Abstract and Applied Analysis

Volume 2012 (2012), Article ID 126018, 18 pages

doi:10.1155/2012/126018

Research Article

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

Huidong Cheng, Tongqian Zhang, and Fang Wang

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

Received 16 April 2012; Accepted 13 August 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 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.

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.

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:

where represents the densities for prey's density while is the predator's; is the unit rate of prey density in lack of predators. is the Holling functional response. Holling type-I functional response is

where is a constant; when the amount of prey is greater than certain threshold value , predatory rate is a constant.

Holling type-II and type-III functional responses are as follows, respectively:

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:

where are all positive constants, and represent the densities of prey (pest) and predator (natural enemy) at time , respectively; is the intrinsic growth rate of the prey; denotes the death rate of the predator; is the economic threshold ET; represent the proportion of killed prey and predator by spraying pesticides, respectively, is the number of natural enemies released at this time. When the amount of prey reaches the threshold at time , controlling measures are taken and the amount of prey and predator abruptly turns to and , respectively. , .

This paper is organized as follows. In the next section, we present some basic definitions and important lemmas as preliminaries. In Section 3, 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

We consider the following function:

and we can easily know that is positive definite in the first quartile and fits for all conditions of Liapunov function.

We can get that

It is easily proved that on condition that , so all solutions of model (1.4) which form a set are closed trajectory , where .

Since on condition that , so the trajectory of system (2.1) passes through closed curve when it is out of the curve .

Therefore, we observe the straight line:

The derivative of along (2.1) is that

We have on condition that . Therefore, we can get the following lemma.

Lemma 2.1. The system (2.1) possesses the following.

(I)Two steady states -saddle point, and -stable centre under the condition and .(II)The trajectory of system (2.1) goes across the straight line from the right to the left under condition and , and intersects with the straight line .

Definition 2.2. A triple is said to be a semi-dynamical system if is a metric space is the set of all non-negative real and is a continuous map such that;

(i) for all ;(ii) is continuous for and ;(iii) for all and . Sometimes a semi-dynamical system is denoted by .

Definition 2.3. Assume that

(i) is a semi-dynamical system;(ii) is a nonempty subset of ;(iii)function is continuous, and, for any , there exists a such that for any , .

Then, is called an impulsive semi-dynamical system.

For any the function defined as is continuous and we call the trajectory passing through point . The set is called positive semitrajectory of point . The set is called the negative semi-trajectory of point .

Definition 2.4. We consider state-dependent impulsive differential equations:

where and represent the straight line or curve line on the plane and is called impulsive set. The function is continuous mapping, and is called the impulse function. 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 .

Definition 2.5. Suppose that the impulse set and the phase set are both lines, as shown in Figure 1. Define the coordinate in the phase set as follows: denote the point of intersection between and -axis as , then the coordinate of any point in is defined as the distance between and and is denoted by . Let denote the point of intersection between the trajectory starting from and the impulse set , and let denote the phase point of after impulse with coordinate . Then we define as the successor point of , and then the successor function of point is that .

Figure 1

Definition 2.6. A trajectory is called order one periodic solution with period if there exists a point and such that and .

We get these lemmas from the continuity of composite function and the property of continuous function.

Lemma 2.7. Successor function defined in Definition 2.2 is continuous.

Lemma 2.8. In system (1.4), if there exist satisfying successor function , then there must exist a point satisfying the function between the point of and the point of , 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

Phase set of set is that

Isoclinic line is denoted, respectively, by lines

For the convenience, if , is defined as the first point of intersection of and , that is, there exists a such that , and for ; if is defined as the first point of intersection of and , that is there exists a such that , and for .

For any point , we denote as its ordinate. If the point , pulse shall occur at the point , and the impulsive function transfers the point into . Without loss of generality, we assume that the initial point of the trajectory lies in phase set unless otherwise specified.

Due to the practical significance, in this paper we assume the set always lies in the left side of stable centre , that is, .

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

Case 1 (). In this case, sets and are both in the left side of stable center . The line intersects with at point , as shown in Figure 2. The trajectory passing through point which tangents to the set at point intersects with the set at point . Since the point , impulse occurs at point , supposing point is subject to impulsive effects to point , where , and the position of the point has the following three cases.

Figure 2

Subcase    . In this case, the point lies below the point , thus the successor function of the point is .

Take another point , where is small enough. Then there must exist a trajectory passing through the point which intersects at point . Supposing that the point is subject to impulsive effects to point , , so we have .

By Lemma 2.8, there exists an order one periodic solution of system (1.4), whose initial point is between the point and the point in set .

Subcase 1.2   (as shown in Figure 3). In this case, the point lies above the point ; there must exist a trajectory passing through the point which intersects the set at a point under . Denote the coordinates of impulsive point corresponding to the point , then . So we obtain .

Figure 3

Take that another point above , where is small enough. Then there must exist a trajectory passing through the point which intersects at a point . In view of continuous dependence of the solution on initial value and time, we know and the point is close to enough, then we obtain .

From Lemma 2.8, there exists an order one periodic solution of system (1.4), whose initial point is between and in set .

Subcase . coincides with , and the successor function of is , so there exists an order one periodic solution of system (1.4) which is just a part of the trajectory passing through the point .

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

Theorem 3.1. Assuming that , there exists an order one periodic solutions of the system (1.4).

Case 2   (). In this case, the set is in the right side of and set is in the left side of . Let the line intersects the sets and at points and , respectively (Figure 4), then there exists a unique closed trajectory of system (1.4) which is contained inside the point and is tangent to the set at the point .

Figure 4

Since is a closed trajectory, we take their minimal value of abscissas at the trajectory , namely, holds for any abscissas of .

In light of the different positions of the set , we consider the following three subject cases.

Subcase 2.1 . In this case, there is a trajectory, which is contained inside the point and is tangent to the set at point which intersects at the point . Supposing that point is subject to impulsive effects to point , here . 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. If , there exists an order one periodic solution in system (1.4).

Subcase 2.2 . In this case, let the closed trajectory of system (1.4) intersects the set at two points and , as shown in Figure 5. Since , impulse occurs at point . Supposing that point is subject to impulsive effects to point , here .

Figure 5

If or , coincides with or coincides with ; the successor function of or is that or that , so there exists an order one periodic solution of system (1.4) which is just a part of the trajectory .

If , the point lies below the point , 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 , the point is above the point . 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 that , if or , then there exists an order one periodic solution to the system (1.4).

Subcase 2.3  . In this case, we note that the point must lie between the point and the point (as shown in Figure 6). Taking a point such that jumps to after the impulsive effect, we denote . Since , we have . Let , taking a point such that jumps to after the impulsive effects, then we have . This process continues until there exists a satisfying . So we obtain a sequence of set and a sequence of set satisfying . In the following, we will prove the trajectory of system (1.4) with any initiating point of set which will ultimately stay in .

Figure 6

From the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point between the points and will be free from impulsive effect and ultimately will stay in .

For any point below , it must lie between and , where and . After times' impulsive effects, the trajectory with this initiating point will arrive at some point of the set which must be between and , and then ultimately stay in .

The trajectory with any initiating point above will intersect set at some point below with time increasing, so just like the analysis above we obtain that it will ultimately stay in .

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

Theorem 3.4. Assuming that and , if +, there is no periodic solution in system (1.4) and the trajectory with any initiating point of set will stay in .

Case 3  ( ).   In this case, denote the intersection of the line and the set by point , and the intersection of the line and the set by point (as shown in Figure 7). Through Lemma 2.8 and qualitative analysis, there exists a unique closed trajectory of system (1.4) which is tangent to the set at the point and has minimal value at the line . In the light of the different position of the set , we consider the following two cases.

Figure 7

Subcase 3.1  . In this case, there exists a unique trajectory of system (1.4) which is tangent to the set at the point . Set , then pulse occurs at point , and the impulsive function transfers the point into . Like the analysis of Subcase 2.1, we can prove there exists an order one periodic solution in system (1.4) in this case.

Subcase 3.2  . In this case, let the closed trajectory of system (1.4) intersect with at two points and . 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 will stay in .

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 , 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 , then

(I)there exists an odd number of order one periodic solutions of system (1.4) with initial value between and in the set (II)if the periodic solution is unique, then the periodic solution is attractive in region .

Proof. (I) According to the Subcase 1.3, and (see Figure 2); the successor function is continuous, so there exists an odd number of root satisfying , then we can get that there exists an odd number of order one periodic solutions of system (1.4) with initial value between and in the set .

(II) By the derivation of Theorem 3.1, we know there exists an order one periodic solution of system (1.4). We assume trajectory and segment formulate a unique order one periodic solution of system (1.4) with initial point (Figure 8).

On the one hand, take a point satisfying and . The trajectory passes through the point which intersects set at point , that is, , then we have , thus . Since , so we obtain ; set , because ; we know , then we have and . This process is continuous, then we get a sequence of set satisfying

and . Series increase monotonously and have upper bound, so that exists.

Next we will prove . Setting , we will prove . Otherwise , then there is a trajectory passing through the point which intersects the set at point , then we have . Since and , according to the uniqueness of the periodic solution, then we have , thus hold. Set . In view of vector field and disjointness of any two trajectories, we know and , then we have ; this contradicts the fact that is a limit of sequence , so we obtain and .

On the other hand, set , then jumps to under the impulsive effects. Since , we have , thus we obtain that and that . Set , then jumps to under the impulsive effects. We have , ; this process is continuous, and we can obtain a sequence of set satisfying

and . Series decrease monotonously and have lower bound, so that exists. Similarly, we can prove that .

Since the trajectory initiating any point of will certainly intersect with set , next we only need to prove the trajectory initiating any point of will ultimately tend to be the unique order one periodic solution.

Any point below must be in some interval , . Without loss of generality, we assume the point . The trajectory with initiating point moves between trajectories and and intersects with at some point between and , and under the impulsive effects it jumps to the point of which is between , then trajectory continues to move between trajectories and . This process can be continued unlimitedly. Since , the intersection sequence of trajectory and set will ultimately tend to point . Similarly, if , we can also get that the intersection sequence of trajectory and set will ultimately tend to point . Thus the trajectory initiating any point below ultimately tends to be the unique order one periodic solution .

The trajectory with any initiating point above of will intersect with set at some point below with time increasing, so like the analysis above we obtain the trajectory initiating any point above will ultimately tend to be the unique order one periodic solution .

From the above analysis, we know the trajectory initiating any point of will ultimately tend to be order one periodic solution . Therefore, the unique order one periodic solution is attractive in the region . This completes the proof.

Figure 8

Remark 4.2. If system (1.4) satisfies the condition of Theorem 3.1 and that , then order one periodic solution with initial point between and is unattractive.

Theorem 4.3. If system (1.4) satisfies the condition of Theorem 3.1 and , there exists a unique order one periodic solution of system (1.4) which is attractive in the region .

Proof. Through the derivation of Theorem 3.1, we know there exists an order one periodic solution of system (1.4), whose initial point is between and in set . We assume trajectory and segment formulate an order one periodic solution of system (1.4), and is its initial point satisfying (Figure 9). First we will prove the uniqueness of the order one periodic solution.

We take any two points satisfying , then we obtain two trajectories, whose initiate points are and intersecting with the set and two points and , respectively (Figure 10). In view of the vector field of system (1.4) and the disjointness of any two trajectories, we know . Suppose the points and are subject to impulsive effect to points and respectively, then we have and , so we get ; thus we obtain that the successor function decreases monotonously on interval of , so there is a unique point satisfying . The trajectory is a unique order one periodic solution of system (1.4).

Next we prove the attractiveness of the order one periodic solution in the region .

Denote the first intersection point of the trajectory with initiating point and the impulsive set by and that the corresponding consecutive points are , , , and so forth, respectively. Consequently, under the effect of impulsive function , the corresponding points after pulse are , , , and so forth. In view of and disjointness of any two trajectories, we have that

so and hold. As is proved in Theorem 4.1, we can prove .

Since the trajectory initiating any point of will certainly intersect with set , next we only need to prove the trajectory initiating any point of will ultimately tend to be order one periodic solution .

The trajectory with initiating point between and will intersect impulsive set with time increasing; under the impulsive effects it arrives at a point of which is between or . Then like the analysis of Theorem 4.1, we know the trajectory initiating any point between and will ultimately tend to be order one periodic solution .

Assume a point of the set jumps to under the impulsive effect. Set . Assume a point of the set jumps to under the impulsive effect. Set . This process is continuing until there exists a satisfying . So we obtain a sequence of the set and a sequence of the set satisfying . For any point of the set below , it must lie between and where . After times' impulsive effects, the trajectory from this initiating point will arrive at some point of the set which must be between and and then will ultimately tend to order one periodic solution . There is no order one periodic solution with the initial point below .

The trajectory with any initiating point above of will intersect with the set at some point below with time increasing, so the trajectory with any initiating point above will ultimately tend to be order one periodic solution . There is no order one periodic solution with the initial point above .

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 will ultimately tend to be order one periodic solution . Therefore, order one periodic solution is attractive in the region . This completes the proof.

Figure 9

Figure 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:

where . 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 set , initiating points are (red), (green), and (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.

(a)

(b)

(c)

Figure 11: The time series and phase diagram for system (1.4) starting from initial value (blue), (green), (red),,.

Example 5.2. Existence and attractiveness of positive periodic solutions.

We set , initiating points are (red), (green), and (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.

(a)

(b)

(c)

Figure 12: The time series and phase diagram for system (1.4) starting from initial value (blue), (green), (red), ,.

Example 5.3. Existence and attractiveness of positive periodic solutions.

We set ; initiating points is . Figure 13 shows that results of Case 3 are valid.

(a)

(b)

(c)

Figure 13: The time series and phase diagram for system (1.4) starting from initial value (0.4,0.5), .

Acknowledgment

This project was supported by the National Natural Science Foundation of China (No. 10872118).

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