Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

*e phytoplankton-fish model for catching fish with impulsive feedback control is established in this paper. Firstly, the Poincaré map for the phytoplankton-fish model is defined, and the properties of monotonicity, continuity, differentiability, and fixed point of Poincaré map are analyzed. In particular, the continuous and discontinuous properties of Poincaré map under different conditions are discussed. Secondly, we conduct the analysis of the necessary and sufficient conditions for the existence, uniqueness, and global stability of the order-1 periodic solution of the phytoplankton-fish model and obtain the sufficient conditions for the existence of the order-k(k≥ 2) periodic solution of the system. Numerical simulation shows the correctness of our results which show that phytoplankton and fish with the impulsive feedback control can live stably under certain conditions, and the results have certain reference value for the dynamic change of phytoplankton in aquatic ecosystems.


Introduction
Fisheries can provide people with quality food resources for the survival and development of human beings. erefore, the healthy development of fishery resources is the focus of attention. If we cannot regulate the capture of fish regularly, it will lead to the depletion of fishery resources. People would be faced with increasing shortages of fish resources. erefore, it is imperative to formulate effective fishing strategies, maintain the ecological balance of fishery resources, and protect the ecological environment [1][2][3][4].
Due to the interaction of energy conversion and the nutrient cycle between plankton and herbivores (such as fish), they play an significant role in the most terrestrial and aquatic ecological system. In [5], a phytoplankton and zooplankton model is established by the interaction of nutrients, and the dynamics properties such as limit cycle are investigated in this paper. In [6], a commercially valuable model of phytoplankton and zooplankton predation is proposed, which analyzes the stability of the equilibrium point, explores methods to maintain the ecological balance of the population at different harvest levels, and discusses the impact of selective harvesting on fisheries.
Recently, threshold state pulsed dynamic systems have been widely used [7][8][9][10][11][12][13][14]. e geometric theory of impulse dynamical system has been well-developed [15][16][17][18][19][20][21][22][23]. Many pulse equations have been studied which simulate the ecological processes of populations [24][25][26][27][28][29][30][31][32]. However, with the further development of the state feedback control model, we need more new methods to find out the complete dynamic properties and control strategies of dynamic systems and to discuss its biological significance. e primary purpose of this paper is to provide a comprehensive qualitative analysis of the global dynamics through analyzing a phytoplankton-fish model with impulse feedback control using the Poincaré map. e central arrangement of this paper is as follows. In Section 2, we establish a phytoplankton-fish model for catching fish based on impulse feedback control. In Section 3, the Poincaré map for the phytoplankton-fish model is defined and the properties of monotonicity, continuity, and differentiability of the Poincaré map are analyzed. In Section 4, we discuss the existence, uniqueness, and global stability of the order-1 periodic solution of the model and obtain the conditions for the existence of order-k(k ≥ 2) periodic solution of the model. In Section 5, we perform numerical simulations.

Model Establishment
In the ecosystem of lakes, phytoplankton is considered to be the most favorable source of food for fish or other aquatic animals. Wang et al. [33] propose a predator model of continuously harvested phytoplankton and herbivorous fish: where u and v represent the population density of phytoplankton and fish at the moment, respectively, and E is the effort for continuous harvesting. For details of other parameters, see [34][35][36].
System (1) shows that, without considering the number of phytoplankton and herbivore, continuous capture will lead to resource depletion. We will seek an integrated capture strategy to achieve ecological stability, for which we make the following assumption: (i) Assuming that phytoplankton and fish stocks are evenly distributed within the lake (ii) Let r be the intrinsic growth rate of phytoplankton, α be the absorption rate of phytoplankton by fish, β be the conversion rate of biomass, and d denote the mortality rate of fish (iii) Formula au/b + u represents the death number of fish due to the distribution of phytoplankton toxicants, where b represents semisaturation constant and a represents the rate at which phytoplankton releases toxins Let H denote the threshold at which the fish are allowed to be caught, that is, when the density of the fish is lower than the threshold H, it is unreasonable to catch fish; however, only when the density of fish reaches, the predetermined value H can catch the fish. e amount of phytoplankton is affected when the fish is caught. e numbers of the fish and the amount of phytoplankton are updated to v(t) − (τ/1 + θv(t)) and u(t)(1 − (δu(t)/u(t) + c)), respectively.
We have established the following impulse feedback control model based on the abovementioned assumptions: System (2) is called a semicontinuous dynamic system [38][39][40].
When there is no impulse, system (2) becomes Lemma 1 (see [37]). System (3) has two equilibrium points, O(0,0) and E * (u * , v * ), where the boundary equilibrium point (0, 0) is the saddle point and the internal balance point (u * , v * ) is a stable center: It is easy to calculate, the first integral of system (3) is where H 0 is the constant related to the initial value. For ease of expression, two isoclines are defined as To study the dynamic properties of system (2), the following research methods are given.

Poincaré Map of System (2) and Its Properties
We intersect the dynamic properties of model (2) } based on biological significance. In order to accurately define the impulse set and phase set of the pulsed semidynamic system (2), the following set is given first: Apparently, within R + 2 , Σ N and Σ M represent two straight lines of the vertical v-axis, respectively, and we assume Σ N and Σ M intersect the line L 2 at point A + and point A − , respectively, since the internal balance point E * is the center point. According to the size of H, we can divide the pulse set and phase set into the following cases: Case I: H < v * When H < v * , the trajectory Γ A + must intersect the Σ M at point A 11 (u 11 , H) (see Figure 1(a)). In this case, impulse sets and the corresponding phase set is 2 Complexity Case II: H > v * When H > v * , if the trajectory Γ A + intersects the Σ M at a point, we assume this intersect is A 22 (u 22 , H) (see Figure 1(b)). In this case, the impulse set and the phase set are the same as those in Case I. It is easy to infer the impulse set and the corresponding phase set If the trajectory Γ A + and Σ M does not intersect (see Figure 1(c)), the trajectory Γ A − intersects the Σ N at point A 31 (u 31 ,H − (τ/1 + θH)) and point A 32 (u 32 ,H − (τ/1 + θH)), respectively, where u 32 >u 31 .
In this case, the impulse set and the corresponding phase set Based on the abovementioned discussion, we define the Poincaré map as follows. Let where 0 < u + k < +∞; the trajectory π t, t 0 , u k which goes through point A + k will reach the Σ M at point after time t 1 , and here en, there is which indicates that the ordinate u + k+1 is determined by u + k . Since point A k+1 is on the impulse set, A k+1 jumps to point where Consider the scalar differential equation of model (3): Let v + 0 � H − (τ/1 + θH) and u + 0 � S, then we have According to model (21), From (20) and (23), the Poincare � map expression of system (2) is , and monotonically increasing on (u * , +∞). Figure 2(b)). When G m (u * ) < u * , there is a unique fixed point on (0, u * ) (see Figure 2(a)).
from the uniqueness of the solution of the differential equation, it can be known that and by the definition of the Poincare � map we can It can be seen from the uniqueness of the solution of the differential equation that G m (u + ; therefore, G m (s) is monotonically decreasing on (0, u * ). Note 1. Because G m (s) is monotonically decreasing on (0, u * ] and monotonically increasing on (u * , +∞), it is easy to know that G m (s) ≥ G m (u * ) is true for any s ∈ (0, +∞), and G m (s) � G m (u * ) if and only if s � u * . (ii) It is easy to know that w(u, v) is continuously differentiable in the first quadrant from (21), and from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the Poincare � map G m (s) is continuously differentiable in the first quadrant. (iii) Considering the Poincare � map at the position of image G m (u * ) of u * , there are three cases: 4 Complexity according to the continuous differentiable of the closed interval.
In conclusion, G m (s) has at least one fixed point. en, we prove the fixed point is unique. We assume that system (2) has two fixed points, u 1 and u 2 , that is to say G m (u 1 ) � u 1 and G m (u 2 ) � u 2 . Let u 1 < u 2 , we define en, take the derivative of the abovementioned formula: Let . (29) en, that is, and From system (2), that is, It is easy to know that (u 1 /μ(u 1 )) < 1 if H < v * , so It is contradictory with d u 1 u 2 (H − (τ/1 + θH)) > d u 1 u 2 (H), so the fixed point is unique.   Figure 3(a)). When G m (u 31 ) > u 31 , there is no fixed point (see Figure 3(b)).
Proof For any u + k 1 , u + k 2 ∈ [u 32 , +∞) and u + k 1 < u + k 2 , from the uniqueness of the solution of the differential equation, it can be known that and by the definition of the Poincare � map we get will pass through the isocline L 2 which intersects the Σ N at point A + k 11 (H − (τ/1 + θH), u + k 11 ) and point A + k 21 (H − (τ/1 + θH), u + k 21 ), respectively. en, u + k i1 (i � 1, 2) ∈ [u 32 , +∞) and u + k 11 > u + k 21 . It can be seen from the uniqueness of the solution of the differential equation that G m (u + k 1 ) > G m (u + k 2 ); therefore, G m (s) is monotonically decreasing on (0, u 31 ].
(ii) From (21), we can know that w(u, v) is continuously differentiable in the first quadrant; from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the G m (s) is (a) When G m (u 31 ) ≤ u 31 (see Figure 3(a)), we assume G m (u 31 ) � u 1 ≤ u 31 , and we know that G m (s) is monotonically decreasing on (0, u 31 ], so G m (u 1 ) ≥ G m (u 31 ) � u 1 , and because G m (u 31 ) ≤ u 31 , so there is a point u ∈ (u 1 , u 31 ], and it satisfies G m (u) � u, according to the continuous differentiability of the closed interval. (b) When G m (u 31 ) > u 31 (see Figure 3 For any u k ∈ (u 32 , +∞), the trajectory of point A k (u k , H − (τ/1 + θH)) is tangent to the straight line Σ M at point A + k (u + k , H); A + k (u + k , H − (τ/1 + θH)) will be pulsed to A k+1 (u k+1 , H − (τ/1 + θH)). Easy to get u k+1 < u + k < u k , that is, u k+1 ≠ u k , so there is no u ∈ (u 32 , +∞) and satisfies G m (u) � u.
In a word, when G m (u 31 ) ≤ u 31 , there is a unique fixed point on (0, u 31 ]. When G m (u 31 ) > u 31 , there is no fixed point. □

The Order-k(k ≥ 1) Periodic Solution of the Semicontinuous Dynamic System (2) and Its Stability
From eorem 1, we know that system (2) has a unique fixed point under certain conditions. at is, system (2) has a unique order-1 periodic solution, and the following are the dynamic properties of system (2). Proof. From (iii) of eorem 1, we know that when G m (u * ) > u * ⊗, G m (s) has a unique fixed point u on (u * , +∞), i.e., G m (u) � u. For any point p + 0 (H − (τ/1 + θ H), u + 0 ) in Σ N , where u + 0 > u * , the trajectory of the point p + 0 will intersect the impulse set, then reach the point p + 1 (H − (τ/1 + θH), u + 1 ), which is G m (u + 0 ) � u + 1 , repeating the abovementioned process: that is, Furthermore, e following two cases are discussed below according to the size of u + 0 : 1) When u * < u + 0 ≤ u, since G m (u * ) > u * and G m (s) is monotonously increasing on (u * , +∞), let G m (u + i ) � u + i+1 , then 6 Complexity Repeating the abovementioned process, we can obtain From the monotonous boundedness of the sequence, we can obtain lim n⟶+∞ G m n u 0 (2) When u < u + 0 < + ∞, from the known conditions, By mathematical induction, (3) When 0 < u + 0 < u * , since G m (u * ) > u * and G m (s) is monotonously decreasing on (0, u * ), so for any So, we can conclude that G m (u + 0 ) > u * , and we can convert this to two cases based on the size of G m (u + 0 ). When u * < G m (u + 0 ) < u, this is the same as case (1) above; when G m (u + 0 ) > u, this is the same as case (2) above. In both cases, lim n⟶+∞ G n m u 0 e order-1 periodic solution of system (2) is globally asymptotically stable. □ Theorem 4. When H < v * , G m (u * ) < u * and G 2 m (u * ) < u * are true, and then the semicontinuous dynamical system (2) has a stable order-1 periodic solution or order-2 periodic solution.
is means that, for any u + 0 ∈ (0, +∞), there is always be a positive integer i which satisfies So, we only need to take account of the initial point

Complexity 7
Let G m (u + 0 ) ≠ u + 0 and G 2 m (u + 0 ) ≠ u + 0 , that is, the trajectory with the initial point p + 0 is not the order-1 periodic solution or order-2 periodic solution of system (2). We consider the following four situations: According to the monotonicity of the Poincare � map, us, there is Proved by mathematical induction, Case II: en, so Case III: u * ≥ u + 1 > u + 2 > u + 0 ≥ G m (u * ). In the same way, we can deduce Case IV: . We can perform the procedure similar to Case I, which can yield\ Considering Cases I and III, there must exist u ∈ (u * , G m (u * )) so that lim n⟶∞ u 2n which means that system (2) has a stable order-1 periodic solution.
For Cases I and IV, there are two points u 1 ≠ u 2 and lim n⟶∞ u 2n− 1 is shows that system (2) has a stable order-2 periodic solution with the initial points (u 1 , H − (τ/1 + θH)) and(u 2 , H − (τ/1 + θH)). eorem 4 illustrates that system (2) has a stable order-1 or order-2 periodic solution under certain conditions. However, sufficient and necessary conditions for global stability are not given. en, we give below theorem. □ Theorem 5. Let H 1 < v * and G m (u * ) < u * , then the necessary and sufficient conditions for the global stability of the order-1 periodic solution of system (2) is G 2 m (u + ) < u + , for any u + ∈ (0, u * ].
Proof. Sufficiency: It can be seen from eorem 4 that when For any u + ∈ (u, u * ), let u + 1 � G m (u + ) and u + 2 � G m (u + 1 ) � G 2 m (u + ) because of G 2 m (u + ) < u + < u * , and from the monotonicity of G m (u), we can get u >u + 1 > G m (u * ); furthermore, By the monotonic boundness of the sequence, erefore, the order-1 periodic solution of system (2) is globally asymptotically stable. Necessity: We assume that the order-1 periodic solution of system (2) is globally asymptotically stable. en, the following part proves that G 2 m (u + ) < u + is right for any u + ∈ [u, u * ], which is proved by the counterevidence method below.
If G 2 m (u + ) < u + is not true for any u + ∈ [u, u * ], then there exists a maximum u 0 ∈ [u, u * ], which satisfies G m 2 (u 0 ) ≥ u 0 . For any ε > 0, there exist u 1 and G 2 m (u 1 ) < u 1 , which is true for any u − ε < u 1 < u + ε from eorem 4. From the continuity of G 2 m (u) on the interval [u 0 , u 1 ], it follows that there is at least one number u → ∈ [u 0 , u 1 ] and G m 2 ( u → ) � u → , which indicate that the trajectory with ( u → , H − (τ/1 + θH)) as the initial point is the order-2 periodic solution, and this is contradictory. □ Theorem 6. If G m (u * ) < u * and there exist u + m � min u + : G m (u + ) � u * , when G 2 m (u * ) < u + m , then system (2) has an order-3 periodic solution.
then system (2) has an order-k(k ≥ 2) periodic solution. □ We simulated the order-1 periodic solution and order-2 periodic solution in the case of Figure 3(a) (see Figure 5). In the simulated Poincare � map, we found the order-1 periodic solution (the intersection point of the red and yellow line and blue line in Figure 5(a)) and the order-2 periodic solution (the intersection point of the black line and red and yellow line in Figure 5(a)). Figure 5       order-1 periodic solution are both changing periodically, in which the number of plankton in the order-2 periodic solution has a period of two. e period of the number of phytoplankton in the order-1 periodic solution is one.

Simulations
In Figure 7, the blue and red lines indicate the trajectory of the system with or without pulses. is shows that the number of phytoplankton and fish can be kept in a stable range.
As can be revealed in Figure 8, different initial points will eventually converge to the same order-1 periodic solution and tend to be stable. is indicates the global asymptotic stability of the order-1 periodic solution.

Conclusion
How to rationally develop and utilize fishery resources has become an essential issue for the sustainable development of lake resources. In this paper, we introduce the impulse feedback control into the phytoplankton-fish model. It is of great significance to study the dynamics of phytoplankton and fish.
Compared with [1], we use the Poincare � map as a tool to give a more comprehensive qualitative analysis and to prove the dynamics of the phytoplankton-fish model, for example, the necessary and sufficient conditions for the global asymptotic stability of the order-1 periodic solution and the existence conditions of the order-k(k ≥ 2) periodic solution.
For the biological significance, the order-k(k ≥ 1) periodic solution of system with state impulsive feedback control indicates that the number of phytoplankton and fish populations can maintain periodic oscillations under certain conditions with proper capture of fish, that is, the number of phytoplankton and fish can be kept in a stable range. ese results have some reference values to the dynamic changes in aquatic ecosystem research of fish and phytoplankton.

Data Availability
e data used to support the findings of this study are available upon request from the corresponding author.