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

Volume 2012, Article ID 181752,21pages doi:10.1155/2012/181752

Research Article

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

Huidong Cheng, Fang Wang, and Tongqian Zhang

College of Science, Shandong University of Science and Technology, Qingdao 266510, China

Correspondence should be addressed to Huidong Cheng,chd900517@sdust.edu.cn Received 18 February 2012; Accepted 10 April 2012

Academic Editor: Recai Kilic

Copyrightq2012 Huidong Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 research1–5. In population dynamical system, Tang et al.6discussed 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 Liu10discussed 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.12also 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, Holling15proposes three different pre- dations with functional response based on experiments; the average predator-prey system with Holling response is as follows:

xt xgx−yφx,

yt −dy eyφx, 1.1

wherexrepresents the prey’s density, whileyis the predator’s;gxis the unit rate of prey density in lack of predators;φxis the Holling functional response, among which Holling type I functional response is

φx

cx, x≤x₀,

cx0, x > x0, 1.2

wherecis a constant; when the amount of prey is greater than certain threshold valuex0, predatory rate is a constant. Referring to15for 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:

xt rxt−cxtyt,

yt −dyt ecxtyt, x≤x₀, xt rxt−cx₀yt,

yt −dyt ecx₀yt, x > x0,

x /h₁, h₂ orxh₁, y > y^∗,

Δxt 0,

Δyt δ, xh₁, y≤y^∗, Δxt −αxt,

Δyt −βyt q, xh₂,

1.3

wherer, c, d, e, h1, andh2 are all positive constants,xtandytrepresent the densities of preypestand predatornatural enemy, respectively;r is the intrinsic growth rate of the prey;ddenotes the death rate of the predator;α, β∈0,1represent the proportion of killed prey and predator by spraying pesticides respectively,δ >0 is the number of natural enemies released at this timet_h1, when the amount of the prey reaches the threshold h₁ at timet_h1, control measures are takenreleasing natural enemiesand the amount of predator abruptly turns to yth1 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 to1−αh2 and

1−βyth2, respectively.Δxxt −xt,Δyyt −yt,xt limw→0xt w,yt lim_w→0yt w. Referring to12for details.

2. Preliminaries

We first consider the model1.3without impulse effects:

xt rxt−cxtyt,

yt −dyt ecxtyt, x≤x₀, xt rxt−cx₀yt,

yt −dyt ecx0yt. x > x₀.

2.1

We consider the following function:

V x, y

_x

x^∗

−d eφs

φs ds

_y

y^∗

s−y^∗

s ds, 2.2

we can easily know thatVx, yis positive definite in the first quartile and fits for all con- ditions of Lyapunov function.

We can get that

V x, y

exy^∗ φx

φx−φx^∗φx^∗ x^∗ −φx

x

. 2.3

It is easily proved thatVx, y≡0 on condition thatx≤x₀; so all solutions of model 1.3form a set{x, y/Vx, y≤Vx0, y^∗} are closed trajectoryVx, y C, where 0< C <

Vx0, y^∗.

SinceVx, y > 0 on condition thatx > x₀; so the trajectory of system2.1passes through closed curveVx, y Cwhen it is out of the curveVx, y Vx0, y^∗.

Therefore, we observe the straight line:

L x, y

y x−n, n >0, x₀< x≤h. 2.4

The derivative ofLx, yalong2.1is that L

x, y

/_L0x y−dy ecx0y rx−ecx0

−dn−ecx₀n cx₀n cx₀h−dx₀−ecx0−r−cx₀

≤dh−ecx²₀ rh cx₀h−d−ecx₀ cx₀n.

2.5

We have thatL/_L0 < 0 on condition thatn > dh−ecx²₀ rh cx₀h/d−ecx₀ cx₀. Therefore, we can get the following Lemma.

Lemma 2.1. The system2.1possesses:

Y

N M

X

0 Q

yB

yA

C

Figure 1

Itwo steady states 00,0—saddle point, andRd/ec, r/c Rx^∗, y^∗—stable centre on the condition thatx≤x₀and thatd≤ecx₀;

IIthe trajectory of system2.1goes across the straight liney x−n0 from the right to the left on condition thatx0≤x≤hand thatn >dh−ecx²₀ rh cx0h/d−ecx0 cx0 and intersects with the straight linexx₀.

Definition 2.2. Suppose that the impulse setMand the phase setNare both lines, as shown in Figure 1. Define the coordinate in the phase setNas follows: denote the point of intersection Q betweenN and x-axis asO, then the coordinate of any pointAin N is defined as the distance between A and Q and is denoted by y_A. Let C denote the point of intersection between the trajectory starting fromAand the impulse setM, and letBdenote the phase point ofCafter impulse with coordinatey_B. Then, we defineBas the successor point ofA, and then the successor function16of pointAis thatfA y_B−y_A.

Lemma 2.3. In system1.3, if there existA∈N, B∈Nsatisfying successor functionfAfB<

0, then there must exist a pointP P ∈NsatisfyingfP 0 the function between the point ofA and the point ofB, thus there is an order one periodic solution in system1.3.

In this paper, we assume that the condition d ≤ ecx0 holds. By the biological background of system1.3, we only considerD{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. InSection 3, we prove the existence for an order one periodic solution of system1.3. The sufficient conditions for the attractiveness of order one periodic solutions of system 1.3 are obtained inSection 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 1.3by using the successor function defined in this paper. For this goal, we denote

M₁

x, y

x h₁,0≤y≤ r

c δ

, M₂ x, y

|xh₂, y≥0 , N₁IM1

x, y

|xh₁, r

c < y≤ r

c δ

, N2IM2 x, y

|x 1−αh2, y≥0 .

3.1

Isoclinic line is denoted respectively by lines:

L1 x, y

|y r

c, 0≤x≤x0

, L₂

x, y

|x d

ec, 0≤x≤x₀, y≥0

, L₃

x, y

|y r cx0

x, x > x₀, y≥ r c

.

3.2

For the convenience, if P ∈ Ω− M, FP is defined as the first point of intersection of C P and M, that is, there exists a t1 ∈ R such that FP ΠP, t1 ∈ M, and for 0 < t < t₁,ΠP, t ∈/ M; if B ∈ N, RB is defined as the first point of intersection of C⁻P and N, that is, there exists a t₂ ∈ R such that RB ΠB,−t2 ∈ N, and for

−t < t <0,ΠB, t∈/N.

For any pointP, denotey_Pas its ordinate. If the pointPh, yP∈M, pulse occurs at the pointP, the impulsive function transfers the pointPintoP ∈N. Without loss of generality, we assume the initial point of the trajectory lies in phase setNunless otherwise specified.

According to the practical significance, in this paper we assume that the setNalways lies in the left side of stable centreR, that is,h₁< r/c,1−αh2< r/c.

In the light of the different position of the setN1 and the set N2, we consider the following three cases.

Case 10 < h₁ < d/ec. In this case, setM₁andN₁are both in the left side of stable center R as shown inFigure 2. Take a point Bh1, r/c ε ∈ N1 above A, whereε > 0 is small enough, then there must exist a trajectory passing throughBwhich intersects with the set M₁ at point P₁h1, y_p1, we havey_p1 < r/c. Since p₁ ∈ M₁, pulse occurs at the point P₁, the impulsive function transfers the pointP1 intoP₁h1, yp1 δandP₁ must lie aboveB;

therefore, inequationa/b ε < y_p1 δholds, thus the successor function ofBis thatfB y_p1 δ−r/c ε>0.

On the other hand, the trajectory with the initial point P₁ intersects with M1 at pointP₂h1, y_p2, in view of vector field and disjointness of any two trajectories, we know

Y

N1

M1

L1

L2

B1

R

P1

P2

P₁⁺

P₂⁺ P₂ⁱ⁺

X 0

A

Figure 2

y_p2 < y_p1< r/c. Supposing that the pointP₂is subject to impulsive effects to pointP₂h1, y_p

2, wherey_p2 y_p2 δ, the position ofP₂ has the following two cases.

Subcase 1.1r/c < yp2 δ < yp1 δ. In this case, the point P₂ lies above the pointAand underP₁, we havefP₁ y_p2 δ−yp1 δ<0.

ByLemma 2.3, there exists an order one periodic solution of system1.3, whose initial point is betweenBandP₁ in setN1.

Subcase 1.2r/c≥y_p2 δas shown inFigure 2. The pointP₂ lies below the pointA, that is,P₂ ∈M₁, then pulse occurs at the pointP₂, the impulsive function transfers the pointP₂ intoP₂ h1, yp2 2δ.

Ifr/c < y_p2 2δ < y_p1 δ, like the analysis of Subcase 1.1, there exists an order one periodic solution of system1.3.

Ifr/c > yp2 2δ, that is,P₂ ∈ M1; we repeat the above process until there exists k ∈Z such thatP₂ jumps toP₂ⁱ h1, y_p2 k 2δafterktimes’ impulsive effects which satisfiesr/c < y_p2 k 2δ < yp1 δ. Like the analysis of Subcase 1.1, there exists an order one periodic solution of system1.3.

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 Theorem 3.1 that the number of natural enemies should be selected appropriately, which aims to reduce releasing impulsive times to save manpower and resources.

Case 2h2< d/ec. In this case, setsM2andN2are both in the left side of stable centerR, in the light of the different position of the setN₂, we consider the following two cases.

Y N1

N2 M2

M1

L1

B

B

R

1

P1

P0

P2

P₂⁺ P₁⁺

P₀⁺

X 0

A S

Figure 3

Subcase 2.10 < h₁ < 1−αh2 < h₂. In this case, the setN₂ is in the right side ofM₁ as shown inFigure 3. The trajectory passing through pointAwhich tangents toN₂at pointA intersects with the setM2at pointP0h2, yp0. Since the pointP0 ∈M2, then impulse occurs at pointP₀, supposing the pointP₀ is subject to impulsive effects to pointP₀1−αh2, y_P

0, wherey_P0 1−βyP0 q, the position ofP₀ has the following three cases.

1 1 −βyP0 q > r/c: Take a pointB11−αh2, 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 P₁h2, y_P1. In view of continuous dependence of the solution on initial value and time, we know thatyP1 < yP0, and the point P₁ is close to P₀ enough, so we have the point P₁ close to P₀ enough and y_P1 < y_P0, then we obtainfB1 y_P1 −y_B1 > 0. On the other hand, the trajectory passing through pointBwhich tangents toN1 at pointBintersect with N₂ at point S. SetFS P₂h2, y_P2 ∈ M₂. Denote the coordinates of impulsive point P₂1−αh2, y_P2corresponding to the point P₂h2, y_P2. If y_S ≥ y_P0, then yP₂ < yP₀. So we obtainfS yP₂ −yS < 0. There exists an order one periodic solution of system1.3, whose initial point is between the pointB₁and the pointS in setN₂. Ify_S< y_P0 andy_P2 > y_S, from the vector field of system1.3, we know that the trajectory of system1.3with any initiating point on theN2will ultimately stay inΓ1 after one impulsive effectas shown inFigure 4. Therefore, there is no an order one periodic solution of system1.3;

2 1−βyP0 q < r/cas shown inFigure 5: In this case, the pointP₀ lies below the pointA, that is,1−βyP0 q < r/c, thus the successor function of the point A isfA 1−βyP0 q−r/c < 0. Take another point B₁1−αh2, ε ∈ N₂, whereε >0 is small enough. Then there must exist a trajectory passing through the pointB₁which intersects withM₂at a pointP₁h2, y_P1∈M₂. Supposing the point

Y

N2

N1

M1

M2

L1

B

B

1

R S

P1

P2

P0

P₁⁺ P₂⁺

P₂⁺ P₀⁺

X 0

A

Figure 4

Y

N2

M2

L1

B1

R

P0

P1

P₁⁺ P₀⁺

X 0

A

Figure 5

P₁h2, y_P1is subject to impulsive effects to pointP₁1−αh2, y_P

1, then we have y_P1 > ε, so we havefC1 y_P1 −ε >0. FromLemma 2.3, there exists an order one periodic solution of system1.3, whose initial point is betweenB1andAin setN2; 3 1−βyP0 q r/c:P₀ coincides withA, and the successor function ofAis that fA 0, so there exists an order one periodic solution of system1.3which is just a part of the trajectory passing through theA.

Y N1

N2

M1

M2

L1

L2

B R

P1

P2

P₁⁺

P₁⁺ P₂⁺ P₂⁺

X 0

S

A Γ1

Figure 6

Now we can summarize the above results in the following theorem.

Theorem 3.3. Assuming thatd < ecx0,0< h1<1−αh2< h2< d/ec.

If1−βyP0 q≤r/c, there exists an order one periodic solutions of the system1.3.

If1−βyP0 q > r/candy_S ≥ y_P0 ory_S < y_P0 andy_P2 ≤ y_S, there exists an order one periodic solutions of the system1.3.

Subcase 2.20 < 1−αh2 < h₁ < h₂. In this case, the setN₂ is in the left side ofN₁. Any trajectory from initial pointx₀, y₀ ∈ N2 will intersect with M1 at some point with time increasing. Like the analysis of Case1, the trajectory from initial pointx₀, y₀ ∈N₂ on the setN2will stay in the regionΩ1 {x, y| x≥ 0, y ≥ 0, x ≤ h1}. Similarly, any trajectory from initial pointx₀, y₀∈Ω0{x, y|x≥0, y≥0, x≤h₂}will stay in the regionΩ1after one impulsive effect or free from impulsive effect.

Theorem 3.4. Ifd < ecx0 and 0 < 1−αh2 < h1 < h2 < d/ec, there is no order one periodic solutions to the system 1.3, and the trajectory with initial pointx₀, y₀ ∈ Ω0 {x, y | x ≥ 0, y≥0, x≤h₂}will stay in the regionΩ1{x, y|x≥0, y≥0, x≤h₁}.

Case 3d/ec < h2 ≤x0. In this case, the setM2is in the right side of stable centerR. In the light of the different position ofN2, we consider the following two subject cases.

Subcase 3.1h1<1−αh2. In this case, the setM₂is in the right side ofR. Then there exists a unique closed trajectoryΓ1of system1.3which contains the pointRand is tangent toM2

at the pointA.

Since Γ1 is a closed trajectory, we take the minimal value δmin of abscissas at the trajectoryΓ1, namely,δ_min ≤xholds for any abscissas ofΓ1.

1h1<1−αh2< δmin: In this case, there is a trajectory, which contains the pointRand is tangent to theN₂at the pointBintersecting withM₂at a pointP₁h2, y_P1∈M₂. Suppose point P1 is subject to impulsive effects to point P₁1−αh2, yP₁, here yP₁ 1−βyP1 q. Like the analysis of Subcase 2.1, we can prove that there exists an order one periodic solution to the system1.3in this caseas shown inFigure 6;

Y

N1 N2

M1

M2

L1

R

P1

P2

L2

P₀⁺

P₁⁺ P₆⁺

X 0

C A

S

B A2

A1

Γ1

Figure 7

S Y

N1

N2

M1

M2

L1

B R

P1

L2

P₁⁺ P₁⁺

P₀⁺

P₀⁺ P₀⁺

X 0

A A1

A2

Γ1

Figure 8

2h1 < δmin < 1−αh2: In this case, denote the closed trajectoryΓ1 of system1.3 intersecting with the setN₂two pointA₁1−αh2, y_A1andA₂1−αh2, y_A2 as shown inFigure 7. SinceA∈M₂, impulse occurs at the pointA. Suppose pointA is subject to impulsive effects to pointP₀1−αh2, yP₀, hereyP₀ 1−βr/c q.

If1−βr/c q < y_A2, the pointP₀ lies below the pointA₂. Like the analysis of 2of Subcase 2.1, we can prove that there exists an order one periodic solution to the system1.3in this case. If1−βr/c q > yA1, the pointP₀ is above the point A₁. Suppose the trajectory passing through pointBwhich tangents toN₁at point Bintersects withN₂at a pointS. Like the analysis of1of Subcase 2.1, we obtain sufficient conditions of existence of order one periodic solution to the system1.3;

3y_A2<1−βr/c q < y_A1: In this case, we note that the pointP₀ must lie between the pointA₁ and the pointA₂as shown inFigure 8. Take a pointB₁ ∈ M₂ such thatB1jumps toA2after the impulsive effect and denoteA2 B₁. SinceyP₀ > yB₁, we havey_A > y_B1. LetRB1 B₂ ∈N₂, take a pointB₂ ∈M₂such thatB₂ jumps

Y

N2 M2

L1 B

R

P1

P2

L2

L3

P₁⁺

P₁⁺

P₂⁺

X 0

A

C

x0

Γ2

Figure 9

toB₂ after the impulsive effects, then we have y_B

1 > y_B

2, y_B1 > y_B2. This process continues until there exists aB_K ∈ N₂K ∈ Z satisfyingy_B

k < q. So we obtain a sequence{B_k}_k1,2,...,Kof setM2and a sequence{Bk}_k1,2,...,Kof setN2 satisfying RBk−1 B_k ∈N₂, y_Bk−1 > y_B

k. In the following, we will prove that the trajectory of system1.3with any initiating point of setN₂ will ultimately stay inΓ1. From the vector field of system 1.3, we know the trajectory of system 1.3 with any initiating point between the pointA1andA2will be free from impulsive effect and ultimately will stay inΓ1. For any point belowA₂, it must lie betweenB_k andB_k−1, herek 2,3. . . , K 1 andA2 B₁. Afterktimes’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set N2 which must be betweenA₁andA₂, and then ultimately stay inΓ1. Denote the intersection of the trajectory passing through the pointB which tangents toN1 at pointBwith the setN2at pointS1−αh2, yS Figure 7. The trajectory of system1.3with any initiating point on segmentA₁Sintersects with the setN₂ at some point belowA₂ with time increasing, so just like the analysis above we obtain it will ultimately stay inΓ1. Therefore, for any point belowSwill ultimately stay in regionΓ1with time increasing.

Now we can summarize the above results as the following theorem.

Theorem 3.5. Assuming that d < ecx0, h1 < δmin < 1−αh2 < d/ec < h2 ≤ x0 andyA2 <

1−βr/c q < y_A1, there is no periodic solution in system 1.3 and the trajectory with any initiating point belowSwill stay inΓ1or in the regionΩ1 {x, y|x≥0, y≥0, x≤h₁}.

Case 4 x0 < h2. In this case, denote the intersection of the line L1 with the set N at point B1−αh2, r/c, and the intersection between the line L₃ and the set M₂ at point Ah2, rh₂/cx₀ as shown inFigure 9. By Lemma 2.3 and means of qualitative analysis, there exists a unique closed trajectoryΓ2of system1.3which is tangent to the setM2at the point Aand has minimal valueλminat the lineL1. In the light of the different position of the setN2, we consider the following two cases.

Subcase 4.10< h1 <1−αh2< λmin. In this case, there exists a unique trajectory of system 1.3which is tangent to the setN₂ at the pointB. SetFB P₁ ∈ M₂, then pulse occurs

at pointP1, the impulsive function transfers the pointP1 intoP₁. Like the analysis of1of Subcase 2.1, we can prove that there exists an order one periodic solution in system1.3in this case.

Subcase 4.2h1< λmin<1−αh < x0 < h2. In this case, let the closed trajectoryΓ2of system 1.3intersectsN₂at two pointA₁1−αh, yA1andA₂1−αh, yA2. Like the analysis of2 of Subcase 3.1, we can prove that there exists an order one periodic solution in system1.3in this case; like the analysis of3of Subcase 3.1we can prove that there is no periodic solution in system1.3.

4. Attractiveness of the Order One Periodic Solution

In this section, under the condition of existence of order one periodic solution to system1.3 and the initial value of pest populationx0≤h₂, we discuss its attractiveness. We focus on Cases1and2; by similar method we can obtain similar results about Cases3and4.

Theorem 4.1. Assuming thatd < cex0, h1 < h2 < r/c andδ ≥ r/c. IfyP₀ > yP₂ > yP or y_P

0 < y_P

2 < y_P , then

Ithere exists a unique order one periodic solution of system1.3;

IIIf 1−αh2 < h₁, order one periodic solution of system1.3is attractive in the region Ω0{x, y|x≥0, y≥0, x≤h2}.

Proof. By the derivation of Theorem 3.1, we know that there exists an order one periodic solution of system1.3. We assume that trajectoryP P and segmentP P formulate an order one periodic solution of system1.3, that is, there exists aP ∈ N₂such that the successor function ofP satisfiesfP 0. First, we will prove the uniqueness of the order one periodic solution.

We take any two pointsC1h1, yC1 ∈N1, C2h1, yC2∈N1satisfyingyC2 > yC1 > yA, then we obtain two trajectories whose initiate points areC₁andC₂intersect with the setM₁ at two pointsD1h1, yD1andD2h1, yD2, respectivelyFigure 10. In view of the vector field of system1.3and the disjointness of any two trajectories without impulse, we know that yD1 > yD2. Suppose the pointsD1andD2are subject to impulsive effect to pointsD₁h1, yD₁ andD₂h2, y_D

2, respectively, then we havey_D

1 > y_D

2andfC1 y_D

1−yC1, fC2 y_D

2−yC2, so we getfC1−fC2 < 0, thus we obtain that the successor functionfxis decreasing monotonously inN₁; therefore there is a unique pointP ∈N₁satisfyingfP 0, and the trajectoryPP P is a unique order one periodic solution of system1.3.

Next we prove the attractiveness of the order one periodic solutionPP P in the re- gionΩ0 {x, y | x ≥ 0, y ≥ 0, x ≤ h₂}. We focus on the casey_P

0 > y_P

2 > y_P ; by similar method we can obtain similar results about casey_P0 < y_P2 < y_P Figure 11.

Take any point P₀h1, yP₀ ∈ N1 above P . Denote the first intersection point of the trajectory from initiating point P₀h1, y_P0 with the set M₁ at P₁h1, y_P1, and the corresponding consecutive points areP2h1, yP2, P3h1, yP3, P4h1, yP4, . . ., respectively.

Consequently, under the effect of impulsive functionI, the corresponding points after pulse areP₁h1, y_P

1, P₂h1, y_P

2, P₃h1, y_P

3, . . .

Y

N1

M1

L1

R L2

X 0

A B2

D⁺₁

D⁺₂

D1

D2

C1

Figure 10

Y

N1

M1

L1

R P₁⁺ P⁺

P⁺ P0₂⁺

X 0

A

L4

P_2k−2⁺ P_2k⁺ P_2k+1⁺ P_2k−1⁺

P2

P2k

P2k+2

PP2k+1

P2k−1

P3

P1

Figure 11

Due to conditionsy_P0 > y_P2 > y_P , y_P

k y_Pk δ, δ≥a/band disjointness of any two trajectories, we get a sequence{P_k}_k1,2,...of the setN1satisfying

y_P

1 < y_P3 <· · ·< y_P2k−1 < y_{P2k 1}<· · ·< y_P <· · ·< y_P

2k< y_P2k−2 <· · ·< y_P2 < y_P0. 4.1

S N2

B N1

A

M1

H C E

R L1

G M2

P0

D1

D2

Dk

Dk+1

P Ck+2

Ck+1

C3

C2

X Y

P₀⁺ D⁺₁ D⁺_k D⁺_k+1Q P⁺ C⁺_k+1 C_k⁺ C⁺₂ C₁⁺ 0

Figure 12

So the successor functionfP_2k−1 y_P

2k−y_P2k−1>0 andfP_2k y_{P2k 1}−y_P

2k <0 hold. Series {yP_2k−1}_k1,2,...increases monotonously and has upper bound, so limk→ ∞yP_2k−1exists. Next we will prove lim_{k→ ∞}y_P2k−1 y_P . Set lim_{k→ ∞}P_2k−1 C , we will proveP C . Otherwise P /C , then there is a trajectory passing through the pointC which intersects the setM₁ at pointC, then we have y_C > y_P, y_C > y_P . SincefC ≥ 0 andP /C , according to the uniqueness of the periodic solution, then we havefC y_C −y_C >0, thusy_C < y_P < y_C hold. Analogously, let trajectory passing through the pointC which intersects the setM1at pointC, and the corresponding consecutive points is C, then y_C > y

C > y_p> y

C, y_C > y

C >

y

C > y_p > y_C , then we havefC y

C −y

C >0, this contradicts to the fact thatC is a limit of sequence{P_2k−1}_k1,2,..., so we obtainP C . Therefore, we have lim_{k→ ∞}yP_2k−1 yP . Similarly, we can prove limk→ ∞yP_2k yP .

From above analysis, we know that there exists a unique order one periodic solution in system1.3, and the trajectory from initiating any point of theN₁will ultimately tend to be order one periodic solutionPP P .

Any trajectory from initial pointx₀, y₀ ∈ Ω0 {x, y | x ≥ 0, y ≥ 0, x ≤ h₂}will intersect withN1 at some point with time increasing on the condition that1−αh2 < h1 <

h2< d/bλ−dh; therefore, the trajectory from initial point onN1ultimately tends to be order one periodic solutionPP P . Therefore, order one periodic solutionPP P is attractive in the regionΩ0. This completes the proof.

Remark 4.2. Assuming thatd < ecx0, h1 < h2 < d/ecand δ ≥ r/c, if yP < yP₀ < yP₂ or y_P > y_P0 > y_P2, then the order one periodic solution is unattractive.

Theorem 4.3. Assuming thatd < ecx0, h1 <1−αh2 < h2 < d/ecandyP₀ < yA (as shown in Figure 12), then

Ithere exists an odd number of order one periodic solutions of system1.3with initial value betweenC₁ andAin setN₂;

IIif1−αh2< h₁and the periodic solution is unique, then the periodic solution is attractive in regionΩ2, hereΩ2 is open region which is constituted by trajectoryGB, segmentBH, segmentHE, and segmentEG.

Proof. Idue to the2of Subcase 2.1,fA<0 andfC₁>0 and the continuous successor functionfx, there exists an odd number of root satisfyingfx 0, then we can get that there exists an odd number of order one periodic solutions of system1.3with initial value betweenC₁ andAin setN₂;

IIby the derivation ofTheorem 3.3, 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 trajectoryPPand segmentP P formulate the unique order one periodic solution of system 1.3with initial pointP ∈N2.

On the one hand, take a pointC₁1−αh2, y_C

1 ∈ N₂ satisfyingy_C

1 ε < q and yC₁ < yP . The trajectory passing through the pointC₁1−αh2, εwhich intersects with set M2at pointC2h2, yC2, that is,FC₁ C2 ∈ M2, then we haveyC2 < yP, thusyC₂ < yP . Sincey_C

2 1−βyC2 q > ε, so we obtainfC₁ y_C

2−yC₁ y_C

2−ε >0; setFC₂ C₃∈M₂, becauseyC₁ < yC₂ < yP , we know thatyC2 < yC3 < yP, then we haveyC₂ < yC₃ < yP and fC₂ yC₃ −yC₂ > 0. This process is continuing, then we get a sequence{C_k}_k1,2,...of the setN₂satisfying

y_C1 < y_C2 <· · ·< y_C

k <· · ·< y_P 4.2 andfC_k yC_{k 1}−yC_k >0. Series{yC_k}_k1,2,...increases monotonously and has upper bound, so limk→ ∞yC_k exists. Like the proof ofTheorem 4.1, we can prove limk→ ∞yC_k yP .

On the other hand, setFP₀ D₁ ∈ M₂, then D₁ jumps toD₁ ∈ N₂ under the impulsive effects. SinceyP < yP₀ < yA, we have yP < yD1 < yP0, thus we obtain yP <

yD₁ < yP₀, fP₀ yD₁ −yP₀ <0. SetFD₁ D2 ∈ M2, thenD2 jumps toD₂ ∈N2 under the impulsive effects. We havey_P < y_D

2 < y_D

1. This process is continuing, we can obtain a sequence{D_k}_k1,2...of the setN₂satisfying

y_P

0 > y_D

1 > y_D

2 >· · ·> y_D

k >· · ·> y_P 4.3 andfD_k y_{Dk 1}−yD_k <0. Series{yD_k}_k1,2,...decreases monotonously and has lower bound, so lim_{k→ ∞}y_D

k exists. Similarly, we can prove lim_{k→ ∞}y_D

k y_P .

Any point Q ∈ N2 below A must be in some intervalyD_{k 1}, yD_kk1,2,..., yD₁, yP₀, yP₀, yA, yC_k, yC_{k 1k1,2,...}. Without loss of generality, we assume that the point Q ∈ yD_{k 1}, yD_k. The trajectory with initiating point Q moves between trajectory D_kD_{k 1} and D_{k 1}D_{k 2}and intersects withM2at some point betweenD_{k 2}andD_{k 1}, under the impulsive effects it jumps to the point of N₂ which is between yD_{k 2}, y_{Dk 1}, then trajectory ΠQ, t continues to move between trajectoryD_{k 1}D_{k 2}andD_{k 2}D_{k 3}. This process can be continued

unlimitedly. Since limk→ ∞yD_k yP , the intersection sequence of trajectoryΠQ, t with the setN₂ will ultimately tend to be the pointP . Similarly, ifQ ∈yC_k, y_{Ck 1}, we also can get the intersection sequence of trajectoryΠQ, t, and the set N2will ultimately tend to be point P . Thus the trajectory from initiating any point belowAultimately tends to be the unique order one periodic solutionPP P .

Denote the intersection of the trajectory passing through the pointBwhich tangents to N₁ at the point B, and the set N₂ by a pointS1−αh2, y_S. The trajectory from any initiating point on segmentASwill intersect with the setN2at some point belowAwith time increasing, so like the analysis above we obtain that the trajectory from any initiating point on segmentASwill ultimately tend to be the unique order one periodic solutionPP P .

Since the trajectory with any initiating point of theΩ2will definitely intersect with set N₂. From the above analysis, we know that the trajectory with any initiating point on segment ASwill ultimately tend to be order one periodic solutionPP P . Therefore, the unique order one periodic solutionPP P is attractive in the regionΩ2. This completes the proof.

Remark 4.4. Assuming thatd < ecx₀, h₁<1−αh2 < h₂ < d/ecandy_C

1 < y_A< y_P0, the order one periodic solution with initial point betweenAandP₀ 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 system1.3exists 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:

xt 0.4xt−0.6xtyt,

yt −0.2yt 0.3xtyt, x≤0.8, xt 0.4xt−0.48yt,

yt −0.6yt 0.24yt, x >0.8,

x /h₁, h₂ orxh₁, y > y^∗,

Δxt 0,

Δyt 0.8, xh₁, y≤y^∗, Δxt −0.5xt,

Δyt −0.2yt 0.5, xh₂,

5.1

where 0< h₁ < h₂ < x^∗. Now, we consider the impulsive effects influences on the dynamics of system5.1.

Example 5.1. Existence and attractiveness of order one periodic solution.