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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 204642, 18 pages
http://dx.doi.org/10.1155/2013/204642
Research Article

Dynamical Analysis of a Pest Management Model with Saturated Growth Rate and State Dependent Impulsive Effects

1College of Science, Shandong University of Science and Technology, Qingdao 266590, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received 11 March 2013; Accepted 20 May 2013

Academic Editor: Tonghua Zhang

Copyright © 2013 Wencai Zhao 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

A new pest management mathematical model with saturated growth is proposed. The integrated pest management (IPM) strategy by introducing two state dependent pulses into the model is considered. Firstly, we analyze singular points of the model qualitatively and get the condition for focus point. Secondly, by using geometry theory of impulsive differential equation, the existence and stability of periodic solution of the system are discussed. Lastly, some examples and numerical simulations are given to illustrate our results.

1. Introduction

Forest diseases and insect pests are the most destructive natural disasters to forestry; they are always an important constraint to forestry production worldwide and cause major economic losses in forestry. For example, spruce budworm, mountain pine beetle, and gypsy moth have led to significant losses in Canadian forests [1]. Therefore, how to prevent the occurrence and spreading of forest diseases and insect pests and reduce the economic loss is always an important subject in theory and practice. Mathematical models have been established to study change rule of the number of forest pests; these models include ordinary differential equation (ODE) system, difference equation system, partial differential equation (PDE) system, and so forth. In order to control pests, integrated pest management (IPM) [24] including two measures will be taken: one is chemical control (spraying pesticide) and the other is biological control (releasing natural enemy). In the process of pest management, if the amount of pests is small, biological control will be implemented, and if the amount of pests is large, integrated pest control will be applied. Usually, system with sudden disturbance always leads to impulsive system. In the forest pest management system, human’s interference can cause a sudden change of the number or density of pests. So impulsive differential equation (IDE) can be established to explain this phenomenon. Many biological models based on time impulse [513] or one state impulse [14, 15] have been studied by many authors. The model with two state impulses is very rare. In [16, 17], two models with two state impulses are established to study the existence and the stability of periodic solution of the models, respectively, by Poincaré map and the Lambert function. In [18], a model with two state impulses is built up and the geometrical analysis of the model is made by Bendixson theorem of impulsive differential equations and the successor function. Motivated by [15, 17, 18], we get the following reasonable model by introducing two state dependent impulses: where , is the density of pests at time , and is the density of natural enemies. is saturated growth rate, is predation rate of natural enemies, and is the growth rate of natural enemies, after digestion and absorption. is the death rate of natural enemies. and are the threshold with slight damage and serious damage to the forest, respectively. is the ordinate of intersection of the line and . are the amount of natural enemies released one time. and are death rates of pests and natural enemies due to pesticide, respectively. According to the biological significance, we only discuss .

The paper is organized as follows. In Section 2, some definitions and theorems are given for the later use. In Section 3, the stability of system (1) without impulsive effect is investigated qualitatively. In Section 4, the existence and stability of periodic solution to system (1) are investigated using the successor function, Bendixson theorem, and stability theorem of periodic solution of impulsive differential equations. In Section 5, an example and some simulations are exerted to prove the results.

2. Preliminaries

We consider the state impulsive differential equation where is one-dimensional line or curve in . is called impulsive set, and the function is continuous mapping, . is called the impulse function, and is called the corresponding image set.

Definition 1. If there exists a point in the image set such that , and the impulsive mapping , then one calls the periodic solution of system (2).

Definition 2. Function is called a successor function of the point (see Figure 1).

204642.fig.001
Figure 1: The successor function of system (2).

Theorem 3 (Bendixson theorem of impulsive differential equations). Assume is a Bendixson region of system (2); if does not contain any critical points of system (2), then system (2) has a closed orbit in .

Theorem 4. Assume that, in continuous dynamic system , there exist two points , in the pulse phase concentration such that the successor function and ; then there exists a point between the two points , that makes ; that is, the system has periodic solution.

Theorem 5 (see [19, 20]). Consider the state impulsive differential equation where the function is corresponding to the impulsive set . The -periodic solution , of the model is orbitally asymptotically stable if the multiplier is calculated by the following formula: where and , , , , , , , are calculated at the point and , .

3. The Stability of System (1) without Impulsive Effect

In this section, we consider system (1) without impulsive effect. Let ; then we get the following system: The system has two equilibrium points and , where and . Assume that the variational matrix of the system (7) is as follows: At , Obviously, is saddle point. At , The characteristic equation of satisfies , where , and . Analyse ; we have the following conclusion: (i)if , that is, , then is stable node; (ii)if , that is, , then is stable critical node; (iii)if , that is, , then is stable focus. Denote : ; we have the following theorem.

Theorem 6. The positive equilibrium point of system (7) is stable focus provided is true.

Theorem 7. The solution of system (7) is bounded.

Proof. Let be solution of system (7) satisfied initial conditions , . From the second equation of system (7), it follows that for . And there exists , such that orbit intersects with isoclinic line at point . Thus, we have for . Let ; we have for .
On the other hand, making three lines in the first quadrant, : , : and : , we can acquire that Then we have for and large enough. Similarly, Then we have for . According to the previous discussion, there exists an area which is composed of , , , , . For all initial points , , for , we can acquire . Thus, the solution of system (7) satisfied initial conditions is bounded.

Theorem 8. The positive equilibrium point of system (7) is globally asymptotically stable provided condition is true.

Proof. Firstly, we claim that the positive equilibrium point of system (7) is stable focus point and around it there does not exist closed rail line provided condition is true. In fact, let Dulac function be ; in regard to the system (7), it follows that Due to the Bendixson-Dulac theorem, around , there does not exist closed rail line. Synthesize Theorems 6 and 7; therefore we have that the positive equilibrium point of system (7) is globally asymptotically stable provided condition is true.

Vector graph of system (7) with a stable focus can be seen in Figure 2.

204642.fig.002
Figure 2: Vector graph of system (7) with a stable focus, where , , , , and .

4. The Analysis of System (1) with State Dependent on Pulse

4.1. The Existence of Periodic Solutions of the System

In this section, we will discuss the existence of periodic solution of system (1) by using the successor function defined in this paper and qualitative analysis. Let , ; we know the curve : and -axis are -isoline and the line : and -axis are -isoline. From the previous discussion, we know is a stable focus provided is true. Denote the first impulsive set as , the second impulsive set as , the image set corresponding to set as , and the image set corresponding to set as . Let points , , and be the intersection of line with lines , , and , respectively. Due to the practical significance, in this paper we assume the set always lies in the left side of stable focus ; that is, . We will start our discussion about the existence of periodic solution for (1) with the initial point . The structure graph of system (1) can be seen in Figure 3.

204642.fig.003
Figure 3: The structure graph of system (1).
4.1.1. The Orbit Starting from the Point on above

Starting from , the orbit intersects with at point and then produces pulse to the point on . According to (1), the following is got: . After time impulse, the point should satisfy . Thus, we only need to consider case , and three cases should be discussed.

Case 1. If the impulsive point corresponding to is exactly , thus, the curve shall constitute a periodic orbit of (1) (see Figure 4).

204642.fig.004
Figure 4: The orbit starting from the point above (Case 1 in Section 4.1.1).

Case 2. If the impulsive point corresponding to is below on , obviously, the successor function of satisfies . On the other hand, choose a point next to on (i.e., ). The orbit of (1) starting from point intersects with at point and then jumps to point in the line . Because is next to , is next to , and , thus, the successor function of satisfies . By Theorem 4, there exists a period solution (see Figure 5).

204642.fig.005
Figure 5: The orbit starting from the point (Case 2 in Section 4.1.1).

Case 3. If the impulsive point corresponding to is above on , obviously . On the other hand, the orbit starting from point intersects with at point and then jumps to point in the line . According to system (1), the following is got: ; then the successor function of satisfies . Based on the previous discussion, according to Theorem 4, there exists a periodic orbit of (1) (see Figure 6).

204642.fig.006
Figure 6: The orbit starting from the point (Case 3 in Section 4.1.1).
4.1.2. The Orbit Starting from the Point

Starting from , the orbit intersects with at point and then produces pulse to the point on . According to (1), the following is got: By selecting , three cases should be discussed.

Case 1. If the impulsive point corresponding to is exactly , thus, the curve shall constitute a periodic orbit of (1) (see Figure 7).

204642.fig.007
Figure 7: The orbit starting from the point (Case 1 in Section 4.1.2).

Case 2. If the impulsive point corresponding to is above on , thus, . Choose a point above in the straight line . Only when it intersects with at , can the orbit of (1) starting from point be vertical. After intersecting with at point through from left to right, it jumps to point in the line . According to the existence and uniqueness of impulsive differential equations, must be definitely below on , and must be below on ; therefore the successor function of is . Based on the previous discussion, we can see that the region surrounded by the closed curve is a positive invariant set of (1) and it contains no equilibrium point. According to Theorem 3, there exists a periodic orbit of (1) in (see Figure 8).

204642.fig.008
Figure 8: The orbit starting from the point (Case 2 in Section 4.1.2).

Case 3. If the impulsive point corresponding to is below on , we shall have the successor function .
In the meantime, take a point next to on (.) Starting from , the orbit intersects with at point and then produces pulse to the point on . According to (1), the following is got: Thus, must be above , and the successor function of is .
As a result, the region surrounded by the closed curve involves a periodic solution of (1) (see Figure 9).

204642.fig.009
Figure 9: The orbit starting from the point (Case 3 in Section 4.1.2).
4.1.3. The Orbit Starting from the Point between the Second Impulsive Set and Its Image Set

The orbit starting from point , between line and line , intersects with line at and jumps onto point on . There are the following cases with changing .

Case 1. If on is exactly , the orbit from point moves to the point on . For , there are the following cases.(a) If is exactly , then the curve forms a periodic orbit of (1) (see Figure 10). (b) If on is below , the successor function of is .In the meantime, take a point between and next to -axis. After starting from , the orbit hits the point on , then jumps onto the point on , and then moves to the point on . Obviously, is above ; thus, the successor function of is .As a result, we get a periodic orbit in the region encircled by the closed curve (see Figure 11).(c) If on is above , which does not exist periodic orbit, otherwise the curve and must be crossed, which leads to the conflict with the existence and uniqueness of solution to impulsive differential equations.

204642.fig.0010
Figure 10: The orbit starting from the point (Case 1(a) in Section 4.1.3).
204642.fig.0011
Figure 11: The orbit starting from the point (Case 1(b) in Section 4.1.3).

Case 2. If on is below , the same conclusion can be got in the similar discussion previously mentioned (see Figures 12 and 13).

204642.fig.0012
Figure 12: The orbit starting from the point (Case 2 in Section 4.1.3).
204642.fig.0013
Figure 13: The orbit starting from the point (Case 2 in Section 4.1.3).

Case 3. If on is above , for the orbit starting from , there exist the following cases.(a) If the orbit starting from crosses from the right to the left and becomes vertical only when it goes through the line , the point is the intersection of the orbit and . The same conclusion can be got as the discussion in Section 4.1.1 (see Figure 14). (b) The orbit starting from becomes vertical only when it crosses , and then it moves on automatically to on , which is exactly , or below , or above .If on is exactly , then the curve forms a periodic orbit (see Figure 15). If on is below , then the successor function of is .On the other hand, take a point between and near to -axis, and the orbit starting from hits the point on , then jumps onto the point on , and then returns to the point on . Obviously, on is above . Thus, the successor function of is .Based on the previous discussion above, there exists a periodic orbit in the region encircled and (see Figure 16).(c) If on is above , which does not exist periodic orbit, the reason is the same as in Case 1(c) in Section 4.1.3.

204642.fig.0014
Figure 14: The orbit starting from the point (Case 3(a) in Section 4.1.3).
204642.fig.0015
Figure 15: The orbit starting from the point (Case 3(b) in Section 4.1.3).
204642.fig.0016
Figure 16: The orbit starting from the point (Case 3(b) in Section 4.1.3).
4.2. The Stability Analysis of Periodic Solutions of the System (1)
4.2.1. The Stability of Periodic Solution about Impulsive Set

Theorem 9. Let , be the periodic solution of system (1), and , ; if , the periodic solution is stable.

Proof. Let and , , ; one gets According to Theorem 5, we have Obviously, if , that is, , we have . Thus the periodic solution to the system (1) is stable. This completes the proof.

4.2.2. The Stability of Periodic Solution about Impulsive Set

Theorem 10. Let , be the periodic solution of system (1), and ; if or , the periodic solution to the system (1) is stable.

Proof. Let , be the periodic solution of system (1), and ; one gets Thus, If or we have ; then the periodic solution to the system (1) is stable. This completes the proof.

5. Example and Numerical Simulation

To verify the validity of our results, we give an example. Let , , , , , , , , and . By calculation, we obtain , , , and . Then we have Numerical analysis of system (24) is being done using Maple 14.0. We have the following cases.

Case 1. Initial point is in the line , where .(a) Let be above (see Figure 17). (b)Let be below (see Figure 18).

fig17
Figure 17: Illustration of basic behavior of system (24).
fig18
Figure 18: Illustration of basic behavior of system (24).

Case 2. Initial point is in the line . Let be .(a)The impulsive point corresponding to is exactly , where (see Figure 19). (b)The impulsive point corresponding to is above on , where (see Figure 20). (c)The impulsive point corresponding to is below on , where (see Figure 21).

fig19
Figure 19: Illustration of basic behavior of system (24).
fig20
Figure 20: Illustration of basic behavior of system (24).
fig21
Figure 21: Illustration of basic behavior of system (24).

Case 3. Initial point is between the second impulsive set and its image set . Let be .(a)The impulsive point corresponding to is exactly , where (see Figure 22).(b)The impulsive point corresponding to is below on , where (see Figure 23).(c) The impulsive point corresponding to is above on , where (see Figure 24). The impulsive point corresponding to is above on , where and (see Figure 25).

fig22
Figure 22: Illustration of basic behavior of system (24).
fig23
Figure 23: Illustration of basic behavior of system (24).
fig24
Figure 24: Illustration of basic behavior of system (24).
fig25
Figure 25: Illustration of basic behavior of system (24).

All the simulations show agreement with the results in this paper.

6. Conclusion

Certainly, the actual pest management will be very complicated; such biological models may be described by more state dependent pulses and time impulse, but they are difficult to study by the use of successor functions. In this paper, a pest management model with saturated growth rate is constructed. We proposed two state dependent impulsive effects in the model according to quantity of pests and natural enemies. Our primary results are to explain the effects of human interference in the actual pest management on dynamics of the system. If the quantity of pests is less than , but the quantity of natural enemies is enough relatively (>), human interference is unnecessary. If the quantity of pests reaches a certain amount , but the quantity of natural enemies is less than or equal to , only biological control (releasing natural enemy) is implemented, and if the quantity of pests is large (reach ), integrated pest control (spraying pesticide and releasing natural enemy) is applied. We think our results will offer help to the actual pest management.

Acknowledgments

This work is supported by the Shandong Provincial Natural Sciences Funds of China (Grant no. ZR2012AM012) and a Project of Shandong Province Higher Educational Science and Technology Program of China (Grant no. J13LI05).

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