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Abstract and Applied Analysis
Volume 2012, Article ID 101386, 17 pages
http://dx.doi.org/10.1155/2012/101386
Research Article

The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

Department of Mathematics Education, Catholic University of Daegu, Gyeongsan 712-702, Republic of Korea

Received 26 August 2011; Accepted 28 October 2011

Academic Editor: Agacik Zafer

Copyright © 2012 Hunki Baek. 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 Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations.

1. Introduction

In the last decades, some impulsive systems have been studied in population dynamics such as impulsive birth [1, 2], impulsive vaccination [3, 4], and chemotherapeutic treatment of disease [5, 6]. In particular, the impulsively controlled prey-predator population systems have been investigated by a number of researchers [715]. Thus the field of research of impulsive differential equations seems to be a new growing interesting area in recent years. Many authors in the articles cited above have shown theoretically and numerically that prey-predator systems with impulsive control are more efficient and economical than classical ones to control the prey (pest) population. However, the majority of these studies only consider impulsive control at fixed time intervals to eradiate the prey (pest) population. Such control measure of prey (pest) management is called fixed-time control strategy, modeled by impulsive differential equations. Although this control measure is better than classical one, it has shortcomings, regardless of the growth rules of the prey (pest) and the cost of management. In recent years, in order to overcome such drawbacks, several researchers have started paying attention to another control measure based on the state feedback control strategy, which is taken only when the amount of the monitored prey (pest) population reaches a threshold value [2, 1619]. Obviously, the latter control measure is more reasonable and suitable for prey (pest) control.

In order to investigate the dynamic behaviors of a population model with the state feedback control strategy, an autonomous Lotka-Volterra system, which is one of the most basic and important models, is considered. Actually, the principles of Lotka-Volterra models have remained valid until today and many theoretical ecologists adhere to their principles (cf. [8, 2022]).

Thus, in this paper, we consider the following Lotka-Volterra type prey-predator system with impulsive state feedback control: where all parameters except and are positive constants. Here, and are functions of the time representing population densities of the prey and the predator, respectively, is the inherent net birth rate per unit of population per unit time of the prey, is the self-inhibition coefficient, is the per capita rate of predation of the predator, denotes the death rate of the predator, is the rate of conversion of a consumed prey to a predator, presents the fraction of the prey which die due to the harvesting or pesticide, and so forth, and and represent the amount of immigration or stock of the predator. We denote by the economic threshold and and . When the amount of the prey reaches the threshold at time , controlling measures are taken and hence the amounts of the prey and predator immediately become and , respectively.

The main purpose of this research is to investigate theoretically and numerically the dynamical behaviors of system (1.1).

This paper is organized as follows. In the next section, we present a useful lemma and notations and construct a Poincaré map to discuss the dynamics of the system. In Section 3, the sufficient conditions for the existence of a semi-periodic solution of system (1.1) with are established via the Poincaré criterion. On the other hand, in Section 4, we find out some conditions for the existence and stability of stable positive period-one solutions of system (1.1). Further, under some conditions, we show that there exists a stable positive periodic solution of period 1 or 2; however, there is no positive periodic solutions with period greater than and equal to three. In order to testify our theoretical results by numerical simulations, in Section 5, we give some numerical examples and the bifurcation diagrams of solutions that show the existence of a chaotic solution of system (1.1). Finally, we have a discussion in Section 6.

2. Preliminaries

Many considerable investigators have studied the dynamic behaviors of system (1.1) without the state feedback control. (cf. [23, 24].) It has a saddle , one locally stable focus and a saddle if the condition holds. Since the carrying capacity of the prey population is , so it is meaningful that the economical threshold is less than . Thus, throughout this paper, we set up the following two assumptions: From the biological point of view, it is reasonable that system (1.1) is considered to control the prey population in the biological meaning space .

The smoothness properties of , which denotes the right hand of (1.1), guarantee the global existence and uniqueness of a solution of system (1.1) (see [25, 26] for the details).

Let and . Firstly, we denote the distance between the point and the set by and define, for any solution of system (1.1), the positive orbit of through the point as Now, we introduce some definitions (cf. [27]).

Definition 2.1 (orbital stability). is said to be orbitally stable if, given , there exists such that, for any other solution of system (1.1) satisfying , then for .

Definition 2.2 (asymptotic orbital stability). is said to be asymptotically orbitally stable if it is orbitally stable and for any other solution of system (1.1), there exists a constant such that, if , then .

In order to discuss the orbital asymptotical stability of a positive periodic solution of system (1.1), a useful lemma, which follows from Corollary  2 of Theorem  1 given in Simeonov and Bainov [28], is considered as follows.

Lemma 2.3 (analogue of the Poincaré criterion). The -periodic solution of system is orbitally asymptotically stable if the multiplier satisfies the condition , where where denotes and denotes and , , , , , , , and are calculated at the point , , and . Also is a sufficiently smooth function on a neighborhood of the points such that and is the moment of the jump, where .

From now on, we construct two Poincaré maps to discuss the dynamics of system (1.1). For this, we introduce two cross-sections and . In order to establish the Poincaré map of via an approximate formula, suppose that system (1.1) has a positive period-1 solution with period and the initial condition , where . Then the periodic trajectory intersects the Poincaré section at the point and then jumps to the point due to the impulsive effects with and . Thus

Now, we consider another solution with the initial condition . Suppose that this trajectory which starts form first intersects at the point when and then jumps to the point on . Then we have Set and , then and . Let and . It is well known that, for , the variables and are described by the relation where the fundamental solution matrix satisfies the matrix equation with (the identity matrix). Set and . We can express the perturbed trajectory in a first-order Taylor expansion It follows from that Since and , we obtain . So, we can construct a Poincaré map of as follows: where and are calculated according to (2.7).

Now we construct another type of Poincaré maps. Suppose that the point is on the section . Then is on due to the impulsive effects, and the trajectory with the initial point intersects at the point , where is determined by and the parameters and . Thus we can define a Poincaré map as follows: The function is continuous on , and because of the dependence of the solutions on the initial conditions.

Definition 2.4. A trajectory of system (1.1) is said to be order -periodic if there exists a positive integer such that is the smallest integer for .

Definition 2.5. A solution of system (1.1) is called a semitrivial solution if its one component is zero and another is nonzero.

Note that, for each fixed point of the map in (2.12), there is an associated periodic solution of system (1.1), and vice versa.

3. The Existence and Stability of a Periodic Solution When

In this section, we consider system (1.1) with as follows:

First, let to calculate a semitrivial periodic solution of system (3.1). Then system (3.1) can be changed into the following impulsive differential equation: Under the initial value , the solution of the equation can be obtained as , where . Assume that and in order to get a periodic solution of (3.2). Then we have the period of a semitrivial periodic solution of (3.1). Thus system (1.1) with has a semitrivial periodic solution with the period as follows: where .

Using the Poincaré map defined in (2.12), we will have a criterion for the stability of this semitrivial periodic solution .

Theorem 3.1. The semitrivial periodic solution of system (1.1) with is locally stable if the condition holds, where .

Proof. We already discussed the existence of the semitrivial periodic solution . It follows from (2.8) that Let . Then we can infer from (3.5) that, for , Since and , we obtain that . Thus it is only necessary to calculate . From the fourth equation of (3.6), we obtain . Since and , so we obtain . Therefore, Note that is a fixed point of and Under condition (3.4), we get . So system (1.1) with has a stable semitrivial periodic solution.

Remark 3.2. From the proof of Theorem 3.1, we note that if . It means that the semitrivial periodic solution system (1.1) with is unstable if .

Now, we discuss the existence of a positive periodic solution of the system (3.1) with .

Theorem 3.3. System (1.1) with has a positive period-one solution if the condition holds, where .

Proof. It follows from Theorem 3.1 that the semitrivial periodic solution passing through the points and is stable if , where . Now, define , where is the Poincaré map. From now on, we will show that there exist two positive numbers and such that and by following two steps.
Step 1. We will show that for some . First, consider the trajectory starting with the point for a sufficiently small number . This trajectory meets the Poincaré section at the point and then jumps to the point and reaches the point . Since , the semitrivial solution is unstable by Remark 5.4. So we can choose an such that for . Thus the point is above the point . So we have . From (2.12), we know that Thus we know that .Step 2. We will show that for some . To do this, suppose that the line meets at . The trajectory of system (1.1) with the initial point meets the line at then jumps to the point and then reaches the point on the Poincaré section again. However, for any , the point is not above the point in view of the vector field of system (1.1). Thus . So we have only to consider the following two cases.Case (i): If , that is, , then system (1.1) has a positive period-one solution.Case (ii): If , then Thus, it follows from (3.10) and (3.11) that the Poincaré map has a fixed point, which corresponds to a positive period-one solution for system (1.1) with . Thus we complete the proof.

Remark 3.4. Under the condition , we show that the semitrivial periodic solution of system (1.1) is stable when and there exists a positive period-one solution of system (1.1). Since , a fold bifurcation takes place at . Furthermore, from the proof of Theorem 3.3, we know that system (1.1) with has a positive period-one solution passing through the points and and satisfying the condition for some .

Now we discuss the stability of the positive periodic solution of system (1.1).

Theorem 3.5. Assume that . Let be the positive period-one solution of system (1.1) with period passing through the points and . Then the positive periodic solution is orbitally asymptotically stable if the condition holds, where and .

Proof. In order to discuss the stability of the positive periodic solution of system (1.1), we will use the Lemma 2.3. First, we note that Since we obtain that Thus we have . By Remark 3.4, for , we have , and so we get when which means that this periodic solution is stable. In addition, for , we know due to and . Since the derivative with respect to is negative, so we know that when . Further, we can find such that . Therefore, if the condition (3.12) holds, then we obtain , which implies from Lemma 2.3 that the positive periodic solution is orbitally asymptotically stable.

Remark 3.6. System (1.1) has a stable periodic semitrivial solution and a stable positive period-1 solution if and , respectively. We already know from Remark 3.4 that a fold bifurcation occurs at . Thus, from the facts, we can suppose that a flip (period-doubling) bifurcation occurs at . Moreover, we can figure out that system (1.1) might have a chaotic solution via a cascade of period doubling.

4. The Existence and Stability of a Positive Periodic Solution When

In this section we will take into account the existence and stability of positive periodic solutions in the two cases of and . In fact, under the condition , the trajectories starting from any initial point with intersects the section infinite times. However, under the condition , the trajectories starting from any initial point with do not intersect the section .

4.1. The Case of

Theorem 4.1. Assume that , , and . Then the system (1.1) has a positive period-one solution. Moreover, if this periodic solution has a period and passes through the points and , then it is asymptotically orbitally stable provided with where and and .

Proof. We will use the similar method to Theorem 3.3 to prove the existence of a periodic solution of system (1.1).
Firstly, in order to show for some , let be in the Poincaré section , where is small enough such that . The trajectory of system (1.1) with the initial point intersects the point on the Poincaré section , then jumps to the point , and then reaches the point on again. From the choice of the value , we know that and hence the points and are above the points and , respectively. Thus we have . It follows from (2.12) that
Secondly, to find a positive number such that suppose that the line meets at . The trajectory of system (1.1) with the initial point meets the line at then jumps to the point and then reaches the point on the line again. Suppose that there exists a such that . Then the point is just the point if . The point lies above the point if , while it lies under if . However, for any , the point is not above the point in view of the vector field of the system (1.1). Thus and hence .
Therefore, we have a periodic solution by the similar method to Theorem 3.3. Further, the stability condition for this period-one solution can be obtained by using the same method used in the proof of Theorem 3.5. Thus we complete the proof.

4.2. The Case of

Theorem 4.2. Assume that , , and . Then there exists such that system (1.1) has a stable positive solution of period 1 or 2 if , where depends on the value . Moreover, system (1.1) has no periodic solutions of period ( ).

Proof. First, assume that the orbit, which just touches at the point with , meets at the two points and , where . We will prove this theorem by the following five steps.
Step 1. We will show that if , then any trajectory of system (1.1) intersects with infinite times. Note that every trajectory passing through the point with cannot intersect with as time goes to infinite and tends to the focus eventually. Therefore, if all trajectories of system (1.1) pass through the points with after finite times impulsive effects on , they all tend to the focus and there are no positive periodic solutions. From this fact, we know that the condition , in which depends on the value as a function , is a sufficient condition for a trajectory of system (1.1) which intersects with infinite times in view of the impulsive effects and .From now on, let the condition hold.Step 2. Next, we will show that for , where is the next point of that touches . Note that for any point with , the point is above the point . Thus, for any two points and , where , the points and lie above the point and, further, it follows from the vector field of the system (1.1) that , that is, Thus, from the Poincaré map and , we obtain , , and for given . Therefore, we have only to consider three cases as follows:Case (i): ,Case (ii): ,Case (iii): , .Step 3. In order to show the existence of a positive solution of period 1 or 2, consider the Cases (i) and (ii). First, if Case (i) is satisfied, then it is easy to see that system (1.1) has a positive period-one solution. Now, suppose that Case (ii) is satisfied. Then without loss of generality, we can say that . It follows form (4.3) that . Furthermore, if , then there exists a positive period-two solution of system (1.1).Step 4. Now, we will prove that system (1.1) cannot have periodic solutions of period () if Case (iii) holds. For this, assume that , which means that system (1.1) has a positive period- solution. However, we will show that this is impossible. If , then from (4.3), we obtain that and then or . If , then from (4.3), we have and then and . So the relation of , , and is one of the following: (a) If , then from (4.3), we have . It is also true that . We again obtain and then . By means of induction, we have Similar to (a), for Cases (b), (c), and (d), we obtain respectively. If there exists a positive period- solution in the system (1.1), then , , which is a contradiction to (4.5)–(4.6). Thus there is no positive period- solution () if .Step 5. From Step 4, we can show that there exists a stable period-1 or-2 solution in these cases. In fact, it follows from (4.5) that where . Therefore, and . Thus system (1.1) has a positive period-2 solution in the case (a). Moreover, it is easily proven from (4.5) and (4.7) that this positive period-2 solution is local stable. Similarly, we have system (1.1) has a stable positive period-1 solution in cases (b) and (c) and has a stable positive period-2 solution in case (d).

5. Numerical Examples

In this section, we will present some numerical examples to discuss the various dynamical aspects of system (1.1) and to testify the validity of our theoretical results obtained in the previous sections.

Example 5.1. In order to exhibit the dynamical complexity as varies, let and fix the other parameters as follows: In this example, we set an initial value as . It is from Theorem 3.1 that the periodic semitrivial solution is stable if (see Figures 1 and 2). We display the bifurcation diagram in Figure 2(a). From the Remark 3.4, we know that a fold bifurcation takes place at . Figure 2(a) shows that a positive period-one solution bifurcates from the periodic semitrivial solution at and a positive period-two solution bifurcates from the positive period-one solution via a flip bifurcation at , which leads to the period-doubling bifurcation and then chaos (see Figures 2(b) and 3). It follows from Theorem 4.2 that system with cannot have positive period-3 solution under some conditions. However, if , a period-3 solution can exist (see Figure 4).

101386.fig.001
Figure 1: (a) The trajectory of system (1.1) with when . (b-c) Time series.
fig2
Figure 2: (a) The bifurcation diagram of system (1.1) . (b) A chaotic solution of system (1.1) with when .
fig3
Figure 3: (a) A period-4 solution when and . (b) A period-8 solution when and .
fig4
Figure 4: (a) A period-3 solution when and . (b) The enlarged part of (a) for .

Example 5.2. Under the condition , we know that there is no semitrivial solution in system (1.1). In this case, set the parameters as follows: Throughout this example, we regard the point as an initial value. Figure 5(a) shows the bifurcation diagrams of system (1.1) with as a bifurcation parameter when . It follows from Theorem 4.1 that there exists a period-one solution for any and this solution is stable when as shown in Figure 5(a). It is easy to see that there are no fold bifurcations. However, at , a flip bifurcation occurs and the cascade of the flip bifurcation leads to chaotic solutions like the previous example. Thanks to Figure 5(a), we know that system (1.1) undergoes the complex dynamical behaviors including periodic doubling, chaotic behaviors, and periodic windows.

fig5
Figure 5: (a) The bifurcation diagram of system (1.1) with . (b) The bifurcation diagram of system (1.1) with .

Example 5.3. It follows from Theorem 4.2 that if the value satisfies the condition , there exists some such that, for all , system (1.1) has a stable positive period-one or-two solution if , but does not have period- () solutions. To substantiate these theoretical results by numerical simulation, let and , and let the other parameters be the same as in Example 5.2. Then we obtain . Figure 5(b) of the bifurcation diagram of system (1.1) numerically displays that there exist no period- solutions () except stable positive period-1 or-2 solutions. Thus the value is also an important parameter in the dynamical aspects of system (1.1). For this reason, we investigate the effects of the parameter on system (1.1). For this, let and , and let be a bifurcation parameter. It is easy to see from Figure 6 that the parameter causes various dynamical behaviors of system (1.1) such as a cascade of reverse period-doubling bifurcations, also called period halving, period windows, chaotic regions, stable period-2 solutions, and so forth.
From a biological point of view, as mentioned in Section 1, the value represents the amount of immigration or releasing of the predator. Particularly, from Figure 6, one can figure out that the number of the predator cannot be easily estimated when the amount of is small due to chaotic behaviors of solutions to the system; on the contrary, if sufficient amount of the predator is released impulsively, then the number of the predator (eventually, the number of the prey) can be predictable due to periodic behaviors of solutions to the system.

101386.fig.006
Figure 6: The bifurcation diagram of system (1.1) with and with respect to .

Remark 5.4. Now, we will demonstrate the superiority of the state-dependent feedback control in comparison with the fixed-time control via an example. For this, assume that , , , , , , , , and in system (1.1) with an initial value . Figure 7(b) shows that the prey population cannot be controlled below the threshold value if we take the impulsive control measure at fixed time . However, it is seen from Figure 7(a) that only after several attempts of control does the solution approach the periodic solution. Thus this example shows that the impulsive state feedback measure is more effective in real biological control.

fig7
Figure 7: The trajectories of system (1.1) with (a) under the state feedback control and (b) under the fixed time control when .

6. Conclusion

In this paper, a state-dependent impulsive dynamical system concerning control strategy has been proposed and analyzed. Particularly, a state feedback measure for controlling the prey population is taken when the amount of the prey reaches a threshold value. The dynamical behaviors have been investigated, including the existence of periodic solutions with period 1 and 2 and their stabilities. In addition, we have numerically shown that system (1.1) has various dynamical aspects including a chaotic behavior. Based on the main theorems of this paper, the amount of the prey population can be completely controlled below the threshold value by one, two, or at most finite number of applying impulsive effects. From a biological point of view, it will be very helpful and useful to control the prey population.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011–0006087).

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