! y Lemma 4.5. There exists a constant M > 0 such that x1 ðtÞ 6 M; x2 ðtÞ 6 M; yðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (3) for all t large enough. Proof. Due to the positivity of x1 ðtÞ; x2 ðtÞ; yðtÞ, we define ðtÞ ¼ x1 ðtÞ þ x2 ðtÞ þ yðtÞ: When t 6¼ ns, we have Dþ UðtÞ þ kUðtÞ ¼ ðr þ kÞx2 ðtÞ ðl kÞx1 ðtÞ ðd kÞyðtÞ bx22 ðtÞ að1 kÞx2 ðtÞyðtÞ 6 M0 ; 2
where k ¼ minfl; dg, and M0 ¼ ðrþkÞ . 4b When t ¼ ns; Uðnsþ Þ ¼ UðnsÞ þ p. By Lemma 4.3, for t 2 ðns; ðn þ 1Þs we have M 0 kt pð1 enks Þ kðtnsÞ M 0 e þ : e þ UðtÞ 6 Uð0þ Þ 1 eks k k So UðtÞ is uniformly ultimately bounded by a positive constant and there exists a constant M > 0 such that x1 ðtÞ 6 M; x2 ðtÞ 6 M; yðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (3) for all t large enough. The proof is complete. h Lemma 4.6 [15]. Let V : Rþ R2 ! R and V 2 V 0 . Assume that ( t 6¼ nT; n ¼ 1; 2; . . . ; Dþ Vðt; XÞ 6 gðt; Vðt; XÞÞ; Vðt; Xðt þ ÞÞ 6 Wn ðVðt; XðtÞÞÞ;
t ¼ nT; n ¼ 1; 2; . . . ;
ð5Þ
where g : Rþ Rþ ! R is continuous in ðnT; ðn þ 1ÞT Rþ and for each v 2 R2þ ; n 2 Z þ lim
ðt;yÞ!ðnT þ ;vÞ
gðt; yÞ ¼ gðnT þ ; vÞ
exists and is finite, Wn : Rþ ! Rþ is nondecreasing. Let RðtÞ be the maximal solution of the scalar impulsive differential equation: 8 0 t 6¼ nT; n ¼ 1; 2; . . . ; > < U ðtÞ ¼ gðt; UÞ; Uðtþ Þ ¼ Wn ðUðtÞÞ; t ¼ nT; n ¼ 1; 2; . . . ; > : Uð0þ Þ ¼ U 0 ; defined on ½0; 1Þ. Then Vð0þ ; X 0 Þ 6 U 0 implies that Vðt; XðtÞÞ 6 RðtÞ, t P 0, where XðtÞ is any solution of (5).
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Next, we investigate the stability of the pest-eradication periodic solution of system (3) with initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. Denote p1 ¼
drds : aðl þ dÞ
~ðtÞÞ is locally asymptotically stable if p > p1 . Theorem 4.7. The pest-eradication periodic solution ð0; 0; y Proof. The local stability of pest-eradication periodic solution may be determined by considering the behaviors of small ~ðtÞ, then the linearized equations of amplitude perturbation of the solution. Define uðtÞ ¼ x1 ðtÞ; vðtÞ ¼ x2 ðtÞ; wðtÞ ¼ yðtÞ y system (3) read as 8 9 u0 ðtÞ ¼ rvðtÞ ðl þ dÞuðtÞ; > > > = > > > ~ðtÞvðtÞ; > t 6¼ ns; n ¼ 1; 2; 3; . . . ; v0 ðtÞ ¼ duðtÞ ay > > > > < w0 ðtÞ ¼ kay ~ðtÞvðtÞ dwðtÞ; ; 9 ð6Þ > uðnsþ Þ ¼ uðnsÞ; > > = > > > > vðnsþ Þ ¼ vðnsÞ; t ¼ ns; n ¼ 1; 2; 3; . . . > > > > ; : wðnsþ Þ ¼ wðnsÞ; Let UðtÞ be the fundamental solution matrix of system (6), then UðtÞ must satisfy 0 1 ðl þ dÞ r 0 dUðtÞ B C ~ðtÞ ¼@ d ay 0 AUðtÞ ¼ AUðtÞ dt ~ðtÞ d 0 kay and Uð0Þ ¼ I, the identity matrix. We can see that one of the eigenvalues of matrix A is d, and the other two eigenvalues are determined by the 2 2 matrix B, where ðl þ dÞ r B¼ : ~ðtÞ d ay ~ðtÞ < 0, k1 k2 ¼ aðl þ dÞy ~ðtÞÞ rd. When p > p1 , Denote the eigenvalues of B as k1 , k2 , then we have k1 þ k2 ¼ ðl þ dÞ ay ~ðtÞ is locally asymptotically stable. The proof is complete. h we get k1 k2 > 0. Therefore, by the Floquet’s theorem, ð0; 0; y We shall now prove the global stability of the pest-eradication period solution. Denote p2 ¼
drðeds 1Þ : aðl þ dÞ
~ðtÞÞ is globally asymptotically stable, if p > p2 . Theorem 4.8. The pest-eradication period solution ð0; 0; y Proof. Let f ðxÞ ¼ ex 1 x: Because f 0 ðxÞ ¼ ex 1 > 0;
whenever x > 0;
we have eds 1 > ds and drðeds 1Þ drds > ; aðl þ dÞ aðl þ dÞ
i:e:; p2 > p1 :
~ðtÞÞ is locally asymptotically stable if p > p2 . In the following, we shall prove its global By Theorem 4.7, we know that ð0; 0; y attraction. Let VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ: Then we get V 0 ðtÞjð3Þ ¼ ½rd bðl þ dÞx2 ðtÞ aðl þ dÞyðtÞx2 ðtÞ:
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When p > p2 , we can select > 0 small enough, such that r1 ¼ dr þ apðlþdÞ < 0. From the third and sixth equations of syseds 1 tem (3), we have 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ P dyðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð7Þ yðnsþ Þ ¼ yðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . ds
pe ~ðtÞ aðlþdÞ P 1e By Lemmas 4.4 and 4.6, we know that there exists a t1 > 0, such that yðtÞ P y ds aðlþdÞ > 0, for all t P t 1 . Thus, when t P t 1 , we have apðl þ dÞ x2 ðtÞ ¼ r1 x2 ðtÞ < 0: V 0 ðtÞjð3Þ 6 ðdr þ Þ ds e 1
So VðtÞ ! 0, and x1 ðtÞ ! 0; x2 ðtÞ ! 0 as t ! 1. Notice that the limit system of system (3) is exactly system (4), together with ~ðtÞÞ is globally attractive. The proof is complete. h Lemma 4.4, we know that the pest-eradication period solution ð0; 0; y In the above two theorems we discuss the condition for the eradication of the pest. Permanence of the system implies that the pests can exist. Thus, it is necessary to investigate the permanence of the system (3) next. Denote p3 ¼
drð1 eds Þ : aðl þ dÞ
Theorem 4.9. If p < p3 , then system (3) is permanent. Proof. Suppose ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ is a solution of system (3) with initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. From Lemdr mas 4.2 and 4.5, we may assume 0 < x1 ðtÞ 6 M; 0 < x2 ðtÞ 6 M; 0 < yðtÞÞ 6 M, for all t P 0, and M > ðaþbÞðlþdÞ . Let peds m4 ¼ 1e > 0; > 0. From Lemmas 4.4 and 4.6, we clearly have yðtÞ P m for all t large enough. We shall next find 2 2 4 ds an m > 0, such that x1 ðtÞ þ x2 ðtÞ P m for all t large enough. We shall do it in two steps. Step 1: Since p < p3 , we can choose m2 > 0; 1 > 0 small enough such that p<
ðdr 1 bðl þ dÞm2 Þð1 eðdkam2 Þs Þ : aðl þ dÞ
Denote q ¼ dr 1 bðl þ dÞm2 r2 ¼
aðd þ lÞp > 0; 1 eðdkam2 Þs
dr ða þ bÞM < 0: lþd
Now, we consider the Liapunov function as follows VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ: We claim that VðtÞ < ðl þ dÞm2 ¼ m3 cannot hold for all t P 0. Otherwise, we have that x2 ðtÞ < m2 for all t P 0. Then, from the third and the sixth equations of system (3), we get 0 y ðtÞ 6 ðkam2 dÞyðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð8Þ yðnsþ Þ ¼ yðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . 1 ~ ðtÞ þ aðlþdÞ , for all t P t 1 . Where From Lemmas 4.4 and 4.6, we know that there exists a t 1 > 0, such that yðtÞ 6 u peðdkam2 ÞðtnsÞ ~ uðtÞ ¼ 1eðdkam2 Þs ; t 2 ðns; ðn þ 1Þs; n 2 Z þ is the unique periodic solution of the following system: 0 u ðtÞ ¼ ðkam2 dÞuðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð9Þ uðnsþ Þ ¼ uðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . .
Thus, yðtÞ 6
p 1 þ 1 eðdkam2 Þs aðl þ dÞ
and (3) V 0 ðtÞjð3Þ ¼ ½dr bðl þ dÞx2 ðtÞ aðd þ lÞyðtÞx2 ðtÞ P dr 1 bðl þ dÞm2 for all t P t 1 . Thus, V 0 ðtÞjð3Þ P qx2 ðt1 Þ ¼ 3 > 0
aðd þ lÞp x2 ðtÞ ¼ qx2 ðtÞ > 0 1 eðdkam2 Þs
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and VðtÞ ! þ1;
x2 ðtÞ ! þ1
as t ! þ1:
It is a contradiction, since VðtÞ is ultimately bounded. Therefore, there exists t 1 > 0, such that Vðt1 Þ P m3 . Step 2: If VðtÞ P m3 hold for all t P t 1 , then we can select m ¼ m2 , and our aim is obtained. We consider those solutions which leave the region D ¼ fðx1 ; x2 ; yÞ 2 R3þ : VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ < m3 g, and reenter it again. Let t ¼ inf tPt1 fVðtÞ < m3 g. Then, VðtÞ P m3 for t 2 ½t1 ; t Þ, and Vðt Þ ¼ m3 . Suppose t 2 ½n1 s; ðn1 þ 1ÞsÞ, n1 2 N. Select n2 ; n3 such that 1 1 ; ln d kam2 Mþp q1 n3 s þ r2 ðn2 þ 1Þs > 0; n2 s >
q1 ¼ qx2 ððn1 þ 1 þ n2 Þsþ Þ: Let T ¼ ðn2 þ n3 Þs. We claim that there must be a t2 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T, such that Vðt 2 Þ P m3 . Otherwise, VðtÞ < m3 and x2 ðtÞ < m2 , for all t 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T. Consider system 8 0 > < u ðtÞ ¼ ðkam2 dÞuðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð10Þ uðnsþ Þ ¼ uðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . ; > : uððn1 þ 1Þsþ Þ ¼ yððn1 þ 1Þsþ Þ: We have uðtÞ ¼ ðuððn1 þ 1Þsþ Þ
p ~ ðtÞ Þeðdkam2 Þðtðn1 þ1ÞsÞ þ u 1 eðdkam2 Þs
for t 2 ðns; ðn þ 1Þs; n1 þ 1 6 n 6 n1 þ 1 þ n2 þ n3 . Then ~ ðtÞj 6 ðM þ pÞeðdkam2 Þn2 s < 1 juðtÞ u and ~ ðtÞ þ yðtÞ 6 uðtÞ 6 u
1 p 1 6 þ aðl þ dÞ 1 eðdkam2 Þs aðl þ dÞ
for ðn1 þ 1 þ n2 Þs 6 t 6 ðn1 þ 1Þs þ T. Which implies that V 0 ðtÞjð3Þ P qx2 ððn1 þ 1 þ n2 Þsþ Þ ¼ q1 > 0 and Vððn1 þ 1Þs þ TÞ P Vððn1 þ 1 þ n2 ÞsÞeq1 n3 s :
ð11Þ
When t 2 ½t ; ðn1 þ 1 þ n2 Þs, we have V 0 ðtÞjð3Þ ¼ ½dr bðl þ dÞx2 ðtÞ aðl þ dÞyðtÞx2 ðtÞ P ½dr ða þ bÞðl þ dÞMx2 ðtÞ dr ða þ bÞM ðl þ dÞx2 ðtÞ P r2 VðtÞ: ¼ lþd Thus, Vððn1 þ 1 þ n2 ÞsÞ P Vðt Þer2 ðn2 þ1Þs > 0:
ð12Þ
From Eqs. (11) and (12), we get Vððn1 þ 1Þs þ TÞ P Vðt Þer2 ðn2 þ1Þsþq1 n3 s > m3 : It is a contradiction. Thus, there exists t 2 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T, such that Vðt2 Þ P m3 . For t 2 ½t ; t 2 Þ, we have : VðtÞ P Vðt Þer2 ðtt Þ P m3 er2 ð1þn1 þn2 þn3 Þs ¼ m1 : m1 , then x1 ðtÞ þ x2 ðtÞ P m for all t P t1 . For t > t2 , the same arguments can be continued since Vðt2 Þ P m3 . If we select m ¼ lþd By Definition 4.1, the system is permanence. The proof is complete. h
Remark 2. From Theorem 4.9, we can see that if the releasing number of natural enemy is less than p3 , then the system (3) is permanent. That is, the prey (pests) will persist and they will do harm to the crops. Under this condition, we can increase the releasing amount until p > p2 , and by the results of Theorem 4.8, we know that the prey (pests) will be doomed. 5. Discussion In this paper, two Lotka–Volterra predator–prey models with stage structure are proposed and investigated. In the first model, continuous biological control is taken, and we get the condition for the existence and local stability of two
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equilibriums (trivial equilibrium and positive equilibrium). The global asymptotical stability for the positive equilibrium (pest-eradication equilibrium) is also proved by use of Liapunov function. In the second model, we study the existence and stability of the pest-eradication periodic solution of the system. The condition under which system (3) will be permanent is also obtained. In both models, we get the sufficient conditions for the prey (pest) to extinction. In the real world, people can take continuous biological control or impulsive biological control to suppress the pest population. However, there are still some interesting problems: ð1Þ in the continuous biological control model, how about the global stability of equilibriums and the existence of nontrivial periodic solutions, when p < p0 ; (2) if p < p3 , then system (3) is permanent; if p > p2 , then the ~ðtÞÞ is globally asymptotically stable; but we have not consider the case that pest-eradication periodic solution ð0; 0; y p2 > p > p3 . Wether there exist some nontrivial periodic solutions, and how about the dynamical behaviors of system (3). In this paper, our aim is to control the prey (pest) population, so we omit the above problems. But from the view point of mathematics, they are interesting, and we shall study these problems in the future. Acknowledgements This work is partly supported by the National Natural Science Foundation of China (10471117). We would like to thank the referees very much for their valuable comments and suggestions. References [1] X.Y. Song, L.S. Chen, Optimal harvesting policy and stability for single-species growth model with stage structure, Journal of Systems Science and Complexity 15 (2002) 194–201. [2] Y.N. Xiao, L.S. Chen, Stabilizing effect of cannibalism on a structured competitive system, Acta Mathematics 22A (2) (2002) 210–216 (in Chinese). [3] X.Y. Song, L.S. Chen, Modelling and analysis of a single-species system with stage structure and harvesting, Mathematics and Computer Modelling 36 (2002) 67–82. [4] X.Y. Song, L.S. Chen, Optimal harvesting and stability for a predator–prey system with stage structure, Acta Mathematical Applicate Sinica 18 (2002) 423–430. [5] Z.H. Lu, L.S. Chen, Global attractivity of nonautonomous inshore–offshore fishing models with stage-structure, Applicable Analysis 81 (2002) 589–605. [6] S.Q. Liu, L.S. Chen, R. Agarwal, Recent progress on stage-structured population dynamics, Mathematics and Computer Modelling 36 (2002) 1319–1360. [7] S.Q. Liu, L.S. Chen, Z.J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, Journal of Mathematical Analysis and Applications 274 (2002) 667–684. [8] S.Q. Liu, L.S. Chen, G.L. Luo, Extinction and permance in competitive stage-structured system with time delays, Nonlinear Analysis 51 (2002) 1347– 1361. [9] S.Y. Tang, L.S. Chen, Multiple attractors in stage-structured population models with birth pulses, Bulletin of Mathematical Biology 65 (2003) 479–495. [10] Y.N. Xiao, L.S. Chen, On an SIS epidemic model with stage structure, Journal of Systems Science and Complexity 16 (2003) 275–288. [11] Y.N. Xiao, L.S. Chen, An SIS epidemic models with stage structure and a delay, Acta Mathematical Applicate Sinica (English Series) 18 (4) (2002) 607– 618. [12] W.G. Aiello, H.I. Freedman, A time delay model of single-species growth with stage structure, Mathematical Biosciences 101 (1990) 139–153. [13] W.G. Aiello, H.I. Freedman, J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal on Applied Mathematics 52 (1992) 855–869. [14] W.D. Wang, L.S. Chen, A predator–prey system with stage structure for predator, Computers Mathematics with Applications 33 (1997) 83–91. [15] V. Lakshmikantham, D.D. Bainov, P. Simeonov (Eds.), Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [16] D. Bainov, P. Simeonov (Eds.), Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, 1993. [17] Y.N. Xiao, L.S. Chen, F.V.D. Bosh, Dynamical behavior for stage-structured SIR infectious disease model, Nonlinear Analysis: RWA 3 (2002) 175–190. [18] Z.H. Lu, S.J. Gang, L.S. Chen, Analysis of an SI epidemic with nonlinear transmission and stage structure, Acta Mathematics Sinica 4 (2003) 440–446. [19] 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.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Staged-structured Lotka–Volterra predator–prey models for pest management Ruiqing Shi a,b,*, Lansun Chen a a b
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, People’s Republic of China
a r t i c l e
i n f o
Keywords: Stage structured Biological control Impulsive Pest-eradication equilibrium Pest-eradication periodic solution
a b s t r a c t In this paper, two predator–prey models with stage structure are constructed and investigated. In the first model, continuous biological control is taken. The existence and local stability of two equilibriums are studied. By the Liapunov stability theorem, we obtain the condition for the global asymptotical stability of the trivial equilibrium (i.e., pest-eradication equilibrium). In the second model, impulsive biological control is taken. By use of the Floquet’s theorem, small-amplitude perturbation method and comparison techniques, we get the condition which guarantees the global asymptotical stability of the pest-eradication periodic solution. The sufficient condition for the permanence of the impulsive system is also obtained. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction According to reports of Food and Agriculture Organization of the United Nations, the warfare between human and pests (such as locust, Aphis, cotton bollworm, etc.) has sustained for thousands of years. With the development of society and the progress of science and technology, man has adopted some advanced and modern weapons such as chemical pesticides, biological pesticides, remote sensing and measuring, computers, atomic energy and so on. At last, some brilliant achievements have been obtained. However, the warfare is not over, and will continue. A great deal of and a large variety of pesticides were used to control pests. Pesticides are useful because they can quickly kill a significant portion of a pest population and sometimes provide the only feasible method for preventing economic loss. However, pesticide pollution is also recognized as a major health hazard to human beings and beneficial insects. In the natural world, many pest individuals have a life story that takes them through two stages, immature and mature. The natural enemy (predator) only catch the mature pests (prey), for the immature pests are protected by their eggshells. There are some literatures about single-species models with stage structure [1,3,5,12]; while other models are about competitive systems or predator–prey systems with stage structure [2,4,7,8,14]. There are still some other models concerning stage structure [3,6,9–11,13,17,18]. Liu and Chen [19] developed the Holling II Lotka–Volterra predator–prey (natural enemy–pest) system by periodic constant impulsive immigration of natural enemy. They gave the conditions for extinction of the pest and permanence of the system and studied the influences on the inherent oscillation caused by the impulsive perturbations. But the paper did not consider the stage structure. Wang and Chen [14] studied a predator–prey system with stage structure for the predator, but they did not consider biological control. In this paper, we consider the Lotka–Volterra predator–prey models with stage structure for the prey (pest). For the purpose of pest management, we shall release natural
* Corresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China. E-mail addresses: [email protected] (R. Shi), [email protected] (L. Chen). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.032
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enemy (predator), then more of the pests (prey) will be captured. The natural enemy may be cultivated in the laboratory, or migrate from other areas. We can release the natural enemy continuously, or impulsively. The main purpose of this paper is to construct two realistic models of biological control systems for pest management, to investigate their dynamics and to compare the results obtained for the ordinary differential model, corresponding to the continuous biological control, with those obtained for the impulsive differential model, corresponding to the impulsive biological control. The present paper is organized as follows. In Section 2, the main biological assumptions are formulated, and two models are constructed. In Section 3, by using the qualitative theory of ordinary differential equations, we investigate the behavior of the ordinary system which models the process of continuous release of natural enemy. In Section 4, we study the impulsive system which models the process of periodic release of natural enemy at fixed moments. By use of the Floquet’s theory for impulsive differential equations, small-amplitude perturbation methods and comparison techniques, we get the global asymptotic stability of the pest-eradication periodic solution and the condition for the permanence of the system. Finally, a brief discussion is provided in the last section. 2. Models formulation The basic stage-structured Lotka–Volterra predator–prey model is 8 0 > < x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > : 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ;
ð1Þ
where x1 ðtÞ and x2 ðtÞ denote the densities of immature and mature prey (pest) population, respectively, yðtÞ denotes the density of the predator (natural enemy) population. All coefficients are positive constants. The model is derived with the following assumptions. ðH1 Þ At any time t P 0 birth into the immature prey population is proportional to the existing mature prey population, with proportionality constant r. ðH2 Þ The immature prey population has the natural death rate l, and the predator has a natural death rate d. The death rate of the mature prey population has a logistic nature, that is, proportional to the square of the population with proportionality constant b. ðH3 Þ The immature prey population enter into mature population with proportionality constant d. ðH4 Þ Only the mature preys are preyed by the predator, and the immature preys are protected by their eggshells, where a is the per-capita rate of predation of the predator, and 0 < k < 1 is the rate of conversing prey into predator.In this paper, we construct the following two models, and the parameters are of the same with those in model (1), with the assumption follows. ðH5 Þ The predator population are released with constant rate of p > 0 in model (2), or with constant number p > 0 periodically with period s in model (3), respectively: 8 0 > < x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; ð2Þ x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > : 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ þ p: And 8 0 9 x1 ðtÞ ¼ rx2 ðtÞ lx1 ðtÞ dx1 ðtÞ; > > > = > > > > x02 ðtÞ ¼ dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ; > > > > ; > < y0 ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ; > Dx1 ðtÞ ¼ 0; 9 > > > = > > > Dx ðtÞ ¼ 0; > t ¼ ns; n ¼ 1; 2; 3; . . . > 2 > > > ; : DyðtÞ ¼ p;
t 6¼ ns; n ¼ 1; 2; 3; . . . ;
3. Qualitative analysis for system (2) Denote p0 ¼
drd : aðl þ dÞ
By simple calculation, we have Lemma 3.1. (1) System (2) always has a trivial equilibrium (pest-eradication equilibrium) E0 ¼ ðx01 ; x02 ; y0 Þ ¼ ð0; 0; pdÞ.
ð3Þ
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(2) If p < p0 , then system (2) has a unique positive equilibrium (natural enemy–pest coexistence equilibrium) E1 ¼ ðx1 ; x2 ; y Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr
ay
r x2 , x2 ¼ lþd b where x1 ¼ lþd
, and y ¼
ðkadr lþd dbÞþ
2
2 ðkadr lþd dbÞ þ4ka pb 2
2ka
.
Theorem 3.2. If p > p0 , then the trivial equilibrium E0 is locally asymptotically stable. If p < p0 , then E0 is unstable, and the positive equilibrium E1 is locally asymptotically stable. Proof. The characteristic equation at E0 is ðk þ dÞ½k2 þ ðl þ d þ ay0 Þk þ aðl þ dÞy0 dr ¼ 0: If p > p0 holds, then all of the eigenvalues of the above characteristic equation have negative real part. Thus, E0 is locally asymptotically stable. If p < p0 holds, then one of the eigenvalues of the above characteristic equation has positive real part. Thus, E0 is unstable. By Lemma 3.1, we know that the positive equilibrium E1 exists when p < p0 . The characteristic equation at E1 is k3 þ a1 k2 þ a2 k þ a3 ¼ 0; where dr p þ bx2 þ > 0; a1 ¼ ðl þ dÞ þ lþd y p dr p 2 þ bx2 a2 ¼ ðl þ dÞ bx2 þ þ þ ka x2 y > 0; y lþd y p 2 a3 ¼ ðl þ dÞ bx2 þ ka x2 y > 0 y and a1 a2 a3 ¼
dr p p dr p dr p 2 8 p þ bx2 þ ðl þ dÞ bx2 þ þ þ bx þ Þ þ ka x y þ ðbx 2 2 2 lþd y y l þ d y lþd y y p dr p þ ðl þ dÞ ðl þ dÞ bx2 þ þ > 0: y l þ d y
By the Hurwitz criterion, we get that all of the three eigenvalues of the above characteristic equation have negative real part. Thus, E1 is locally asymptotically stable, whenever it exists. This completes the proof. h Theorem 3.3. The trivial equilibrium E0 is globally asymptotically stable if p P p0 . Proof. Define the Liapunov function as follows: yðtÞ VðtÞ ¼ c1 x1 ðtÞ þ c2 x2 ðtÞ þ c3 yðtÞ y0 y0 ln 0 ; y where ci are positive constants to be determined (i ¼ 1; 2; 3), then dVðtÞ y0 ½kax2 ðtÞyðtÞ dyðtÞ þ p ¼ c1 ½rx2 ðtÞ ðl þ dÞx1 ðtÞ þ c2 ½dx1 ðtÞ bx22 ðtÞ ax2 ðtÞyðtÞ þ c3 1 dt ð2Þ yðtÞ c3 d y0 ðyðtÞ y0 Þ2 þ c1 rx2 ðtÞ ½c1 ðl þ dÞ c2 dx1 ðtÞ ax2 ðtÞyðtÞ c2 c3 k 1 c2 bx22 ðtÞ: ¼ yðtÞ yðtÞ If we set c2 ¼ kc3 ¼ l þ d, and c1 ðl þ dÞ ¼ c2 d, then dVðtÞ c3 d apðl þ dÞ 0 2 2 ðyðtÞ y x2 ðtÞ: ¼ Þ c bx ðtÞ þ dr 2 2 dt ð2Þ yðtÞ d < 0. Therefore E0 is globally asymptotically stable. The proof is complete. If p P p0 , then we easily get dVðtÞ dt
h
ð2Þ
Remark 1. From the above theorems, we can see that if the coefficients satisfy the condition p < p0 , then, the prey (pests) will persist, and they will do harm to the crops. Under this condition, we can release predator (natural enemy) with rate p P p0 , and by the result of Theorem 3.3, we know that the prey (pests) will be controlled. 4. Qualitative analysis for system (3) Definition 4.1. System (3) is said to be permanent if there are constants m; M > 0 (independent of initial value) and a finite time T 0 such that for all solutions ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ with all initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0,
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m 6 x1 ðtÞ þ x2 ðtÞ 6 M; m 6 yðtÞ 6 M x2 ð0þ Þ; yð0þ ÞÞ.
hold for all t P T 0 . Here T 0
261
may depend on the initial values ðx1 ð0þ Þ;
Lemma 4.2 [15,16]. Suppose ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ is a solution of system (3) with initial values x1 ð0þ Þ P 0; x2 ð0þ Þ P 0; yð0þ Þ P 0, then x1 ðtÞ P 0; x2 ðtÞ P 0; yðtÞ P 0, for all t P 0. And further x1 ðtÞ > 0; x2 ðtÞ > 0; yðtÞ > 0, if x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. Lemma 4.3 [15]. Let the function m 2 PC 0 ½Rþ ; R satisfies the inequalities ( m0 ðtÞ 6 pðtÞmðtÞ þ qðtÞ; t P t0 ; t 6¼ tk ; k ¼ 1; 2; . . . ; mðt þ k Þ 6 dk mðt k Þ þ bk ;
t ¼ tk ;
where p; q 2 PC½Rþ ; R and dk P 0; bk are constants, then 0 1 Rt Rt Z Y Y X pðsÞds pðsÞds t t0 @ Abk þ dk e þ dj e k mðtÞ 6 mðt0 Þ t 0
t 0
t k
t
Y
Rt pðrÞdr dk e s qðsÞds
t 0 s
for all t P t 0 . If x1 ðtÞ ¼ x2 ðtÞ ¼ 0, for all t P 0, then we get the subsystem of system (3) as follows: 0 y ðtÞ ¼ dyðtÞ; t¼ 6 ns; n ¼ 1; 2; . . . ; yðnsþ Þ ¼ yðnsÞ þ p;
t ¼ ns; n ¼ 1; 2; . . .
ð4Þ
dðtnsÞ
~ð0þ Þ ¼ 1epds is a positive periodic solution of system (4). The ~ðtÞ ¼ pe1eds ; t 2 ðns; ðn þ 1Þs; n 2 Z þ ¼ f1; 2; 3; . . .g; y Obviously y þ p dt ~ solution of system (4) is yðtÞ ¼ ðyð0 Þ 1eds Þe þ yðtÞ, t 2 ðns; ðn þ 1Þs; n 2 Z þ . Therefore, the complete expression for the dðtnsÞ ~ðtÞÞ ¼ ð0; 0; pe1eds Þ; t 2 ðns; ðn þ 1Þs; n 2 Z þ . And we pest-eradication periodic solution of system (3) is obtained as ð0; 0; y derive ~ðtÞ as t ! 1. Lemma 4.4 [16]. For every solution of system (4) with initial condition yð0þ Þ > 0, it follows that yðtÞ ! y Lemma 4.5. There exists a constant M > 0 such that x1 ðtÞ 6 M; x2 ðtÞ 6 M; yðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (3) for all t large enough. Proof. Due to the positivity of x1 ðtÞ; x2 ðtÞ; yðtÞ, we define ðtÞ ¼ x1 ðtÞ þ x2 ðtÞ þ yðtÞ: When t 6¼ ns, we have Dþ UðtÞ þ kUðtÞ ¼ ðr þ kÞx2 ðtÞ ðl kÞx1 ðtÞ ðd kÞyðtÞ bx22 ðtÞ að1 kÞx2 ðtÞyðtÞ 6 M0 ; 2
where k ¼ minfl; dg, and M0 ¼ ðrþkÞ . 4b When t ¼ ns; Uðnsþ Þ ¼ UðnsÞ þ p. By Lemma 4.3, for t 2 ðns; ðn þ 1Þs we have M 0 kt pð1 enks Þ kðtnsÞ M 0 e þ : e þ UðtÞ 6 Uð0þ Þ 1 eks k k So UðtÞ is uniformly ultimately bounded by a positive constant and there exists a constant M > 0 such that x1 ðtÞ 6 M; x2 ðtÞ 6 M; yðtÞ 6 M for each solution ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ of system (3) for all t large enough. The proof is complete. h Lemma 4.6 [15]. Let V : Rþ R2 ! R and V 2 V 0 . Assume that ( t 6¼ nT; n ¼ 1; 2; . . . ; Dþ Vðt; XÞ 6 gðt; Vðt; XÞÞ; Vðt; Xðt þ ÞÞ 6 Wn ðVðt; XðtÞÞÞ;
t ¼ nT; n ¼ 1; 2; . . . ;
ð5Þ
where g : Rþ Rþ ! R is continuous in ðnT; ðn þ 1ÞT Rþ and for each v 2 R2þ ; n 2 Z þ lim
ðt;yÞ!ðnT þ ;vÞ
gðt; yÞ ¼ gðnT þ ; vÞ
exists and is finite, Wn : Rþ ! Rþ is nondecreasing. Let RðtÞ be the maximal solution of the scalar impulsive differential equation: 8 0 t 6¼ nT; n ¼ 1; 2; . . . ; > < U ðtÞ ¼ gðt; UÞ; Uðtþ Þ ¼ Wn ðUðtÞÞ; t ¼ nT; n ¼ 1; 2; . . . ; > : Uð0þ Þ ¼ U 0 ; defined on ½0; 1Þ. Then Vð0þ ; X 0 Þ 6 U 0 implies that Vðt; XðtÞÞ 6 RðtÞ, t P 0, where XðtÞ is any solution of (5).
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Next, we investigate the stability of the pest-eradication periodic solution of system (3) with initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. Denote p1 ¼
drds : aðl þ dÞ
~ðtÞÞ is locally asymptotically stable if p > p1 . Theorem 4.7. The pest-eradication periodic solution ð0; 0; y Proof. The local stability of pest-eradication periodic solution may be determined by considering the behaviors of small ~ðtÞ, then the linearized equations of amplitude perturbation of the solution. Define uðtÞ ¼ x1 ðtÞ; vðtÞ ¼ x2 ðtÞ; wðtÞ ¼ yðtÞ y system (3) read as 8 9 u0 ðtÞ ¼ rvðtÞ ðl þ dÞuðtÞ; > > > = > > > ~ðtÞvðtÞ; > t 6¼ ns; n ¼ 1; 2; 3; . . . ; v0 ðtÞ ¼ duðtÞ ay > > > > < w0 ðtÞ ¼ kay ~ðtÞvðtÞ dwðtÞ; ; 9 ð6Þ > uðnsþ Þ ¼ uðnsÞ; > > = > > > > vðnsþ Þ ¼ vðnsÞ; t ¼ ns; n ¼ 1; 2; 3; . . . > > > > ; : wðnsþ Þ ¼ wðnsÞ; Let UðtÞ be the fundamental solution matrix of system (6), then UðtÞ must satisfy 0 1 ðl þ dÞ r 0 dUðtÞ B C ~ðtÞ ¼@ d ay 0 AUðtÞ ¼ AUðtÞ dt ~ðtÞ d 0 kay and Uð0Þ ¼ I, the identity matrix. We can see that one of the eigenvalues of matrix A is d, and the other two eigenvalues are determined by the 2 2 matrix B, where ðl þ dÞ r B¼ : ~ðtÞ d ay ~ðtÞ < 0, k1 k2 ¼ aðl þ dÞy ~ðtÞÞ rd. When p > p1 , Denote the eigenvalues of B as k1 , k2 , then we have k1 þ k2 ¼ ðl þ dÞ ay ~ðtÞ is locally asymptotically stable. The proof is complete. h we get k1 k2 > 0. Therefore, by the Floquet’s theorem, ð0; 0; y We shall now prove the global stability of the pest-eradication period solution. Denote p2 ¼
drðeds 1Þ : aðl þ dÞ
~ðtÞÞ is globally asymptotically stable, if p > p2 . Theorem 4.8. The pest-eradication period solution ð0; 0; y Proof. Let f ðxÞ ¼ ex 1 x: Because f 0 ðxÞ ¼ ex 1 > 0;
whenever x > 0;
we have eds 1 > ds and drðeds 1Þ drds > ; aðl þ dÞ aðl þ dÞ
i:e:; p2 > p1 :
~ðtÞÞ is locally asymptotically stable if p > p2 . In the following, we shall prove its global By Theorem 4.7, we know that ð0; 0; y attraction. Let VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ: Then we get V 0 ðtÞjð3Þ ¼ ½rd bðl þ dÞx2 ðtÞ aðl þ dÞyðtÞx2 ðtÞ:
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When p > p2 , we can select > 0 small enough, such that r1 ¼ dr þ apðlþdÞ < 0. From the third and sixth equations of syseds 1 tem (3), we have 0 y ðtÞ ¼ kax2 ðtÞyðtÞ dyðtÞ P dyðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð7Þ yðnsþ Þ ¼ yðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . ds
pe ~ðtÞ aðlþdÞ P 1e By Lemmas 4.4 and 4.6, we know that there exists a t1 > 0, such that yðtÞ P y ds aðlþdÞ > 0, for all t P t 1 . Thus, when t P t 1 , we have apðl þ dÞ x2 ðtÞ ¼ r1 x2 ðtÞ < 0: V 0 ðtÞjð3Þ 6 ðdr þ Þ ds e 1
So VðtÞ ! 0, and x1 ðtÞ ! 0; x2 ðtÞ ! 0 as t ! 1. Notice that the limit system of system (3) is exactly system (4), together with ~ðtÞÞ is globally attractive. The proof is complete. h Lemma 4.4, we know that the pest-eradication period solution ð0; 0; y In the above two theorems we discuss the condition for the eradication of the pest. Permanence of the system implies that the pests can exist. Thus, it is necessary to investigate the permanence of the system (3) next. Denote p3 ¼
drð1 eds Þ : aðl þ dÞ
Theorem 4.9. If p < p3 , then system (3) is permanent. Proof. Suppose ðx1 ðtÞ; x2 ðtÞ; yðtÞÞ is a solution of system (3) with initial values x1 ð0þ Þ > 0; x2 ð0þ Þ > 0; yð0þ Þ > 0. From Lemdr mas 4.2 and 4.5, we may assume 0 < x1 ðtÞ 6 M; 0 < x2 ðtÞ 6 M; 0 < yðtÞÞ 6 M, for all t P 0, and M > ðaþbÞðlþdÞ . Let peds m4 ¼ 1e > 0; > 0. From Lemmas 4.4 and 4.6, we clearly have yðtÞ P m for all t large enough. We shall next find 2 2 4 ds an m > 0, such that x1 ðtÞ þ x2 ðtÞ P m for all t large enough. We shall do it in two steps. Step 1: Since p < p3 , we can choose m2 > 0; 1 > 0 small enough such that p<
ðdr 1 bðl þ dÞm2 Þð1 eðdkam2 Þs Þ : aðl þ dÞ
Denote q ¼ dr 1 bðl þ dÞm2 r2 ¼
aðd þ lÞp > 0; 1 eðdkam2 Þs
dr ða þ bÞM < 0: lþd
Now, we consider the Liapunov function as follows VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ: We claim that VðtÞ < ðl þ dÞm2 ¼ m3 cannot hold for all t P 0. Otherwise, we have that x2 ðtÞ < m2 for all t P 0. Then, from the third and the sixth equations of system (3), we get 0 y ðtÞ 6 ðkam2 dÞyðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð8Þ yðnsþ Þ ¼ yðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . 1 ~ ðtÞ þ aðlþdÞ , for all t P t 1 . Where From Lemmas 4.4 and 4.6, we know that there exists a t 1 > 0, such that yðtÞ 6 u peðdkam2 ÞðtnsÞ ~ uðtÞ ¼ 1eðdkam2 Þs ; t 2 ðns; ðn þ 1Þs; n 2 Z þ is the unique periodic solution of the following system: 0 u ðtÞ ¼ ðkam2 dÞuðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð9Þ uðnsþ Þ ¼ uðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . .
Thus, yðtÞ 6
p 1 þ 1 eðdkam2 Þs aðl þ dÞ
and (3) V 0 ðtÞjð3Þ ¼ ½dr bðl þ dÞx2 ðtÞ aðd þ lÞyðtÞx2 ðtÞ P dr 1 bðl þ dÞm2 for all t P t 1 . Thus, V 0 ðtÞjð3Þ P qx2 ðt1 Þ ¼ 3 > 0
aðd þ lÞp x2 ðtÞ ¼ qx2 ðtÞ > 0 1 eðdkam2 Þs
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and VðtÞ ! þ1;
x2 ðtÞ ! þ1
as t ! þ1:
It is a contradiction, since VðtÞ is ultimately bounded. Therefore, there exists t 1 > 0, such that Vðt1 Þ P m3 . Step 2: If VðtÞ P m3 hold for all t P t 1 , then we can select m ¼ m2 , and our aim is obtained. We consider those solutions which leave the region D ¼ fðx1 ; x2 ; yÞ 2 R3þ : VðtÞ ¼ dx1 ðtÞ þ ðl þ dÞx2 ðtÞ < m3 g, and reenter it again. Let t ¼ inf tPt1 fVðtÞ < m3 g. Then, VðtÞ P m3 for t 2 ½t1 ; t Þ, and Vðt Þ ¼ m3 . Suppose t 2 ½n1 s; ðn1 þ 1ÞsÞ, n1 2 N. Select n2 ; n3 such that 1 1 ; ln d kam2 Mþp q1 n3 s þ r2 ðn2 þ 1Þs > 0; n2 s >
q1 ¼ qx2 ððn1 þ 1 þ n2 Þsþ Þ: Let T ¼ ðn2 þ n3 Þs. We claim that there must be a t2 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T, such that Vðt 2 Þ P m3 . Otherwise, VðtÞ < m3 and x2 ðtÞ < m2 , for all t 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T. Consider system 8 0 > < u ðtÞ ¼ ðkam2 dÞuðtÞ; t 6¼ ns; n ¼ 1; 2; 3; . . . ; ð10Þ uðnsþ Þ ¼ uðnsÞ þ p; t ¼ ns; n ¼ 1; 2; 3; . . . ; > : uððn1 þ 1Þsþ Þ ¼ yððn1 þ 1Þsþ Þ: We have uðtÞ ¼ ðuððn1 þ 1Þsþ Þ
p ~ ðtÞ Þeðdkam2 Þðtðn1 þ1ÞsÞ þ u 1 eðdkam2 Þs
for t 2 ðns; ðn þ 1Þs; n1 þ 1 6 n 6 n1 þ 1 þ n2 þ n3 . Then ~ ðtÞj 6 ðM þ pÞeðdkam2 Þn2 s < 1 juðtÞ u and ~ ðtÞ þ yðtÞ 6 uðtÞ 6 u
1 p 1 6 þ aðl þ dÞ 1 eðdkam2 Þs aðl þ dÞ
for ðn1 þ 1 þ n2 Þs 6 t 6 ðn1 þ 1Þs þ T. Which implies that V 0 ðtÞjð3Þ P qx2 ððn1 þ 1 þ n2 Þsþ Þ ¼ q1 > 0 and Vððn1 þ 1Þs þ TÞ P Vððn1 þ 1 þ n2 ÞsÞeq1 n3 s :
ð11Þ
When t 2 ½t ; ðn1 þ 1 þ n2 Þs, we have V 0 ðtÞjð3Þ ¼ ½dr bðl þ dÞx2 ðtÞ aðl þ dÞyðtÞx2 ðtÞ P ½dr ða þ bÞðl þ dÞMx2 ðtÞ dr ða þ bÞM ðl þ dÞx2 ðtÞ P r2 VðtÞ: ¼ lþd Thus, Vððn1 þ 1 þ n2 ÞsÞ P Vðt Þer2 ðn2 þ1Þs > 0:
ð12Þ
From Eqs. (11) and (12), we get Vððn1 þ 1Þs þ TÞ P Vðt Þer2 ðn2 þ1Þsþq1 n3 s > m3 : It is a contradiction. Thus, there exists t 2 2 ½ðn1 þ 1Þs; ðn1 þ 1Þs þ T, such that Vðt2 Þ P m3 . For t 2 ½t ; t 2 Þ, we have : VðtÞ P Vðt Þer2 ðtt Þ P m3 er2 ð1þn1 þn2 þn3 Þs ¼ m1 : m1 , then x1 ðtÞ þ x2 ðtÞ P m for all t P t1 . For t > t2 , the same arguments can be continued since Vðt2 Þ P m3 . If we select m ¼ lþd By Definition 4.1, the system is permanence. The proof is complete. h
Remark 2. From Theorem 4.9, we can see that if the releasing number of natural enemy is less than p3 , then the system (3) is permanent. That is, the prey (pests) will persist and they will do harm to the crops. Under this condition, we can increase the releasing amount until p > p2 , and by the results of Theorem 4.8, we know that the prey (pests) will be doomed. 5. Discussion In this paper, two Lotka–Volterra predator–prey models with stage structure are proposed and investigated. In the first model, continuous biological control is taken, and we get the condition for the existence and local stability of two
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equilibriums (trivial equilibrium and positive equilibrium). The global asymptotical stability for the positive equilibrium (pest-eradication equilibrium) is also proved by use of Liapunov function. In the second model, we study the existence and stability of the pest-eradication periodic solution of the system. The condition under which system (3) will be permanent is also obtained. In both models, we get the sufficient conditions for the prey (pest) to extinction. In the real world, people can take continuous biological control or impulsive biological control to suppress the pest population. However, there are still some interesting problems: ð1Þ in the continuous biological control model, how about the global stability of equilibriums and the existence of nontrivial periodic solutions, when p < p0 ; (2) if p < p3 , then system (3) is permanent; if p > p2 , then the ~ðtÞÞ is globally asymptotically stable; but we have not consider the case that pest-eradication periodic solution ð0; 0; y p2 > p > p3 . Wether there exist some nontrivial periodic solutions, and how about the dynamical behaviors of system (3). In this paper, our aim is to control the prey (pest) population, so we omit the above problems. But from the view point of mathematics, they are interesting, and we shall study these problems in the future. Acknowledgements This work is partly supported by the National Natural Science Foundation of China (10471117). We would like to thank the referees very much for their valuable comments and suggestions. References [1] X.Y. Song, L.S. Chen, Optimal harvesting policy and stability for single-species growth model with stage structure, Journal of Systems Science and Complexity 15 (2002) 194–201. [2] Y.N. Xiao, L.S. Chen, Stabilizing effect of cannibalism on a structured competitive system, Acta Mathematics 22A (2) (2002) 210–216 (in Chinese). [3] X.Y. Song, L.S. Chen, Modelling and analysis of a single-species system with stage structure and harvesting, Mathematics and Computer Modelling 36 (2002) 67–82. [4] X.Y. Song, L.S. Chen, Optimal harvesting and stability for a predator–prey system with stage structure, Acta Mathematical Applicate Sinica 18 (2002) 423–430. [5] Z.H. Lu, L.S. Chen, Global attractivity of nonautonomous inshore–offshore fishing models with stage-structure, Applicable Analysis 81 (2002) 589–605. [6] S.Q. Liu, L.S. Chen, R. Agarwal, Recent progress on stage-structured population dynamics, Mathematics and Computer Modelling 36 (2002) 1319–1360. [7] S.Q. Liu, L.S. Chen, Z.J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, Journal of Mathematical Analysis and Applications 274 (2002) 667–684. [8] S.Q. Liu, L.S. Chen, G.L. Luo, Extinction and permance in competitive stage-structured system with time delays, Nonlinear Analysis 51 (2002) 1347– 1361. [9] S.Y. Tang, L.S. Chen, Multiple attractors in stage-structured population models with birth pulses, Bulletin of Mathematical Biology 65 (2003) 479–495. [10] Y.N. Xiao, L.S. Chen, On an SIS epidemic model with stage structure, Journal of Systems Science and Complexity 16 (2003) 275–288. [11] Y.N. Xiao, L.S. Chen, An SIS epidemic models with stage structure and a delay, Acta Mathematical Applicate Sinica (English Series) 18 (4) (2002) 607– 618. [12] W.G. Aiello, H.I. Freedman, A time delay model of single-species growth with stage structure, Mathematical Biosciences 101 (1990) 139–153. [13] W.G. Aiello, H.I. Freedman, J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal on Applied Mathematics 52 (1992) 855–869. [14] W.D. Wang, L.S. Chen, A predator–prey system with stage structure for predator, Computers Mathematics with Applications 33 (1997) 83–91. [15] V. Lakshmikantham, D.D. Bainov, P. Simeonov (Eds.), Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [16] D. Bainov, P. Simeonov (Eds.), Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, 1993. [17] Y.N. Xiao, L.S. Chen, F.V.D. Bosh, Dynamical behavior for stage-structured SIR infectious disease model, Nonlinear Analysis: RWA 3 (2002) 175–190. [18] Z.H. Lu, S.J. Gang, L.S. Chen, Analysis of an SI epidemic with nonlinear transmission and stage structure, Acta Mathematics Sinica 4 (2003) 440–446. [19] 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.