Dynamics of Unilateral and Bilateral Control Systems with State Feedback for Renewable Resource Management

In this paper, mathematical models for the management of biological resources based on a given predator-prey relationship are proposed, and two types of control strategies, unilateral and bilateral control with impulsive state feedback, are studied. )e existence of the order-1 homoclinic orbit, order-1 periodic solution, and bifurcation of homoclinic of the unilateral control system are obtained, and the attraction region of this system is also discussed. Besides, sufficient conditions for the existence and stability of the order-1 and order-2 periodic solutions of the bilateral control system are also gained. A series of numerical simulations including bifurcation diagrams of periodic solution are performed, which not only verify the theoretical results we get but also reveal some peculiar dynamical phenomena, such as the appearance of high-order periodic solutions and existence of parameter intervals with drastic order change of periodic solution. By comparing the two management strategies, our study encourages bilateral control rather than unilateral control for the risk of predator extinction.


Introduction
Reasonable exploitation of renewable resources has been a topic drawing close attention of researchers, resource owners, and resource management departments [1]. Enormous contributions have been made on sustainable management and exploitation of renewable resources [1][2][3][4][5][6][7][8][9][10]. However, the previous studies focused more on the fluctuating environment and different policies. Srinivasu et al. proposed optimization models for resource management and investigated the impact of periodic varying environment and harvest policies on the revenue [1]. Emilie Lindkvist et al. presented a new method to investigate the influence mechanism of several different factors to the management strategies of renewable resources by applying a computational form of adaptive management [11]. Stochastic factors and resource-constrained control tasks were also considered in the management process [12][13][14].
A dynamic model for renewable resource management has played an important role in promoting the sustainable development of a region or a country due to its ability to anticipate future trend. In order to design management strategies of sustainable development, a variety of mathematical models have been constructed to study the impacts of different significant influence factors. Clark studied mathematical models of commercial fishing taking into consideration the economic and environmental factors [2]. en with some investigations, Leung and Wang [15] presented a commercial fishing model incorporating capital investment. Ragozin and Brown [16] studied a model for a predator-prey system where the prey is supposed to have no commercial value while the predator can benefit humans. In contrast, Mesterton-Gibbons investigated a predator-prey system where the two species are both harvested with an optimal policy [17]. A nonautonomous model with periodic coefficients was proposed by Fan and Wang in [18], and they studied the positive periodic solution and its stability.
Feedback control was also used in the general renewable resource management. Sandal and Steinshamn in [19] studied the optimal exploitation of renewable natural resources by a feedback model. ey presented some feedback rules in computing optimal yields and depletion rates by applying perturbation theory. ey also found the analytical expressions of the optimal harvest under an explicit feedback control law [12]. Fuertes et al. gave an optimal resource allocation policy aiming at maximizing the control performance and also provided experimental comparisons to show the priority of this policy [14].
With the rapid development of theories and methods of the semicontinuous dynamic system in recent years, a large number of mathematical models of renewable resource management considering state feedback control are formulated and studied. Xiao and Dai studied a predator-prey system with Allee effect and state feedback and discussed the coexistence mode of the two biological resources [20]. Huang et al. constructed a population system with two types of harvesting, and they exploited the impact of the harvest effort on the development trend of population [21]. Liang et al. formulated a predator-prey system with mutual interference and also investigated the dynamical behaviors of the system [22]. Besides, Chen and Zhao proposed a delayed singlespecies model with impulsive control, and they mainly discussed the existence and stability of order-1 periodic solutions [23]. Furthermore, Guo et al. in [24] investigated a state feedback for the algae-fish system while Fu and Chen in [25] formulated an ecological system for water hyacinth and studied two types of state-dependent impulsive controls.
Overexploitation or excessive use of a renewable resource that has economic value can be a threat to biodiversity. As we all know, all species do not exist in isolation, and they will interact with other species around them. A species may be the prey, or the predator, or a competitor of another species. So when it is overexploited, what will happen to the other one? And what should humans do to protect the biodiversity of the ecological system? Based on these considerations, in this paper by constructing semicontinuous dynamic systems, we present mathematical models for the management of two kinds of biological resources based on a given predator-prey relationship. Since the quantity of the prey population can be easily obtained by some modern devices and measurement techniques, we design two control strategies, unilateral and bilateral control, with the state feedback of the prey population. e paper is organized as follows. In Section 2, we formulate two mathematical models for the unilateral and bilateral state feedback control of the two kinds of biological resources. In Section 3, we give qualitative analysis for the unilateral and bilateral control systems, and dynamical behaviors such as the existence and stability of order-1 and order-2 periodic solutions and bifurcation phenomenon corresponding to control parameters are investigated. In Section 4, series of numerical simulations including bifurcation diagrams of periodic solution are performed, which not only verify the theoretical results we get but also reveal some peculiar dynamical phenomena. We finish this paper with a brief discussion in Section 5.

Basic Model.
e basic model studied in this paper is the following predation system [26]: where x(t) and y(t) represent the densities of the prey and predator, respectively. r, K, α, β, μ, and D are the positive constants. αx(t)e − βx(t) denotes the functional response when the predation confronts group defense from the prey species.
For model (1), there are always an unstable saddle point O(0, 0) and a boundary equilibrium E 0 (K, 0). To get positive equilibria, we need to solve the following algebra equations: By simple calculation, we have the following lemma.
Proof. From Lemma 1, we know that if αμ > eβD and x 2 < K, the system (1) has two positive steady states E 1 (x 1 , y 1 ) and E 2 (x 2 , y 2 ) besides two boundary equilibria O(0, 0) and E 0 (K, 0). To study the stability of these equilibria, we give the Jacobi matrix of the linearized system of (1) in the following:

Complexity
By simple calculation, we have Obviously, O(0, 0) is an unstable saddle and E 0 (K, 0) is a stable node if x 2 < K. At points E i (x i , y i ), i � 1, 2, the characteristic equation has the form where Because we can get a 2 < 0 when x i � x 2 and a 2 > 0 when x i � x 1 ; then, E 2 (x 2 , y 2 ) is a saddle point. Besides, x i � x 1 implies a 1 < 0, so E 1 (x 1 , y 1 ) is a stable node or focus. e proof is completed. □ Theorem 2. If r(β − (1/K)) + αμ < 2 ��������� ((βrD)/K), then there are no closed orbits for system (1) provided that the conditions listed in eorem 1 are satisfied.
Proof. In consideration of the unstable saddle E 2 (x 2 , y 2 ), if there is a close orbit (a periodic orbit or a homoclinic loop) of system (1), it must be in the following domain: Let dt � e βx dτ, then system (1) is changed into an equivalent form: Applying Dulac function B(x, y) � x − 1 , we have Note that − (βr/K)x − (D/x) ≤ − 2 ��������� ((βrD)/K), then we can get In addition, we know that e βx ≥ 1, x ∈ (0, x 2 ], and if By Dulac theory, system (1) does not have a closed orbit in Ω 1 . is completes the proof. □ 2.2. Model Formulation. Li et al. [27] investigated impulsive control tactics of system (1) at periodic fixed moments, gained the existence and local stability of the prey-extinction periodic solution, and also exploited several kinds of bifurcation phenomena. He [28] studied system (1) incorporating impulsive state feedback controls. However, the study just focused on a special case, that is to say, system (1) has a unique positive equilibrium E 1 (x 1 , y 1 ) which is globally stable.
In this work, we study system (1) under the conditions provided in eorems 1 and 2; that is to say, system (1) has two positive equilibria: a hyperbolic saddle E 2 (x 2 , y 2 ) and a stable node or focus E 1 (x 1 , y 1 ) (see Figure 1).
Assume that the prey and predator are two kinds of biological resources which have economic value, while the quantity of the prey population can be easily monitored by some modern devices and measurement techniques. Our research object is providing scientific strategies for the sustainable development and reasonable utilization of these two kinds of resources.
From Figure 1, we can see that if the prey density is less than x L (x L is the distance between the stable manifold of E 2 (x 2 , y 2 ) and y-axis) or greater than x 2 , there is a high probability that the predator species goes extinct. So in the following, for the biodiversity of this ecological system, we choose a lower control threshold h 1 > x 1 > x L and an upper control threshold h 2 < x 2 for the prey density. To avoid overexploitation of the prey population, we first choose h 1 as the control threshold. With the state feedback, once the density of the prey drops down to h 1 , then people release a certain amount of the prey and harvest the predator by a specific ratio at the same time. en, we formulate a model for the unilateral control with state feedback of the prey population: where Besides, for any prey size x � n, x 1 < n < x 2 , the predator species itself has a sustainable range [n mi , n mx ]. If the predator resource is overexploited, for example, if its density is less than n mi , then the prey size will exceed x 2 and Complexity the predator will become extinct. For this reason, we set another upper control threshold h 2 besides the lower one.
en, when the density of the predator is less than (h 2 ) mi and the density of the prey increases to h 2 , people release a certain amount of the predator and harvest a certain amount of the prey at the same time. For this case, we formulate another management strategy with bilateral control: Here, m 1 , m 2 , p, and τ are the positive constants that satisfy e aim of this paper is to investigate the management strategies that can maintain the sustainable development of the two species with rational replenishment and exploitation by studying systems (11) and (12).

Preliminary.
Before the study of the above two systems, some notations and definitions of semidynamical systems are introduced. ese will be used in the following discussion.
Consider the general planar semidynamical system: be an arbitrary solution of (5). Denote π t (X 0 , X) � X(t) to be the solution corresponding to initial value condition X(0) � X 0 . en, its forward orbit can be described by π(X 0 , X) � {π t (X 0 , X) | t ≥ 0}, which is also denoted by π(X) or π + (X) for simplicity. We denote Definition 1 (see [29,30]). An orbit π(X) of (13) is said to be a periodic orbit if there is a k ∈ N + such that X k � X 0 . Denote k 0 � min{k ∈ N + , X k � X 0 }, then the orbit π(X) is called an order-k 0 periodic orbit.

en, (ξ(t), η(t)) is orbitally asymptotically stable.
As previously mentioned, if the prey density is less than x L or greater than x 2 , there is a high probability that the predator species goes extinct. erefore, to get a mathematical model which is more nearing to the practice, we assume that the initial pest density is larger than h 1 and less than x 2 .
In the following, we will provide a series of qualitative analysis for systems (11) and (12) in the region Ω that is surrounded by x-axis, the threshold line x � h 1 , and the unstable flow of E 2 (x 2 , y 2 ).

Periodic Solution, Homoclinic Bifurcation, and Attraction
Region of System (11). For system (11), we denote the impulsive set by while the corresponding phase set is then there exists two fixed parameter values 0 < p 0 < p * < 1 such that (1) If p � p * , then system (11) admits an order-1 homoclinic cycle; (2) If p 0 < p < p * , then system (11) admits a unique order-1 periodic orbit in Ω; (3) If p * < p < 1, then system (11) has no periodic solution in region Ω.
Proof. Let A and D denote the first intersection points of the unstable flow of E 2 (x 2 , y 2 ) (in the direction of time increasing) with straight lines x � h 1 and x � h 1 + m 1 , respectively. However, point B is the last intersection point of the stable flow of E 2 (x 2 , y 2 ) (in the direction of time increasing) and the straight line x � 0 at points C and G, respectively (see Figure 2). For every point M ∈ Ω, we denote its coordinate by M(x M , y M ) and the forward orbit from it by π + (M). Because the phase set N 1 is parallel to yaxis, for any point S ∈ N, we define its coordinate as y S and denote the successor function of point S by f sor (S).
Since 0 < p < 1, there has two fixed parameter values p 0 and p * such that (1 − p 0 )y A � y C and (1 − p * )y A � y B (see the left figure in Figure 2).
en, there is f sor (B) = 0, and the orbit BE 2 ∪ E 2 A ∪ AB forms a homoclinic cycle, that is to say, an order-1 circle which has a saddle on it.
(2) p 0 < p < p * . To prove that system (11) Since f sor is continuous, there must exist a point S ∈ MM 1 ⊂ N 1 such that f sor (S) � 0 (see the right figure in Figure 2). In the following, we discuss the uniqueness of the order-1 periodic orbit. For any point P(x P , y P ) ∈ N 1 , y P < y B , the forward orbit π + (P) tends to the stable node E 0 (K, 0). Besides, for every point P ∈ C D ⊂ N 1 , we have f sor (P) < 0. erefore, if the order-1 periodic orbit exists, it must begin from BC. We arbitrarily choose two points B 1 , B 2 ∈ BC, y B < y B 1 < y B 2 < y C . Since different trajectories are disjoint, it is easy to obtain f sor (B 1 ) > f sor (B 2 ).
at is to say, f sor is a monotonically decreasing function defined on the line segment BC, and there is a unique point S ∈ BC ⊂ N 1 such that f sor (S) � 0. From the above, system (11) has a unique order-1 periodic orbit in Ω (see Figure 3).
Besides, the forward orbit π + (P) tends to the stable node E 0 (K, 0) for any point P(x P , y P ) ∈ N 1 , y P < y B . us, system (11) has no periodic orbit in Ω. is completes the proof.

Remark 1.
Choose p as the bifurcation parameter. According to eorem 3, we can get that if parameter p equals to the critical value p * , there is an order-1 homoclinic cycle of system (11); when parameter p gradually decreases from p � p * to p 0 < p < p * , the homoclinic cycle disappears while an order-1 periodic orbit appears; when parameter p gradually increases from p � p * to p * < p < 1, the homoclinic cycle disappears but no new order-1 periodic cycle appears.

then system (11) has an attraction region in Ω, where Ω is surrounded by the two separatrices of E 2 (x 2 , y 2 ) and the threshold line x � h 1 .
Proof. Let the order-1 periodic orbit Γ S intersect the impulsive set x � h 1 at point S′. Denote point B by point L + 0 . ere is f sor (L + 0 ) > 0 and the successor point L + 1 is above point S (see Figure 4).
For π + (L + 1 ), we need to determine the position of its successor point L + 2 . For this purpose, consider the forward orbit π + (L + 1 ) and it meets the impulsive set x � h 1 at point L 1 ′ . Obviously, L 1 ′ is between two points S′ and G. According to the conditions p 0 < p < p * and (1 − p)y G > y B , we get (1 − p)y G < (1 − p)y L 1 ′ , so the successor point of L + 1 (denoted by L + 2 ) is above the point B and we have Repeating the above steps, two point sequences L + 2k and L + 2k+1 , k = 0, 1, 2, . . ., separated by point S are obtained, where sequence y L + 2k k � 0, 1, 2, . . . is monotonically increasing with y L + 2k ≤ y S while y L + 2k+1 } k�0,1,2,... is monotonically decreasing with y S ≤ y L + 2k+1 . We further suppose that y L 2k ⟶ y S 0 ≤ y S , as k ⟶ ∞, And two points S 0 and point S 1 may coincide with each other, that is to say, points S 0 , S 1 , and S are the same one.
Let the forward orbits π + (S 0 ), π + (S 1 ) meet the impulsive set x � h 1 at points S 1 ′ and S 1 ′ , respectively. Denote the region circled by the closed curve S 0 S 0 ′ ∪ S 0 ′ S 1 ′ ∪ S 1 ′ S 1 ∪ S 1 S 0 by △Ω, and we will prove that it is the attraction region of system (11) in Ω.
Furthermore, for any point P + 0 ∈ S 1 D, the forward orbit π + (P + 0 ) will intersect the phase set x � h 1 + m 1 on BS 0 after impulsive jumps once or twice and then is eventually attracted to the region △Ω. To sum up, we get that △Ω is the attraction region of system (11) in Ω. is completes the proof. □ Remark 3. If three points S 0 , S 1 , and S coincide with each other, then the attraction region △Ω is exactly the order-1 cycle Γ S we obtained in eorem 3. In this case, Γ S is globally orbitally asymptotically stable in Ω. Of course, these three points S 0 , S 1 , and S may not coincide with each other; then, the attraction region △Ω can be circled by an order-2 orbit. (12). For system (12), two impulsive sets can be defined as

Analysis of System
However, the corresponding phase sets are the same which is denoted by For the convenience of the following discussion, let the unstable flow of E 2 (x 2 , y 2 ) meet the straight lines x � h 1 , x � n, and x � h 2 for the first time at points A, D, and E, while the stable flow of E 2 (x 2 , y 2 ) meet the three lines for the last time at points H, B, and K (in the direction of time increasing). Besides, the three lines intersect the isoclinic line _ x � 0 at points G, C, and F. For the forward orbit π + (F) and backward orbit π − (F), we denote their intersection points with x � h 1 and x � n by N and Q, respectively (see Figure 5).
Proof. Consider the point L ∈ N 2 . If y L < y Q , then the forward orbit π + (L) will firstly meet the second impulsive set M 22 and goes through the corresponding impulsive jump, while if y L > y Q , then the forward orbit π + (L) will meet the first impulsive set M 21 first and goes through the corresponding impulsive jump.
Similar to the analysis in eorem 3, we can get the existence of the order-1 periodic solution Γ S for system (12), where S ∈ N 2 satisfies y Q < y S < y C . e proof is completed. □ Remark 4. Similar to the discussion in eorem 4, we can obtain that if (1 − p)y G > y Q , then system (12) has a local attraction region which is circled by the closed curve AG ∪ GC ∪ CQ ∪ QF ∪ FE ∪ EDA. Theorem 6. If (r(1 − (n/K))/αe − βn ) < τ < y D − y F , (1 − p) y A < y Q , and x 1 < h 1 < n < h 2 < x 2 , then system (12) admits an order-2 periodic orbit in Ω provided that conditions in eorems 1 and 2 are satisfied.
Proof. Since τ > (r(1 − (n/K))/αe − βn ), we have y D − y F > τ > y C . Choose a point A 1 ∈ N 2 next to Q satisfying y A 1 < y Q , and let the forward orbit π + (A 1 ) meet the impulsive set M 22 at point A 1 ′ with y A 1 ′ < y F ; then, A 1 ′ is mapped to B 1 ∈ N 2 . ere is y C < y B 1 < y D for y D − y F > τ > y C . e forward orbit π + (B 1 ) meets the impulsive set M 21 at point C 1 with y N < y C 1 < y A ; then, C 1 is mapped to D 1 ∈ N 2 . Obviously, we have y D 1 < y A 1 . en, f 2 sor (A 1 ) � y D 1 − y A 1 < 0 (see the left figure in Figure 6).
Choose another point A 2 ∈ N 2 satisfying 0 < y A 2 ≪ 1, and let the forward orbit π + (A 2 ) meet the impulsive set M 22 at point A 2 ′ with y A 2 ′ < y K ; then, A 2 ′ is mapped to B 2 ∈ N 2 . ere is y C < y B 2 < y B 1 for y D − y F > τ > y C . e forward orbit π + (B 2 ) meets the impulsive set M 21 at point C 2 with y G < y C 2 < y C 1 ; then, C 2 is mapped to D 2 ∈ N 2 . Obviously, Due to the continuity of f 2 sor , there must exist a point S + ∈ A 1 A 2 such that f 2 sor (S) � 0. en, system (12) admits an order-2 periodic orbit. e proof is completed. □ Remark 5. According to the proof of eorem 5, system (12) may have order-k(k > 2) periodic orbits provided that the parameters satisfy certain conditions.

Numerical Simulations
In this section, a series of numerical simulations are carried out, which will on the one hand confirm the theoretical results and, on the other hand, reveal some peculiar dynamical phenomena.

Complexity 9
E 2 (x 2 , y 2 ) are unstable saddle points while E 0 (K, 0) and E 1 (x 1 , y 1 ) are stable ones. For unilateral control system (11), we choose the lower control threshold h 1 � 0.66 > x 1 and m 1 � 0.05, then there are a fixed parameter values p * such that system (11) admits an order-1 periodic orbit for some p < p * (see Figure 7), while it has no periodic orbit if p > p * (see Figure 8). Obviously, the order-1 periodic orbit in Figure 7 is orbitally stable. For bilateral control system (12), we choose the two control thresholds h 1 � 0.61 > x 1 and h 2 � 0.71 < x 2 , while m 1 � 0.05 and m 2 � 0.05 such that the shared phase set is x � n � 0.66. By changing the impulse parameter, we show that system (12) can have different types of periodic solution (see . When the order-2 periodic orbit of system (12) exists, we investigate its stability. Choose different initial values; we can see that this periodic solution is orbitally stable (see Figure 13). (12). According to the discussion in Section 4.1, when we fix p � 0.6, system (12) can exhibit different types of periodic solution with the change of parameter τ. We now exploit the bifurcation phenomenon with respect to τ. For this end, we first set p � 0.6, and Figure 14 shows the bifurcation diagram of system (12) with respect to τ (τ ∈ [0, 1]). From Figure 14, we find that system (12) can exhibit several types of periodic solution, such as Complexity order-k periodic solution, k � 1, 2, 3, 4, 5. However, the order bifurcation cannot lead to chaos. is is somewhat similar to a discrete dynamical system. Besides, from Figure 14, we can see that orders 1, 2, 3, 4, and 5 periodic orbits exist for wide ranges of τ. With the increase of τ, there is an order-1 periodic orbit when τ ∈ [0, 0.2401) and then a positive order-2 periodic solution bifurcates from the order-1 periodic solution at τ ≈ 0.2401 through a fold bifurcation. e order-2 periodic orbit exists when τ ∈ (0.2401, 0.4031); then, a positive order-3 periodic solution bifurcates from the order-2 periodic solution at τ ≈ 0.4031. Analogously, a positive order-4 periodic solution bifurcates from the order-3 periodic solution at τ ≈ 0.6165 and a positive order-5 periodic solution bifurcates from the order-4 periodic solution at τ ≈ 0.8952.

Bifurcation Diagram of System
In addition, we also set τ = 0.7 and observe the bifurcation phenomenon with respect to p. Figure 15 shows the bifurcation diagram of system (12) with respect to p (p ∈ (0, 1)). Similarly, we find that system (12) also exhibits many types of periodic solution, and the order bifurcation does not lead to chaos. Furthermore, for wide ranges of p, system (12) has periodic solutions of order 4 (for example p ∈ (0.3, 0.7) (see case p = 0.6 in Figure 12) or order 5 (for example, p ∈ (0.18, 0.25] ∪ (0.75, 0.82), see case p = 0.25 in Figure 16(c)).
Besides, from Figure 15, we can see that, with the change of τ, there are two small subintervals in which the order of the periodic solution rises and falls dramatically. Figures in Figure 17 are two zoomed parts in the parameter interval p ∈ (0, 0.3] and p ∈ [0.7, 1). From the left one, we can see that the order of the periodic solution is very high when p is near zero and then gradually drops down to 4 (for example, 0.28 < p < 0.3). However, from the right one, we can see that the order of the periodic solution rises from 4 (0.7 < τ < 0.73) to a high level. e time series of system (12) are present in Figure 16 for four different values of p ∈ (0, 0.3]. We can see easily that the order of the periodic solution changes greatly. Similarly, the time series of the system (12) for p ∈ [0.9, 0.95] are present in Figure 18 and the order also changes dramatically. Obviously, these profiles of system (12) are associated with Figures 15 and 17.

Conclusion
In this paper, mathematical models for the management of two kinds of biological resources based on a given predatorprey relationship are proposed, and two types of control strategies, unilateral and bilateral control with the state feedback of the prey population, are investigated both theoretically and numerically. 14 Complexity For the basic predator-prey system (1), we extend and renew the study of [26] and find conditions under which system (1) has two positive equilibria while E 1 (x 1 , y 1 ) is a stable saddle and E 2 (x 2 , y 2 ) is an unstable saddle ( eorem 1). Besides, we also provide conditions that can guarantee that the system (refer to system (1)) has no closed orbits (refer to eorem 2). rough analysis of the phase diagram, we find that the basic system has risk of resource depletion without control measures. en, we introduce two types of control strategies, unilateral and bilateral control, both of which apply impulsive state feedback control technique. By applying the method of successor function, the existence of the order-1 homoclinic orbit and order-1 periodic orbit of the unilateral control system is obtained (refer to eorem 3) and the attraction region of this system is also exploited (refer to eorem 4). e phenomenon of homoclinic bifurcation is discussed by choosing p as the bifurcation parameter. We first verify the existence of the bifurcation point p * and then show that if p � p * , system (11) admits an order-1 homoclinic cycle; when parameter p gradually decreases from p � p * to p < p * , the homoclinic cycle disappears while an order-1 periodic orbit appears; when parameter p gradually increases from p � p * to p * < p < 1, the homoclinic cycle disappears but no new order-1 periodic cycle appears. Besides, by using geometry theory and Analogue of Poincaré criterion, we investigate the existence and stability of positive periodic solutions of the bilateral control system (refer to eorem 5-7). Furthermore, a series of numerical simulations (including bifurcation diagram of periodic solution) are performed which not only confirm the theoretical results but also reveal some peculiar dynamical phenomena, such as the appearance of high-order periodic solutions and existence of parameter intervals with drastic order change of periodic solution.
Our study show that both unilateral control and bilateral control are beneficial to the biodiversity of the ecological system we studied in this paper. ey can reduce the risk of resource depletion to some extent. From the unilateral control, we can see that the risk of extinction of the predator population still exists.
at is, when the density of the predator is low enough while the prey population continues to increase, the predator will go extinct. However, the bilateral control can avoid the extinction of both species. erefore, when comparing the two management strategies, our study encourages bilateral control rather than unilateral control for the risk of predator extinction.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.