Dynamical Properties of a Herbivore-Plankton Impulsive Semidynamic System with Eating Behavior

In this paper, an impulsive semidynamic system of the relationship between plankton and herbivore is established, and the Poincaré map method is used to extend the new properties of the model. We define the Poincaré map of the impulsive point series in phase concentration and analyze the characteristics. A comprehensive and detailed analysis of the periodic solution is performed. In addition, the numerical simulations illustrate the correctness of our arguments. The results show that plankton and herbivore can survive stably under effective control.


Introduction
In terrestrial and aquatic ecosystems, the energy conversion and nutrient cycle between plankton and herbivores play an important role [1]. Since plankton can provide energy for herbivores, studies on the population model of plankton and herbivores have drawn increasing attention from scholars [2][3][4]. In Zhong et al.'s paper [5], the interaction model of nutrient solution, plankton, and herbivores was established, and the limit cycle and other dynamic properties were studied. Sharma et al. [6] put forward a predator model of plankton and herbivores, made a qualitative analysis of the model system, discussed the stability of the equilibrium point, and explored ways to maintain the ecological balance of the population at different harvest levels.
Recently, the impulsive semidynamical system with threshold has been widely applied [7][8][9][10][11], such as in the diagnosis and treatment of the antivirus system [12][13][14][15], the integrated control strategy of ecological resources and plant diseases and insect pests [16][17][18][19][20], vaccination strategies, and epidemic control [21][22][23][24][25]. e research and application of geometric theory in pulsed semidynamic systems have also yielded good results [26][27][28][29][30]; the existence, uniqueness, and stability of system periodic solutions are discussed. Also, the successor function, Lyapunov function, and coincidence theory were used to study the dynamics of biological systems [31][32][33][34][35][36]. However, with the further development of the state feedback control model and the complexity of global dynamics, there are still many unresolved problems in model research. For example, when the control parameters are changed, the global dynamics are not well processed; the periodic solutions of the model are not thoroughly studied; and the biological meanings of the complex dynamics are not well analyzed and revealed. In order to solve these problems, more advanced qualitative techniques and new methods are needed to reveal the complete dynamic properties and control strategies of biological dynamic systems. erefore, we propose a herbivore-plankton pulse semidynamic system, which uses the Poincaré map method to comprehensively analyze and study the complex dynamics of the model.
In previous studies, the main assumptions are the number of organisms fished or destroyed during the control process is a linear relationship with the population density or number; when the biological populations reach the threshold, the number of natural enemies released during the pulse is constant and regardless of density. However, in actual situations, in order to better control the biological population and obtain benefits, it is necessary to make appropriate control in combination with factors such as biological population density and fishing rate. erefore, in this paper, for the proposed herbivore-plankton impulsive semidynamic system, the amount of plankton released is a function of its density, and the number of herbivores caught is related to the maximum fishing rate. is allows both plankton and herbivores to be effectively controlled, more in line with actual conditions. is article is arranged in the following sections: in Section 2, the useful definition of the pulsed semidynamic system and related lemmas are presented. In Section 3, we introduce the plankton-herbivore model and give its qualitative properties. e definition of Poincaré map and its main properties are studied in Section 4. In Section 5, the stability conditions of the boundary periodic solution and the order-1 periodic solution are given, and the existence of the order-k (k ≥ 2) periodic solution is further proved. Finally, the numerical simulation is performed to verify the conclusion.

Preliminaries
In practical biological problems, state feedback control strategies can be broadly defined, usually modeled by impulsive semidynamic systems. Control strategies (harvesting, applying pesticides, treatment, etc.) are only implemented when a particular species attains the previously defined threshold. In many papers, a common assumption is that all solutions from a phase set must undergo an unlimited number of pulses. In order to better research the paper, we propose the following useful definitions and lemma: Definition 1. A generalized plane pulse semidynamic system with the state-dependent feedback control can be depicted as the following: dp(t) dt � F(p, y), where F, G, α, and β are continuous funtions and M ⊂ R 2 represents the impulsive set. (p, y) ∈ R 2 , Δp � p + − p, and Δy � y + − y. For any w(p, y) ∈ M, w + indicates the impulsive point; thus, the impulsive function I is defined as w + � I(ω) � (p + α(p, y), y + β(p, y)) � (p + , y + ) ∈ R 2 . We define N � I(w) as the phase set; then, for all Define (X, Π, R + ) or (X, Π) as the semidynamical system, then X represents a metric space and R + represents all nonnegative reals. When w ∈ X, the function Π w (t) � Π(w, t) is clearly continuous such that Π(w, 0) � w and Π(Π(w, t), s) � Π(w, t + s) for all w ∈ X and t, s ∈ R + . e set U + � Π(w, t) | t ∈ R + is described as the positive orbit of w.

Herbivore-Plankton Interaction Model and Its Properties
Ntr et al. [42] proposed the following herbivore-plankton interaction model: 2 Complexity where P and Y are the density of plankton and herbivores, respectively. K is the constant carrying capacity without herbivores, D is the rate of biological conversion, B/(C + P) indicates the function response of the herbivore to change in the density of plankton, C is the semisaturation constant, and B represents the maximum consumption of herbivores. For more biological definitions, see [42].
We nondimensionalize system (5) and get the following state feedback control model [43]: , where p and y denote the density of plankton and herbivore at time t, respectively, and a indicates the effect of density dependence on herbivores. αp and βy indicate the quantity of plankton and herbivore reduced by catching or exhaustion, respectively. System (6) shows that when the quantity of herbivores reaches certain threshold y, without considering the quantity of plankton, harvesting both herbivores and plankton causes excessive depletion of plankton and leads to a lack of food for herbivores, which in turn affects their production. And at this time, the control of plankton and herbivores is only related to the density, but in actual situations, we need to control the biological population based on the fishing rate and actual factors. erefore, we establish the state feedback control model as follows: , , system (7) shows that whenever the quantity of plankton reaches certain threshold p, people will catch herbivores and increase the amount of plankton to (1 − δy(t)/y(t) + β)y(t) and p(t) + τ/1 + θp(t), respectively, where δ represents the maximum fishing rate, τ indicates the maximum amount of plankton added, and θ represents the morphological parameter. For more definition of biological parameters, see [44]. Define f(y) � (1 − (δy/y + β))y, and by calculating, we get the function f(y) is increasing in (0, +∞). e qualitative analysis of system (8) is discussed as follows when system (7) has no pulse: , [43] (Figure 1).

and has a limit cycle
In addition, Hopf branch occurs in the case of k � 2p * + 2, where p * meets the following condition: To facilitate the study of the dynamics of system (7), we define the following two isoclines:

Poincaré Map and Its Main Properties
We first construct the Poincaré map to study the dynamics of system (7). According to the biological significance, this article only needs to be discussed in Ω, where erefore, we first define the following two sets:

Complexity
It is easy to know set S M and S N represents two parallel lines L 3 : p � p and L 4 : p � p + (τ/1 + θp) in Ω. We represent the intersection of L 4 and L 1 as A(p + (τ/1 + θp), y A ) and the intersection of L 3 and L 1 as and . And there exists a curve Γ that starts at point A and intersects with L 3 at point C(p, y C ).
According to the above definition, we construct the impulsive semidynamical system of system (7).
Define the impulsive set as while the continuous function uses the expression as follows: and the impulsive function f(y) � (1 − (δy/y + β))y is increasing in (0, +∞). So, we represent the phase set as follows ( Figure 2):

Complexity
Unless otherwise specified, in this article, we assume that the initial point Next, we define the Poincaré map of system (6) in the case (H 1 ).
Since the internal equilibrium point D * is the stable point, we assume that p > p * . en, for any point A + k � (p + (τ/1 + θp), y + k ) ∈ S N , the trajectory passing through point A + k will reach the impulse set M through time t 1 , so where is indicates that the ordinate y k+1 of the point at which the initial point on the phase set reaches the pulse set is determined by y + k . en, we define y k+1 ≜ φ(y + k ). erefore, the Poincaré map can be expressed as follows: en, system (8) is represented as the following scalar differential equation: dy dp Obviously, l(p, y) is continuous and differentiable in Ω. We make p + 0 � p + (τ/1 + θp) and According to model (20), function y(x, μ) with initial value of (p + (τ/1 + θp), μ) can be expressed as erefore, the form of the Poincaré map is Theorem 2. We assume that system (8) has a stable internal equilibrium point which meets k < 2(p * + 1) and p > p * ; then, the Poincaré map F M (μ) of system (8) contains the following properties ( Figure 3):

μ) takes the minimum value at y A and has no maximum value
Proof.
(i) According to eorem 1 and p > p * , for any point (ii) We first prove the monotonicity of the function F M (μ) on the interval [y A , +∞). Choose any two points with μ k . From the properties of the solution of the differential equation, we get that Complexity y k 1 +1 < y k 2 +1 ; then, from system (19) the trajectory Π(t, p + (τ/1 + θp), μ k + i ) across the isocline L 1 and reach S N at points A k 1 ′ (p + (τ/1 + θp), μ k 1 ′ ) and A k 2 ′ (p + (τ/1 + θp), μ k 2 ′ ). From the Cauchy-Lipschitz theorem, we have μ k 2 ′ > μ k 1 ′ and μ k i ′ ∈ [y A , +∞). According to the above proof, (20), it is clear that l(p, y) is continuous and differentiable in the first quadrant. According to the Cauchy-Lipschitz theorem and system (19), the function F M (μ) is continuously differentiable in R + 2 . (iv) First, we consider the value of the Poincaré map F M (μ) of y A and prove the existence of the fixed point. If F M (y A ) � y A , then the Poincaré map F M (μ) possess a fixed point.
In the light of property (iii) and the theorem of zero existence, there exists at least a point y ∈ (y + q , y A ), which meets F M (y) � y. From property (ii), F M (μ) has at least a fixed point on (0, y A ]. If F M (y A ) > y A , then F M (y A ) − y A > 0. When y + 0 ⟶ ∞, from model (19), we obtain y 1 < y + 0 and y + 1 � y 1 (1 − (δy 1 /y 1 + β)) < y 1 . en, F M (y + 0 ) � y + 1 < y + 0 . From the monotonicity and continuity of F M (μ), F M (μ) has at least a fixed point y ∈ [y A , +∞) in which F M (y) � y.
us, the fixed point of system (7) is unique when k < (p 2 − ay 2 /p).
(v) From property (ii), we obtain that F M (μ) is decreasing on (0, y A ] and increasing on [y A , +∞). us, for any μ ∈ (0, +∞), there is F M (μ) ≥ F M (y A ). So, when μ � y A , the function F M (μ) reaches the minimal value which is also the minimum value. And the minimum value is F M (y A ) � y(p, y A ) − δy 2 (p, y A )/y(p, y A ) + β.
Since F M (μ) is increasing on [y A , +∞), then F M (μ) has only a minimum value on (0, +∞) and no maximum value.

Order-k Periodic Solution and Its Stability Analysis.
According to eorem 2 (iv), we conclude that system (7) has a unique fixed point, which implies that the system possesses an order-1 periodic solution. In this section, we prove the stability of periodic solution and the existence of order-k (k ≥ 2) periodic solution. So, we make the following provisions.
For A + 0 (p + (τ/1 + θp), y + 0 ) ∈ N, the trajectory of the initial point A + 0 will reach the impulse set M, which is applied to the point A + 1 (p + (τ/1 + θp), y + 1 ) by the impulse. In the light of the definition of the Poincaré map, we have F M (y + 0 ) � y + 1 , and repeating the process above, then Further promotion, e sequence y + k is available, k � 0, 1, 2, . . ..

For any trajectory in which the initial point is
is increasing on [y A , +∞) and there exists a unique fixed point, then y A < F M (y A ) < F M (y + 0 ) � y + 1 < F M (y) � y and y + 1 < F M (y + 1 ) � y + 2 <y. It can be inferred from this that According to the monotonically bounded sequence, we get lim n⟶+∞ F n M (y + 0 ) � y. When y < y + 0 , by using a similar approach, we have y � F M (y) < F M (y + 0 ) < y + 0 and F M (y + 1 ) � y + 2 < y + 1 � F M (y + 0 ). Furthermore, we get y < . . . < F n M (y + 0 ) < F n− 1 M (y + 0 ) < . . . < y + 0 . en, lim n⟶+∞ F n M (y + 0 ) � y. erefore, the order-1 periodic solution of system (7) is globally asymptotic stable. □ Theorem 6. If conditions (H 1 ), F M (y A ) < y A , and F 2 M (y A ) < y A are established, then model (7) has a stable order-1 or order-2 periodic solution.
From situations 1 and 4, lim n⟶∞ y + 2n+1 � y * 1 ≤ y A and lim n⟶∞ y + 2n− 1 � y * 2 ≥ F M (y A ). erefore, in situations 1 and 4, system (7) has stable order-2 periodic solution. eorem 6 shows that system (7) has stable order-1 or order-2 periodic solutions under certain conditions. However, this theorem does not give sufficient and necessary conditions for the global stability of order-1 periodic solution. erefore, eorem 7 is given as follows: Theorem 7. When F M (y A ) < y A , then the sufficient and necessary conditions for the global asymptotic stability of the order-1 periodic solution of system (7) Proof. From eorem 2, a stable order-1 periodic solution is available in system (7) when F M (y A ) < y A . en, system (7) has y ∈ [F M (y A ), y A ] in which F M (y) � y. For any y + 0 ∈ (y, y A ], make y + n � F n M (y + 0 ), n � 1, 2, . . ., and according to the properties of Poincaré map F M (μ), we have F M (y A ) < y + 1 < y < y + 2 < y + 0 < y A ; furthermore, we get F M (y A ) < y + 2n+1 < y < y + 2n < y A . From the monotone boundedness of the sequence, we know lim n⟶∞ y + 2n � lim n⟶∞ y + 2n+1 � y. erefore, the order-1 periodic solution of system (7) is globally asymptotically stable.
We assume the order-1 periodic solution is globally stable, and the following proves that for any y + 0 ∈ (y, y A ] satisfies F 2 M (y + 0 ) < y + 0 . If this condition is not met, then there exists minimum y + 0 ∈ (y, y A ] which makes F 2 M (y + 0 ) ≥ y + 0 . From eorem 6, we get that, for any small ε > 0, there exists y + 1 ∈ (y − ε, y) which makes F 2 M (y + 1 ) < y + 1 . From the continuity of F M (μ), there exists y ∈ (y + 1 , y + 0 ) which makes F 2 M (y) � y. erefore, system (7) has order-2 periodic solution, which is contradictory to the global stability of order-1 periodic solution. □ Theorem 8. When F M (y A ) < y A and F 2 M (y A ) < y + m , where F M (y + m ) � y A , y + m � min y + , F M (y + ) � y A , then system (7) has the order-3 periodic solution.
Proof. Since F M (y A ) < y A , from eorem 2, we get that system (7) Figure 4: Path curve of system (7).
In the same way, we can prove that if F k− 1 M (y A ) < y + m , F M (y + m ) � y A Z, system (7) possesses the order-k (k ≥ 2) periodic solution.

Numerical Simulations
An impulsive semidynamic system of plankton and herbivore is established in this paper. e numerical simulation illustrates the correctness of our arguments.
We set the parameter value as k � 1.8, a � 0.15, τ � 0.6, p � 0.6, θ � 0.8, δ � 0.25, and β � 0.8, and obtain that system (7) possesses a globally asymptotically stable order-1 periodic solution if F M (y A ) > y A (Figure 5(a)). Furthermore, let k � 1.8, a � 0.15, τ � 0.6, p � 0.6, θ � 0.8, δ � 0.65, and β � 0.4. We get that model (7) has unique order-1 periodic solution when F M (y A ) < y A , and it is globally asymptotically stable (Figure 6(a)). Figures 5(b) and 5(c) are time series diagram when F M (y A ) > y A , and Figures 6(b) and 6(c) are time series diagram when F M (y A ) < y A . In addition, the red trajectories represent the phase portrait and time series of system (7) without pulses. By comparison, we get that plankton and herbivore can survive stably under effective control.
From Figure 7, we get that although the trajectories start from different initial points, they all converge to the same order-1 periodic solution. And the trajectory eventually stabilizes, which implies that the order-1 periodic solution is globally asymptotically stable.
When herbivores are absent and δ � 0, we get the expression of the boundary period solution. rough analysis, the boundary periodic solution is asymptotically stable under certain conditions. us, the rationality of the plankton release method can be obtained. After no longer harvesting herbivores, that is, when δ � 0, we get that the number of herbivores increases and eventually approaches a stable value, and the plankton population will oscillate periodically from Figures 8(a)-8(c)). Although herbivores will increase temporarily, they will be in equilibrium after a period of time, and the number will no longer increase. However, pulse control allows herbivores to grow periodically, resulting in sustainable harvests and thus greater gains.  us, we get the rationality and necessity of taking pulse control of this paper.

Conclusion
In this paper, an impulsive semidynamic system for the relationship between plankton and herbivores is proposed. We adopt the method of Poincaré map to comprehensively analyze the dynamic properties of system (7). Moreover, state feedback control is adopted to enable plankton and herbivores to survive stably. Firstly, the model is introduced, and the conditions for the existence and stability of the equilibrium point of the model are analyzed. Secondly, we define the Poincaré map of the model and study its main properties such as monotonicity, continuity, fixed point, and extremum. en, we discuss the existence and local and global stability of order-1 periodic solution and give sufficient conditions for the local stability of the boundary periodic solution. In addition, the existence of the order-k (k ≥ 2) periodic solution is obtained through the proof of the existence of the order-3 periodic solution. Finally, our theoretical results are verified by numerical simulation. From biological perspective, periodic solutions and state feedback control have important roles in ecology and fisheries. When the plankton is reduced to a certain amount, a state feedback control strategy of increasing plankton and hunting herbivores is adopted. In the end, a periodic solution can be formed, and the plankton and herbivore are stabilized. e analysis of the boundary periodic solution verifies the rationality and necessity of the pulse form used in this paper. In particular, if not controlled, the number of