# Multi-State Dependent Impulsive Control for Holling I Predator-Prey Model

Authors: Huidong Cheng; Fang Wang; Tongqian Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

According to the different effects of biological and chemical control, we propose a model for Holling I functional response predator-prey system concerning pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove that the existence of order one periodic solution of such system and 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 which 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

More and more scholars have paid close attention, and studied impulsive differential equation since the 1980s. Impulsive differential equation theory, especially the one in a fixed time, has been deeply developed and widely applied in various fields through years of research [1–5]. In population dynamical system, Tang et al. [6] discussed the stage-structure system for single population with birth pulse and got the existence and stability of periodic solution; Liu et al. [7–9] studied the impulse control strategy of Lotka-Volterra system; he also set up and discussed the Holling type II predator-prey model with impulse control strategy. Ballinger and Liu [10] discussed the persistence of population model with impulse effect. Tang and Cheke [11] first proposed the “Volterra” model with state-dependence, and they applied this model to pest management and proved existence and stability of periodic solution of first and second orders. Then, Liu et al. [12] also proposed bait-dependent digestive model with state pulse, and the model had the existence of positive periodic solution and stability of orbit. Recently, Jiang and Liu et al. [12–14] have proposed pest management model with state pulse and phase structure and several predator-prey models with state pulse and had the existence of semitrivial periodic solution and positive periodic solution and stability of orbit.In consideration of predator-prey capacity, Holling [15] proposes three different predations with functional response based on experiments; the average predator-prey system with Holling response is as follows:(1.1)x′(t)=xg(x)-yϕ(x),y′(t)=-dy+eyϕ(x),
where x represents the prey’s density, while y is the predator’s; g(x) is the unit rate of prey density in lack of predators; ϕ(x) is the Holling functional response, among which Holling type I functional response is(1.2)ϕ(x)={cx,x≤x0,cx0,x>x0,
where c is a constant; when the amount of prey is greater than certain threshold value x0, predatory rate is a constant. Referring to [15] for details.As the Lotka-Volterra predator-prey system with Holling functional response is more practical, many authors have studied it [12, 14, 15]. The researches mostly focus on Lotka-Volterra predator-prey model with Holling type II or Holling type III functional response in contrast to the model with Holling type I. This paper sets up a state-dependent impulsive mathematical model concerning pest control which adopts different control methods at different thresholds and adopts new mathematic method to study existence and attractiveness of order one periodic solution of such system; thus the following pest-control model with Holling type-I functional response is set up:(1.3)x′(t)=rx(t)-cx(t)y(t),y′(t)=-dy(t)+ecx(t)y(t),x≤x0,x′(t)=rx(t)-cx0y(t),y′(t)=-dy(t)+ecx0y(t),x>x0,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 r,c,d,e,h1, and h2 are all positive constants, x(t) and y(t) represent the densities of prey (pest) and predator (natural enemy), respectively; r is the intrinsic growth rate of the prey; d denotes the death rate of the predator; (α,β)∈(0,1) represent the proportion of killed prey and predator by spraying pesticides respectively, δ>0 is the number of natural enemies released at this time th1, when the amount of the prey reaches the threshold h1 at time th1, control measures are taken (releasing natural enemies) and the amount of predator abruptly turns to y(th1)+δ. When the amount of prey reaches the threshold h2 at time th2, control measures are taken and the amount of prey and predator abruptly turns to (1-α)h2 and (1-β)y(th2), respectively. Δx=x(t+)-x(t), Δy=y(t+)-y(t), x(t+)=limw→0x(t+w), y(t+)=limw→0y(t+w). Referring to [12] for details.


## 2. Preliminaries

We first consider the model (1.3) without impulse effects:(2.1)x′(t)=rx(t)-cx(t)y(t),y′(t)=-dy(t)+ecx(t)y(t),x≤x0,x′(t)=rx(t)-cx0y(t),y′(t)=-dy(t)+ecx0y(t).x>x0.We consider the following function:(2.2)V(x,y)=∫x*x-d+eϕ(s)ϕ(s)ds+∫y*ys-y*sds,we can easily know thatV(x,y) is positive definite in the first quartile and fits for all conditions of Lyapunov function.We can get that(2.3)V′(x,y)=exy*ϕ(x)(ϕ(x)-ϕ(x*))(ϕ(x*)x*-ϕ(x)x).It is easily proved thatV′(x,y)≡0 on condition that x≤x0; so all solutions of model (1.3) form a set {(x,y)/V(x,y)≤V(x0,y*)}are closed trajectory V(x,y)=C, where 0<C<V(x0,y*).SinceV′(x,y)>0 on condition that x>x0; so the trajectory of system (2.1) passes through closed curve V(x,y)=C when it is out of the curve V(x,y)=V(x0,y*).Therefore, we observe the straight line:(2.4)L(x,y)=y+x-n,n>0,x0<x≤h.
The derivative of L(x,y) along (2.1) is that(2.5)L′(x,y)/L=0=x′+y′=-dy+ecx0y+rx-ecx0=-(dn-ecx0n+cx0n+cx0h-dx0)-(ecx0-r-cx0)≤dh-ecx02+rh+cx0h-(d-ecx0+cx0)n.
We have that L′/L=0<0 on condition that n>((dh-ecx02+rh+cx0h)/(d-ecx0+cx0)). Therefore, we can get the following Lemma.Lemma 2.1.
The system (2.1) possesses:
(I)
two steady states0(0,0)—saddle point, and R(d/ec,r/c)=R(x*,y*)—stable centre on the condition that x≤x0 and that d≤ecx0;(II)
the trajectory of system (2.1) goes across the straight line y+x-n=0 from the right to the left on condition that x0≤x≤h and that n>((dh-ecx02+rh+cx0h)/(d-ecx0+cx0)) and intersects with the straight line x=x0.Definition 2.2.
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 [16] of point A is that f(A)=yB-yA.Figure 1Lemma 2.3.
In system (1.3), 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 (1.3).In this paper, we assume that the conditiond≤ecx0 holds. By the biological background of system (1.3), we only consider D={(x,y):x≥0,y≥0}.This paper is organized as follows. In the next section, we present some basic definitions and important lemmas as preliminaries. In Section3, we prove the existence for an order one periodic solution of system (1.3). The sufficient conditions for the attractiveness of order one periodic solutions of system (1.3) 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 Order One Periodic Solution

In this section, we shall investigate the existence of an order one periodic solution of system (1.3) by using the successor function defined in this paper. For this goal, we denote (3.1)M1={(x,y)x=h1,0≤y≤rc+δ},M2={(x,y)∣x=h2,y≥0},N1=I(M1)={(x,y)∣x=h1,rc<y≤rc+δ},N2=I(M2)={(x,y)∣x=(1-α)h2,y≥0}.
Isoclinic line is denoted respectively by lines:(3.2)L1={(x,y)∣y=rc,0≤x≤x0},L2={(x,y)∣x=dec,0≤x≤x0,y≥0},L3={(x,y)∣y=rcx0x,x>x0,y≥rc}.
For the convenience, if P∈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 pointP, denote yP as its ordinate. If the point P(h,yP)∈M, pulse occurs at the point P, the impulsive function transfers the point P into P+∈N. Without loss of generality, we assume the initial point of the trajectory lies in phase set N unless otherwise specified.According to the practical significance, in this paper we assume that the setN always lies in the left side of stable centre R, that is, h1<r/c,(1-α)h2<r/c.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/ec).
In this case, setM1 and N1 are both in the left side of stable center R (as shown in Figure 2). Take a point B(h1,r/c+ε)∈N1 above A, where ε>0 is small enough, then there must exist a trajectory passing through B which intersects with the set M1 at point P1(h1,yp1), we have yp1<r/c. 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 B; therefore, inequation a/b+ε<yp1+δ holds, thus the successor function of B is that f(B)=yp1+δ-(r/c+ε)>0.
On the other hand, the trajectory with the initial pointP1+ intersects with M1 at point P2(h1,yp2), in view of vector field and disjointness of any two trajectories, we know yp2<yp1<r/c. Supposing that 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.Figure 2Subcase 1.1 (r/c<yp2+δ<yp1+δ).
In this case, the pointP2+ lies above the point A and under P1+, we have f(P1+)=yp2+δ-(yp1+δ)<0.
By Lemma2.3, there exists an order one periodic solution of system (1.3), whose initial point is between B and P1+ in set N1.Subcase 1.2 (r/c≥yp2+δ (as shown in Figure 2)).
The pointP2+ 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).
Ifr/c<yp2+2δ<yp1+δ, like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.3).
Ifr/c>yp2+2δ, that is, P2++∈M1; we repeat 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 r/c<yp2+(k+2)δ<yp1+δ. Like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.3).Now we can summarize the above results in the following theorem.Theorem 3.1.
Ifd<ec,0<h1<d/ec, then there exists an order one periodic solution of the system (1.3).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/ec).
In this case, setsM2 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).
In this case, the setN2 is in the right side of M1 (as shown in Figure 3). The trajectory passing through point A which tangents to N2 at point A intersects with the set 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.Figure 3(1)
(1-β)yP0+q>r/c: Take a point B1((1-α)h2,ε+r/c)∈N2 above A, where ε>0 is small enough. Then there must exist a trajectory passing through the point B1 which intersects with M2 at point P1(h2,yP1). In view of continuous dependence of the solution on initial value and time, we know that  yP1<yP0, and the point P1 is close to P0 enough, so we have the point P1+ close to P0+ enough and yP1+<yP0+, then we obtain f(B1)=yP1+-yB1>0. On the other hand, the trajectory passing through point B which tangents to N1 at point B intersect with N2 at point S. 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). If yS≥yP0+, then yP2+<yP0+. So we obtain f(S)=yP2+-yS<0. There exists an order one periodic solution of system (1.3), whose initial point is between the point B1 and the point S in set N2. If yS<yP0+ and yP2+>yS, from the vector field of system (1.3), we know that the trajectory of system (1.3) with any initiating point on the N2 will ultimately stay in Γ1 after one impulsive effect (as shown in Figure 4). Therefore, there is no an order one periodic solution of system (1.3);(2)
(1-β)yP0+q<r/c (as shown in Figure 5): In this case, the point P0+ lies below the point A, that is, (1-β)yP0+q<r/c, thus the successor function of the point A is f(A)=(1-β)yP0+q-r/c<0. Take another point B1((1-α)h2,ε)∈N2, where ε>0 is small enough. Then there must exist a trajectory passing through the point B1 which intersects with M2 at a point P1(h2,yP1)∈M2. Supposing 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(C1)=yP1+-ε>0. From Lemma 2.3, there exists an order one periodic solution of system(1.3), whose initial point is between B1 and A in set N2;(3)
(1-β)yP0+q=r/c: 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 (1.3) which is just a part of the trajectory passing through the A.Figure 4Figure 5Now we can summarize the above results in the following theorem.Theorem 3.3.
Assuming thatd<ecx0,0<h1<(1-α)h2<h2<d/ec.
If(1-β)yP0+q≤r/c, there exists an order one periodic solutions of the system (1.3).
If(1-β)yP0+q>r/c and yS≥yP0+ or yS<yP0+ and yP2+≤yS, there exists an order one periodic solutions of the system (1.3).Subcase 2.2 (0<(1-α)h2<h1<h2).
In this case, the setN2 is in the left side of N1. Any trajectory from initial point (x0+,y0+)∈N2 will intersect with M1 at some point with time increasing. Like 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 after one impulsive effect or free from impulsive effect.Theorem 3.4.
Ifd<ecx0 and 0<(1-α)h2<h1<h2<d/ec, there is no order one periodic solutions to the system (1.3), 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/ec<h2≤x0).
In this case, the setM2 is in the right side of stable center R. In the light of the different position of N2, we consider the following two subject cases.Subcase 3.1 (h1<(1-α)h2).
In this case, the setM2 is in the right side of R. Then there exists a unique closed trajectory Γ1 of system (1.3) which contains the point R and is tangent to M2 at the point A.
SinceΓ1 is a closed trajectory, we take 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: In this case, there is a trajectory, which contains the point R and is tangent to the N2 at the point B intersecting with 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. Like the analysis of Subcase 2.1, we can prove that there exists an order one periodic solution to the system (1.3) in this case (as shown in Figure 6);(2)
h1<δmin<(1-α)h2: In this case, denote the closed trajectory Γ1 of system (1.3) intersecting with the set N2 two point A1((1-α)h2,yA1)and A2((1-α)h2,yA2)(as shown in Figure 7). 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)(r/c)+q. If (1-β)(r/c)+q<yA2, the point P0+ lies below the point A2. Like the analysis of (2) of Subcase 2.1, we can prove that there exists an order one periodic solution to the system (1.3) in this case. If (1-β)(r/c)+q>yA1, the point P0+ is above the point A1. Suppose the trajectory passing through point B which tangents to N1 at point B intersects with N2 at a point S. Like the analysis of (1) of Subcase 2.1, we obtain sufficient conditions of existence of order one periodic solution to the system (1.3);(3)
yA2<(1-β)(r/c)+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 8). Take a point B1∈M2 such that B1 jumps to A2 after the impulsive effect and 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 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 that the trajectory of system (1.3) with any initiating point of set N2 will ultimately stay in Γ1. From the vector field of system (1.3), we know the trajectory of system (1.3) 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 below A2, 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 point B which tangents to N1 at point B with the set N2 at point S((1-α)h2,yS)(Figure 7). The trajectory of system (1.3) with any initiating point on segment A1S¯ intersects with the set N2 at some point below A2 with time increasing, so just like the analysis above we obtain it will ultimately stay in Γ1. Therefore, for any point below S will ultimately stay in region Γ1 with time increasing.Figure 6Figure 7Figure 8Now we can summarize the above results as the following theorem.Theorem 3.5.
Assuming thatd<ecx0,h1<δmin<(1-α)h2<d/ec<h2≤x0 and yA2<(1-β)(r/c)+q<yA1, there is no periodic solution in system (1.3) and the trajectory with any initiating point below S will stay in Γ1 or in the region Ω1={(x,y)∣x≥0,y≥0,x≤h1}.Case 4 (x0<h2).
In this case, denote the intersection of the lineL1 with the set N at point B((1-α)h2,r/c), and the intersection between the line L3 and the set M2 at point A(h2,rh2/cx0) (as shown in Figure 9). By Lemma 2.3 and means of qualitative analysis, there exists a unique closed trajectory Γ2 of system (1.3) which is tangent to the set M2 at the point A and has minimal value λmin at the line L1. In the light of the different position of the set N2, we consider the following two cases.Figure 9Subcase 4.1 (0<h1<(1-α)h2<λmin).
In this case, there exists a unique trajectory of system (1.3) which is tangent to the set N2 at the point B. Set F(B)=P1∈M2, then pulse occurs at point P1, the impulsive function transfers the point P1 into P1+. Like the analysis of (1) of Subcase 2.1, we can prove that there exists an order one periodic solution in system (1.3) in this case.Subcase 4.2 (h1<λmin<(1-α)h<x0<h2).
In this case, let the closed trajectoryΓ2 of system (1.3) intersects N2 at two point A1((1-α)h,yA1) and A2((1-α)h,yA2). Like the analysis of (2) of Subcase 3.1, we can prove that there exists an order one periodic solution in system (1.3) in this case; like the analysis of (3) of Subcase 3.1 we can prove that there is no periodic solution in system (1.3).


## 4. Attractiveness of the Order One Periodic Solution

In this section, under the condition of existence of order one periodic solution to system (1.3) and the initial value of pest population x(0)≤h2, we discuss its attractiveness. We focus on Cases 1 and 2; by similar method we can obtain similar results about Cases 3 and 4.Theorem 4.1.
Assuming thatd<cex0,h1<h2<r/c and δ≥r/c. If yP0+>yP2+>yP+ or yP0+<yP2+<yP+, then(I)
there exists a unique order one periodic solution of system (1.3);(II)
If(1-α)h2<h1, order one periodic solution of system (1.3) is attractive in the region Ω0={(x,y)∣x≥0,y≥0,x≤h2}.Proof.
By the derivation of Theorem3.1, we know that there exists an order one periodic solution of system (1.3). We assume that trajectory PP+̂ and segment PP+¯ formulate an order one periodic solution of system (1.3), 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 intersect with the set M1 at two points D1(h1,yD1) and D2(h1,yD2), respectively (Figure 10). In view of the vector field of system (1.3) and the disjointness of any two trajectories without impulse, we know that  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 that the successor function f(x) is decreasing monotonously in N1; therefore 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 (1.3).
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 11).
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 I, 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, 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 function f(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 C̃, then we have yC̃>yP,yC̃+>yP+. Since f(C+)≥0 and P+≠C+, according to the uniqueness of the periodic solution, then we have f(C+)=yC̃+-yC+>0, thus yC+<yP+<yC̃+ hold. Analogously, let trajectory passing through the point C+ which intersects the set M1 at point C̃̃, and the corresponding consecutive points is C̃̃̃, then yC̃>yC̃̃̃>yp>yC̃̃,yC̃+>yC̃̃̃+>yC̃̃+>yp+>yC+, then we have f(C̃̃+)=yC̃̃̃+-yC̃̃+>0, this contradicts to the fact that C+ is a limit of sequence {P2k-1+}k=1,2,…, so we obtain P+=C+. Therefore, we have limk→∞yP2k-1+=yP+. Similarly, we can prove limk→∞yP2k+=yP+.
From above analysis, we know that there exists a unique order one periodic solution in system (1.3), 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 10Figure 11Remark 4.2.
Assuming thatd<ecx0,h1<h2<d/ec and δ≥r/c, if yP+<yP0+<yP2+ or yP+>yP0+>yP2+, then the order one periodic solution is unattractive.Theorem 4.3.
Assuming thatd<ecx0,h1<(1-α)h2<h2<d/ec and yP0+<yA (as shown in Figure 12), then(I)
there exists an odd number of order one periodic solutions of system (1.3) with initial value between C1+ and A in set N2;(II)
if(1-α)h2<h1 and the periodic solution is unique, then the periodic solution is attractive in region Ω2, here Ω2 is open region which is constituted by trajectory GB̂, segment BH¯, segment HE¯, and segment EG¯.Figure 12Proof.
(I) due to the (2) of Subcase2.1, 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 that there exists an odd number of order one periodic solutions of system (1.3) with initial value between C1+ and A in set N2;
(II) by the derivation of Theorem3.3, we know that there exists an order one periodic solution of system (1.3) whose initial point is between C1+ and P0+ in the set N2. Assume trajectory P+P̂ and segment PP+¯ formulate the unique order one periodic solution of system (1.3) 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 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 that  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,… increases monotonously and has 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 that the point Q∈[yDk+1+,yDk+). The trajectory with initiating point Q moves between trajectory Dk+Dk+1̂ and Dk+1+Dk+2̂ 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 Π̃(Q,t) continues to move between trajectory Dk+1+Dk+2̂ and Dk+2+Dk+3̂. This process can be continued unlimitedly. Since limk→∞yDk+=yP+, the intersection sequence of trajectory Π̃(Q,t) with the set N2 will ultimately tend to be the point P+. Similarly, if Q∈[yCk+,yCk+1+), we also can get the intersection sequence of trajectory Π̃(Q,t), and the set N2 will ultimately tend to be point P+. Thus the trajectory from initiating any point below A ultimately tends to be 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, and the set N2 by 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 that 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 definitely intersect with set N2. From the above analysis, we know that 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 thatd<ecx0,h1<(1-α)h2<h2<d/ec and yC1+<yA<yP0+, the order one periodic solution with initial point between A and P0+ is unattractive.


## 5. Conclusion

In this paper, a state-dependent impulsive dynamical model with Holling I functional response predator-prey 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 and demonstrate the sufficient conditions that system (1.3) 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. In order to testify the validity of our results, we consider the following example:
(5.1)x′(t)=0.4x(t)-0.6x(t)y(t),y′(t)=-0.2y(t)+0.3x(t)y(t),x≤0.8,x′(t)=0.4x(t)-0.48y(t),y′(t)=-0.6y(t)+0.24y(t),x>0.8,x≠h1,h2orx=h1,y>y*,Δx(t)=0,Δy(t)=0.8,x=h1,y≤y*,Δx(t)=-0.5x(t),Δy(t)=-0.2y(t)+0.5,x=h2,
where 0<h1<h2<x*. 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 seth2=0.6,(1-α)h2<h2<x*, initiating points are (0.45,0.25) (red), (0.4,0.2) (green), and (0.3,0.1) (blue), respectively. Figure 13 shows that the conditions of Theorems 3.1 and 4.1 hold, system (1.3) exists order one periodic solution, 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 (1.3) starting from initial value (0.45,0.25) (red), (0.4,0.2) (green), and (0.3,0.1) (blue) 0<h1<h2<x*.
(a)(b)(c)Example 5.2.
Existence and attractiveness of positive periodic solution.
We seth1=0.3,h2=0.6,h1<(1-α)h2<x*<h2<x0, initiating points are (0.45,0.25) (red), (0.4,0.2) (green), and (0.3,0.1) (blue), respectively. Figure 14 shows that the conditions of Theorems 3.3 and 4.3 hold, system (1.3) exists order one periodic solution, 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 (1.3) starting from initial value (0.45,0.25)(red), (0.4,0.2)(green), and (0.3,0.1) (blue) h1<(1-α)h2<x*<h2<x0.
(a)(b)(c)Example 5.3.
Existence and attractive of positive periodic solutions.
We seth1=0.3,h2=0.8,h1<(1-α)h2<x*<h2, initiating points are (0.4,0.1) (red), (0.4,0.2) (green), and (0.4,0.15) (blue) as shown in Figure 15. Therefore, the conditions of Theorem 3.3 and Theorem 4.3 hold, so system (1.3) exists order one periodic solution, and it is attractive.The time series and phase diagram for system (1.3) starting from initial value (0.4,0.1) (red), (0.4,0.2) (green), and (0.4,0.15) (blue) h1=0.3, h2=0.8, h1<(1-α)h2<x*<h2.
(a)(b)(c)Example 5.4.
Existence and attractive of positive periodic solutions.
We seth1=0.3,h2=1.5,h1<(1-α)h2<x*<x0<h2, initiating points are (0.9,0.2) Figure 16 shows that results of Case 4 are valid.
These results show that the state-dependent impulsive effects contribute significantly to the richness of the dynamics of the model. 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 to the models with impulsive state feedback control [12–15], so they are deserved further promotion.The time series and phase diagram for system (1.3) starting from initial value (0.9,0.2)h1=0.3,h2=1.5,h1<(1-α)h2<x*<x0<h2.
(a)(b)(c)



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*Source: 181752-2012-06-17.xml*