Dynamic Analysis of Beddington–DeAngelis Predator-Prey System with Nonlinear Impulse Feedback Control

Li, Dezhao; Cheng, Huidong; Liu, Yu

doi:https://doi.org/10.1155/2019/5308014

Complexity

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Abstract Introduction Preliminaries Numerical Simulation Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Geometric and Polynomial Approaches of Complex Systems and Control in Mathematics and Applied Sciences

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Research Article | Open Access

Volume 2019 | Article ID 5308014 | https://doi.org/10.1155/2019/5308014

Dynamic Analysis of Beddington–DeAngelis Predator-Prey System with Nonlinear Impulse Feedback Control

Dezhao Li,¹Huidong Cheng,¹and Yu Liu²

Guest Editor: Alejandro Hossian

Received04 Sept 2019

Revised15 Oct 2019

Accepted28 Oct 2019

Published22 Nov 2019

Abstract

In this paper, a predator-prey system with pesticide dose-responded nonlinear pulse of Beddington–DeAngelis functional response is established. First, we construct the Poincaré map of the impulsive semidynamic system and discuss its main properties including the monotonicity, differentiability, fixed point, and asymptote. Second, we address the existence and globally asymptotic stability of the order-1 periodic solution and the sufficient conditions for the existence of the order-k(k ≥ 2) periodic solution. Thirdly, we give the threshold conditions for the existence and stability of boundary periodic solutions and present the parameter analysis. The results show that the pesticide dosage increases with the extension of the control period and decreases with the increase of the threshold. Besides, the state pulse feedback control can manage the pest population at a certain level and avoid excessive application of pesticides.

1. Introduction

The predator-prey system plays an important role in the relationship of biological populations, so many predator-prey systems with different functional responses have been studied, such as the Monod type [1–5], the Holling type [6–13], and the Ivlev type [14–18]. Currently, the use of chemicals is more and more widespread in agriculture. Also, when the number of pests reaches a critical value, we can release natural enemies. Therefore, the feedback control of pulse state is proposed [17, 19–22]. The Beddington–DeAngelis functional response was introduced by Beddington [23] and DeAngelis et al. [24]. The Beddington–DeAngelis functional response avoided some of the singular behaviors of the ratio-dependent model at low density [25]. Cantrell and Cosner discussed the following predator-prey system with Beddington–DeAngelis functional response [23, 24, 26]:where are positive constants. and represent the population density of prey and predator at time t, K is the environmental carrying capacity of the prey, and r is the intrinsic growth rate of prey. Function indicates the Beddington–DeAngelis functional response, and stands for mutual interference between the predators. The constants ε and μ represent the rate of conversion and death rate of predators, respectively.

For simplicity, we determine dimensionless system (1) and scale it as follows:

Then, we getwhere

In recent years, many pulse equations have been studied that simulate the ecological processes of populations, and most of these studies are pulse differential equations at the fixed time [1, 27–32]. However, feedback control of time pulse has certain defects, which may reduce crop yield and possibly increase management costs. Therefore, we can choose to spray the pesticide when the quantity of pests reaches a certain threshold instead of spraying the pesticide at a fixed time. This measure avoids the possibility of the explosive growth of the number of pests and is more suitable for pest control. This paper studies the Beddington–DeAngelis system with pulse state feedback control strategies:

Capturing or using chemicals on predators and prey may impulsively reduce the density of the predators and prey. and represent the survival rate of prey and predator populations when a given dose D of insecticides is applied and and . We assume that insecticides have different insecticidal rates for these two populations, where and , is the constant number of natural enemies released [33, 34].

Some published articles focus on the property of the successor function and Poincaré map to discuss the existence of order-1 periodic solution and the local stability. Besides, if the proposed model has the first integrals, the existence of order-2 periodic solutions can be discussed [35–40]. However, due to the complexity of such model as the Beddington–DeAngelis system in this paper, the problem of global dynamics such as the global stability of the model and the existence of order- periodic solution has not been well solved. Also, there a few researches on the property of Poincaré map while it is applied. So the main arrangement of this paper is as follows. In Section 2, some preliminaries about the pulse semidynamic system and system (3) are given. In Section 3, we construct the Poincaré map, deduce the expression of Poincaré map function of system (5), and then give some of its properties such as the monotonicity, differentiability, fixed point, and asymptote. In Section 4, we prove the existence and stability of the boundary periodic solution and order- periodic solution of system (5). In Section 5, we conducted a numerical simulation.

2. Preliminaries

2.1. Preliminaries of Pulse Semidynamic Systems

A pulsed semipowered system with state-dependent feedback control can be expressed as [41, 42]

Here, , P, Q, α, and β are continuous functions from to R. Let be the impulse set of system (6), and for any , the impulse occurs; the map I is defined aswhere is the impulse point for f. We define as the phase set of the system and , is the metric space, and is the set of all nonnegative reals; we call as a semidynamic system. For any , , , where [43]. The setis called the positive orbit of f. Furthermore, for any set , let

Next, we give the definition of impulsive semidynamic system and order-k periodic solution.

Definition 1 (see [44, 45]). The pulsed semidynamic system consists of the nonempty closed subset M of X, the continuous semidynamic system , and the continuous function I.
We denote the points of discontinuity of by and call an impulsive point of . We define a function from X into the extended positive reals as follows: let ; if , we set ; otherwise , and we set , where for but .

Definition 2 (see [46]). For trajectory in , if there are nonnegative integers and , k is the minimum integer satisfying and ; then, the period of is T, and there is a period of order-k.

Definition 3 (see [42, 47]). The T-periodic solution of thwe systemis orbitally asymptotically stable and enjoys the property of asymptotic phase if satisfies , wherewithand ϕ is continuously differentiable with respect to u and . can also be denoted by . P, Q, , , , , , and are calculated at the point , , and .

2.2. Preliminaries of System (3)

We know that the solution of system (3) is positive and bounded for all t. If , then the equilibrium point is globally asymptotically stable [25].

If , system (3) has three equilibrium points, , , and , respectively, where and are all positive and satisfy the following conditions:

For completeness, we summarize global results for system (3) in Lemma 1 [25, 48, 49].

Lemma 1. (i)If , is globally asymptotically stable(ii)If and , the is globally asymptotically stable(iii)If and , there is an exact limit cycle

3. Poincaré Map

3.1. Domains of the Poincaré Map

In the following parts, we only discuss the case of as the globally asymptotically stable point of system (5).

System (5) has two isoclinal lines, which are defined as and :

Next, the lines associated with the phase set and impulse set are defined as and :

In this case, the value range of is and lines and always intersect with line . The intersection point of and presents as , and the intersection point of and presents as , where and are, respectively,

The open set defined in is as follows:

The impulse set M is the part of line above the U-axis and below point G:

The continuous function I is expressed asso the phase set N iswhere . We assume that the initial point is always on the in the following sections.

3.2. Construction of Poincaré Map

The Poincaré map of system (5) can be defined in different ways. In this work, we choose the to define the Poincaré map.

Because is chosen as the globally asymptotically stable point in system (5), and the value range of is defined in the impulse set as , any trajectory starting from the point on the phase set must intersect with the impulse set at the point . From Cauchy–Lipschitz theorem, we know the value of is only determined by ; in order to discuss fluently, we make . The point on the impulse set is mapped to the point on after experiencing an impulse. is obtained by the impulse function of system (5). is the initial point of the next impulse function on the phase set.

So we can express the Poincaré map of system (5) as

We can determine the Poincaré map from the points on the phase set. Next, we infer the expression of Poincaré map function φ and discuss its properties according to the expression of φ.

According to the following formula of system (5),

We can rewrite system (5) as a scalar differential equation on the phase set:

For model (23), we only focus on the region

The function is continuous and differentiable in the region . Besides, let and , where , , and , so

From (23), we get

According to (21) and (26), we can obtain the definition of the Poincaré map φ:

We simulate a numerical simulation of the Poincaré map function of model (5) (see Figure 1). The two cases are (Figure 1(a)) and (Figure 1(b)).

(a)

(b)

3.3. The Main Properties of Poincaré Map

Through our analysis of the expression and numerical model of the Poincaré map φ, and assuming , the following properties of Poincaré map are given.

Theorem 1. The domain of φ is , and the range of φ is , where . φ monotonically increases on and monotonically decreases on , and as the value of increases continuously, φ approaches the asymptote .

Proof. We first prove the domain of φ is . Because in system (3) is globally asymptotically stable, and because , any initial point from the will reach the impulse set M; so the definition domain of φ is .
Next, we divide into two parts, which are and . First of all, we can choose two points and and assume , then the trajectory of these two points will intersect with at the two points and . According to Cauchy–Lipschitz theorem, is true for any , and the formula of φ isSo it is easy to find for any and on and , is always true; therefore, φ monotonically increases on .
And then, we prove that the φ is monotonically decreasing on . Similarly, we take two numbers and on and assume , then the trajectories of the two initial points and from will cross the isoclinal line and then intersect at and , respectively, where . Then, these two curves will intersect at two points and , respectively. According to Cauchy–Lipschitz theorem, is always true, and the expression of φ is obtained in (27), so for any on , is always true, and φ is monotonically decreasing on .
Then, we prove that as the value of increases, φ tends to be stable and approaches the asymptote . We define the closure of asSince φ increases monotonically on and decreases monotonically on , is the invariant set of system (5). LetifHere is the scalar product of two vectors, and then, the vector field will eventually reach the boundary , so is an invariant set, and by calculation, one obtainsSince φ is monotonically increases on and monotonically decreases on , is bounded for any and . From the Cauchy–Lipschitz Theorem, is only determined by and can be expressed by . And any point on the phase set N is always going to be ; hence, . Then, we haveSo as the value of increases, φ tends to be stable and approaches the asymptote .
Since φ increases monotonically on and decreases monotonically on , ; thus, φ takes the maximum value at and takes the minimum value at 0, where and , so the range of value is .

Theorem 2. φ is continuously differentiable.

Proof. Here, we can use the initial conditions of continuity and differentiability theorem which is to take parameters of Cauchy theorem and Lipschitz theorem to determine the continuity and differentiability of φ; by system (5), we can get that both and functions are continuously differentiable in the first quadrant; so by Cauchy theorem and Lipschitz theorem with parameters, we can get that the φ is a continuously differentiable function.

Theorem 3. φ always has at least one fixed point if .

Proof. From Theorem 1, we know that φ is monotonically increasing on and monotonically decreasing on . Then, we divide it into two cases to discuss the existence of the fixed point for φ. Case I: when , on the one hand, , so φ has at least one number on such that ; on the other hand, since φ is monotonically decreasing on and , φ has no fixed point on . In conclusion, when , the φ has at least one fixed point. Case II: when , because φ monotonically decreases on and as the value of increases continuously, φ approaches asymptote , so φ has only one point on such that . And the number of fixed points on is unknown. So when , the φ has at least one fixed point.In conclusion, φ always has at least one fixed point.

4. Study on the Periodic Solutions of System (5)

4.1. Boundary Periodic Solutions of System (5)

For system (5), if the predator population becomes extinct and the predator also terminates its release, then system (5) has a boundary periodic solution, which produces the following system:

Solving (34) with initial value ,

The trajectory from the initial point will eventually intersect with the straight line of the impulse set over time:

Solving the equation of T and D, we getwhere T is the period of the boundary periodic solution and D is the insecticide dose required to control the number of pests below the . Then, the boundary periodic solution of system (5) with a period of T is

Theorem 4. The boundary periodic solution of system (5) is asymptotically stable if

Proof. By Definition 3, we obtainFrom the above formula, we can getsoIn addition,The expression of isIf condition (39) is true, then . It means the periodic solution of the boundary is asymptotically stable.

4.2. Existence and Stability of Periodic Solutions of Order-k at

From Theorem 3, the Poincaré map function φ of system (5) has at least one fixed point; that is to say, system (5) must have at least one order-1 periodic solution.

Theorem 5. The order-1 periodic solution is orbitally asymptotically stable if and only ifwhere

Proof. We use and to represent the start point and the endpoint of the order-1 periodic solution, respectively. From Theorem 4, we know that the Floquet multiplierIf (45) is true, then , so the order-1 periodic solution is always orbitally asymptotically stable.

Theorem 6. If , there is at least one locally asymptotically stable order-1 periodic solution in system (5). Furthermore, if there is only one fixed point on , then there is a globally asymptotically stable order-1 periodic solution of system (5).

Proof. If , Poincaré map φ has at least one fixed point. This also proves that there is at least one order-1 periodic solution in system (5). From Theorem 5, we know that the periodic solution is always asymptotic stable if . So if , then there is at least one locally asymptotically stable order-1 periodic solution in system (5).
If , there is only one fixed point on . This proves that there is a unique order-1 periodic solution in system (5). According to Theorem 5, we can conclude that the periodic solution is asymptotic stable.
For any trajectory starting from , if , then . After n times pulses, monotonically increases, so .
In contrary, if , we need to discuss this according to different cases. On the one hand, if is always holding, we can conclude that is monotonically decreasing because and . On the other hand, is not true for all n. We make the smallest which satisfies . Then, there must be a positive integer and monotonically increases as increases, so . Therefore, there is a globally asymptotically stable order-1 periodic solution of system (5).

Theorem 7. If , , and there are no fixed points on of φ; then, system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution.

Proof. If there are no fixed points on of φ, then there is a positive constant i which makes and . Based on the definition of the Poincaré map φ, we get and . Because the φ is decreasing on for any , after one pulse, there is . Therefore, holds for all . Also, is monotonically increasing on so thatNext, the existence of the periodic solution is discussed. First of all, for any and make and . If and , then is the fixed point of φ which proves system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution. So the relations about , , ,, and are needed to be discussed. (Figure 2(a)). Then, and . It can be obtained by mathematical induction that (Figure 2(b)). Then, and ; i.e., it can be obtained by mathematical induction that (Figure 2(c)). In the same way, about case , we can obtain (Figure 2(d)). By using the same method as case , we can obtainFor case (ii), is monotonically increasing and is monotonically decreasing; for case (iii), is monotonically decreasing and is monotonically increasing. It is concluded that for case (ii) and case (iii), there exists either a unique fixed point such thator exists two distinct values and and such thatHowever, for cases (i) and (iv), only the later case can be true.
These results verify that there exists either an order-1 periodic solution or a periodic solution+ (5).

(a)

(b)

(c)

(d)

Theorem 8. If , then satisfies , and if there is no fixed point on , when , then system (5) has an order-3 periodic solution.

Proof. If , and there is no fixed point on , it can be seen from Theorem 1 that there is a unique order-1 period solution in :because Poincaré map φ is continuous on closed intervals andAccording to the intermediate value theorem, there exists , and .
Furthermore,According to the properties of continuous functions on closed intervals, there must be at least one value of to enableThis means that system (5) has an order-3 periodic solution.
If we replace condition in Theorem 8 with condition , where , the order-k periodic solution of system (5) can be obtained by a similar method of Theorem 8.

5. Numerical Simulation

In the state impulse feedback control, we assign appropriate thresholds for and D (see Figures 3 and 4). In Figures 3 and 4, the red line shows the trajectory of the system without the impulse, and the green line shows the trajectory of the system with the impulse; this suggests that populations of predators and pests can be kept within a stable range.

(a)

(b)

(c)

(a)

(b)

(c)

It can be seen from Figure 5 that different initial points will eventually converge to the same order-1 periodic solution and tend to be stable; this indicates the global asymptotic stability of the order-1 periodic solution.

The above numerical simulation also shows that the number of pests can be controlled in the state pulse feedback control, which verifies the feasibility of state pulse feedback control.

In Section 4.1, when the predators disappear and the pests reach , we obtain the expression of the boundary period solution and the expression of the pesticide dose. Next, we discuss which key factors can affect the pesticide dose D. We gave some reasonable parameters, as shown in Figure 6. The results show that as decreases, dose D must also be increased (Figure 6(a)). Furthermore, as the T of chemical control increases, the dose D increases (Figure 6(b)). Biologically, we need to consider both the threshold and the period T in the process of pest control.

(a)

(b)

According to condition (39), we can judge whether the chemical control can stabilize the boundary periodic solution alone. means that chemical control by D dose alone can control the pest population below the , and vice versa. Therefore, how much the dose D and the threshold can affect has drawn our attention. For these, we have carried out numerical simulations, as shown in Figure 7(a). The results show that when a single chemical control method is used, high dose D can control the pest population. In addition, as shown in Figure 7(b), for a relatively small , we have , and once is greater than a certain value, . The results show that under the fixed parameter values, the smaller the value, the better the prevention and control of pests. In the process of pesticide management, as long as we choose a reasonable threshold under pulse state feedback control, we can avoid excessive use of pesticides and reduce some negative effects of pesticides.

(a)

(b)

6. Conclusion

Compared with previous studies on state-dependent feedback control, we mainly do the following work: the global dynamics of complex models are studied according to the Poincaré map, and the main properties of Poincaré map are studied to prove the existence of fixed points and the existence of order-k periodic solutions. Besides, we study the effect of pesticide dose on single chemical control or chemical control combined with biological control. The results show that the pest population density can not only be controlled below the threshold under the state pulse feedback control but also avoid excessive application of pesticides and reduce some negative effects of pesticides.

Data Availability

The data used to support the findings of this study are available upon request to the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371230), SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001), Joint Innovative Center for Safe And Effective Mining Technology and Equipment of Coal Resources, SDUST Innovation Fund for Graduate Students (SDKDYC190351).

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Copyright © 2019 Dezhao Li 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.

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