# Multi-State Dependent Impulsive Control for Pest Management

Authors: Huidong Cheng; Fang Wang; Tongqian Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

According to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution of such system, and further, the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results. Our results show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.


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## Body

## 1. Introduction

It is of great value to study pest management method applied in agricultural production; entomologists and the whole society have been paying close attention to how to control pests effectively and to save manpower and material resources. In agricultural production, pesticides-spraying (chemical control) and release of natural enemies (biological control) are the ways commonly used for pest control. But if we implement chemical control as soon as pests appear, many problems are caused: the first is environmental pollution; the second is increase of costs including human and material resources and time; the third is killing natural enemies, such as parasitic wasp; the last is pests’ resistance to pesticides, which brings great negative effects instead of working as well as had been expected [1–3]. The second way, which controls pests with the help of the increasing natural enemies, can avoid problems caused by chemical control and gets more and more attention. So many scholars have been studying and discussing it [4–8]. Considering the effectiveness of the chemical control and nonpollution and limitations of the biological one, people have proposed the method of integrated pest management (IPM), which is a pest management system integrating all appropriate ways and technologies to control economic injury level (EIL) caused by pest populations in view of population dynamics and its relevant environment. In the process of practical application, people usually implement the following two schemes for the integrated pest management: one is to implement control at a fixed time to eradicate pests [9, 10]; the other is to implement measures only when the amount of pests reaches a critical level, which is to make the amount less than certain economic impairment level, not to wipe out pests [11–13]. Salazar conducted an experiment of broad bean being damaged by bean sprouts worm in 1976 and found “crops’ compensation to damage of pests”, that is, yields of crops which had been damaged a little by pests in the early growth are actually higher than those without damage. In other words, we do not want to wipe out pests but to control them to a certain economic injury level (EIL). So, the second is used most in the process of agricultural industry. Tang and Cheke [14] first proposed the “Volterra” model in the from of a state-dependent impulsive model:
(1.1)x′(t)=x(t)(a-by(t)),y′(t)=y(t)(-d+cx(t)),x≠ET,Δx(t)=-αx(t),Δy(t)=q,x=ET,
and they applied this model to pest management and proved existence and stability of periodic solution of first and second order. Then Tang and Cheke [14] also proposed bait-dependent digestive model with state pulse:
(1.2)x′(t)=x(t)(a-by(t)),y′(t)=y(t)(λbx(t)1+bhx(t)-d),x≠hmax,Δx(t)=-αx(t),Δy(t)=q,x=hmax,
they had the existence of positive periodic solution and stability of orbit. Recently Jiang and Lu et al. [15–17] have proposed pest management model with state pulse and phase structure and several predator-prey models with state pulse and had the existence of semi-trivial periodic solution and positive periodic solution and stability of orbit.It is worth mentioning that the vast majority of research on population dynamics system with state pulse considers single state pulse, which is to say, only when the amount of population reaches the same economic threshold can measures be taken (e.g., chemical control and biological control); but this single state-pulse control does not confirm to reality. In fact, we often need to use different control methods under different states in real life. For example, in the process of pest management, when the amount of pests is small, biological control is implemented; when the amount is large, combination control is applied. Tang et al. [18] have investigated and developed a mathematical model with hybrid impulsive model:
(1.3)x′(t)=rx(t)(1-δx(t))-bx(t)y(t),x<ET,y′(t)=y(t)(cx(t)-a),t=λm,x(t+)=(1-p1)x(t),y(t+)=(1-p2)y(t),x(t)=ET,y(λm+)=(1+p3)y(λ+)+q,t=λm.
Motivated by Tang, on the basis of the above analysis, we set up the following predicator-prey system with different control methods in different thresholds:
(1.4)x′(t)=x(t)(a-by(t)),y′(t)=y(t)(λbx(t)1+bhx(t)-d),x≠h1,h2orx=h1,y>y*,Δx(t)=0,Δy(t)=δ,x=h1,y⩽y*,Δx(t)=-αx(t),Δy(t)=-βy(t)+q,x=h2,
where x(t) and y(t) represent, respectively, the prey and the predator population densities at time t; a, b, λ, h1, h2 and d are all positive constants and h1<h2; y*=a/b. α,β∈(0,1) represent the fraction of pest and predator, respectively, which die due to the pesticide when the amount of prey reaches economic threshold h2 and q is the release amount of predator. λbx(t)/(1+bhx(t)) is the per capita functional response of the predator. When the amount of the prey reaches the threshold h1 at time th1, controlling measures are taken (releasing natural enemies) and the amount of predator abruptly turns to y(th1)+δ. When the amount of the prey reaches the threshold h2 at time th2, spraying pesticide, and releasing natural enemies and the amount of prey and predator abruptly turn to (1-α)x(th2) and (1-β)y(th2)+q, respectively. Refer to [17] Liu et al. for details.


## 2. Preliminaries

First, we give some basic definitions and lemmas.Definition 2.1.
A triple(X,Π,R+) is said to be a semidynamical system if X is a metric space, R+ is the set of all nonnegative real, and Π(P,t):X×R+→X is a continuous map such that:(i)
Π(P,0)=P for all P∈X;(ii)
Π(P,t) is continuous for t and s;(iii)
Π(Π(P,t))=Π(P,t+s) for all P∈X and t,s∈R+. Sometimes a semi-dynamical system (X,Π,R+) is denoted by (X,Π).Definition 2.2.
Assuming that(i)
(X,Π) is a semi-dynamical system;(ii)
M is a nonempty subset of X;(iii)
functionI:M→X is continuous and for any P∈M, there exists a ε>0 such that for any 0<|t|<ε, Π(P,t)∉M.
Then,(X,Π,M,I) is called an impulsive semi-dynamical system.
For anyP, the function ΠP:R+→X defined as ΠP(t)=Π(P,t) is continuous, and we call ΠP(t) the trajectory passing through point P. The set C+(P)={Π(P,t)/0≤t<+∞} is called positive semitrajectory of point P. The set C-(P)={Π(P,t)/-∞<t≤0} is called the negative semi-trajectory of point P.Definition 2.3.
One considers state-dependent impulsive differential equations:(2.1)x′(t)=P(x,y),y′(t)=Q(x,y),(x,y)∉M(x,y),Δx(t)=α(x,y),Δy(t)=β(x,y),(x,y)∈M(x,y),
where M(x,y) and N(x,y) represent the straight line or curve line on the plane, M(x,y) is called impulsive set. The function I is continuous mapping, I(M)=N, I is called the impulse function. N(x,y) is called the phase set. We define “dynamic system” constituted by the definition of solution of state impulsive differential equation (2.1) as “semicontinuous dynamic systems”, which is denoted as (Ω,f,I,M).Definition 2.4.
Suppose that the impulse setM and the phase set N are both lines, as shown in Figure 1. Define the coordinate in the phase set N as follows: denote the point of intersection Q between N and x-axis as O, then the coordinate of any point A in N is defined as the distance between A and Q and is denoted by yA. Let C denote the point of intersection between the trajectory starting from A and the impulse set M, and let B denote the phase point of C after impulse with coordinate yB. Then, we define B as the successor point of A, and then the successor function of point A is that f(A)=yB-yA.Figure 1
Successor function defined.Definition 2.5.
A trajectoryΠ~(P0,t) is called order one periodic solution with period T if there exists a point P0∈N and T>0 such that P=Π(P0,t)∈M and P+=I(P)=P0.We get these lemmas from the continuity of composite function and the property of continuous function.Lemma 2.6.
Successor function defined in Definition2.1 is continuous.Lemma 2.7.
In system (1.4), if there exist A∈N, B∈N satisfying successor function f(A)f(B)<0, then there must exist a point P (P∈N) satisfying f(P)=0 the function between the point of A and the point of B, thus there is an order one periodic solution in system (1.4).
Next, we consider the model (1.4) without impulse effects:
(2.2)x′(t)=x(t)(a-by(t)),y′(t)=y(t)(λbx(t)1+bhx(t)-d).
It is well known that the system (2.2) possesses(I)
two steady statesO(0,0)-saddle point, and R(d/b(λ-dh),a/b)=R(x*,y*)(λ>dh)-stable centre;(II)
a unique closed trajectory through any point in the first quadrant contained inside the pointR.In this paper, we assume that the conditionλ>dh holds. By the biological background of system (1.4), we only consider D={(x,y):x≥0,y≥0}. Vector graph of system (2.2) can be seen in Figure 2.Figure 2
Illustration of vector graph of system (2.2).This paper is organized as follows. In the next section, we present some basic definitions and an important lemmas as preliminaries. In Section3, we prove existence for an order one periodic solution of system (1.4). The sufficient conditions for the attractiveness of order one periodic solutions of system (1.4) are obtained in Section 4. At last, we state conclusion, and the main results are carried out to illustrate the feasibility by numerical simulations.


## 3. Existence of the Periodic Solution

In this section, we will investigate the existence of an order one periodic solution of system (1.4) by using the successor function defined in this paper and qualitative analysis. For this goal, we denote that M1={(x,y)/x=h1,0≤y≤a/b}, and that M2={(x,y)∣x=h2,y≥0}. Phase set of set M is that N1=I(M1)={(x,y)∣x=h1,a/b<y≤(a/b)+δ} and that N2=I(M2)={(x,y)∣x=(1-α)h2,y≥q}. Isoclinic line is denoted, respectively, by lines: L1={(x,y)∣y=a/b,x≥0} and L2={(x,y)∣x=d/b(λ-dh),y≥0}.For the convenience, ifP∈Ω-M, F(P) is defined as the first point of intersection of C+(P) and M, that is, there exists a t1∈R+ such that F(P)=Π(P,t1)∈M, and for 0<t<t1,Π(P,t)∉M; if B∈N,R(B) is defined as the first point of intersection of C-(P) and N, that is, there exists a t2∈R+ such that R(B)=Π(B,-t2)∈N, and for -t<t<0,Π(B,t)∉N. For any point P, we denote yP as its ordinate. If the point P(h,yP)∈M, then pulse occurs at the point P, the impulsive function transfers the point P into P+∈N. Without loss of generality, unless otherwise specified we assume the initial point of the trajectory lies in phase set N.Due to the practical significance, in this paper we assume the set always lies in the left side of stable centreR, that is, h1<d/b(λ-dh) and (1-α)h2<d/b(λ-dh).In the light of the different position of the setN1 and the set N2, we consider the following three cases.Case 1 (0<h1<d/b(λ-dh)).
In this case, setM1 and N1 are both in the left side of stable center R (as shown in Figure 3). Take a point B1(h1,(a/b)+ε)∈N1 above A, where ε>0 is small enough, then there must exist a trajectory passing through B1 which intersects with M1 at point P1(h1,yp1), we have yp1<a/b. Since p1∈M1, pulse occurs at the point P1, the impulsive function transfers the point P1 into P1+(h1,yp1+δ) and P1+ must lie above B1, therefore inequation (a/b)+ε<yp1+δ holds, thus the successor function of B1 is f(B1)=yp1+δ-((a/b)+ε)>0.
On the other hand, the trajectory with the initial pointP1+ intersects M1 at point P2(h1,yp2), in view of vector field and disjointness of any two trajectories, we know yp2<yp1<a/b. Supposing the point P2 is subject to impulsive effects to point P2+(h1,yp2+), where yp2+=yp2+δ, the position of P2+ has the following two cases.
Subcase 1.1 (a/b<yp2+δ<yp1+δ). In this case, the point P2+ lies above the point A and below P1+, then we have f(P1+)=yp2+δ-(yp1+δ)<0.
By Lemma2.7, there exists an order one periodic solution of system (1.4), whose initial point is between B1 and P1+ in set N1.
Subcase 1.2 (a/b≥yp2+δ (as shown in Figure 4)). The point P2+ lies below the point A, that is, P2+∈M1, then pulse occurs at the point P2+, the impulsive function transfers the point P2+ into P2++(h1,yp2+2δ).Ifa/b<yp2+2δ<yp1+δ, like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4).Ifa/b>yp2+2δ, that is, P2++∈M1, then we repent the above process until there exists k∈Z+ such that P2++ jumps to P2i+((h1,yp2+(k+2)δ) after k times’ impulsive effects which satisfies a/b<yp2+(k+2)δ<yp1+δ. Like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4).
Now, we can summarize the above results as the following theorem.Figure 3
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δ.Theorem 3.1.
Ifλ>dh,0<h1<d/b(λ-dh), then there exists an order one periodic solution of the system (1.4).Remark 3.2.
It shows from the proved process of Theorem3.1 that the number of natural enemies should be selected appropriately, which aims to reduce releasing impulsive times to save manpower and resources.Case 2 (h2<d/b(λ-dh)).
In this case, setM2 and N2 are both in the left side of stable center R, in the light of the different position of the set N2, we consider the following two cases.
Subcase 2.1 (0<h1<(1-α)h2<h2<d/b(λ-dh)). In this case, the set N2 is in the right side of M1 (as shown in Figure 5). The trajectory passing through point A which tangents to N2 at point A intersects with M2 at point P0(h2,yp0). Since the point P0∈M2, then impulse occurs at point P0. Supposing the point P0 is subject to impulsive effects to point P0+((1-α)h2,yP0+), where yP0+=(1-β)yP0+q, the position of P0+ has the following three cases:
(1) ((1-β)yP0+q>a/b). Take a point B1((1-α)h2,ε+a/b)∈N2 above A, where ε>0 is small enough. Then there must exist a trajectory passing through the point B1 which intersects with the set M2 at point P1(h2,yP1). In view of continuous dependence of the solution on initial value and time, we know yP1<yP0 and the point P1 is close to P0 enough, so we have the point P1+ is close to P0+ enough and yP1+<yP0+, then we obtain f(B1)=yP1+-yB1>0.
On the other hand, the trajectory passing through pointB tangents to N1 at point B. Set F(S)=P2(h2,yP2)∈M2. Denote the coordinates of impulsive point P2+((1-α)h2,yP2+) corresponding to the point P2(h2,yP2).IfyS≥yP0+ then yP2+<yP0+<yS. So we obtain f(S)=yP2+-yS<0. There exists an order one periodic solution of system (1.4), whose initial point is between the point B1 and the point S in set N2 (Figure 5).IfyS<yP0+ and yP2+≤yS, we have f(S)=yP2+-yS≤0, we conclude that there exists an order one periodic solution of system (1.4).IfyS<yP0+ and yP2+>yS, from the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point on the N2 will ultimately stay in Ω1={(x,y)/0≤x≤h1,y≥0} after one impulsive effect. Therefore there is no an order one periodic solution of system (1.4).
(2) ((1-β)yP0+q<a/b (as shown in Figure 6)). In this case, the point P0+ lies below the point A, that is, (1-β)yP0+q<a/b, thus the successor function of the point A is f(A)=(1-β)yP0+q-a/b<0.
Take another pointB1((1-α)h2,ε)∈N2, where ε>0 is small enough. Then there must exist a trajectory passing through the point B1 which intersects M2 at point P1(h2,yP1)∈M2. Suppose the point P1(h2,yP1) is subject to impulsive effects to point P1+((1-α)h2,yP1+), then we have yP1+>ε. So we have f(B1)=yP1+-ε>0.
From Lemma2.7, there exists an order one periodic solution of system (1.4), whose initial point is between B1 and A in set N2.
(3) ((1-β)yP0+q=a/b). P0+ coincides with A, and the successor function of A is that f(A)=0, so there exists an order one periodic solution of system (1.4) which is just a part of the trajectory passing through the A.Figure 5
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Assuming thatλ>dh and 0<h1<(1-α)h2<h2<d/b(λ-dh).IfyP0+≤yA, there exists an order one periodic solutions of the system (1.4).IfyP0+>yA, if yS≥yP0+ or yS<yP0+, and yS>yP2+, there exists an order one periodic solutions of the system (1.4).IfyP0+>yA, yS<yP0+ and yS<yP2+, there is no an order one periodic solutions of the system (1.4). The trajectory of system (1.4) with any initiating point on the N2 will ultimately stay in Ω1={(x,y)/0≤x≤h1,y≥0} after one impulsive effect.Subcase 2.2
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). In this case, the set N2 is on the left side of N1. Any trajectory from initial point (x0+,y0+)∈N2 will intersect with M1 at some point with time increasing. By the analysis of Case 1, the trajectory from initial point (x0+,y0+)∈N2 on the set N2 will stay in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1}. Similarly, any trajectory from initial point (x0+,y0+)∈Ω0={(x,y)∣x≥0,y≥0,x≤h2} will stay in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1} after one impulsive effect or free from impulsive effect.Theorem 3.4.
Ifλ>dh, 0<(1-α)h2<h1<h2<d/b(λ-dh), there is no an order one periodic solutions to the system (1.4), and the trajectory with initial point (x0+,y0+)∈Ω0={(x,y)∣x≥0,y≥0,x≤h2} will stay in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1}.Case 3 (d/b(λ-dh)<h2).
In this case, the setM2 is on the right side of stable center R. In the light of the different position of N2, we consider the following two subcases.
Subcase 3.1 (h1<(1-α)h2<d/b(λ-dh)<h2). In this case, the set M2 is in the right side of R. Then there exists a unique closed trajectory Γ1 of system (1.4) which contains the point R and is tangent to M2 at the point A.
SinceΓ1 is closed trajectory, we take their the minimal value δmin of abscissas at the trajectory Γ1, namely, δmin≤x holds for any abscissas of Γ1.
(1) (h1<(1-α)h2<δmin<d/b(λ-dh)<h2). In this case, there is a trajectory, which contains the point R(d/b(λ-dh),a/b) and is tangent to the N2 at the point B intersects M2 at a point P1(h2,yP1)∈M2. Suppose point P1 is subject to impulsive effects to point P1+((1-α)h2,yP1+), here yP1+=(1-β)yP1+q. The position of P1+ has the following three sub-cases.If(1-β)yP1+q<a/b (Figure 7), the point P1+ lies below the point B. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case.If(1-β)yP1+q>a/b, the point P1+ lies above the point B; the trajectory from initiating point P1+ intersects with the line L1 at point C. If h1≤yC (Figure 8), we have yP1>yP2 and yP1+>yP2+, then the successor function of P1+ is that f(P1+)=yP2+-yP1+<0. Then, we know that there exists an order one periodic solution of system (1.4), whose initial point is between the point P1+ and B in set N2. If h1>yC  (Figure 9), there is a trajectory which is tangent to the N1 at a point D intersects with M2 at a point P3(h2,yP3)∈M2, P3 jumps to P3+ after the impulsive effects. If yP3+≤yB1, we can easily know that there exists an order one periodic solution of system (1.4). If yP3+>yB1, by the qualitative analysis of the system (1.4), we know that trajectory with any initiating point on the N2 will ultimately stay in Γ1 after a finite number of impulsive effects.If(1-β)yP1+q=a/b, the point P1+ coincides with the point B, and the successor function of the point B is that f(B)=0; then there exists an order one periodic solution which is just a part of the trajectory passing through the point B.Figure 7
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b, h1>yc.Now, we can summarize the above results as the following theorem.Theorem 3.5.
Assuming thatλ>dh and h1<(1-α)h2<δmin<d/b(λ-dh)<h2.If(1-β)yP1+q≤a/b, there exists an order one periodic solution to the system (1.4).If(1-β)yP1+q>a/b and h1≤yC, then there exists an order one periodic solution to the system (1.4).If(1-β)yP1+q>a/b, h1>yC and yP3+≤yB1, then there exists an order one periodic solution to the system (1.4).
(2)  (h1<δmin<(1-α)h2<d/b(λ-dh)<h2). In this case, denote the closed trajectory Γ1 of system (1.4) intersects with N2 two points A1((1-α)h2,yA1) and A2((1-α)h2,yA2) (as shown in Figure 10). Since A∈M2, impulse occurs at the point A. Suppose point A is subject to impulsive effects to point P0+((1-α)h2,yP0+), here yP0+=(1-β)(a/b)+q.If(1-β)(a/b)+q=yA1 or (1-β)(a/b)+q=yA2, then P0+ coincides with A1 or P0+ coincides with A2, and the successor function of A1 or A2 is that f(A1)=0 or f(A2)=0. So, there exists an order one periodic solution of system (1.4) which is just a part of the trajectory Γ1.If(1-β)(a/b)+q<yA2, the point P0+ lies below the point A2. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case.If(1-β)(a/b)+q>yA1 (as shown in Figure 11), the point P0+ is above the point A1. Like the analysis of Subcase 3.1(1), we obtain sufficient conditions of existence of order one periodic solution to the system (1.4).Figure 10
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.Theorem 3.6.
Assuming thatλ>dh,h1<(1-α)h2<δmin<d/b(λ-dh)<h2.If(1-β)(a/b)+q≤yA2, there exists an order one periodic solution to the system (1.4).If(1-β)(a/b)+q>yA1 and h1≤yC, then there exists an order one periodic solution to the system (1.4).If(1-β)(a/b)+q>yA1,(1-β)yP1+q>a/b,h1>yC, and yP3+≤yB1, then there exists an order one periodic solution to the system (1.4).
(3)  (yA2<(1-β)(a/b)+q<yA1). In this case, we note that the point P0+ must lie between the point A1 and the point A2 (As shown in Figure 12). Taking a point B1∈M2 such that B1 jumps to A2 after the impulsive effect, denote A2=B1+. Since yP0+>yB1+, we have yA>yB1. Let R(B1)=B2+∈N2, take a point B2∈M2 such that B2 jumps to B2+ after the impulsive effects, then we have yB1+>yB2+, yB1>yB2. This process continues until there exists a BK+∈N2 (K∈Z+) satisfying yBk+<q. So we obtain a sequence {Bk+}k=1,2,…,K of the set M2 and a sequence {Bk}k=1,2,…,K of set N2 satisfying R(Bk-1)=Bk+∈N2,yBk-1+>yBk+. In the following, we will prove the trajectory of system (1.4) with any initiating point of set N2 will ultimately stay in Γ1.Figure 12
y
A
2
<
(
1
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(
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/
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+
q
<
y
A
1.From the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point between the point A1 and A2 will be free from impulsive effect and ultimately will stay in Γ1.For any point belowA2, it must lie between Bk+ and Bk-1+, here k=2,3,…,K+1 and A2=B1+. After k times’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set N2 which must be between A1 and A2, and then ultimately stay in Γ1.Denote the intersection of the trajectory passing through the pointB which tangents to N1 at point B with the set N2 at S((1-α)h2,yS). With time increasing, the trajectory of system (1.4) from any initiating point on segment A1S¯ intersect with the set N2 at some point which is below A2; so just like the analysis above we obtain, it will ultimately stay in Γ1. So for any point below S, will ultimately stay in region Γ1 with time increasing.Now, we can summarize the above results as the following theorem.Theorem 3.7.
Assuming thatλ>dh,h1<δmin<(1-α)h2<d/b(λ-dh)<h2, and yA2<(1-β)(a/b)+q<yA1, there is no periodic solution in system (1.4), and the trajectory with any initiating point on the set N2 will stay in Γ1 or in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1}.Subcase 3.2 (0<(1-α)h2<h1<d/b(λ-dh)<h2). In this case, the set N2 is on the left side of the set N1 and M2 in the right side of R. Like the analysis of Subcase 2.2, we can know that any trajectory with initial point (x0+,y0+)∈Ω0={(x,y)∣x≥0,y≥0,x≤h2} will stay in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1} after one impulsive effect or free from impulsive effect.


## 4. Attractiveness of the Order One Periodic Solution

In this section, under the condition of existence of order one periodic solution to system (1.4) and the initial value of pest population x(0)≤h2, we discuss its attractiveness. We focus on   Case 1 and   Case 2; by similar method, we can obtain similar results about Case 3.Theorem 4.1.
Assuming thatλ>dh,h1<d/(b(λ-dh)) and δ≥a/b.
IfyP0+>yP2+>yP+ or yP0+<yP2+<yP+ (Figure 14), then(I)
there exists a unique order one periodic solution of system (1.4),(II)
if(1-α)h2<h1, order one periodic solution of system (1.4) is attractive in the region Ω0={(x,y)∣x≥0,y≥0,x≤h2}.Proof.
By the derivation of Theorem3.1, we know there exists an order one periodic solution of system (1.4). We assume trajectory PP+^ and segment PP+¯ formulate an order one periodic solution of system (1.4), that is, there exists a P+∈N2 such that the successor function of P+ satisfies f(P+)=0. First, we will prove the uniqueness of the order one periodic solution.
We take any two pointsC1(h1,yC1)∈N1,C2(h1,yC2)∈N1 satisfying yC2>yC1>yA, then we obtain two trajectories whose initiate points are C1 and C2 intersects the set M1 two points D1(h1,yD1) and D2(h1,yD2), respectively, (Figure 13). In view of the vector field of system (1.4) and the disjointness of any two trajectories without impulse, we know yD1>yD2. Suppose the points D1 and D2 are subject to impulsive effect to points D1+(h1,yD1+) and D2+(h2,yD2+), respectively, then we have yD1+>yD2+ and f(C1)=yD1+-yC1,f(C2)=yD2+-yC2, so we get f(C1)-f(C2)<0, thus we obtain the successor function f(x) is decreasing monotonously of N1, so there is a unique point P+∈N1 satisfying f(P+)=0, and the trajectory P+PP+^ is a unique order one periodic solution of system (1.4).
Next, we prove the attractiveness of the order one periodic solutionP+PP+^ in the region Ω0={(x,y)∣x≥0,y≥0,x≤h2}. We focus on the case yP0+>yP2+>yP+; by similar method, we can obtain similar results about case yP0+<yP2+<yP+ (Figure 14).
Take any pointP0+(h1,yP0+)∈N1 above P+. Denote the first intersection point of the trajectory from initiating point P0+(h1,yP0+) with the set M1 at P1(h1,yP1), and the corresponding consecutive points are P2(h1,yP2),P3(h1,yP3),P4(h1,yP4),…, respectively. Consequently, under the effect of impulsive function, the corresponding points after pulse are P1+(h1,yP1+),P2+(h1,yP2+),P3+(h1,yP3+),….
Due to conditionsyP0+>yP2+>yP+, yPk+=yPk+δ, δ≥a/b and disjointness of any two trajectories, then we get a sequence {Pk+}k=1,2,… of the set N1 satisfying
(4.1)yP1+<yP3+<⋯<yP2k-1+<yP2k+1+<⋯<yP+<⋯<yP2k+<yP2k-2+<⋯<yP2+<yP0+.
So the successor functionf(P2k-1+)=yP2k+-yP2k-1+>0 and f(P2k+)=yP2k+1+-yP2k+<0 hold. Series {yP2k-1}k=1,2,… increases monotonously and has upper bound, so limk→∞yP2k-1+ exists. Next, we will prove limk→∞yP2k-1+=yP+. Set limk→∞P2k-1=C+, we will prove P+=C+. Otherwise P+≠C+, then there is a trajectory passing through the point C+ which intersects the set M1 at point C~, then we have yC~>yP, yC~+>yP+. Since f(C+)≥0 and P+≠C+, according to the uniqueness of the periodic solution, then we have f(C+)=yC~+-yC+>0, thus yC+<yP+<yC~+ hold. Analogously, let trajectory passing through the point C+ which intersects the set M1 at point C~~, and the corresponding consecutive points is C~~~, then yC~>yC~~~>yp>yC~~,yC~+>yC~~~+>yC~~+>yp+>yC+, then we have f(C~~+)=yC~~~+-yC~~+>0, this is, contradict to the fact that C+ is a limit of sequence {P2k-1+}k=1,2,…, so we obtain P+=C+. So, we obtain limk→∞yP2k-1+=yP+. Similarly, we can prove limk→∞yP2k+=yP+.
From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from initiating any point of the N1 will ultimately tend to be order one periodic solution P+PP+^.
Any trajectory from initial point(x0+,y0+)∈Ω0={(x,y)∣x≥0,y≥0,x≤h2} will intersect with N1 at some point with time increasing on the condition that (1-α)h2<h1<h2<d/b(λ-dh); therefore the trajectory from initial point on N1 ultimately tends to be order one periodic solution P+PP+^. Therefore, order one periodic solution P+PP+^ is attractive in the region Ω0. This completes the proof.Figure 13
There is a unique order one periodic solution (Theorem4.1).Figure 14
Order one periodic solution is attractive (Theorem4.1).Remark 4.2.
Assuming thatλ>dh,h1<h2<d/b(λ-dh) and δ≥a/b, if yP+<yP0+<yP2+ or yP+>yP0+>yP2+ then the order one periodic solution is unattractive.Theorem 4.3.
Assuming thatλ>dh,h1<(1-α)h2<h2<d/b(λ-dh) and yP0+<yA (as shown in Figure 15), then(I)
There exists an odd number of order one periodic solutions of system (1.4) with initial value between C1+ and A in set N2.(II)
If the periodic solution is unique, then the periodic solution is attractive in regionΩ2, here Ω2 is open region which is constituted by trajectory GB^, segment BH¯, segment HE¯, and segment EG¯.Figure 15
Attractneness of order one periodic solution (Theorem4.3).Proof.
(I) According to the Subcase2.1(2), f(A)<0 and f(C1+)>0, and the continuous successor function f(x), there exists an odd number of root satisfying f(x)=0, then we can get there exists an odd number of order one periodic solutions of system (1.4) with initial value between C1+ and A in set N2.
(II) By the derivation of Theorem3.3, we know there exists an order one periodic solution of system (1.4) whose initial point is between C1+ and P0+ in the set N2. Assume trajectory P+P^ and segment PP+¯ formulate the unique order one periodic solution of system (1.4) with initial point P+∈N2.
On the one hand, take a pointC1+((1-α)h2,yC1+)∈N2 satisfying yC1+=ε<q and yC1+<yP+. The trajectory passing through the point C1+((1-α)h2,ε) which intersects with the set M2 at point C2(h2,yC2), that is, F(C1+)=C2∈M2, then we have yC2<yP, thus yC2+<yP+, since yC2+=(1-β)yC2+q>ε. So, we obtain f(C1+)=yC2+-yC1+=yC2+-ε>0; Set F(C2+)=C3∈M2, because yC1+<yC2+<yP+, we know yC2<yC3<yP, then we have yC2+<yC3+<yP+ and f(C2+)=yC3+-yC2+>0. This process is continuing, then we get a sequence {Ck+}k=1,2,… of the set N2 satisfying
(4.2)yC1+<yC2+<⋯<yCk+<⋯<yP+
and f(Ck+)=yCk+1+-yCk+>0. Series {yCk+}k=1,2,… increase monotonously and have upper bound, so limk→∞yCk+ exists. Like the proof of Theorem 4.1, we can prove limk→∞yCk+=yP+.
On the other hand, setF(P0+)=D1∈M2, then D1 jumps to D1+∈N2 under the impulsive effects. Since yP+<yP0+<yA, we have yP<yD1<yP0, thus we obtain yP+<yD1+<yP0+,f(P0+)=yD1+-yP0+<0. Set F(D1+)=D2∈M2, then D2 jumps to D2+∈N2 under the impulsive effects. We have yP+<yD2+<yD1+; this process is continuing, we can obtain a sequence {Dk+}k=1,2,…, of the set N2 satisfying
(4.3)yP0+>yD1+>yD2+>⋯>yDk+>⋯>yP+
and f(Dk+)=yDk+1+-yDk+<0. Series {yDk+}k=1,2,… decreases monotonously and has lower bound, so limk→∞yDk+ exists. Similarly, we can prove limk→∞yDk+=yP+.
Any pointQ∈N2 below A must be in some interval [yDk+1+,yDk+)k=1,2,…, [yD1+,yP0+), [yP0+,yA), [yCk+,yCk+1+)k=1,2,…. Without loss of generality, we assume the point Q∈[yDk+1+,yDk+). The trajectory with initiating point Q moves between trajectory Dk+Dk+1^ and Dk+1+Dk+2^ and intersects with M2 at some point between Dk+2 and Dk+1; under the impulsive effects, it jumps to the point of N2 which is between [yDk+2+,yDk+1+), then trajectory Π~(Q,t) continues to move between trajectory Dk+1+Dk+2^ and Dk+2+Dk+3^. This process can be continued unlimitedly. Since limk→∞yDk+=yP+, the intersection sequence of trajectory Π~(Q,t), and the set N2 will ultimately tend to the point P+. Similarly, if Q∈[yCk+,yCk+1+), we also can get the intersection sequence of trajectory Π~(Q,t) and the set N2 will ultimately tend to point P+. Thus, the trajectory initiating any point below A ultimately tend to the unique order one periodic solution P+PP+^.
Denote the intersection of the trajectory passing through the pointB which tangents to N1 at the point B with the set N2 at a point S((1-α)h2,yS). The trajectory from any initiating point on segment AS¯ will intersect with the set N2 at some point below A with time increasing. So like the analysis above, we obtain the trajectory from any initiating point on segment AS¯ will ultimately tend to be the unique order one periodic solution P+PP+^.
Since the trajectory with any initiating point of theΩ2 will certainly intersect with the set N2, then from the above analysis, we know the trajectory with any initiating point on segment AS¯ will ultimately tend to be order one periodic solution P+PP+^. Therefore, the unique order one periodic solution P+PP+^ is attractive in the region Ω2. This completes the proof.Remark 4.4.
Assuming thatλ>dh, h1<(1-α)h2<h2<d/b(λ-dh), and yC1+<yA<yP0+, then the order one periodic solution with initial point between A and P0+ is unattractive.Theorem 4.5.
Assuming thatλ>dh, h1<(1-α)h2<h2<d/b(λ-dh), yP0+>yP1+>yA (Figure 16) then, there exists a unique order one periodic solution of system (1.4) which is attractive in the region Ω2, here Ω2 is open region which enclosed by trajectory GB^, segment BH¯, segment HE¯ and segment EG¯.Figure 16
Attractneness of order one periodic solution (Theorem4.5).Proof.
By the derivation of Theorem3.3, we know there exists an order one periodic solution of system (1.4). We assume trajectory P+P^ and segment PP+¯ formulate an order one periodic solution of system (1.4), that is, P+∈N2 is its initial point satisfying f(P+)=0. Like the proof of Theorem 4.1, we can prove the uniqueness of the order one periodic solution of system (1.4).
Next, we prove the attractiveness of the order one periodic solutionP+PP+^ in the region Ω2.
Denote the first intersection point of the trajectory from initiating pointP0+ with the impulsive set M2 at P1(h,yP1), and the corresponding consecutive points are P2(h,yP2),P3(h,yP3),P4(h,yP4)⋯ respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are P1+(h,yP1+),P2+(h,yP2+),P3+(h,yP3+), …. In view of yP0+>yP1+>yA and disjointness of any two trajectories, we have
(4.4)yP1+<yP3+<⋯<yP2k-1+<yP2k+1+<⋯yP2k+<yP2k-2+<yP2+<yP0+.
So f(P2k-1+)=yP2k+-yP2k-1+>0 and f(P2k+)=yP2k+1+-yP2k+<0 hold. Like the proof of Theorem 4.1, we can prove limk→∞yP2k-1+=limk→∞yP2k+=yP+.
The trajectory from initiating point betweenB0+ and P0+ will intersect with impulsive set N2 with time increasing, under the impulsive effects it arrives at a point of N2 which is between [yP2k-1+,yP2k+1+) or [yP2k+,yP2k-2+). Then like the analysis of Theorem 4.3, we know the trajectory from any initiating point between B0+ and P0+ will ultimately tend to be order one periodic solution P+PP+^.
Denote the intersection of the trajectory passing through pointB which tangents to N1 at point B with the set N2 at S. Since the trajectory from initiating any point below S of the set N2 will certain intersect with set N2, next we only need to prove the trajectory with any initiating point below S of the set N2 will ultimately tend to be order one periodic solution P+PP+^.
Assume a pointB0 of set M2 jumps to B0+ under the impulsive effect. Set R(B0)=B1+∈N2. Assume point B1 of set N2 jumps to B1+ under the impulsive effect. Set R(B1)=B2+∈N2. This process is continuing until there exists a BK0+∈N (K0+∈N2) satisfying yBK0+<q. So we obtain a sequence {Bk}k=0,1,2,…,K0 of set M2 and a sequence {Bk+}k=0,1,2,…,K0 of set N2 satisfying R(Bk-1)=Bk+,yBk+<yBk-1+. For any point of set N2 below B0+, it must lie between Bk+1+ and Bk+ here k=1,2,…,K0. After K0+1 times’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set N2 which must be between B0+ and P0+, and then will ultimately tend to order one periodic solution P+PP+^. There is no order one periodic solution with the initial point below B0+.
The trajectory with any initiating point in segmentAS¯ will intersect with the set N2 at some point below B0+ with time increasing. Like the analysis above, we obtain the trajectory initiating any point on segment AS¯ will ultimately tend to be the unique order one periodic solution P+PP+^.
From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from any initiating point below S will ultimately tend to be order one periodic solution P+PP+^. Therefore, order one periodic solution P+PP+^ is attractive in the region Ω2. This completes the proof.


## 5. Conclusion

In this paper, a state-dependent impulsive dynamical model concerning different control methods at different thresholds is proposed, we find a new method to study existence and attractiveness of order one periodic solution of such system. We define semicontinuous dynamical system and successor function, demonstrate the sufficient conditions that system (1.4) exists order one periodic solution with differential geometry theory and successor function. Besides, we successfully prove the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis.The method can be also extended to mechanical dynamical systems with impacts, for example [19, 20].These results show that the state-dependent impulsive effects contribute significantly to the richness of the dynamics of the model. The conditions of existence of order one periodic solution in this paper have more extensively applicable scope than the conditions given in [14]. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. The methods of the theorems are proved to be new in this paper, and these methods are more efficient and easier to operate than the existing research methods which have been applied the models with impulsive state feedback control [16–18, 21], so they are deserved further promotion. In this paper, according to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds, the corresponding control is exerted, which leads to the two state impulses in model. Certainly, many biological systems will always be described by three or more state variables, which are the main work in the future.In order to testify the validity of our results, we consider the following example.(5.1)x′(t)=x(t)(0.4-0.5y(t)),y′(t)=y(t)(0.25x(t)1+0.1x(t)-0.6),x≠h1,h2orx=h1,y>y*,Δx(t)=0,Δy(t)=δ,x=h1,y⩽y*,Δx(t)=-αx(t),Δy(t)=-βy(t)+q,x=h2,
where α,β∈(0,1), δ>0, q>0, 0<h1<h2. Now, we consider the impulsive effects influences on the dynamics of system (5.1).Example 5.1.
Existence and attractiveness of order one periodic solution.
We seth1=1, α=0.6, β=0.8, q=0.8, h2=1.8, initiating points are (0.85,0.2) (red), (0.8,0.5) (green), and (0.75,0.11) (blue), respectively. Figure 17 shows that the conditions of Theorems 3.1 and 4.1 hold, system (5.1) exists order one periodic solution. The trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.The time series and phase diagram for system (1.4) starting from initial value (0.85,0.2) (red), (0.8,0.5) (green), and (0.75,0.11) (blue),  δ=0.6, h1=1<x*.
(a)
(b)
(c)Example 5.2.
Existence and attractiveness of positive periodic solution.
We seth1=0.7, α=0.6, β=0.8, q=0.8, h2=1.8, h1<(1-α)h2<x*, initiating points are (0.8,0.1) (red), (0.7,0.5) (green), and (0.75,0.3) (blue), respectively. Figure 18 shows that the conditions of Theorems 3.3 and 4.3 hold, there exists order one periodic solution of the system (5.1), and the trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.The time series and phase diagram for system (1.4) starting from initial value (0.8,0.1) (red), (0.7,0.5) (green), and (0.75,0.3) (blue) α=0.6, β=0.3, q=0.8, h2=1.8, h1<h2<x*.
(a)
(b)
(c)Example 5.3.
Existence and attractive of positive periodic solutions.
We seth1=0.7, α=0.6, β=0.8, q=0.8, h2=3.5, h1<(1-α)h2<x*<h2, initiating points are (1,0.7) (red), (1.4,0.5) (green), and (1.2,1) (blue) as shown in Figure 19. Therefore, the conditions of Theorems 3.6 and 4.5 hold, then system (5.1) exists order one periodic solution, and it is attractive.The time series and phase diagram for system (1.4) starting from initial value (1, 0.7) (red), (1.4, 0.5) (green), and  (1.2,1) (blue) α=0.6, β=0.3, q=0.8, h2=3.5, h1<x*<h2.
(a)
(b)
(c)



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*Source: 381503-2012-09-19.xml*