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

Authors: Yufei Wang; Huidong Cheng; Qingjian Li
Journal: Complexity
 (2020)

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## Abstract

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.


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## Body

## 1. 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 [2–4]. 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 [7–11], such as in the diagnosis and treatment of the antivirus system [12–15], the integrated control strategy of ecological resources and plant diseases and insect pests [16–20], vaccination strategies, and epidemic control [21–25]. The research and application of geometric theory in pulsed semidynamic systems have also yielded good results [26–30]; 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 [31–36]. 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. Therefore, 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. Therefore, 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. This allows both plankton and herbivores to be effectively controlled, more in line with actual conditions.This article is arranged in the following sections: in Section2, 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. The 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.


## 2. 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:(1)dptdt=Fp,y,dytdt=Gp,y,p,y∉M,Δpt=αp,y,Δyt=βp,y,p,y∈M,where F,G,α,andβ are continuous funtions and M⊂R2 represents the impulsive set. p,y∈R2, Δp=p+−p, and Δy=y+−y. For any wp,y∈M, w+ indicates the impulsive point; thus, the impulsive function I is defined as w+=Iω=p+αp,y,y+βp,y=p+,y+∈R2. We define N=Iw as the phase set; then, for all w∈M,Iw=w+∈N, while N∩M=∅.
DefineX,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 Πwt=Πw,t is clearly continuous such that Πw,0=w and ΠΠw,t,s=Πw,t+s for all w∈X and t,s∈R+. The set U+=Πw,t∣t∈R+ is described as the positive orbit of w. For any t≥0, we have Vw,t=z:Πz,t=w, and for all set M⊂X, then M+w=U+w∩M−w.Definition 2.
A trajectoryΠw in X,Π,M,I is called the periodic of period Tk and order k if there are integers m≥0 and k≥1 which make k the smallest integer of wm+=wm+k+ and Tk=∑i=mm+k−1ϕwi=∑i=mm+k−1si.
More details on the concept and nature of the impulsive semidynamical system can be found in [37–41].Lemma 1.
The order-k periodic solution p,y=ξt,ηt of system(2)dptdt=Fp,y,dytdt=Gp,y,p+=p+αp,y,y+=y+βp,y,is orbitally asymptotically stable and enjoys the property of the asymptotic phase if the Floquet multiplier χ1 satisfies the condition χ1<1, where(3)χ1=∏k=1qΔkexp∫0T∂F∂pξt,ηt+∂G∂yξt,ηtdt,with(4)Δk=F+∂β/∂y∂Φ/∂p−∂β/∂p∂Φ/∂y+∂Φ/∂p+G+∂α/∂p∂Φ/∂y−∂α/∂y∂Φ/∂p+∂Φ/∂yF∂Φ/∂p+G∂Φ/∂y,where F,G,∂α/∂p,∂α/∂y,∂β/∂p,∂β/∂y,∂Φ/∂p,∂Φ/∂y are calculated at the point ξtk,ηtk, F+=Fξtk+,ηtk+, and G+=Gξtk+,ηtk+. And Φ is continuously differentiable with respect to p,y. p,y∉M is equivalent to ϕp,y≠0.


## 3. Herbivore-Plankton Interaction Model and Its Properties

Ntr et al. [42] proposed the following herbivore-plankton interaction model:(5)dPtdt=PtK−Pt−BYtC+Pt,dYtdt=DYtPtC+Pt−AYt,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]:(6)dptdt=ptk−pt−yt1+pt,dytdt=ytpt1+pt−ayt,y<y¯,Δpt=−αpt,Δyt=−βyt,y=y¯,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.Therefore, we establish the state feedback control model as follows:(7)dptdt=ptk−pt−yt1+pt,dytdt=ytpt1+pt−ayt,p>p¯,△pt=pt+−pt=τ1+θpt,△yt=yt+−yt=−δy2tyt+β,p=p¯,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−δyt/yt+βyt and pt+τ/1+θpt, 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 fy=1−δy/y+βy, and by calculating, we get the function fy is increasing in 0,+∞.The qualitative analysis of system (8) is discussed as follows when system (7) has no pulse:(8)dptdt=ptk−pt−yt1+pt,dytdt=ytpt1+pt−ayt.Theorem 1.
The extinction equilibrium pointO0,0 of system (8) is unstable, the boundary equilibrium point D0k,0 is the saddle point, and D∗p∗,y∗ is the internal equilibrium point [43] (Figure 1).(i)
IfH1:k<2p∗+2, then internal equilibrium point D∗ is stable(ii)
IfH2:k>2p∗+2, then internal equilibrium point D∗ is unstable and has a limit cycleIn addition, Hopf branch occurs in the case of k=2p∗+2, where p∗ meets the following condition:(9)p∗=a1+p∗2k−p∗,y∗=p∗a1+p∗.Figure 1
(a) Phase diagram of system (7) with k=1.8anda=0.15. (b) The existence of the limit cycle with k=3anda=0.08. (c) Stability of the limit cycle.
(a)
(b)
(c)To facilitate the study of the dynamics of system (7), we define the following two isoclines:(10)L1:y=1+pk−p,L2:y=pa1+p.


## 4. 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(11)Ω=p,y∣p≥0,y≥0.Therefore, we first define the following two sets:(12)SM=p,y∣p=p¯,y≥0,SN=p,y∣p=p¯+τ1+θp¯,y≥0.It is easy to know setSM and SN represents two parallel lines L3:p=p¯ and L4:p=p¯+τ/1+θp¯ in Ω. We represent the intersection of L4 and L1 as Ap¯+τ/1+θp¯,yA and the intersection of L3 and L1 as Bp¯,yB, where yA=1+p¯+τ/1+θp¯k−p¯−τ/1+θp¯ and yB=1+p¯k−p¯. And there exists a curve Γ that starts at point A and intersects with L3 at point Cp¯,yC.According to the above definition, we construct the impulsive semidynamical system of system (7).Define the impulsive set as(13)M=p,y∣p=p¯,y≥yC,while the continuous function uses the expression as follows:(14)I:p¯,y∈M⟶p+,y+=p¯+τ1+θp¯,1−δyy+βy,and the impulsive function fy=1−δy/y+βy is increasing in 0,+∞. So, we represent the phase set as follows (Figure 2):(15)N=IM=p+,y+∣p+=p¯+τ1+θp¯,y+≥1−δyCyC+βyC.Figure 2
Phase set of system (7) with k=2anda=0.1.Unless otherwise specified, in this article, we assume that the initial pointA0+p0+,y0+ belongs to N. The trajectory through the point A0+ is expressed as follows:(16)Πt,t0,A0+=Πpt,t0,p0+,y0+,yt,t0,p0+,y0+.Next, we define the Poincaré map of system (6) in the case H1.Since the internal equilibrium pointD∗ is the stable point, we assume that p¯>p∗. Then, for any point Ak+=p¯+τ/1+θp¯,yk+∈SN, the trajectory passing through point Ak+ will reach the impulse set M through time t1, so(17)Πt1,p¯+τ1+θp¯,yk+=Πt1,p¯,yk+1=Πp+t1,p¯+τ1+θp¯,yk+,y+t1,p¯+τ1+θp¯,yk+=Πp¯,yk+1,where(18)yk+1=y+t1,p¯+τ1+θp¯,yk+.This indicates that the ordinateyk+1 of the point at which the initial point on the phase set reaches the pulse set is determined by yk+. Then, we define yk+1≜φyk+.Therefore, the Poincaré map can be expressed as follows:(19)yk+1+=1−δyk+1yk+1+βyk+1=1−δφyk+φyk++βφyk+=FMyk+.Then, system (8) is represented as the following scalar differential equation:(20)dydp=yp/1+p−aypk−p−y/1+p≜lp,y,yp¯+τ1+θp¯=y0+.Obviously,lp,y is continuous and differentiable in Ω. We make p0+=p¯+τ/1+θp¯ and y0+=μ with μ∈0,+∞ for which p0+,y0+∈Ω.Define(21)yx=yx;p¯+τ1+θp¯,μ=yx,μ,p¯≤x≤p¯+τ1+θp¯.According to model (20), function yx,μ with initial value of p¯+τ/1+θp¯,μ can be expressed as(22)yx,μ=μ+∫xp¯+τ/1+θp¯lt,yt,μdt.Therefore, the form of the Poincaré map is(23)FMμ=yp¯,μ−δy2p¯,μyp¯,μ+β.Theorem 2.
We assume that system (8) has a stable internal equilibrium point which meets k<2p∗+1 and p¯>p∗; then, the Poincaré map FMμ of system (8) contains the following properties (Figure 3):(i)
The domain ofFMμ is 0,+∞(ii)
FMμ is increasing on yA,+∞ and decreasing on 0,yA(iii)
Poincaré mapFMμ is continuously differentiable(iv)
FMμ has a unique fixed point(v)
FMμ takes the minimum value at yA and has no maximum valueFigure 3
Image of Poincaré mapFMμ and the parameters fixed as r=2.5,φ1=0.2,a1=0.3,φ2=0.01,a2=0.05,andET=2.5. (a) b = 0.7. (b) b = 0.98.
(a)
(b)Proof.
(i)
According to Theorem1 and p¯>p∗, for any point A+=p¯+τ/1+θp¯,μ∈SN, where μ≥0 and the trajectory through point A+ always intersects the impulse set M. Therefore, the function FMμ is meaningful when μ∈0,+∞. So, the domain of FMμ is 0,+∞.(ii)
We first prove the monotonicity of the functionFMμ on the interval yA,+∞. Choose any two points with μk1+, μk2+∈yA,+∞, where μk1+<μk2+. Then, after time ti, the trajectory Πti,p¯+τ/1+θp¯,μki+=Πti,p¯,yki+1∈M,i=1,2, where yki+1∈yC,+∞. From the properties of the solution of the differential equation, we get that yk1+1<yk2+1; then, from system (19), the Poincaré map FMμk1+<FMμk2+. Therefore, FMμ is increasing on yA,+∞.
Ifμki+∈0,yA with μk1+<μk2+, the trajectory Πt,p¯+τ/1+θp¯,μki+ across the isocline L1 and reach SN at points Ak1′p¯+τ/1+θp¯,μk1′ and Ak2′p¯+τ/1+θp¯,μk2′. From the Cauchy–Lipschitz theorem, we have μk2′>μk1′ and μki′∈yA,+∞. According to the above proof, FMμk1+>FMμk2+; therefore, FMμ is decreasing on 0,yA.(iii)
From system (20), it is clear that lp,y is continuous and differentiable in the first quadrant. According to the Cauchy–Lipschitz theorem and system (19), the function FMμ is continuously differentiable in R2+.(iv)
First, we consider the value of the Poincaré mapFMμ of yA and prove the existence of the fixed point. If FMyA=yA, then the Poincaré map FMμ possess a fixed point.
IfFMyA<yA, let yq+=FMyA∈0,yA. From property (ii), we get when yq+<yA, then FMyq+>FMyA=yq+. In the light of property (iii) and the theorem of zero existence, there exists at least a point y^∈yq+,yA, which meets FMy^=y^. From property (ii), FMμ has at least a fixed point on 0,yA.
IfFMyA>yA, then FMyA−yA>0. When y0+⟶∞, from model (19), we obtain y1<y0+ and y1+=y11−δy1/y1+β<y1. Then, FMy0+=y1+<y0+. From the monotonicity and continuity of FMμ, FMμ has at least a fixed point y^∈yA,+∞ in which FMy^=y^. Thus, FMμ possess the fixed point on 0,+∞.
Next, the uniqueness of the fixed point is proved as follows. We assume that system (7) admits two fixed points y^1 and y^2, where y^1>y^2. Define Φy^1y^2p=y^1p−y^2p. Since(24)=Φy^1y^2′p=y^1′p−y^2′p=y^1p/1+p−ay^1pk−p−y^1/1+p−y^2p/1+p−ay^2pk−p−y^2/1+p=g′ξy^1−y^2,where(25)gy=yp/1+p−aypk−p−y/1+p,g′y=pkp−p2+ay2/1+p+2aypp−kpk−p−y/1+p2,therefore, when k<p2−ay2/p, we have g′y<0. Then, Φy^1y^2′p<0 which means that Φy^1y^2′p is a monotonically decreasing function and Φy^1y^2p¯+τ/1+θp¯<Φy^1y^2p¯. Then,(26)β=δy^12y^1−y^1+−y^1=δy^2+Φy^1y^2p¯+τ/1+θp¯2y^2+Φy^1y^2p¯+τ/1+θp¯−y^2+Φy^1y^2p¯−y^2+Φy^1y^2p¯+τ/1+θp¯>δy^22y^2−y^2+−y^2=β,which leads to a contradiction. Thus, the fixed point of system (7) is unique when k<p2−ay2/p.(v)
From property (ii), we obtain thatFMμ is decreasing on 0,yA and increasing on yA,+∞. Thus, for any μ∈0,+∞, there is FMμ≥FMyA. So, when μ=yA, the function FMμ reaches the minimal value which is also the minimum value. And the minimum value is FMyA=yp¯,yA−δy2p¯,yA/yp¯,yA+β.
SinceFMμ is increasing on yA,+∞, then FMμ has only a minimum value on 0,+∞ and no maximum value.


## 5. Periodic Solution of the System

### 5.1. Boundary Periodic Solution

In the absence of herbivores andδ=0, system (7) possess the boundary periodic solution. Thus, we represent the system as(27)dptdt=ptk−ptpt>p¯,pt+=pt+τ1+θpt,pt=p¯,where the initial condition is p0+=p¯+τ/1+θp¯, and by calculating the first equation, we have(28)pt=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kt−1+θp¯p¯−τ.Supposing the solution starting frompt,0 reaches the line L3 at time T, one has(29)p¯=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kT−1+θp¯p¯−τ.Solving the above equation onT, then(30)T=−1kln−θkp¯2−θp¯3+kτ−p¯τ+kp¯−p¯2p¯−kθp¯+θp¯2+τ−k+p¯.Thus, the boundary periodic solutionpTt,0 of model (7) with period T is(31)pTt=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kt−1+θp¯p¯−τ.Theorem 3.
When conditionχ1<1 is satisfied, then the boundary periodic solution pTt,0 is orbitally asymptotically stable, and(32)χ1=Δ1exp∫0TQtdt,where(33)Δ1=p¯+τ/1+θp¯k−p¯−τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯.Proof.
LetFp,y=pk−p−y/1+p, Gp,y=yp/1+p−ay, αp,y=τ/1+θp, βp,y=−δy2/y+β, and Φp,y=p−p¯,pTT,yTT=p¯,0,pTT+,yTT+=p¯+τ/1+θp¯,0.
Then,(34)∂F∂p=k−2p−y1+p2,∂G∂y=p1+p−2ay,∂α∂p=−τθ1+θp2,∂β∂y=−δy2+2βδyy+β2,∂Φ∂p=1,∂α∂y=∂β∂p=∂Φ∂y=0.Δ1=F+∂β/∂y∂Φ/∂p−∂β/∂p∂Φ/∂y+∂Φ/∂p+G+∂α/∂p∂Φ/∂y−∂α/∂y∂Φ/∂p+∂Φ/∂yF∂Φ/∂p+G∂Φ/∂y=F+p¯+τ/1+θp¯,01+∂β/∂gFp¯,0=p¯+τ/1+θp¯k−p¯−τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯∫0T∂F∂ppTT,yTT+∂G∂ypTT,yTTdt=∫0Tk−2pTt+pTt1+pTtdt=∫0TQtdt.
Furthermore,(35)χ1=Δ1exp∫0T∂F∂ppTT,yTT+∂G∂gpTT,yTTdt=Δ1exp∫0TQtdt.
Therefore, from Lemma1, we obtain the conclusion of Theorem 3 when χ1<1.


### 5.2. Order-k Periodic Solution and Its Stability Analysis

According to Theorem2 (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.ForA0+p¯+τ/1+θp¯,y0+∈N, the trajectory of the initial point A0+ will reach the impulse set M, which is applied to the point A1+p¯+τ/1+θp¯,y1+ by the impulse. In the light of the definition of the Poincaré map, we have FMy0+=y1+, and repeating the process above, then(36)FMy1+=FMFMy0+≜FM2y0+=y2+.Further promotion,(37)yk+=FMky0+.The sequenceyk+ is available, k = 0, 1, 2, ….Theorem 4.
The order-1 period solutionξt,ηt is orbitally asymptotically stable if(38)p¯+τ/1+θp¯k−p¯−τ/1+θp¯−η0/1+p¯+τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯−η0/1+p¯exp∫0TUtdt<1,where(39)Ut=k−2p−y1+p2+p1+p−2ay.Proof.
We define the starting and the ending point of the order-1 periodic solution asξT,ηT=p¯,η0 and ξT+,ηT+=p¯+τ/1+θp¯,η0−δη02/η0+β, respectively.
Thus,(40)Δ2=F+p¯+τ/1+θp¯,η0−δη02/η0+β1−δy2+2βδy/y+β2Fp¯,η0=p¯+τ/1+θp¯k−p¯−τ/1+θp¯−η0/1+p¯+τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯−η0/1+p¯,∫0T∂F∂p+∂G∂ydt=∫0Tk−2p−y/1+p2+p/1+p−2aydt=∫0TUtdt.
The multiplierχ2 is obtained:(41)χ2=Δ2exp∫0TUtdt.
From (38), we obtain χ2<1; thus, the order-1 periodic solution is locally stable.Theorem 5.
When conditionsH1 and FMyA>yA are established, then the order-1 periodic solution of system (7) is globally asymptotic stable.Proof.
According to Theorem2, when FMyA>yA, system (7) possesses unique fixed point y^, which means system (7) has an order-1 periodic solution, where y^∈yA,+∞. Also, we get the Poincaré map FMμ which increases on yA,+∞.
For any trajectory in which the initial point isA0+p¯+τ/1+θp¯,y0+, when yA<y0+<y^, since FMμ is increasing on yA,+∞ and there exists a unique fixed point, then yA<FMyA<FMy0+=y1+<FMy^=y^ and y1+<FMy1+=y2+<y^. It can be inferred from this that yA<FMy0+<…<FMn−1y0+<FMny0+<…<y^. According to the monotonically bounded sequence, we get limn⟶+∞FMny0+=y^.
Wheny^<y0+, by using a similar approach, we have y^=FMy^<FMy0+<y0+ and FMy1+=y2+<y1+=FMy0+. Furthermore, we get y^<…<FMny0+<FMn−1y0+<…<y0+. Then, limn⟶+∞FMny0+=y^. Therefore, the order-1 periodic solution of system (7) is globally asymptotic stable.Theorem 6.
If conditionsH1, FMyA<yA, and FM2yA<yA are established, then model (7) has a stable order-1 or order-2 periodic solution.Proof.
For any pointA0+p¯+τ/1+θp¯,y0+∈N, where y0+∈0,+∞, we discuss the following two cases of function FMy0+ which are defined by point y0+.(i)
Wheny0+∈yA,+∞, since the function FMμ monotonically increases and has no fixed point when μ∈yA,+∞, therefore, there must be a positive integer m so that ym−1+=FMym−2+>yA and FMym−1+=ym+<yA; then, ym+=FMym−1+>FMyA and ym+∈FMyA,yA(ii)
Wheny0+∈0,yA, since the function FMμ which decreases in 0,yA and FM2yA<yA and FMyA<yA, there must exist m¯ which satisfies FMyA<FMm¯y0+<yA, where m¯≥1
Through the above discussion, wheny0+∈0,+∞, there always exist a positive integer m which satisfies FMyA<FMmy0<yA. Since FMμ is decreasing in FMyA,yA, thus,(42)FMFMyA,yA=FMyA,FM2yA⊂FMyA,yA.
So, we only need to discuss the case of functionFMμ in interval FMyA,yA. A0+p¯+τ/1+θp¯,y0+ is the trajectory of the initial point, where y0+∈FMyA,yA. After the action of the impulse, we get the sequence yn+,n=0,1,2…, where yn+=FMny0+,n=1,2….
According to the magnitude ofyn+, there exist four situations:Situation 1.
y
A
≥
F
M
y
0
+
>
y
0
+
>
F
M
y
1
+
≥
F
M
y
A (Figure 4(a)); then, yA≥y1+>y0+>y2+>FMyA, and thus, we have y3+=FMy2+>FMy0+=y1+ and y4+=FMy3+<FMy1+=y2+, so y3+>y1+>y0+>y2+>y4+. After an infinite number of pulses, we get(43)yA≥…≥y2n+1+>y2n−1+>…>y1+>y0+>y2+>…>y2n+>y2n+2+>…>FMyA.Figure 4
Path curve of system (7). (a)yA≥y1+>y0+>y2+≥FMyA. (b)yA≥y1+>y2+>y0+≥FMgA.
(a)
(b)Situation 2.
y
A
≥
F
M
y
0
+
>
F
M
y
1
+
>
y
0
+
≥
F
M
y
A (Figure 4(b)); furthermore, we have FMy1+=y2+<FMy2+=y3+<FMy0+=y1+ and FMy3+=y4+>FMy1+=y2+. After an infinite number of pulses, we get(44)yA≥y1+>…≥y2n−1+>y2n+1+>…>y2n+2+>y2n+>…>y2+>y0+>FMyA.Situation 3.
y
A
>
y
0
+
>
F
M
y
1
+
>
F
M
y
0
+
≥
F
M
y
A; from the above similar methods, we obtain(45)yA>y0+>y2+>…>y2n+>y2n+2+>…>y2n+1+>y2n−1+>…>y1+≥FMyA.Situation 4.
y
A
>
F
M
y
1
+
>
y
0
+
>
F
M
y
0
+
≥
F
M
y
A; from the above similar methods, we get(46)yA≥…>y2n+2+>y2n+>…>y2+>y0+>y1+>…>y2n−1+>y2n+1+>…≥FMyA.
For situations 2 and 3, sequencesy2n+ and y2n+1+ are monotonically bounded. Then, we have limn⟶∞y2n+=limn⟶∞y2n−1+ or limn⟶∞n⟶∞y2n+=y1^ and limn⟶∞y2n−1+=y2^, where y1^≠y2^. This indicates that, in situations 2 and 3, system (7) has stable order-1 or order-2 periodic solution.
From situations 1 and 4,limn⟶∞y2n+1+=y1∗≤yA and limn⟶∞y2n−1+=y2∗≥FMyA. Therefore, in situations 1 and 4, system (7) has stable order-2 periodic solution.
Theorem6 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. Therefore, Theorem 7 is given as follows:Theorem 7.
WhenFMyA<yA, then the sufficient and necessary conditions for the global asymptotic stability of the order-1 periodic solution of system (7) are FM2y0+<y0+ for any y0+∈y^,yA, where FMy^=y^.Proof.
From Theorem2, a stable order-1 periodic solution is available in system (7) when FMyA<yA. Then, system (7) has y^∈FMyA,yA in which FMy^=y^. For any y0+∈y^,yA, make yn+=FMny0+,n=1,2,…, and according to the properties of Poincaré map FMμ, we have FMyA<y1+<y^<y2+<y0+<yA; furthermore, we get FMyA<y2n+1+<y^<y2n+<yA. From the monotone boundedness of the sequence, we know limn⟶∞y2n+=limn⟶∞y2n+1+=y^. Therefore, 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 anyy0+∈y^,yA satisfies FM2y0+<y0+. If this condition is not met, then there exists minimum y0+∈y^,yA which makes FM2y0+≥y0+. From Theorem 6, we get that, for any small ε>0, there exists y1+∈y^−ε,y^ which makes FM2y1+<y1+. From the continuity of FMμ, there exists y^^∈y1+,y0+ which makes FM2y^^=y^^. Therefore, system (7) has order-2 periodic solution, which is contradictory to the global stability of order-1 periodic solution.Theorem 8.
WhenFMyA<yA and FM2yA<ym+, where FMym+=yA, ym+=miny+,FMy+=yA, then system (7) has the order-3 periodic solution.Proof.
SinceFMyA<yA, from Theorem 2, we get that system (7) has unique y^∈FMyA,yA, which makes FMy^=y^. Since Poincaré map FMμ is continuous in closed intervals 0,y^, FM0>0, and FMy^=y^, from the intermediate value theorem, there is ym+∈0,y^ which makes FMym+=yA, and then FM3ym+=FM2yA<ym+. Thus, we have FM3ym+−ym+<0.
On the contrary, according to the expression of Poincaré map, we obtain thatFM30>0. So, ∃y∈0,ym+ which makes FM3y=y. This indicates that system (7) has order-3 period solution with p¯+τ/1+θp¯,y as the initial point.
In the same way, we can prove that ifFMk−1yA<ym+, FMym+=yA Z, system (7) possesses the order-k k≥2 periodic solution.


## 5.1. Boundary Periodic Solution

In the absence of herbivores andδ=0, system (7) possess the boundary periodic solution. Thus, we represent the system as(27)dptdt=ptk−ptpt>p¯,pt+=pt+τ1+θpt,pt=p¯,where the initial condition is p0+=p¯+τ/1+θp¯, and by calculating the first equation, we have(28)pt=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kt−1+θp¯p¯−τ.Supposing the solution starting frompt,0 reaches the line L3 at time T, one has(29)p¯=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kT−1+θp¯p¯−τ.Solving the above equation onT, then(30)T=−1kln−θkp¯2−θp¯3+kτ−p¯τ+kp¯−p¯2p¯−kθp¯+θp¯2+τ−k+p¯.Thus, the boundary periodic solutionpTt,0 of model (7) with period T is(31)pTt=1+θp¯p¯+τk1+θp¯p¯−kθp¯+τ−kexp−kt−1+θp¯p¯−τ.Theorem 3.
When conditionχ1<1 is satisfied, then the boundary periodic solution pTt,0 is orbitally asymptotically stable, and(32)χ1=Δ1exp∫0TQtdt,where(33)Δ1=p¯+τ/1+θp¯k−p¯−τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯.Proof.
LetFp,y=pk−p−y/1+p, Gp,y=yp/1+p−ay, αp,y=τ/1+θp, βp,y=−δy2/y+β, and Φp,y=p−p¯,pTT,yTT=p¯,0,pTT+,yTT+=p¯+τ/1+θp¯,0.
Then,(34)∂F∂p=k−2p−y1+p2,∂G∂y=p1+p−2ay,∂α∂p=−τθ1+θp2,∂β∂y=−δy2+2βδyy+β2,∂Φ∂p=1,∂α∂y=∂β∂p=∂Φ∂y=0.Δ1=F+∂β/∂y∂Φ/∂p−∂β/∂p∂Φ/∂y+∂Φ/∂p+G+∂α/∂p∂Φ/∂y−∂α/∂y∂Φ/∂p+∂Φ/∂yF∂Φ/∂p+G∂Φ/∂y=F+p¯+τ/1+θp¯,01+∂β/∂gFp¯,0=p¯+τ/1+θp¯k−p¯−τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯∫0T∂F∂ppTT,yTT+∂G∂ypTT,yTTdt=∫0Tk−2pTt+pTt1+pTtdt=∫0TQtdt.
Furthermore,(35)χ1=Δ1exp∫0T∂F∂ppTT,yTT+∂G∂gpTT,yTTdt=Δ1exp∫0TQtdt.
Therefore, from Lemma1, we obtain the conclusion of Theorem 3 when χ1<1.


## 5.2. Order-k Periodic Solution and Its Stability Analysis

According to Theorem2 (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.ForA0+p¯+τ/1+θp¯,y0+∈N, the trajectory of the initial point A0+ will reach the impulse set M, which is applied to the point A1+p¯+τ/1+θp¯,y1+ by the impulse. In the light of the definition of the Poincaré map, we have FMy0+=y1+, and repeating the process above, then(36)FMy1+=FMFMy0+≜FM2y0+=y2+.Further promotion,(37)yk+=FMky0+.The sequenceyk+ is available, k = 0, 1, 2, ….Theorem 4.
The order-1 period solutionξt,ηt is orbitally asymptotically stable if(38)p¯+τ/1+θp¯k−p¯−τ/1+θp¯−η0/1+p¯+τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯−η0/1+p¯exp∫0TUtdt<1,where(39)Ut=k−2p−y1+p2+p1+p−2ay.Proof.
We define the starting and the ending point of the order-1 periodic solution asξT,ηT=p¯,η0 and ξT+,ηT+=p¯+τ/1+θp¯,η0−δη02/η0+β, respectively.
Thus,(40)Δ2=F+p¯+τ/1+θp¯,η0−δη02/η0+β1−δy2+2βδy/y+β2Fp¯,η0=p¯+τ/1+θp¯k−p¯−τ/1+θp¯−η0/1+p¯+τ/1+θp¯1−δy2+2βδy/y+β2p¯k−p¯−η0/1+p¯,∫0T∂F∂p+∂G∂ydt=∫0Tk−2p−y/1+p2+p/1+p−2aydt=∫0TUtdt.
The multiplierχ2 is obtained:(41)χ2=Δ2exp∫0TUtdt.
From (38), we obtain χ2<1; thus, the order-1 periodic solution is locally stable.Theorem 5.
When conditionsH1 and FMyA>yA are established, then the order-1 periodic solution of system (7) is globally asymptotic stable.Proof.
According to Theorem2, when FMyA>yA, system (7) possesses unique fixed point y^, which means system (7) has an order-1 periodic solution, where y^∈yA,+∞. Also, we get the Poincaré map FMμ which increases on yA,+∞.
For any trajectory in which the initial point isA0+p¯+τ/1+θp¯,y0+, when yA<y0+<y^, since FMμ is increasing on yA,+∞ and there exists a unique fixed point, then yA<FMyA<FMy0+=y1+<FMy^=y^ and y1+<FMy1+=y2+<y^. It can be inferred from this that yA<FMy0+<…<FMn−1y0+<FMny0+<…<y^. According to the monotonically bounded sequence, we get limn⟶+∞FMny0+=y^.
Wheny^<y0+, by using a similar approach, we have y^=FMy^<FMy0+<y0+ and FMy1+=y2+<y1+=FMy0+. Furthermore, we get y^<…<FMny0+<FMn−1y0+<…<y0+. Then, limn⟶+∞FMny0+=y^. Therefore, the order-1 periodic solution of system (7) is globally asymptotic stable.Theorem 6.
If conditionsH1, FMyA<yA, and FM2yA<yA are established, then model (7) has a stable order-1 or order-2 periodic solution.Proof.
For any pointA0+p¯+τ/1+θp¯,y0+∈N, where y0+∈0,+∞, we discuss the following two cases of function FMy0+ which are defined by point y0+.(i)
Wheny0+∈yA,+∞, since the function FMμ monotonically increases and has no fixed point when μ∈yA,+∞, therefore, there must be a positive integer m so that ym−1+=FMym−2+>yA and FMym−1+=ym+<yA; then, ym+=FMym−1+>FMyA and ym+∈FMyA,yA(ii)
Wheny0+∈0,yA, since the function FMμ which decreases in 0,yA and FM2yA<yA and FMyA<yA, there must exist m¯ which satisfies FMyA<FMm¯y0+<yA, where m¯≥1
Through the above discussion, wheny0+∈0,+∞, there always exist a positive integer m which satisfies FMyA<FMmy0<yA. Since FMμ is decreasing in FMyA,yA, thus,(42)FMFMyA,yA=FMyA,FM2yA⊂FMyA,yA.
So, we only need to discuss the case of functionFMμ in interval FMyA,yA. A0+p¯+τ/1+θp¯,y0+ is the trajectory of the initial point, where y0+∈FMyA,yA. After the action of the impulse, we get the sequence yn+,n=0,1,2…, where yn+=FMny0+,n=1,2….
According to the magnitude ofyn+, there exist four situations:Situation 1.
y
A
≥
F
M
y
0
+
>
y
0
+
>
F
M
y
1
+
≥
F
M
y
A (Figure 4(a)); then, yA≥y1+>y0+>y2+>FMyA, and thus, we have y3+=FMy2+>FMy0+=y1+ and y4+=FMy3+<FMy1+=y2+, so y3+>y1+>y0+>y2+>y4+. After an infinite number of pulses, we get(43)yA≥…≥y2n+1+>y2n−1+>…>y1+>y0+>y2+>…>y2n+>y2n+2+>…>FMyA.Figure 4
Path curve of system (7). (a)yA≥y1+>y0+>y2+≥FMyA. (b)yA≥y1+>y2+>y0+≥FMgA.
(a)
(b)Situation 2.
y
A
≥
F
M
y
0
+
>
F
M
y
1
+
>
y
0
+
≥
F
M
y
A (Figure 4(b)); furthermore, we have FMy1+=y2+<FMy2+=y3+<FMy0+=y1+ and FMy3+=y4+>FMy1+=y2+. After an infinite number of pulses, we get(44)yA≥y1+>…≥y2n−1+>y2n+1+>…>y2n+2+>y2n+>…>y2+>y0+>FMyA.Situation 3.
y
A
>
y
0
+
>
F
M
y
1
+
>
F
M
y
0
+
≥
F
M
y
A; from the above similar methods, we obtain(45)yA>y0+>y2+>…>y2n+>y2n+2+>…>y2n+1+>y2n−1+>…>y1+≥FMyA.Situation 4.
y
A
>
F
M
y
1
+
>
y
0
+
>
F
M
y
0
+
≥
F
M
y
A; from the above similar methods, we get(46)yA≥…>y2n+2+>y2n+>…>y2+>y0+>y1+>…>y2n−1+>y2n+1+>…≥FMyA.
For situations 2 and 3, sequencesy2n+ and y2n+1+ are monotonically bounded. Then, we have limn⟶∞y2n+=limn⟶∞y2n−1+ or limn⟶∞n⟶∞y2n+=y1^ and limn⟶∞y2n−1+=y2^, where y1^≠y2^. This indicates that, in situations 2 and 3, system (7) has stable order-1 or order-2 periodic solution.
From situations 1 and 4,limn⟶∞y2n+1+=y1∗≤yA and limn⟶∞y2n−1+=y2∗≥FMyA. Therefore, in situations 1 and 4, system (7) has stable order-2 periodic solution.
Theorem6 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. Therefore, Theorem 7 is given as follows:Theorem 7.
WhenFMyA<yA, then the sufficient and necessary conditions for the global asymptotic stability of the order-1 periodic solution of system (7) are FM2y0+<y0+ for any y0+∈y^,yA, where FMy^=y^.Proof.
From Theorem2, a stable order-1 periodic solution is available in system (7) when FMyA<yA. Then, system (7) has y^∈FMyA,yA in which FMy^=y^. For any y0+∈y^,yA, make yn+=FMny0+,n=1,2,…, and according to the properties of Poincaré map FMμ, we have FMyA<y1+<y^<y2+<y0+<yA; furthermore, we get FMyA<y2n+1+<y^<y2n+<yA. From the monotone boundedness of the sequence, we know limn⟶∞y2n+=limn⟶∞y2n+1+=y^. Therefore, 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 anyy0+∈y^,yA satisfies FM2y0+<y0+. If this condition is not met, then there exists minimum y0+∈y^,yA which makes FM2y0+≥y0+. From Theorem 6, we get that, for any small ε>0, there exists y1+∈y^−ε,y^ which makes FM2y1+<y1+. From the continuity of FMμ, there exists y^^∈y1+,y0+ which makes FM2y^^=y^^. Therefore, system (7) has order-2 periodic solution, which is contradictory to the global stability of order-1 periodic solution.Theorem 8.
WhenFMyA<yA and FM2yA<ym+, where FMym+=yA, ym+=miny+,FMy+=yA, then system (7) has the order-3 periodic solution.Proof.
SinceFMyA<yA, from Theorem 2, we get that system (7) has unique y^∈FMyA,yA, which makes FMy^=y^. Since Poincaré map FMμ is continuous in closed intervals 0,y^, FM0>0, and FMy^=y^, from the intermediate value theorem, there is ym+∈0,y^ which makes FMym+=yA, and then FM3ym+=FM2yA<ym+. Thus, we have FM3ym+−ym+<0.
On the contrary, according to the expression of Poincaré map, we obtain thatFM30>0. So, ∃y∈0,ym+ which makes FM3y=y. This indicates that system (7) has order-3 period solution with p¯+τ/1+θp¯,y as the initial point.
In the same way, we can prove that ifFMk−1yA<ym+, FMym+=yA Z, system (7) possesses the order-k k≥2 periodic solution.


## 6. Numerical Simulations

An impulsive semidynamic system of plankton and herbivore is established in this paper. The numerical simulation illustrates the correctness of our arguments.We set the parameter value ask=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,δ=0.25,andβ=0.8, and obtain that system (7) possesses a globally asymptotically stable order-1 periodic solution if FMyA>yA (Figure 5(a)). Furthermore, let k=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,δ=0.65,andβ=0.4. We get that model (7) has unique order-1 periodic solution when FMyA<yA, and it is globally asymptotically stable (Figure 6(a)). Figures 5(b) and 5(c) are time series diagram when FMyA>yA, and Figures 6(b) and 6(c) are time series diagram when FMyA<yA. 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.Figure 5
Phase portrait and time series of system (7) when FMyA>yA. And the parameters are fixed as k=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,δ=0.25,andβ=0.8.
(a)
(b)
(c)Figure 6
Phase portrait and time series of system (7) when FMyA<yA. And the parameters are fixed as k=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,δ=0.65,andβ=0.4.
(a)
(b)
(c)From Figure7, 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.Figure 7
Periodic solution of system (7) at different initial points. And the parameters are fixed as k=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,δ=0.65,andβ=0.4.When herbivores are absent andδ=0, we get the expression of the boundary period solution. Through analysis, the boundary periodic solution is asymptotically stable under certain conditions. Thus, 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. Thus, we get the rationality and necessity of taking pulse control of this paper.Figure 8
Phase portrait and time series when not hunting herbivores. And the parameters are fixed ask=1.8,a=0.15,τ=0.6,p¯=0.6,θ=0.8,andδ=0.
(a)
(b)
(c)


## 7. 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. Then, 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. The 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 plankton will continue to decrease, which will reduce the food for herbivores and affect yield.We summarize some of the innovations in this paper: (1) the impulse semidynamic system studied in this paper uses the new form of pulse, and the control parameters are changed according to the actual situation. This makes the system have more complex dynamic properties, which is worthy of our in-depth study. (2) In this paper, the Poincaré map is defined and used as a tool to carry out more comprehensive dynamic analysis of the system. For example, by analyzing the impulse function, the precise range of the pulse and phase set is obtained, and a more comprehensive and detailed study of periodic solutions is carried out. Compared with [43], we not only prove the existence and stability of the order-1 periodic solution but also give the necessary and sufficient condition for the global stability of the order-1 periodic solution and the existence conditions of the order-k k≥2 periodic solution. (3) Through the analysis and numerical simulation of the model, the practical biological significance of its complex dynamics is revealed. When k>2p∗+2, the system has more complicated dynamic properties. In future research work, we will further discuss the complex dynamics of the system and optimize to obtain maximum benefits.



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