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

and Applied Analysis 3 capacity of the prey population x t is b/a, so it is meaningful that the economical threshold h is less than b/a. Thus, throughout this paper, we set up the following two assumptions: A1 D e < a b , A2 h ≤ b a . 2.1 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 { x, y : x ≥ 0, y ≥ 0}. The smoothness properties of f , 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 R −∞,∞ and R2 { x, y | x ≥ 0, y ≥ 0}. Firstly, we denote the distance between the point p and the set S by d p, S infp0∈S|p − p0| and define, for any solution z t x t , y t of system 1.1 , the positive orbit of z t through the point z0 ∈ R2 as O z0, t0 { z ∈ R2 | z z t , t ≥ t0, z t0 z0 } . 2.2 Now, we introduce some definitions cf. 27 . Definition 2.1 orbital stability . z∗ t is said to be orbitally stable if, given > 0, there exists δ δ > 0 such that, for any other solution z t of system 1.1 satisfying |z∗ t0 −z t0 | < δ, then d z t , O z0, t0 < for t > t0. Definition 2.2 asymptotic orbital stability . z∗ t is said to be asymptotically orbitally stable if it is orbitally stable and for any other solution z t of system 1.1 , there exists a constant η > 0 such that, if |z∗ t0 − z t0 | < η, then limt→∞d z t , O z0, t0 0. 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 T-periodic solution x φ t , y ζ t of system x′ t P ( x, y ) , y′ t Q ( x, y ) , if φ ( x, y ) / 0, Δx α ( x, y ) , Δy β ( x, y ) , if φ ( x, y ) 0, 2.3 is orbitally asymptotically stable if the multiplier μ2 satisfies the condition |μ2| < 1, where


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 7-15 . 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 preypredator 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, [16][17][18][19] . 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, 20-22 . Thus, in this paper, we consider the following Lotka-Volterra type prey-predator system with impulsive state feedback control: x t x t a − bx t − cy t , y t y t −D ex t , x / h, Δx t −px t , Δy t qy t r, x h, where all parameters except q and r are positive constants. Here, x t and y t are functions of the time representing population densities of the prey and the predator, respectively, a is the inherent net birth rate per unit of population per unit time of the prey, b is the self-inhibition coefficient, c is the per capita rate of predation of the predator, D denotes the death rate of the predator, e is the rate of conversion of a consumed prey to a predator, 0 < p < 1 presents the fraction of the prey which die due to the harvesting or pesticide, and so forth, and q > −1 and r ≥ 0 represent the amount of immigration or stock of the predator. We denote by h the economic threshold and Δx t x t −x t and Δy t y t −y t . When the amount of the prey reaches the threshold h at time t h , controlling measures are taken and hence the amounts of the prey and predator immediately become 1 − p h and 1 q y t h r, 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 r 0 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.

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 0, 0 , one locally stable focus D/e, ae−bD /ce and a saddle a/b, 0 if the condition D/e < a/b holds. Since the carrying capacity of the prey population x t is b/a, so it is meaningful that the economical threshold h is less than b/a. 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 { x, y : The smoothness properties of f, 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 R −∞, ∞ and R 2 { x, y | x ≥ 0, y ≥ 0}. Firstly, we denote the distance between the point p and the set S by d p, S inf p 0 ∈S |p − p 0 | and define, for any solution z t x t , y t of system 1.1 , the positive orbit of z t through the point z 0 ∈ R 2 as Now, we introduce some definitions cf. 27 .
Definition 2.1 orbital stability . z * t is said to be orbitally stable if, given > 0, there exists δ δ > 0 such that, for any other solution z t of system 1.1 satisfying Definition 2.2 asymptotic orbital stability . z * t is said to be asymptotically orbitally stable if it is orbitally stable and for any other solution z t of system 1.1 , there exists a constant η > 0 such that, if |z * t 0 − z t 0 | < η, then lim t → ∞ d z t , O z 0 , t 0 0.
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 T-periodic solution x ϕ t , y ζ t of system is orbitally asymptotically stable if the multiplier μ 2 satisfies the condition |μ 2 | < 1, where
From now on, we construct two Poincaré maps to discuss the dynamics of system 1.1 . For this, we introduce two cross-sections 1 { x, y : x 1 − p h, y ≥ 0} and 2 { x, y : x h, y ≥ 0}. In order to establish the Poincaré map of 2 via an approximate formula, suppose that system 1.1 has a positive period-1 solution z t ϕ t , ζ t with period T and the initial condition z 0 A 1 − p h, y 0 ∈ 1 , where y 0 ≡ y 0 > 0. Then the periodic trajectory intersects the Poincaré section 2 at the point A h, y 1 and then jumps to the point A due to the impulsive effects with Δx t −px t and Δy t qy t r. Thus Now, we consider another solution z t ϕ t , ζ t with the initial condition z 0 A 0 1 − p h, y 0 δ y 0 . Suppose that this trajectory which starts form A 0 first intersects 2 at the point A 1 h, y 1 when t T δ t and then jumps to the point A 1 1 − p h, y 2 on 1 . Then we have It is well known that, for 0 < t < T, the variables u t and v t are described by the relation where the fundamental solution matrix Φ t satisfies the matrix equation We can express the perturbed trajectory in a first-order Taylor expansion

2.9
It follows from ϕ T δ t ϕ T h that Abstract and Applied Analysis 5 Since y 2 1 q y 1 r and y 2 − y 0 1 q y 1 − y 1 , we obtain v 1 1 q v * 0 . So, we can construct a Poincaré map F of 1 as follows: where u T and v T are calculated according to 2.7 . Now we construct another type of Poincaré maps. Suppose that the point B k h, y k is on the section 2 . Then B k 1 − p h, 1 q y k r is on 1 due to the impulsive effects, and the trajectory with the initial point B k intersects 2 at the point B k 1 h, y k 1 , where y k 1 is determined by y k and the parameters q and r. Thus we can define a Poincaré map F as follows: The function F is continuous on q, r, and y k because of the dependence of the solutions on the initial conditions.

is called a semitrivial solution if its one component is zero and another is nonzero.
Note that, for each fixed point of the map F in 2.12 , there is an associated periodic solution of system 1.1 , and vice versa.

The Existence and Stability of a Periodic Solution When r 0
In this section, we consider system 1.1 with r 0 as follows:

3.1
First, let y t 0 to calculate a semitrivial periodic solution of system 3.1 . Then system 3.1 can be changed into the following impulsive differential equation:

3.2
Under the initial value x 0 h and x T x 0 in order to get a periodic solution of 3.2 . Then we have the period T 1/a ln a − bh 1 − p / a − bh 1 − p of a semitrivial periodic solution of 3.1 . Thus system 1.1 with r 0 has a semitrivial periodic solution with the period T as follows: Using the Poincaré map F defined in 2.12 , we will have a criterion for the stability of this semitrivial periodic solution ϕ t , ζ t .
Proof. We already discussed the existence of the semitrivial periodic solution ϕ t , 0 . It follows from 2.8 that Then we can infer from 3.5 that, for 0 < t < T 1/a ln 1 −

3.6
Since u 0 u 0 0 and Thus it is only necessary to calculate w 4 t . From the fourth equation of 3.6 , we obtain w 4 t w exp −D eϕ t dt . Since ϕ t dt 1/b ln β b exp at and w 4 0 1, so we obtain Abstract and Applied Analysis 7 Note that v 0 is a fixed point of F q v 0 and

3.8
Under condition 3.4 , we get 0 < D v 0 F q 0 < 1. So system 1.1 with r 0 has a stable semitrivial periodic solution.
Now, we discuss the existence of a positive periodic solution of the system 3.1 with r 0.
Proof. It follows from Theorem 3.1 that the semitrivial periodic solution passing through the points where F is the Poincaré map. From now on, we will show that there exist two positive numbers 1 and ω 0 such that G 1 > 0 and G ω 0 ≤ 0 by following two steps.
Step 1. We will show that G 1 > 0 for some 1 > 0. First, consider the trajectory starting with the point A 1 1 − p h, for a sufficiently small number > 0. This trajectory meets the Poincaré section 2 at the point B 1 h, 1 and then jumps to the point A 2 1 − p h, 1 q 1 and reaches the point B 2 h, 2 . Since q > q 0 , the semitrivial solution is unstable by Remark 5.4. So we can choose an such that 1 q 1 > for q > q 0 . Thus the point B 2 is above the point B 1 . So we have 1 < 2 . From 2.12 , we know that Thus we know that G 1 > 0.
Step 2. We will show that G ω 0 ≤ 0 for some ω 0 > 0. To do this, suppose that the line bx cy − a 0 meets 1 at A 3 1 − p h, a − b 1 − p h /c . The trajectory of system 1.1 with the initial point A 3 meets the line 2 at B 3 h, ω 0 then jumps to the point A 3 1 − p h, 1 q ω 0 and then reaches the point B 4 h, ω 0 on the Poincaré section 2 again. However, for any q > 0, the point B 4 is not above the point B 3 in view of the vector field of system 1.1 . Thus ω 0 ≤ ω 0 . So we have only to consider the following two cases.
Case i : If ω 0 ω 0 , that is, G ω 0 0, then system 1.1 has a positive period-one solution.

Abstract and Applied Analysis
Case ii : If ω 0 < ω 0 , then Thus, it follows from 3.10 and 3.11 that the Poincaré map F has a fixed point, which corresponds to a positive period-one solution for system 1.1 with r 0. Thus we complete the proof. Proof. In order to discuss the stability of the positive periodic solution θ t , ψ t of system 1.1 , we will use the Lemma 2.3. First, we note that

3.13
Since we obtain that 3.15 By Remark 3.4, for q q 1 , we have 1 q 1 ψ 0 a − b 1 − p h /c, and so we get μ 2 0 when q q 1 which means that this periodic solution is stable. In addition, for q q 0 , we know μ 2 1 due to ψ 0 0 and τ 1/a ln a − bh 1 − p / a − bh 1 − p . Since the derivative dμ 2 /dq with respect to q is negative, so we know that 0 < μ 2 < 1 when q 0 < q < q 1 . Further, we can find q 2 > q 1 such that μ 2 g q 2 −1. Therefore, if the condition 3.12 holds, then we obtain −1 < μ 2 < 1, which implies from Lemma 2.3 that the positive periodic solution θ t , ψ t is orbitally asymptotically stable.
Remark 3.6. System 1.1 has a stable periodic semitrivial solution and a stable positive period-1 solution if 0 < q < q 0 and q 0 < q < q 2 , respectively. We already know from Remark 3.4 that a fold bifurcation occurs at q q 0 . Thus, from the facts, we can suppose that a flip perioddoubling bifurcation occurs at q q 2 . Moreover, we can figure out that system 1.1 might have a chaotic solution via a cascade of period doubling.

The Existence and Stability of a Positive Periodic Solution
When r > 0 In this section we will take into account the existence and stability of positive periodic solutions in the two cases of h < D/e and D/e < h. In fact, under the condition h < D/e, the trajectories starting from any initial point x 0 , y 0 with x 0 < h intersects the section 2 infinite times. However, under the condition D/e < h, the trajectories starting from any initial point x 0 , y 0 with x 0 < h do not intersect the section 2 .

The Case of h < D/e
where γ q * 1 and γ q * * −1 and γ q 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 F q, r, r 1 > r 1 for some r 1 > 0, let U 1 1 − p h, r 1 be in the Poincaré section 1 , where r 1 is small enough such that 0 < r 1 < r. The trajectory of system 1.1 with the initial point U 1 intersects the point V 1 h, r 1 on the Poincaré section 2 , then jumps to the point U 2 1 − p h, 1 q r 1 r , and then reaches the point V 2 h, r 2 on 2 again. From the choice of the value r 1 , we know that 1 q r 1 r > r 1 and hence the points U 2 and V 2 are above the points U 1 and V 1 , respectively. Thus we have r 1 < r 2 . It follows from 2.12 that Secondly, to find a positive number m 0 such that m 0 − F q, r, m 0 ≥ 0 suppose that the line bx cy − a 0 meets 1  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. Proof. First, assume that the orbit, which just touches 2 at the point B 0 h, y 1 with y 1 a − bh /c, meets 1 at the two points B 1 − p h, y 2 and B 1 1 − p h, y 3 , where y 3 < a − b 1 − p h /c < y 2 . We will prove this theorem by the following five steps.

The Case of D/e < h
Step 1. We will show that if r > y 2 , then any trajectory of system 1.1 intersects with 2 infinite times. Note that every trajectory passing through the point 1 − p h, y with y ∈ y 3 , y 2 cannot intersect with 1 as time goes to infinite and tends to the focus D/e, ae − bD /ce eventually. Therefore, if all trajectories of system 1.1 pass through the points 1 − p h, y with y ∈ y 3 , y 2 after finite times impulsive effects on 2 , they all tend to the focus and there are no positive periodic solutions. From this fact, we know that the condition r > y 2 , in which y 2 depends on the value h as a function g h , is a sufficient condition for a trajectory of system 1.1 which intersects with 2 infinite times in view of the impulsive effects Δx −px and Δy qy r.
From now on, let the condition r > y 2 hold.
Step 2. Next, we will show that y j 1 < y m 1 for y m < y j , where h, y k 1 is the next point of h, y k that touches 2 . Note that for any point h, y with 0 < y < a − b 1 − p h /c, the point 1 − p h, 1 q y r is above the point B. Thus, for any two points E m h, y m and E j h, y j , where 0 < y m < y j < a − bh /c, the points E m 1 − p h, 1 q y m r and E j 1 − p h, 1 q y j r lie above the point B and, further, it follows from the vector field of the system 1.1 that 0 < y j 1 < y m 1 < a − bh /c, that is, y j 1 < y m 1 for y m < y j .

4.3
Thus, from the Poincaré map and r > y 2 , we obtain y 1 F q, r, y 0 , y 2 F q, r, y 1 , and y n 1 F q, r, y n n 3, 4, . . . for given y 0 ∈ 0, a − bh /c . Therefore, we have only to consider three cases as follows: Case i : y 0 y 1 , Case ii : y 0 / y 1 , 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 y 1 < y 0 . It follows form 4.3 that y 2 > y 1 . Furthermore, if y 2 y 0 , 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 k k ≥ 3 if Case iii holds. For this, assume that y 0 y k , which means that system 1.1 has a positive period-k solution. However, we will show that this is impossible. If y 0 < y 1 , then from 4.3 , we obtain that y 1 < y 2 and then y 2 < y 0 < y 1 or y 0 < y 2 < y 1 . If y 0 > y 1 , then from 4.3 , we have y 1 < y 2 and then y 1 < y 2 < y 0 and y 1 < y 0 < y 2 . So the relation of y 0 , y 1 , and y 2 is one of the following: a y 2 < y 0 < y 1 , b y 0 < y 2 < y 1 , c y 1 < y 2 < y 0 , d y 1 < y 0 < y 2 . 4.4 a If y 2 < y 0 < y 1 , then from 4.3 , we have y 2 < y 1 < y 3 . It is also true that y 2 < y 0 < y 1 < y 3 .
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 0 < y * 0 < y * 1 < a − bh /c. Therefore, y * 1 F q, r, y * 0 and y * 2 F q, r, y * 1 . 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 .

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.  q > −1 and this solution is stable when q * < q < q * * ≈ 4.35 as shown in Figure 5 a . It is easy to see that there are no fold bifurcations. However, at q q * * ≈ 4.35, 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.  Figure 5 b of the bifurcation diagram of system 1.1 numerically displays that there exist no period-k solutions k ≥ 3 except stable positive period-1 or-2 solutions. Thus the value r is also an important parameter in the dynamical aspects of system 1.1 . For this reason, we investigate the effects of the parameter r on system 1.1 . For this, let q 5 and h 0.52, and let r be a bifurcation parameter. It is easy to see from Figure 6 that the parameter r 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 r 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 r 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. 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.

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.