# Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model

Authors: Huidong Cheng; Tongqian Zhang; Fang Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

According to the integrated pest management strategies, a Holling type I functional response predator-prey system concerning state-dependent impulsive control is investigated. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution, and the attractivity 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 than the existing ones for proving the existence and attractiveness of order one periodic solution.


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## Body

## 1. Introduction

It is one of the significant problems in the world today to prevent plant pests and pesticide pollution and to protect ecological balance for the sustainable development of agriculture and forestry, which is also an important research topic demanding prompt solution. In recent decades, plant pests worldwide are increasingly serious with the damage of the world's natural ecosystems. In agricultural production, pesticides spaying (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  [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 [4–6].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 [7, 8]; 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 [9–11].In consideration of predator-prey capacity, Holling [12] 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 densities for 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. Holling type-I functional response is
(1.2)ϕ(x)={cx,x≤x0,cx0,x>x0,whereC is a constant; when the amount of prey is greater than certain threshold value x0, predatory rate is a constant.Holling type-II and type-III functional responses are as follows, respectively:(1.3)ϕ(x)=αx1+ωx,ϕ(x)=αx2β+x2.Refer to [12] for details.As the Lotka-Volterra predator-prey system with Holling functional response is more practical, many authors have studied about it [13–15]. 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 and adopts new mathematic method to study comprehensive control (releasing natural enemies and spraying pesticide) adopted when the amount of predators (pests) reaches the economic test (ET), thus the following pest-control model with Holling type-I functional response is set up:
(1.4)x′(t)=rx(t)-cx(t)y(t),x≤x0,x<h,y′(t)=-dy(t)+ecx(t)y(t),x≤x0,x<h,x′(t)=rx(t)-cx0y(t),x>x0,x<h,y′(t)=-dy(t)+ecx0y(t),x>x0,x<h,Δx(t)=-αx(t),x=h,Δy(t)=-βy(t)+q,x=h,
where r,c,d,e are all positive constants, x(t) and y(t) represent the densities of prey (pest) and predator (natural enemy) at time t, respectively; r is the intrinsic growth rate of the prey; d denotes the death rate of the predator; h is the economic threshold ET; α,β∈(0,1) represent the proportion of killed prey and predator by spraying pesticides, respectively, q>0 is the number of natural enemies released at this time. When the amount of prey reaches the threshold h at time th, controlling measures are taken and the amount of prey and predator abruptly turns to (1-α)h and (1-β)y(th)+q, respectively. Δx=x(t+)-x(t), Δy=y(t+)-y(t),x(t+)=limw→0+x(t+w),y(t+)=limw→0+y(t+w).This paper is organized as follows. In the next section, we present some basic definitions and 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.


## 2. Preliminaries

We first consider the model (1.4) without impulse effects
(2.1)x′(t)=rx(t)-cx(t)y(t),x≤x0,y′(t)=-dy(t)+ecx(t)y(t),x≤x0,x′(t)=rx(t)-cx0y(t),x>x0,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,
and we can easily know that V(x,y) is positive definite in the first quartile and fits for all conditions of Liapunov function.We can get that(2.2)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 (1.4) which 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:(2.3)L(x,y)=y+x-n,n>0,x0<x≤h.
The derivative of L(x,y) along (2.1) is that
(2.4)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 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 the following.(I)
Two steady states0(0,0)-saddle point, and R(d/ec,r/c)=R(x*,y*)-stable centre under the condition x≤x0 and d≤ecx0.(II)
The trajectory of system (2.1) goes across the straight line y+x-n=0 from the right to the left under condition x0≤x≤h and n>(dh-ecx02+rh+cx0h)/(d-ecx0+cx0), and intersects with the straight line x=x0.Definition 2.2.
A triple(X,Π,R+) is said to be a semi-dynamical system if X is a metric space R+ is the set of all non-negative 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 2.3.
Assume 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 ɛ>0 such that for any 0<|t|<ɛ, 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 2.4.
We consider state-dependent impulsive differential equations:(2.5)x′(t)=P(x,y),(x,y)∉M(x,y),y′(t)=Q(x,y),(x,y)∉M(x,y),Δx(t)=α(x,y),(x,y)∈M(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 and M(x,y) is called impulsive set. The function I is continuous mapping, and 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 (2.5) as “semi-continuous dynamic system,” which is denoted as (Ω,f,I,M).Definition 2.5.
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 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 1Definition 2.6.
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 2.7.
Successor function defined in Definition2.2 is continuous.Lemma 2.8.
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).


## 3. Existence of the Order One Periodic Solution

In this section we shall investigate the existence of an order one periodic solution of system (1.4) by using the successor function defined in this paper. For this goal, we denote that
(3.1)M1={(x,y)∈R2+∣x=h,h<x0,y≥0},M2={(x,y)∈R2+∣x=h,h>x0,y≥0}.Phase setN of set M is that
(3.2)N=I(M)={(x,y)∈R2+∣x=(1-α)h,y≤q}.Isoclinic line is denoted, respectively, by lines(3.3)L1={(x,y)(x,y)∈R2+∣y=rc,0≤x≤x0},L2={(x,y)∈R2+∣x=dec,0≤x≤x0,y≥0},L3={(x,y)∈R2+∣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, we denote yP as its ordinate. If the point P(h,yP)∈M, pulse shall occur at the point P, and the impulsive function transfers the point P into P+∈N. Without loss of generality, we assume that the initial point of the trajectory lies in phase set N unless otherwise specified.Due to the practical significance, in this paper we assume the setN always lies in the left side of stable centre R, that is, (1-α)h<d/ec.In the light of the different position of the setM1 and the set M2, we consider the following three cases.Case 1 (0<h<d/ec).
In this case, setsM1 and N are both in the left side of stable center R. The line L1 intersects with N at point A((1-α)h,r/c), as shown in Figure 2. The trajectory passing through point A which tangents to the set N at point A intersects with the set M1 at point P0(h,yP0). Since the point P0∈M1, impulse occurs at point P0, supposing point P0 is subject to impulsive effects to point P0+((1-α)h,yP0+), where yP0+=(1-β)yP0+q, and the position of the point P0+ has the following three cases.Figure 2Subcase
1.1   ((1-β)yP0+q<r/c). In this case, the point P0+ lies below the point A, thus the successor function of the point A is f(A)=(1-β)yP0+q-r/c<0.Take another pointC((1-α)h,ɛ)∈N, where ɛ>0 is small enough. Then there must exist a trajectory passing through the point C which intersects M1 at point C1(h,yC1)∈M1. Supposing that the point C1 is subject to impulsive effects to point C1+((1-α)h,yC1+), yC1+=(1-β)yC1+q>ɛ, so we have f(C)=yC1+-ɛ>0.By Lemma2.8, there exists an order one periodic solution of system (1.4), whose initial point is between the point C and the point A in set N.Subcase1.2  ((1-β)yP0+q>r/c) (as shown in Figure 3). In this case, the point P0+ lies above the point A; there must exist a trajectory passing through the point P0+ which intersects the set M1 at a point P2(h,yP2) under P1. Denote the coordinates of impulsive point P2+((1-α)h,yP2+) corresponding to the point P2(h,yP2), then yP2+<yP0+. So we obtain f(P0+)=yP2+-yP0+<0.Figure 3Take that another pointB1((1-α)h,ɛ+r/c)∈N above A, where ɛ>0 is small enough. Then there must exist a trajectory passing through the point B1 which intersects M1 at a point P1(h,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, then we obtain f(B1)=yP1+-yB1>0.From Lemma2.8, there exists an order one periodic solution of system (1.4), whose initial point is between B1 and P0+ in set N.Subcase
1.3
(
(
1
-
β
)
y
P
0
+
q
=
r
/
c
). P0+ coincides with A, and the successor function of A is 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 point A.Now we can summarize the above results as the following theorem.Theorem 3.1.
Assuming thatd<ecx0,0<h≤d/ec<x0, there exists an order one periodic solutions of the system (1.4).Case 2   (d/ec<h≤x0).
In this case, the setM1 is in the right side of R and set N is in the left side of R. Let the line L1 intersects the sets M1 and N at points A(h,r/c) and B((1-α)h,r/c), respectively (Figure 4), then there exists a unique closed trajectory Γ1 of system (1.4) which is contained inside the point R and is tangent to the set M1 at the point A.Figure 4SinceΓ1 is a closed trajectory, we take their minimal value of abscissas δmin at the trajectory Γ1, namely, δmin≤x holds for any abscissas of Γ1.In light of the different positions of the setN, we consider the following three subject cases.Subcase 2.1
(
0
<
(
1
-
α
)
h
<
δ
min
<
d
/
e
c
). In this case, there is a trajectory, which is contained inside the point R and is tangent to the set N at point B which intersects M1 at the point P1((1-α)h,yP1)∈M1. Supposing that point P1 is subject to impulsive effects to point P1+((1-α)h,yP1+)∈N, here yP1+=(1-β)yP1+q. Like the analysis of Case 1 we can prove there exists an order one periodic solution in system (1.4) in this case.Now we can summarize the above results as the following theorem.Theorem 3.2.
Ifd<ecx0,0<(1-α)h<δmin<d/ec<h≤x0, there exists an order one periodic solution in system (1.4).Subcase 2.2
(
δ
min
<
(
1
-
α
)
h
<
d
/
e
c
). In this case, let the closed trajectory Γ1 of system (1.4) intersects the set N at two points A1((1-α)h,yA1) and A2((1-α)h,yA2), as shown in Figure 5. Since A∈M1, impulse occurs at point A. Supposing that point A is subject to impulsive effects to point P0+((1-α)h,yP0+), here yP0+=(1-β)(r/c)+q.Figure 5If(1-β)(r/c)+q=yA1 or (1-β)(r/c)+q=yA2, P0+ coincides with A1 or P0+ coincides with A2; the successor function of A1 or A2 is that f(A1)=0 or that 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)(r/c)+q<yA2, the point P0+ lies below the point A2, like the analysis of Subcase 1.1, and we can prove there exists an order one periodic solution to the system (1.4) in this case.If(1-β)(r/c)+q>yA1, the point P0+ is above the point A1. Like the analysis of Subcase 1.2, we can prove there exists an order one periodic solution to the system (1.4) in this case.Now we can summarize the above results as the following theorem.Theorem 3.3.
Assuming thatd≤ecx0,δmin<(1-α)h<d/ec<h<x0, if (1-β)(r/c)+q≥yA2 or (1-β)(r/c)+q≥yA1, then there exists an order one periodic solution to the system (1.4).Subcase2.3  (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 6). Taking a point E1∈M1 such that E1 jumps to A2 after the impulsive effect, we denote A2=E1+. Since yP0+>yE1+, we have yA>yE1. Let R(E1)=E2+∈N, taking a point E2∈M1 such that E2 jumps to E2+ after the impulsive effects, then we have yE1+>yE2+,yE1>yE2. This process continues until there exists a EK+∈N(K∈Z+) satisfying yEK+<q. So we obtain a sequence {Ek}k=1,2,…,K of set M1 and a sequence {Ek+}k=1,2,…,K of set N satisfying R(Ek-1)=Ek+∈N,yEk-1+>yEk+. In the following, we will prove the trajectory of system (1.4) with any initiating point of set N which will ultimately stay in Γ1.Figure 6From the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point between the points A1 and A2 will be free from impulsive effect and ultimately will stay in Γ1.For any point belowA2, it must lie between Ek+ and Ek-1+, where k=2,3,…,K+1 and A2=E1+. After k times' impulsive effects, the trajectory with this initiating point will arrive at some point of the set N which must be between A1 and A2, and then ultimately stay in Γ1.The trajectory with any initiating point aboveA1 will intersect set N at some point below A2 with time increasing, so just like the analysis above we obtain that it will ultimately stay in Γ1.Now we can summarize the above results as the following theorem.Theorem 3.4.
Assuming thatd≤ecx0 and δmin<(1-α)h<d/ec<h≤x0, if yA2<(1-β)(r/c)+q<yA1, there is no periodic solution in system (1.4) and the trajectory with any initiating point of set N will stay in Γ1.Case 3  (0<(1-α)h<d/ec<x0<h).
In this case, denote the intersection of the lineL1 and the set N by point B((1-α)h,r/c), and the intersection of the line L3 and the set M2 by point A(h,rh/cx0) (as shown in Figure 7). Through Lemma 2.8 and qualitative analysis, there exists a unique closed trajectory Γ2 of system (1.4) 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 N, we consider the following two cases.Figure 7Subcase3.1  (0<(1-α)h<λmin). In this case, there exists a unique trajectory of system (1.4) which is tangent to the set N at the point B. Set F(B)=P1∈M2, then pulse occurs at point P1, and the impulsive function transfers the point P1 into P1+. Like the analysis of Subcase 2.1, we can prove there exists an order one periodic solution in system (1.4) in this case.Subcase3.2  (λmin<(1-α)h<x0<h). In this case, let the closed trajectory Γ2 of system (1.4) intersect with N at two points A1((1-α)h,yA1) and A2((1-α)h,yA2). Like the analysis of Subcase 2.2 we can prove there exists an order one periodic solution in system (1.4) in this case; like the analysis of Subcase 2.3, we can prove there is no periodic solution in system (1.4) and the trajectory with any initiating point of set N will stay in Γ2.


## 4. Attractiveness of the Order One Periodic Solutions

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)≤h, we discuss its attractiveness. We focus on Case 1, and by similar method we can obtain similar results about Cases 2 and 3.Theorem 4.1.
If system (1.4) satisfies the condition of Theorem 3.1 and yP0+<yA, then(I)
there exists an odd number of order one periodic solutions of system (1.4) with initial value between C and A in the set N(II)
if the periodic solution is unique, then the periodic solution is attractive in regionΩ1={(x,y)∣x≥0,y≥0,x(0)≤h}.Proof.
(I) According to the Subcase 1.3,f(A)<0 and f(c)>0 (see Figure 2); the successor function f(x) is continuous, so 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 (1.4) with initial value between C and A in the set N.
(II) By the derivation of Theorem3.1, we know there exists an order one periodic solution of system (1.4). We assume trajectory P+P^ and segment PP+¯ formulate a unique order one periodic solution of system (1.4) with initial point P+∈N (Figure 8).
On the one hand, take a pointD1+((1-α)h,ɛ)∈N satisfying yD1+=ɛ<q and yD1+<yP+. The trajectory passes through the point D1+((1-α)h,ɛ) which intersects set M1 at point D2(h,yD2), that is, F(D1+)=D2∈M1, then we have yD2<yP, thus yD2+<yP+. Since yD2+=(1-β)yD2+q>ɛ, so we obtain f(D1+)=yD2+-yD1+=yD2+-ɛ>0; set F(D2+)=D3∈M1, because yD1+<yD2+<yP+; we know yD2<yD3<yP, then we have yD2+<yD3+<yP+ and f(D2+)=yD3+-yD2+>0. This process is continuous, then we get a sequence {Dk+}k=1,2… of set N satisfying
(4.1)yD1+<yD2+<⋯<yDk+<⋯<yP+
and f(Dk+)=yDk+1+-yDk+>0. Series {yDk+}k=1,2,… increase monotonously and have upper bound, so that limk→∞yDk+ exists.
Next we will provelimk→∞yDk+=yP+. Setting limk→∞Dk=D+, we will prove P+=D+. Otherwise P+≠D+, then there is a trajectory passing through the point D+ which intersects the set M1 at point D~, then we have yD~<yP,yD~+<yP+. Since f(D+)≥0 and P+≠D+, according to the uniqueness of the periodic solution, then we have f(D+)=yD~+-yD+>0, thus yD+<yD~+<yP+ hold. Set F(D~+)=D~~∈M1. In view of vector field and disjointness of any two trajectories, we know yD~<yD~~<yP and yD~+<yD~~+<yP+, then we have f(D~+)=yD~~+-yD~+>0; this contradicts the fact that D+ is a limit of sequence {Dk+}k=1,2,…, so we obtain P+=D+ and limk→∞yDk+=yP+.
On the other hand, setF(P0+)=C1∈M1, then C1 jumps to C1+∈N under the impulsive effects. Since yP+<yP0+<yA, we have yP<yC1<yP0, thus we obtain that yP+<yC1+<yP0+ and that f(P0+)=yC1+-yP0+<0. Set F(C1+)=C2∈M1, then C2 jumps to C2+∈N under the impulsive effects. We have yP+<yC2+<yC1+, f(C1+)=yC2+-yC1+<0; this process is continuous, and we can obtain a sequence {Ck+}k=1,2,… of set N satisfying
(4.2)yP0+>yC1+>yC2+>⋯>yCk+>⋯>yP+
and f(Ck+)=yCk+1+-yCk+<0. Series {yCk+}k=1,2,… decrease monotonously and have lower bound, so that limk→∞yCk+ exists. Similarly, we can prove that limk→∞yCk+=yP+.
Since the trajectory initiating any point ofΩ1={(x,y)∣x≥0,y≥0,x(0)≤h} will certainly intersect with set N, next we only need to prove the trajectory initiating any point of N will ultimately tend to be the unique order one periodic solution.
Any pointB∈N below A must be in some interval [yCk+1+,yCk+)k=1,2,…,[yC1+,yP0+),[yP0+,yA), [yDk+,yDk+1+)k=1,2,…. Without loss of generality, we assume the point B∈[yCk+1+,yCk+). The trajectory with initiating point B moves between trajectories Ck+Ck+1^ and Ck+1+Ck+2^ and intersects with M1 at some point between Ck+2 and Ck+1, and under the impulsive effects it jumps to the point of N which is between [yCk+2+,yCk+1+), then trajectory Π~(B,t) continues to move between trajectories Ck+1+Ck+2^ and Ck+2+Ck+3^. This process can be continued unlimitedly. Since limk→∞yCk+=yP+, the intersection sequence of trajectory Π~(B,t) and set N will ultimately tend to point P+. Similarly, if B∈[yDk+,yDk+1+), we can also get that the intersection sequence of trajectory Π~(B,t) and set N will ultimately tend to point P+. Thus the trajectory initiating any point below A ultimately tends to be the unique order one periodic solution P+PP+^.
The trajectory with any initiating point aboveA of N will intersect with set N at some point below A with time increasing, so like the analysis above we obtain the trajectory initiating any point above A will ultimately tend to be the unique order one periodic solution P+PP+^.
From the above analysis, we know the trajectory initiating any point ofN will ultimately tend to be order one periodic solution P+PP+^. Therefore, the unique order one periodic solution is attractive in the region Ω1. This completes the proof.Figure 8Remark 4.2.
If system (1.4) satisfies the condition of Theorem 3.1 and that yD1+<yA<yP0+, then order one periodic solution with initial point between A and P0+ is unattractive.Theorem 4.3.
If system (1.4) satisfies the condition of Theorem 3.1 and yP0+>yP1+>yA, there exists a unique order one periodic solution of system (1.4) which is attractive in the region Ω1={(x,y)/x≥0,y≥0,x(0)≤h}.Proof.
Through the derivation of Theorem3.1, we know there exists an order one periodic solution of system (1.4), whose initial point is between A and P0+ in set N. We assume trajectory P+P^ and segment PP+¯ formulate an order one periodic solution of system (1.4), and P+∈N is its initial point satisfying f(P+)=0 (Figure 9). First we will prove the uniqueness of the order one periodic solution.
We take any two pointsC1(h,yC1)∈N,C2(h,yC2)∈N satisfying yC2>yC1>yA, then we obtain two trajectories, whose initiate points are C1 and C2 intersecting with the set M1 and two points D1(h,yD1) and D2(h,yD2), respectively (Figure 10). In view of the vector field of system (1.4) and the disjointness of any two trajectories, we know yD1>yD2. Suppose the points D1 and D2 are subject to impulsive effect to points D1+(h,yD1+) and D2+(h,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) decreases monotonously on interval (yA,yP0+) of N, so there is a unique point P+∈N satisfying f(P+)=0. 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 Ω1.
Denote the first intersection point of the trajectory with initiating pointP0+ and the impulsive set M1 by P1(h,yP1) and that the corresponding consecutive points are P2(h,yP2), P3(h,yP3), P4(h,yP4), and so forth, respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are P1+(h,yP1+), P2+(h,yP2+), P3+(h,yP3+), and so forth. In view of yP0+>yP1+>yA and disjointness of any two trajectories, we have that
(4.3)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. As is proved in Theorem 4.1, we can prove limk→∞yP2k-1+=limk→∞yP2k+=yP+.
Since the trajectory initiating any point ofΩ1 will certainly intersect with set N, next we only need to prove the trajectory initiating any point of N will ultimately tend to be order one periodic solution P+PP+^.
The trajectory with initiating point betweenD0+ and P0+ will intersect impulsive set N with time increasing; under the impulsive effects it arrives at a point of N which is between [yP2k-1+,yP2k+1+) or [yP2k+,yP2k-2+). Then like the analysis of Theorem 4.1, we know the trajectory initiating any point between D0+ and P0+ will ultimately tend to be order one periodic solution P+PP+^.
Assume a pointD0 of the set M1 jumps to D0+ under the impulsive effect. Set R(D0)=D1+∈N. Assume a point D1 of the set M1 jumps to D1+ under the impulsive effect. Set R(D1)=D2+∈N. This process is continuing until there exists a DK0+∈N(K0∈Z+) satisfying yDk0+<q. So we obtain a sequence {Dk}k=0,1,2,…,K0 of the set M1 and a sequence {Dk+}k=0,1,2,…,K0 of the set N satisfying R(Dk-1)=Dk+,yDk+<yDk-1+. For any point of the set N below D0+, it must lie between Dk+1+ and Dk+ where k=1,2,…,K0. After K0+1 times' impulsive effects, the trajectory from this initiating point will arrive at some point of the set N which must be between D0+ 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 D0+.
The trajectory with any initiating point aboveP0+ of N will intersect with the set N at some point below U0+ with time increasing, so the trajectory with any initiating point above P0+ will ultimately tend to be order one periodic solution P+PP+^. There is no order one periodic solution with the initial point above P0+.
From the above analysis, we know there exists a unique order one periodic solution in system (1.4) and the trajectory from any initiating point of N will ultimately tend to be order one periodic solution P+PP+^. Therefore, order one periodic solution P+PP+^ is attractive in the region Ω1. This completes the proof.Figure 9Figure 10


## 5. Conclusion and Numerical Simulations

In this paper, a Holling I predator-prey model with state-dependent impulsive control model concerning different control methods at different thresholds is proposed to find a new method to study existence and attractive of order one periodic solution of such system. We define semicontinuous dynamical system and successor function and demonstrate the sufficient condition that system (1.4) has order one periodic solution with differential geometry theory and successor function; by means of sequence convergence rules and qualitative analysis, we prove the attractiveness of the order one periodic solution. These results show that the state-dependent impulsive effects contribute significantly to the richness of the dynamics of the model. 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 ones that have been applied to the models with impulsive state feedback control [12–15], so they deserve further promotion.In order to testify the validity of our results, we consider the following example:(5.1)dxdt=0.8x-0.6xy,x≤x0,x<h,dydt=-0.2y+0.3xy,x≤x0,x<h,dxdt=0.8x-0.6x0y,x>x0,x<h,dydt=-0.2y+0.3x0y,x>x0,x<h,Δx=-αx,x=h,Δy=-βy+q,x=h,where α,β∈(0,1),q>0,h>0. Now, we consider the impulsive effects on the dynamics of system (5.1).Example 5.1.
Existence and attractiveness of order one periodic solution.
We seth=0.6,α=0.5,β=0.2,q=0.8,h<x*<x0, initiating points are (0.3,0.6) (red), (0.3,0.5) (green), and (0.3,0.1) (blue), respectively. Figure 11 shows that system (5.1) has order one periodic solution when the conditions of Theorems 3.1 and 4.1 hold, 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  (0.3,0.1) (blue), (0.3,0.5) (green),(0.3,0.6) (red),α=0.5,β=0.2,q=0.8,h=0.6,x0=1,h<x*<x0.
(a)
(b)
(c)Example 5.2.
Existence and attractiveness of positive periodic solutions.
We seth1=1,α=0.5,β=0.2,q=0.8,x0=1.5,x*<h<x0, initiating points are (0.3,0.3) (red), (0.3,0.2) (green), and (0.3,0.2.5) (blue), respectively. Figure 12 shows that system (5.1) has order one periodic solution when the conditions of Theorems 3.2 and 4.1 hold, 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 (0.3,0.2) (blue), (0.3,0.1)(green), (0.3,0.15) (red), α=0.5,β=0.2,q=0.8,h=1,x0=1.5,x*<h<x0.
(a)
(b)
(c)Example 5.3.
Existence and attractiveness of positive periodic solutions.
We seth=1.5,α=0.5,β=0.2,q=0.8,x0=0.8,x*<x0<h; initiating points is (0.4,0.5). Figure 13 shows that results of Case 3 are valid.The time series and phase diagram for system (1.4) starting from initial value (0.4,0.5), α=0.5,β=0.2,q=0.8,h=1.5,x0=0.8,x*<x0<h.
(a)
(b)
(c)



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*Source: 126018-2012-10-18.xml*