# Modelling and Qualitative Analysis of Water Hyacinth Ecological System with Two State-Dependent Impulse Controls

Authors: Jin-Bo Fu; Lan-Sun Chen
Journal: Complexity
 (2018)

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## Abstract

Water hyacinth and its ecological invasion have negative impacts on diversity of indigenous species and ecosystems, which becomes one of the hotspots in current ecological research. In this paper, a water hyacinth ecological system with two state-dependent impulse controls is studied. Firstly, we define the successor functions of semicontinuous dynamic system and give existence theorems of order-1 periodic solution and order-2 periodic solution of such system. Secondly, we analyze singular points of the system without impulsive state feedback control qualitatively and get the condition for focus point. Thirdly, we obtain the sufficient condition under which the system has an order-1 or order-2 periodic solution through the method of successor function and prove the stability of the order-1 or order-2 periodic solution by the analogue of Poincaré’s criterion. Furthermore, some examples and numerical simulations are given to illustrate our results.


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## Body

## 1. Introduction

The water hyacinth, a water plant with a showy purple flower, is a native of the Amazon Basin and is now treated as the most important nuisance aquatic plant worldwide. Rapid spread of water hyacinth causes serious environmental problems, which is recognized throughout the world. Water hyacinth, known as one of the top ten creatures into the en-roach on grass, has spread by waters in the south of China in recent years, becoming a great ecological and social harm. The expansive water hyacinth sheets have blocked shipping lanes, prevented commercial ships from entering ports and local fishing vessels from leaving the shore, and interfered with the intake of water by industries. Meanwhile, the invasive weed has created anoxic conditions in the lake, thus raising toxicity and disease levels. The resulting stagnant water is an ideal breeding ground for malarial mosquitoes and schistosomiasis and a detriment for the functioning of fishing industries. As for the ecological harm, biological control, development, and utilization of water hyacinth, see [1–5] and the reference cited therein.Nowadays, biological control measures are widely used to control water hyacinth. Biological control is to introduce natural enemies from the origin, establish populations, and control water hyacinth in the long term. The biological control of water hyacinth has the advantages of long-lasting prevention, low cost, and safe environment.Recently, impulsive state feedback control differential equations have aroused scholars’ enormous interest (see [6–20]). Chen et al. [6] and Chen [7] starting from a practical problem of the pest control established a class of pest prevention model with impulsive feedback control. The geometric theory of semicontinuous dynamical systems is presented, and by applying this theory, it is proved for this model to have at least one order-one periodic solution. Nie et al. [16–18] and Tian et al. [19] put forward a class of predator-prey models with state-dependent impulsive effects. By using Poincaré map and Lambert W function, the criteria for the existence and stability of a semitrivial solution and positive periodic solution of system are obtained. Zhao et al. [20] using successor functions and Poincaré-Bendixson theorem of impulsive differential equations studied the existence of periodical solutions to a predator-prey model with two state impulses. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are obtained. As a continuation, a water hyacinth ecological system with Kuznets curve effect and two state-dependent impulse controls is put forward and numerical simulations are carried out based on the qualitative analysis on the proposed model.This paper is organized as follows. In Section2, a water hyacinth ecological system with Kuznets curve effect and two state-dependent impulses control is put forward, and some basic definitions are given. In Section 3, we first discuss the qualitative analysis of system (3) and then discuss the existence and stability of periodic solutions. In Section 4, we analyze our theoretical results by numerical simulations and give a brief discussion. Finally, conclusions are presented in Section 5.


## 2. Model Formulation and Preliminaries

### 2.1. Model Formulation

In ecosystems, water hyacinth can be farmed with fish. Water hyacinth can not only serve as the diet of fish but can also purify the water quality, which controls the pond water quality well. But there are also disadvantages. Too much water hyacinth, covering the water surface, will reduce the activity space of the fish. Most importantly, it consumes the oxygen in the water and prevents the entry of oxygen in the air; the fish are suffocated, causing the extinction of fish populations in water ecological system. Considering the water hyacinth as feed in fish farms and its characteristic of the ecological damage caused by its rapid spread, inspired by [21–24], we establish the water hyacinth ecological system with Kuznets curve effect.
(1)dxdt=rx1−xK−βxy,dydt=−dy+δm−xxy,where r, K, β, d, δ, and m are all constants; xt and yt denote the water hyacinth (prey) and fish (predator) population, respectively, at any time t; r is the intrinsic growth rate of water hyacinth; K is the carrying capacity of water hyacinth; β is the coefficient of interspecific competition; d denotes the death rate of fish; and δxm−x is the Kuznets curve effect function (see Figures 1–3). When the population of water hyacinth is less than m, it is conducive to the growth of fish; when the population of water hyacinth is greater than m, it will hinder the growth of fish.Figure 1
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=3, and m2=8.Figure 2
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=8, and m2=12.Figure 3
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=12, and m2=14.The rapidly propagating water hyacinth will cover the water surface, causing intense intraspecific competition, leading to death of rot, pollution of water body, and aggravate eutrophication of water body, thus hindering the growth of other aquatic creatures and causing the imbalance of the ecological chain, irreversible damage to the ecosystem, loss of biodiversity, and frequent ecological disasters. The key to solving water hyacinth problem is to develop and make proper use of water hyacinth to avoid harm. At present, many places are trying to develop and utilize water hyacinth, using it as feed for livestock, poultry, grass carp, and so on, so as to turn harm into profit.According to the characteristics of water hyacinth ecosystem, the population size of water hyacinth is set at a controllable economic threshold (ET). When the population size of water hyacinth is less thanET1, it will be detrimental to the growth of fish population. We consider using the methods of manually releasing water hyacinth and fishing fish population to control the ecological balance. When the population size of water hyacinth is greater than ET2, it will lead to the extinction of fish population. We consider using the methods of manually fishing water hyacinth and releasing fish population to control the ecological balance.Now, we take the integrated control tactics into account for water hyacinth ecological system. Once the density of the water hyacinth population reaches the economic threshold (ET), we introduce impulsive state feedback control. Thus, we obtain the following system:(2)dxdt=rx1−xK−βxy,dydt=−dy+δm−xxy,h1<x<h2,△x=1−q1h2−x,△y=−p1y,x=h1,y>rβ1−h1K,△x=−q1x,△y=τ,x=h2,y<rβ1−h2K,Δxt=xt+−xt,Δyt=yt+−yt,h1<x0=x0<h2,y0=y0>0.This model describes that in productive practice for controlling water hyacinth, people always take such a strategy that when the water hyacinth arrives at a given ETh1 or h2, they began to use pulses to control the population of water hyacinths. ET h1 denotes the economic value ET1 with a small population of water hyacinth, and h2 denotes the economic value ET2 with a large population of water hyacinth h1<h2. When the population of water hyacinth is equal to h1 and the population of fish is greater than r/β1−h1/K, the method of impulse control is adopted to obtain economic benefits by harvesting fish, and meanwhile to increase the prey of fish, promoting the population of fish by releasing water hyacinth. 0<p1<1 represents the harvest coefficient of fish, and 1−q1h2−x represents the quantity of water hyacinth. When the population of water hyacinth is equal to h2 and the population of fish is less than r/β1−h2/K, the mass reproduction of water hyacinth is likely to lead to the extinction of fish. At this time, the method of impulse control is adopted to reduce the population of water hyacinth by fishing it and at the same time to increase the population of fish by releasing fish. 0<q1<1 represents the harvest coefficient of water hyacinth, and τ>0 is the release amount of fish.Considering the biology background of system (2), we all assume that x0,y0∈h1<x0<h2,y0>0 and we only discuss system (2) in the region R+2=x,y∣x≥0,y≥0.For system (2), if H1:m2>4d/δ, we change it to x¯=x, y¯=δy, a=r/K, b=β/δ, m1=m−m2−4d/δ/2, and m2=m+m2−4d/δ/2. So we can derive equivalent system (3) from system (2) (let new variables be x, y, and t).
(3)dxdt=xr−ax−by,dydt=yx−m1m2−x,h1<x<h2,△x=1−q1h2−x,△y=−p1δy,x=h1,y>1br−ah1,△x=−q1x,△y=τ,x=h2,y<1br−ah2,h1<x0=x0<h2,y0=y0>0.On the basis of the ecological significance, system (3) should meet r>am2, m1<h1<m2<h2, and 1−q1h2>h1. We can get the relationship between the phase set and isocline as shown in the following figures (see Figures 4–6).Figure 4
The case ofh1<1−q1h2<m2<h2 of system (3).Figure 5
The case ofh1<1−q1h2=m2<h2 of system (3).Figure 6
The case ofh1<m2<1−q1h2<h2 of system (3).We first discuss the same position, i.e.,1−q1h2=m2 (see Figure 5); we change p=p1δ and q1=1−m2/h2. So we can derive equivalent system (4) from system (3).
(4)dxdt=xr−ax−by,dydt=yx−m1m2−x,h1<x<h2,△x=m2−h1,△y=−py,x=h1,y>1br−ah1,△x=m2−h2,△y=τ,x=h2,y<1br−ah2,h1<x0=x0<h2,y0=y0>0.


### 2.2. Preliminaries

In this section, we first introduce some definitions and lemmas of the geometric theory of the semicontinuous dynamic system, which will be useful for the latter discussion. And for more details, we can see [6–15] and the references cited therein.Definition 1.
A tripleX,Π,R+ is said to be a semidynamical system if X is a metric space, R+ is the set of all nonnegative real, and ΠP,t:X×R+→X is a continuous map such that (1) ΠP,0=P for all P∈X; (2) ΠP,t is continuous for t and s; and (3) ΠΠP,s,t=ΠP,s+t for all P∈X and s, t∈R+. Sometimes, a semidynamical system X,Π,R+ is denoted by X,Π.Definition 2.
Define the impulsive setM1=x,y∈R+2∣x=h1,y>r−ah1/b and M2=x,y∈R+2∣x=h2,0<y<r−ah2/b; let M=M1∪M2; M is a closed subset. Define the continuous functions I1:h1,y∈M1→x+,y+=1−q1h2,1−py∈R+2 and I2:h2,y∈M1→x+,y+=1−q1h2,y+τ∈R+2; thus, the phase set N can be defined N=I1M1∪I2M2=x+,y+∈R+2∣x+=1−q1h2,y+>0. Then, Ω,Π,M,I is called an impulsive semidynamic system.Definition 3.
Define the periodic solution of the system (3) (see Figure 7):
(1)
When there is a pointP1 at phase set N and a T1 such that ΠP1,T1=Q1∈M1 and Q1+=I1Q1=P1∈N, then ΠP1,T1 is called order-1 periodic solution
(2)
When there is a pointP2 at phase set N and a T2 such that ΠP2,T2=Q2∈M2 and Q2+=I2Q2=P2∈N, then ΠP2,T2 is called order-1 periodic solution
(3)
When there is a pointP3 at phase set N and a T3 such that ΠP3,T3=Q3∈M2 and Q3+=I2Q3=P4∈N, there exists a T4 such that ΠP4,T4=Q4∈M1 and Q4+=I1Q4=P3∈N, then ΠP3,T3+ΠP4,T4 is called order-2 periodic solutionFigure 7
Define the periodic solution of system (3).Next, we will give the definition of the successor function of system (3). Let L be a coordinate axis defined at N; the origin point is the intersection point of line L:x=1−q1h2 with axis X:y=0; the positive direction is consistent with the positive direction of axis Y:x=0, then we obtain a number axis d. For any x∈d, let dx∈C0 be the coordinate of point x.Definition 4 (see Figure8).
(1)
When there is a pointP1 at phase set N and a T1 such that ΠP1,T1=Q1∈M1 and Q1+=I1Q1∈N, then fP1=dQ1+−dP1 is called the successor function of the point P1
(2)
When there is a pointP2 at phase set N and a T2 such that ΠP2,T2=Q2∈M2 and Q2+=I2Q2∈N, then fP2=dQ2+−dP2 is called the successor function of the point P2
(3)
When there is a pointP3 at phase set N and a T3 such that ΠP3,T3=Q3∈M2 and Q3+=I2Q3=P4∈N, there exists a T4 such that ΠP4,T4=Q4∈M1 and Q4+=I1Q4∈N, then gP3=dQ4+−dP3 is called the successor function of the point P3Remark 1 (see Figure7).
(1)
If the successor functionfP=0, the trajectory ΠP,T with initial point P is an order-1 periodic solution of system (3)
(2)
If the successor functiongP=0, the trajectory ΠP,T with initial point P is an order-2 periodic solution of system (3)Lemma 1.
The successor functionfP and gP is continuous.Lemma 2.
(1)
Let the continuous dynamical system beΠ,X; if there are two points P1∈N and P2∈N at the phase set, they make fP1fP2<0, then there must exist a point P between P1 and P2 so that fP=0; thus, there must exist an order-1 periodic solution by point P
(2)
Let the continuous dynamical system beΠ,X; if there are two points P1∈N and P2∈N at the phase set, they make gP1gP2<0, then there must exist a point P between P1 and P2 so that gP=0; thus, there must exist an order-2 periodic solution by point PLemma 3 (analogue of Poincaré’s criterion) [25].
TheT-periodic solutions x=ξt and y=ηt of the system
(5)dxtdt=Px,y,dytdt=Qx,y,ifφx,y∉M,Δx=αx,y,Δy=βx,y,ifφx,y∈Mare orbitally asymptotically stable if the Floquet multiplier μ2 satisfies the condition μ2<1, where
(6)μ2=∏k=1qΔkexp∫0T∂P∂xξt,ηt+∂Q∂xξt,ηtdt,with
(7)Δk=P+∂β/∂y∂φ/∂x−∂β/∂x∂φ/∂y+∂φ/∂x+Q+∂α/∂x∂φ/∂y−∂α/∂y∂φ/∂x+∂φ/∂yP∂φ/∂x+Q∂φ/∂y,and P, Q, ∂α/∂x, ∂α/∂y, ∂β/∂x, ∂β/∂y, ∂φ/∂x, and ∂ϕ/∂y are calculated at the point ξτk,ητk, and P+=Pξτk+,ητk+, Q+=Qξτk+,ητk+, and φx,y are a sufficiently smooth function with grad φx,y≠0, and τk k∈N are the moments for the jumps.Figure 8
Successor function of system (3).For the convenience of discussion, we need to define the following (see Figure9).Figure 9
Structure diagram inR2+ of system (4).L0:r−ax−by=0dx/dt=0, L3:x−m1=0dy/dt=0, L5:x−m2=0dy/dt=0, L:x−1−q1h2=0, L4:x−h1=0, and L6:x−h2=0.The intersection point ofL0 and L4 is denoted by F1xF1,yF1=F1h1,r−ah1/b.The intersection point ofL0 and L6 is denoted by F6xF6,yF6=F6h2,r−ah2/b. The phase point of F6xF6,yF6 at the phase set L5 is denoted by F4xF4,yF4=F4m2,r−ah2/b+τ. If τ>r−am2/b of system (4), we define a function Wx,y=y+Kx−Km2+r−ah2/b+τ, where m1≤x≤m2. We have dW/dt∣W=0=−K2xm2−x+Kx−m1m2−x2+xah2−x−bτ+r−ah2/b+τx−m1m2−x. We can choose K enough large, such that dW/dt∣W=0<0, so the straight line L7:W=0 is nontangent. The intersection point of L4 and L7 is denoted by F3xF3,yF3=F3h1,Km2−h1+r−ah2/b+τ.Assume thatL1, L2, L8, and L9 are dividing lines of the saddle point E2, where L8 and L9 are the stable flows of E2 and L1 and L2 are the unstable flows of E2. The intersection point of L1 and L4 is denoted by F2xF2,yF2. The intersection point of L2 and L6 is denoted by F5xF5,yF5.


## 2.1. Model Formulation

In ecosystems, water hyacinth can be farmed with fish. Water hyacinth can not only serve as the diet of fish but can also purify the water quality, which controls the pond water quality well. But there are also disadvantages. Too much water hyacinth, covering the water surface, will reduce the activity space of the fish. Most importantly, it consumes the oxygen in the water and prevents the entry of oxygen in the air; the fish are suffocated, causing the extinction of fish populations in water ecological system. Considering the water hyacinth as feed in fish farms and its characteristic of the ecological damage caused by its rapid spread, inspired by [21–24], we establish the water hyacinth ecological system with Kuznets curve effect.
(1)dxdt=rx1−xK−βxy,dydt=−dy+δm−xxy,where r, K, β, d, δ, and m are all constants; xt and yt denote the water hyacinth (prey) and fish (predator) population, respectively, at any time t; r is the intrinsic growth rate of water hyacinth; K is the carrying capacity of water hyacinth; β is the coefficient of interspecific competition; d denotes the death rate of fish; and δxm−x is the Kuznets curve effect function (see Figures 1–3). When the population of water hyacinth is less than m, it is conducive to the growth of fish; when the population of water hyacinth is greater than m, it will hinder the growth of fish.Figure 1
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=3, and m2=8.Figure 2
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=8, and m2=12.Figure 3
The vector diagram of system (8) where r=4, a=0.4, b=0.5, m1=12, and m2=14.The rapidly propagating water hyacinth will cover the water surface, causing intense intraspecific competition, leading to death of rot, pollution of water body, and aggravate eutrophication of water body, thus hindering the growth of other aquatic creatures and causing the imbalance of the ecological chain, irreversible damage to the ecosystem, loss of biodiversity, and frequent ecological disasters. The key to solving water hyacinth problem is to develop and make proper use of water hyacinth to avoid harm. At present, many places are trying to develop and utilize water hyacinth, using it as feed for livestock, poultry, grass carp, and so on, so as to turn harm into profit.According to the characteristics of water hyacinth ecosystem, the population size of water hyacinth is set at a controllable economic threshold (ET). When the population size of water hyacinth is less thanET1, it will be detrimental to the growth of fish population. We consider using the methods of manually releasing water hyacinth and fishing fish population to control the ecological balance. When the population size of water hyacinth is greater than ET2, it will lead to the extinction of fish population. We consider using the methods of manually fishing water hyacinth and releasing fish population to control the ecological balance.Now, we take the integrated control tactics into account for water hyacinth ecological system. Once the density of the water hyacinth population reaches the economic threshold (ET), we introduce impulsive state feedback control. Thus, we obtain the following system:(2)dxdt=rx1−xK−βxy,dydt=−dy+δm−xxy,h1<x<h2,△x=1−q1h2−x,△y=−p1y,x=h1,y>rβ1−h1K,△x=−q1x,△y=τ,x=h2,y<rβ1−h2K,Δxt=xt+−xt,Δyt=yt+−yt,h1<x0=x0<h2,y0=y0>0.This model describes that in productive practice for controlling water hyacinth, people always take such a strategy that when the water hyacinth arrives at a given ETh1 or h2, they began to use pulses to control the population of water hyacinths. ET h1 denotes the economic value ET1 with a small population of water hyacinth, and h2 denotes the economic value ET2 with a large population of water hyacinth h1<h2. When the population of water hyacinth is equal to h1 and the population of fish is greater than r/β1−h1/K, the method of impulse control is adopted to obtain economic benefits by harvesting fish, and meanwhile to increase the prey of fish, promoting the population of fish by releasing water hyacinth. 0<p1<1 represents the harvest coefficient of fish, and 1−q1h2−x represents the quantity of water hyacinth. When the population of water hyacinth is equal to h2 and the population of fish is less than r/β1−h2/K, the mass reproduction of water hyacinth is likely to lead to the extinction of fish. At this time, the method of impulse control is adopted to reduce the population of water hyacinth by fishing it and at the same time to increase the population of fish by releasing fish. 0<q1<1 represents the harvest coefficient of water hyacinth, and τ>0 is the release amount of fish.Considering the biology background of system (2), we all assume that x0,y0∈h1<x0<h2,y0>0 and we only discuss system (2) in the region R+2=x,y∣x≥0,y≥0.For system (2), if H1:m2>4d/δ, we change it to x¯=x, y¯=δy, a=r/K, b=β/δ, m1=m−m2−4d/δ/2, and m2=m+m2−4d/δ/2. So we can derive equivalent system (3) from system (2) (let new variables be x, y, and t).
(3)dxdt=xr−ax−by,dydt=yx−m1m2−x,h1<x<h2,△x=1−q1h2−x,△y=−p1δy,x=h1,y>1br−ah1,△x=−q1x,△y=τ,x=h2,y<1br−ah2,h1<x0=x0<h2,y0=y0>0.On the basis of the ecological significance, system (3) should meet r>am2, m1<h1<m2<h2, and 1−q1h2>h1. We can get the relationship between the phase set and isocline as shown in the following figures (see Figures 4–6).Figure 4
The case ofh1<1−q1h2<m2<h2 of system (3).Figure 5
The case ofh1<1−q1h2=m2<h2 of system (3).Figure 6
The case ofh1<m2<1−q1h2<h2 of system (3).We first discuss the same position, i.e.,1−q1h2=m2 (see Figure 5); we change p=p1δ and q1=1−m2/h2. So we can derive equivalent system (4) from system (3).
(4)dxdt=xr−ax−by,dydt=yx−m1m2−x,h1<x<h2,△x=m2−h1,△y=−py,x=h1,y>1br−ah1,△x=m2−h2,△y=τ,x=h2,y<1br−ah2,h1<x0=x0<h2,y0=y0>0.


## 2.2. Preliminaries

In this section, we first introduce some definitions and lemmas of the geometric theory of the semicontinuous dynamic system, which will be useful for the latter discussion. And for more details, we can see [6–15] and the references cited therein.Definition 1.
A tripleX,Π,R+ is said to be a semidynamical system if X is a metric space, R+ is the set of all nonnegative real, and ΠP,t:X×R+→X is a continuous map such that (1) ΠP,0=P for all P∈X; (2) ΠP,t is continuous for t and s; and (3) ΠΠP,s,t=ΠP,s+t for all P∈X and s, t∈R+. Sometimes, a semidynamical system X,Π,R+ is denoted by X,Π.Definition 2.
Define the impulsive setM1=x,y∈R+2∣x=h1,y>r−ah1/b and M2=x,y∈R+2∣x=h2,0<y<r−ah2/b; let M=M1∪M2; M is a closed subset. Define the continuous functions I1:h1,y∈M1→x+,y+=1−q1h2,1−py∈R+2 and I2:h2,y∈M1→x+,y+=1−q1h2,y+τ∈R+2; thus, the phase set N can be defined N=I1M1∪I2M2=x+,y+∈R+2∣x+=1−q1h2,y+>0. Then, Ω,Π,M,I is called an impulsive semidynamic system.Definition 3.
Define the periodic solution of the system (3) (see Figure 7):
(1)
When there is a pointP1 at phase set N and a T1 such that ΠP1,T1=Q1∈M1 and Q1+=I1Q1=P1∈N, then ΠP1,T1 is called order-1 periodic solution
(2)
When there is a pointP2 at phase set N and a T2 such that ΠP2,T2=Q2∈M2 and Q2+=I2Q2=P2∈N, then ΠP2,T2 is called order-1 periodic solution
(3)
When there is a pointP3 at phase set N and a T3 such that ΠP3,T3=Q3∈M2 and Q3+=I2Q3=P4∈N, there exists a T4 such that ΠP4,T4=Q4∈M1 and Q4+=I1Q4=P3∈N, then ΠP3,T3+ΠP4,T4 is called order-2 periodic solutionFigure 7
Define the periodic solution of system (3).Next, we will give the definition of the successor function of system (3). Let L be a coordinate axis defined at N; the origin point is the intersection point of line L:x=1−q1h2 with axis X:y=0; the positive direction is consistent with the positive direction of axis Y:x=0, then we obtain a number axis d. For any x∈d, let dx∈C0 be the coordinate of point x.Definition 4 (see Figure8).
(1)
When there is a pointP1 at phase set N and a T1 such that ΠP1,T1=Q1∈M1 and Q1+=I1Q1∈N, then fP1=dQ1+−dP1 is called the successor function of the point P1
(2)
When there is a pointP2 at phase set N and a T2 such that ΠP2,T2=Q2∈M2 and Q2+=I2Q2∈N, then fP2=dQ2+−dP2 is called the successor function of the point P2
(3)
When there is a pointP3 at phase set N and a T3 such that ΠP3,T3=Q3∈M2 and Q3+=I2Q3=P4∈N, there exists a T4 such that ΠP4,T4=Q4∈M1 and Q4+=I1Q4∈N, then gP3=dQ4+−dP3 is called the successor function of the point P3Remark 1 (see Figure7).
(1)
If the successor functionfP=0, the trajectory ΠP,T with initial point P is an order-1 periodic solution of system (3)
(2)
If the successor functiongP=0, the trajectory ΠP,T with initial point P is an order-2 periodic solution of system (3)Lemma 1.
The successor functionfP and gP is continuous.Lemma 2.
(1)
Let the continuous dynamical system beΠ,X; if there are two points P1∈N and P2∈N at the phase set, they make fP1fP2<0, then there must exist a point P between P1 and P2 so that fP=0; thus, there must exist an order-1 periodic solution by point P
(2)
Let the continuous dynamical system beΠ,X; if there are two points P1∈N and P2∈N at the phase set, they make gP1gP2<0, then there must exist a point P between P1 and P2 so that gP=0; thus, there must exist an order-2 periodic solution by point PLemma 3 (analogue of Poincaré’s criterion) [25].
TheT-periodic solutions x=ξt and y=ηt of the system
(5)dxtdt=Px,y,dytdt=Qx,y,ifφx,y∉M,Δx=αx,y,Δy=βx,y,ifφx,y∈Mare orbitally asymptotically stable if the Floquet multiplier μ2 satisfies the condition μ2<1, where
(6)μ2=∏k=1qΔkexp∫0T∂P∂xξt,ηt+∂Q∂xξt,ηtdt,with
(7)Δk=P+∂β/∂y∂φ/∂x−∂β/∂x∂φ/∂y+∂φ/∂x+Q+∂α/∂x∂φ/∂y−∂α/∂y∂φ/∂x+∂φ/∂yP∂φ/∂x+Q∂φ/∂y,and P, Q, ∂α/∂x, ∂α/∂y, ∂β/∂x, ∂β/∂y, ∂φ/∂x, and ∂ϕ/∂y are calculated at the point ξτk,ητk, and P+=Pξτk+,ητk+, Q+=Qξτk+,ητk+, and φx,y are a sufficiently smooth function with grad φx,y≠0, and τk k∈N are the moments for the jumps.Figure 8
Successor function of system (3).For the convenience of discussion, we need to define the following (see Figure9).Figure 9
Structure diagram inR2+ of system (4).L0:r−ax−by=0dx/dt=0, L3:x−m1=0dy/dt=0, L5:x−m2=0dy/dt=0, L:x−1−q1h2=0, L4:x−h1=0, and L6:x−h2=0.The intersection point ofL0 and L4 is denoted by F1xF1,yF1=F1h1,r−ah1/b.The intersection point ofL0 and L6 is denoted by F6xF6,yF6=F6h2,r−ah2/b. The phase point of F6xF6,yF6 at the phase set L5 is denoted by F4xF4,yF4=F4m2,r−ah2/b+τ. If τ>r−am2/b of system (4), we define a function Wx,y=y+Kx−Km2+r−ah2/b+τ, where m1≤x≤m2. We have dW/dt∣W=0=−K2xm2−x+Kx−m1m2−x2+xah2−x−bτ+r−ah2/b+τx−m1m2−x. We can choose K enough large, such that dW/dt∣W=0<0, so the straight line L7:W=0 is nontangent. The intersection point of L4 and L7 is denoted by F3xF3,yF3=F3h1,Km2−h1+r−ah2/b+τ.Assume thatL1, L2, L8, and L9 are dividing lines of the saddle point E2, where L8 and L9 are the stable flows of E2 and L1 and L2 are the unstable flows of E2. The intersection point of L1 and L4 is denoted by F2xF2,yF2. The intersection point of L2 and L6 is denoted by F5xF5,yF5.


## 3. Model Analysis

On the basis of the ecological significance, system (3) should meet r>am2, m1<h1<m2<h2, and 1−q1h2>h1. In this section, we will first study the qualitative characteristic of system (3) without the impulsive effect. Secondly, if 1−q1h2=m2, the sufficient conditions for the existence and stability of order-1 positive periodic solutions are analyzed. Thirdly, if 1−q1h2=m2, the sufficient conditions for the existence and stability of order-2 positive periodic solution are analyzed. Finally, if 1−q1h2≠m2, the sufficient conditions for the existence of order-k (k=1,2) positive periodic solution are analyzed.


### 3.1. Simple Qualitative Analysis of System (3) without Impulsive Effect

From system (3) without the impulsive effect, we obtain the following system:
(8)dxdt=xr−ax−by=Px,y,dydt=yx−m1m2−x=Qx,y.Byxr−ax−by=0 and yx−m1m2−x=0, we can obtain two boundary equilibriums E00,0 and E3r/a,0 in system (8).We can see that (i) there is no positive equilibrium ifr<am1; (ii) there is one positive equilibrium E1m1,r−am1/b if am1<r<am2; and (iii) there are two positive equilibriums E1m1,r−am1/b and E2m2,r−am2/b if r>am2.The Jacobian matrix at equilibriumEx,y is given by
(9)JE=r−2ax−by−bxym1+m2−2xx−m1m2−x.Theorem 1.
E00,0 is the saddle point; if am1<r<am2 holds, then E3r/a,0 is the saddle point; if r<am1 or r>am2 holds, then E3r/a,0 is the stable node point.Theorem 2.
E1m1,r−am1/b is the stable node point, if r>am1; E2m2,r−am2/b is the saddle point, if r>am2.Theorem 3.
There exists no limit cycle in system (8) in Ω.Proof 1.
Let the straight linel1:x−r/a−ε=0 ε>0; we have dl1/dt=dx/dt∣x=r/a+ε=−r/a−εaε−by<0, then the straight line segment is nontangent, and the path line of system (8) thereon is always to the left direction. Define a function Fx,y=m1+m2x+by−M, where 0≤x≤r/a+ε. We have dF/dt∣F=0=m1+m2dx/dt+bdy/dt≤−am1+m2x−r+m1m2/2a2+m1+m2r+m1m22/4a−m1m2M. We can choose M enough large, that is M>maxm1+m2r+m1m22/4am1m2,m1+m2r/a+ε such that dF/dt<0, so the straight line l2:F=0 is nontangent, the trajectory of system (8) from the upper right of l2 through l2 into the lower left. The straight line l2 intersects axis y and l1 at points A and B, respectively.
For system (8), we construct a Bendixson ring OABC^O including Ei i=1,2,3,4. Define OA¯, AB¯, BC¯, and CO¯ as the length of line l3:x=0, l2, l1, and l4:y=0, respectively. Since OA¯ and CO¯ are the length of orbit of system (8), the orbits of system (8) go through into the interior of the Bendixson ring from the outer of AB¯ and BC¯. Hence, we know that system (8) is uniformly bounded.
Let the Dulac functionDx,y=x−1y−1; we have ∂PD/∂x+∂QD/∂x=−ay−1<0; by the Bendixson-Dulac theory, system (8) does not exist a limit cycle in Ω=x,y∣0≤x≤r/a+ε,0<m1+m2x+by<M. The proof is completed.The illustration of vector fields of system (8) can be seen in Figures 1–3.


### 3.2. Existence and Stability of Order-1 Periodic Solution of System (4)

Theorem 4.
If0<p<1−r−am2/byF2 and τ>r−am2/b, there exists an order-1 periodic solution in system (4).Proof 2.
If1−pyF2>yE2 (see Figure 10), there exists a point B1m2,yB1∈N, satisfying 0<yB1−yE2<1−pyF2−yE2. The trajectory of system (4) Γ1 over B1 with x=h1 at point C1, which is pulsed to x=m2 and the phase point, is B2m2,yB2, so fB1=yB2−yB1>0.
There exists another pointB3m2,yB3∈N, satisfying yB3≫0. The trajectory of system (4) Γ2 over B3 with x=h1 at point C2, which is pulsed to x=m2 and the phase point, is B4m2,yB4, so fB3=yB4−yB3<0.
Therefore, there must exist a pointBm2,yB∈N, which satisfies yB1<yB<yB3, so that fB=0. By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.Theorem 5.
For system (4), if am2−h1/byF2<p<1−r−am2/byF2, the order-1 periodic solution to it is stable.Proof 3.
According to Lemma3 (see Figure 10), let x=ξ1t and y=η1t be a periodic solution to system (4) and ξ10=m2 and η10=yB; ξ1T=xC=h1 and η1T=yC; we have ξ1+=ξ1T+0=m2 and η1+=η1T+0=1−pη1T=1−pyC=η10=yB
(10)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α1x,y=m2−h1,β1x,y=−py,φ1x,y=x−h1.
Then(11)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α1∂x=∂α1∂y=0,∂β1∂x=0,∂β1∂y=−p,∂φ1∂x=1,∂φ1∂y=0,Δ1=P+∂β1/∂y∂φ1/∂x−∂β1/∂x∂φ1/∂y+∂φ1/∂x+Q+∂α1/∂x∂φ1/∂y−∂α1/∂y∂φ1/∂x+∂φ1/∂yP∂φ1/∂x+Q∂φ1/∂y=P+ξ1T+0,η1T+01−pPξ1T,η1T=PB1−pPC=m2r−am2−b1−pyC1−ph1r−ah1−byC,∫0T∂P∂x+∂Q∂xdt=∫0Tr−2ax−by+x−m1m2−xdt=∫m2h11xdx+∫1−pyCyC1ydy−∫0Taxdx=lnh1m2+ln11−p−∫0Taxdx,μ2=Δ1exp∫0T∂P∂x+∂Q∂xdt=m2r−am2−b1−pyC1−ph1r−ah1−byCexplnh1m2+ln11−p−∫0Taxdx=m2r−am2−b1−pyC1−ph1r−ah1−byCh1m211−pexp−∫0Taxdx≤r−am2−b1−pyCr−ah1−byC.
Sincext,yt is the periodic solution of system (4), if p>am2−h1/byF2>am2−h1/byC and conditions of Theorem 4 hold, then ∣μ2∣<1. Therefore, the order-1 periodic solution of system (4) is stable, if am2−h1/byF2<p<1−r−am2/byF2. Thus, the proof is completed.Theorem 6.
If0<τ<r−am2/b−yF5 and p>1−r−am2/byF3, there exists an order-1 periodic solution in system (4).Proof 4.
If0<τ<r−am2/b−yF5 (see Figure 11), there exists a point A1m2,yA1∈N, satisfying 0<yA1<τ. The trajectory of system (4) Γ3 over A1 with x=h2 at point D1, which is pulsed to x=m2 and the phase point, is A2m2,yA2, so fA1=yA2−yA1>0.
There exists another pointA3m2,yA3∈N, satisfying 0<yE2−yA3≪1. The trajectory of system (4) Γ4 over A3 with x=h2 at point D2, which is pulsed to x=m2 and the phase point, is A4m2,yA4, so fA3=yA4−yA3<0.
Therefore, there must exist a pointAm2,yA∈N and it satisfies yA1<yA<yA3, so that fA=0. By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.Theorem 7.
For system (4), if ah2−m2/b<τ<r−am2/b−yF5, the order-1 periodic solution to it is stable.Proof 5.
According to Lemma3 (see Figure 11), let x=ξ2t and y=η2t be a periodic solution to system (4) and ξ20=xA=m2 and η20=yA; ξ2T=xD=h2 and η2T=yD; we have ξ2+=ξ2T+0=m2 and η2+=η2T+0=η2T+τ=yD+τ=η20=yA
(12)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α2x,y=m2−h2,β2x,y=τ,φ2x,y=x−h2.
Then(13)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α2∂x=0,∂α2∂y=0,∂β2∂x=0,∂β2∂y=0,∂ϕ2∂x=1,∂ϕ2∂y=0,Δ1=P+∂β2/∂y∂φ2/∂x−∂β2/∂x∂φ2/∂y+∂φ2/∂x+Q+∂α2/∂x∂φ2/∂y−∂α2/∂y∂φ2/∂x+∂φ2/∂yP∂φ2/∂x+Q∂φ2/∂y=P+ξ2T+0,η2T+0Pξ2T,η2T=PAPD=m2r−am2−byD+τh2r−ah2−byD,∫0T∂P∂x+∂Q∂xdt=∫0Tr−2ax−by+x−m1m2−xdt=∫m2h21xdx+∫yD+τyD1ydy−∫0Taxdx=lnh2m2+lnyDyD+τ−∫0Taxdx,μ2=Δ1exp∫0T∂P∂x+∂Q∂xdt=m2r−am2−byD+τh2r−ah2−byDexplnh2m2+lnyDyD+τ−∫0Taxdx=m2r−am2−byD+τh2r−ah2−byDh2m2yDyD+τexp−∫0Taxdx≤r−am2−byD+τr−ah2−bη2yDyD+τ<r−am2−byD+τr−ah2−byD.
Sincext,yt is the periodic solution of system (4), if τ>ah2−m2/b and conditions of Theorem 6 hold, then ∣μ2∣<1. Therefore, the order-1 periodic solution of system (4) is stable, if ah2−m2/b<τ<r−am2/b−yF5. Thus, the proof is completed.Figure 10
There exists an order-1 periodic solution of system (4) with 1−pyF2>yE2.Figure 11
There exists an order-1 periodic solution of system (4) with yF5+τ<yE2.


### 3.3. Existence and Stability of Order-2 Periodic Solution of System (4)

Theorem 8.
Ifτ>r−am2/b and p>1−r−am2/byF3, there exists an order-2 periodic solution in system (4).Proof 6.
Ifτ>r−am2/b and p>1−r−am2/byF3 (see Figure 12), there exists a point A3m2,yA3∈N, satisfying yA3=1−pyF3<yE2. The trajectory of system (4) Γ2 over A3 with x=h2 at point D2, which is pulsed to x=m2 and the phase point, is B2m2,yB2∈N. We know that yE2<yB2=yD2+τ<yF4; the trajectory of system (4) Γ4 over B2 with x=h1 at point C2, which is pulsed to x=m2 and the phase point, is A4m2,yA4∈N, so gA3=yA4−yA3<0.
There exists another pointA1m2,yA1∈N, satisfying 0<yA1≪1. The trajectory of system (4) Γ1 over A1 with x=h2 at point D1, which is pulsed to x=m2 and the phase point, is B1m2,yB1∈N. We know that yB1=yD1+τ>yE2; the trajectory of system (4) Γ3 over B1 with x=h1 at point C1, which is pulsed to x=m2 and the phase point, is A2m2,yA2∈N, so gA1=yA2−yA1>0.
So there must exist a pointAm2,yA∈N, satisfying yA1<yA<yA3, so that gA=0. By Lemma 2, we know that system (4) has an order-2 periodic solution. The proof is completed.Theorem 9.
For system (4), if p>1−r−am2/byF3 and τ>r−am2/b+r−ah2/b, the order-2 periodic solution to it is stable.Proof 7.
According to Lemma3 (see Figure 12), let x=ξt and y=ηt be a periodic solution to system (4) and ξ0=xA=m2 and η0=yA; ξT1=xD=h2 and ηT1=yD; we have ξT1+0=m2=xB, ηT1+0=yD+τ=yB, ξT1+T2=h1=xC, and ηT1+T2=yC. We also have ξT1+T2+0=m2=xA and ηT1+T2+0=1−pyC=yA
(14)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α1x,y=m2−h1,β1x,y=−py,φ1x,y=x−h1,α2x,y=m2−h2,β2x,y=τ,φ2x,y=x−h2.
Then(15)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α1∂x=∂α1∂y=0,∂β1∂x=0,∂β1∂y=−p,∂φ1∂x=1,∂φ1∂y=0,∂α2∂x=0,∂α2∂y=0,∂β2∂x=0,∂β2∂y=0,∂φ2∂x=1,∂φ2∂y=0,Δ1=P+∂β1/∂y∂φ1/∂x−∂β1/∂x∂φ1/∂y+∂φ1/∂x+Q+∂α1/∂x∂φ/∂y−∂α1/∂y∂φ1/∂x+∂φ1/∂yP∂φ1/∂x+Q∂φ1/∂y=P+ξT1+T2+0,ηT1+T2+01−pPξT1+T2,ηT1+T2=1−pPAPC=m2r−am2−b1−pyC1−ph1r−ah1−byC,Δ2=P+∂β2/∂y∂φ2/∂x−∂β2/∂x∂φ2/∂y+∂φ2/∂x+Q+∂α2/∂x∂φ2/∂y−∂α2/∂y∂φ2/∂x+∂φ2/∂yP∂φ2/∂x+Q∂φ2/∂y=P+ξT1+0,ηT1+0PξT1,ηT1=PBPD=m2r−am2−byD+τh2r−ah2−byD,∫0T1+T2∂P∂x+∂Q∂xdt=∫0T1+T2r−2ax−by+x−m1m2−xdt=∫0T1r−2ax−by+x−m1m2−xdt+∫T1T1+T2r−2ax−by+x−m1m2−xdt=∫m2h21xdx+∫1−pyCyD1ydy+∫m2h11xdx+∫yD+τyC1ydy−∫0T1+T2axdx,μ2=Δ1Δ2exp∫0T1+T2∂P∂x+∂Q∂xdt=m2r−am2−b1−pyC1−ph1r−ah1−byCm2r−am2−byD+τh2r−ah2−byDh2m2yD1−pyCh1m2yCyD+τ=r−am2−b1−pyCr−ah1−byCr−am2−byD+τr−ah2−byDyDyD+τ<r−am2−b1−pyCr−ah1−byCr−am2−byD+τr−ah2−byD.
Obviously,μ2<1, if p>1−r−am2/byFC and τ>r−am2/b+r−ah2/b and conditions of Theorem 8 hold. Therefore, the order-2 periodic solution of system (4) is stable, if p>1−r−am2/byF3 and τ>r−am2/b+r−ah2/b. Thus, the proof is completed.Figure 12
There exists an order-2 periodic solution of system (4) with 1−pyF3<yE2 and τ>yE2.


### 3.4. Existence of Order-k (k=1,2) Periodic Solution of System (3)

Case I.
1−q1h2<m2 (see Figure 13).
For notation simplicity, let the intersection of the phase setN and L1 be F8xF8,yF8=F81−q1h2,yF8. The intersection point of L8 and L is denoted by F7xF7,yF7=F71−q1h2,yF7.Theorem 10.
If0<p<1−byF7/r−ah1 and τ>yF7, there exists an order-1 periodic solution in system (3).Proof 8.
The proof is similar to that of Theorem4 and omitted thereby.Theorem 11.
Ifp>1−yF7/yF3 and τ<yF7−yF5, there exists an order-1 periodic solution in system (3).Proof 9.
The proof is similar to that of Theorem6 and omitted thereby.Theorem 12.
Ifp>1−yF7/yF3 and τ>yF7, there exists an order-2 periodic solution in system (3).Proof 10.
The proof is similar to that of Theorem8 and omitted thereby.Case II.
1−q1h2>m2 (see Figure 14).
For notation simplicity, let the intersection of the phase setN and L9 be F9xF9,yF9=F91−q1h2,yF9. The intersection point of L2 and L is denoted by F10xF10,yF10=F101−q1h2,yF10.Theorem 13.
If0<p<1−yF9/yF2 and τ>yF9, there exists an order-1 periodic solution in system (3).Proof 11.
The proof is similar to that of Theorem4 and omitted thereby.Theorem 14.
Ifp>1−yF9/yF3 and τ<yF9−yF6, there exists an order-1 periodic solution in system (3).Proof 12.
The proof is similar to that of Theorem6 and omitted thereby.Theorem 15.
Ifp>1−yF9/yF3 and τ>yF9, there exists an order-2 periodic solution in system (3).Proof 13.
The proof is similar to that of Theorem8 and omitted thereby.Figure 13
Structure diagram inR2+ of system (3) with 1−q1h2<m2.Figure 14
Structure diagram inR2+ of system (3) with 1−q1h2>m2.


## 3.1. Simple Qualitative Analysis of System (3) without Impulsive Effect

From system (3) without the impulsive effect, we obtain the following system:
(8)dxdt=xr−ax−by=Px,y,dydt=yx−m1m2−x=Qx,y.Byxr−ax−by=0 and yx−m1m2−x=0, we can obtain two boundary equilibriums E00,0 and E3r/a,0 in system (8).We can see that (i) there is no positive equilibrium ifr<am1; (ii) there is one positive equilibrium E1m1,r−am1/b if am1<r<am2; and (iii) there are two positive equilibriums E1m1,r−am1/b and E2m2,r−am2/b if r>am2.The Jacobian matrix at equilibriumEx,y is given by
(9)JE=r−2ax−by−bxym1+m2−2xx−m1m2−x.Theorem 1.
E00,0 is the saddle point; if am1<r<am2 holds, then E3r/a,0 is the saddle point; if r<am1 or r>am2 holds, then E3r/a,0 is the stable node point.Theorem 2.
E1m1,r−am1/b is the stable node point, if r>am1; E2m2,r−am2/b is the saddle point, if r>am2.Theorem 3.
There exists no limit cycle in system (8) in Ω.Proof 1.
Let the straight linel1:x−r/a−ε=0 ε>0; we have dl1/dt=dx/dt∣x=r/a+ε=−r/a−εaε−by<0, then the straight line segment is nontangent, and the path line of system (8) thereon is always to the left direction. Define a function Fx,y=m1+m2x+by−M, where 0≤x≤r/a+ε. We have dF/dt∣F=0=m1+m2dx/dt+bdy/dt≤−am1+m2x−r+m1m2/2a2+m1+m2r+m1m22/4a−m1m2M. We can choose M enough large, that is M>maxm1+m2r+m1m22/4am1m2,m1+m2r/a+ε such that dF/dt<0, so the straight line l2:F=0 is nontangent, the trajectory of system (8) from the upper right of l2 through l2 into the lower left. The straight line l2 intersects axis y and l1 at points A and B, respectively.
For system (8), we construct a Bendixson ring OABC^O including Ei i=1,2,3,4. Define OA¯, AB¯, BC¯, and CO¯ as the length of line l3:x=0, l2, l1, and l4:y=0, respectively. Since OA¯ and CO¯ are the length of orbit of system (8), the orbits of system (8) go through into the interior of the Bendixson ring from the outer of AB¯ and BC¯. Hence, we know that system (8) is uniformly bounded.
Let the Dulac functionDx,y=x−1y−1; we have ∂PD/∂x+∂QD/∂x=−ay−1<0; by the Bendixson-Dulac theory, system (8) does not exist a limit cycle in Ω=x,y∣0≤x≤r/a+ε,0<m1+m2x+by<M. The proof is completed.The illustration of vector fields of system (8) can be seen in Figures 1–3.


## 3.2. Existence and Stability of Order-1 Periodic Solution of System (4)

Theorem 4.
If0<p<1−r−am2/byF2 and τ>r−am2/b, there exists an order-1 periodic solution in system (4).Proof 2.
If1−pyF2>yE2 (see Figure 10), there exists a point B1m2,yB1∈N, satisfying 0<yB1−yE2<1−pyF2−yE2. The trajectory of system (4) Γ1 over B1 with x=h1 at point C1, which is pulsed to x=m2 and the phase point, is B2m2,yB2, so fB1=yB2−yB1>0.
There exists another pointB3m2,yB3∈N, satisfying yB3≫0. The trajectory of system (4) Γ2 over B3 with x=h1 at point C2, which is pulsed to x=m2 and the phase point, is B4m2,yB4, so fB3=yB4−yB3<0.
Therefore, there must exist a pointBm2,yB∈N, which satisfies yB1<yB<yB3, so that fB=0. By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.Theorem 5.
For system (4), if am2−h1/byF2<p<1−r−am2/byF2, the order-1 periodic solution to it is stable.Proof 3.
According to Lemma3 (see Figure 10), let x=ξ1t and y=η1t be a periodic solution to system (4) and ξ10=m2 and η10=yB; ξ1T=xC=h1 and η1T=yC; we have ξ1+=ξ1T+0=m2 and η1+=η1T+0=1−pη1T=1−pyC=η10=yB
(10)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α1x,y=m2−h1,β1x,y=−py,φ1x,y=x−h1.
Then(11)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α1∂x=∂α1∂y=0,∂β1∂x=0,∂β1∂y=−p,∂φ1∂x=1,∂φ1∂y=0,Δ1=P+∂β1/∂y∂φ1/∂x−∂β1/∂x∂φ1/∂y+∂φ1/∂x+Q+∂α1/∂x∂φ1/∂y−∂α1/∂y∂φ1/∂x+∂φ1/∂yP∂φ1/∂x+Q∂φ1/∂y=P+ξ1T+0,η1T+01−pPξ1T,η1T=PB1−pPC=m2r−am2−b1−pyC1−ph1r−ah1−byC,∫0T∂P∂x+∂Q∂xdt=∫0Tr−2ax−by+x−m1m2−xdt=∫m2h11xdx+∫1−pyCyC1ydy−∫0Taxdx=lnh1m2+ln11−p−∫0Taxdx,μ2=Δ1exp∫0T∂P∂x+∂Q∂xdt=m2r−am2−b1−pyC1−ph1r−ah1−byCexplnh1m2+ln11−p−∫0Taxdx=m2r−am2−b1−pyC1−ph1r−ah1−byCh1m211−pexp−∫0Taxdx≤r−am2−b1−pyCr−ah1−byC.
Sincext,yt is the periodic solution of system (4), if p>am2−h1/byF2>am2−h1/byC and conditions of Theorem 4 hold, then ∣μ2∣<1. Therefore, the order-1 periodic solution of system (4) is stable, if am2−h1/byF2<p<1−r−am2/byF2. Thus, the proof is completed.Theorem 6.
If0<τ<r−am2/b−yF5 and p>1−r−am2/byF3, there exists an order-1 periodic solution in system (4).Proof 4.
If0<τ<r−am2/b−yF5 (see Figure 11), there exists a point A1m2,yA1∈N, satisfying 0<yA1<τ. The trajectory of system (4) Γ3 over A1 with x=h2 at point D1, which is pulsed to x=m2 and the phase point, is A2m2,yA2, so fA1=yA2−yA1>0.
There exists another pointA3m2,yA3∈N, satisfying 0<yE2−yA3≪1. The trajectory of system (4) Γ4 over A3 with x=h2 at point D2, which is pulsed to x=m2 and the phase point, is A4m2,yA4, so fA3=yA4−yA3<0.
Therefore, there must exist a pointAm2,yA∈N and it satisfies yA1<yA<yA3, so that fA=0. By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.Theorem 7.
For system (4), if ah2−m2/b<τ<r−am2/b−yF5, the order-1 periodic solution to it is stable.Proof 5.
According to Lemma3 (see Figure 11), let x=ξ2t and y=η2t be a periodic solution to system (4) and ξ20=xA=m2 and η20=yA; ξ2T=xD=h2 and η2T=yD; we have ξ2+=ξ2T+0=m2 and η2+=η2T+0=η2T+τ=yD+τ=η20=yA
(12)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α2x,y=m2−h2,β2x,y=τ,φ2x,y=x−h2.
Then(13)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α2∂x=0,∂α2∂y=0,∂β2∂x=0,∂β2∂y=0,∂ϕ2∂x=1,∂ϕ2∂y=0,Δ1=P+∂β2/∂y∂φ2/∂x−∂β2/∂x∂φ2/∂y+∂φ2/∂x+Q+∂α2/∂x∂φ2/∂y−∂α2/∂y∂φ2/∂x+∂φ2/∂yP∂φ2/∂x+Q∂φ2/∂y=P+ξ2T+0,η2T+0Pξ2T,η2T=PAPD=m2r−am2−byD+τh2r−ah2−byD,∫0T∂P∂x+∂Q∂xdt=∫0Tr−2ax−by+x−m1m2−xdt=∫m2h21xdx+∫yD+τyD1ydy−∫0Taxdx=lnh2m2+lnyDyD+τ−∫0Taxdx,μ2=Δ1exp∫0T∂P∂x+∂Q∂xdt=m2r−am2−byD+τh2r−ah2−byDexplnh2m2+lnyDyD+τ−∫0Taxdx=m2r−am2−byD+τh2r−ah2−byDh2m2yDyD+τexp−∫0Taxdx≤r−am2−byD+τr−ah2−bη2yDyD+τ<r−am2−byD+τr−ah2−byD.
Sincext,yt is the periodic solution of system (4), if τ>ah2−m2/b and conditions of Theorem 6 hold, then ∣μ2∣<1. Therefore, the order-1 periodic solution of system (4) is stable, if ah2−m2/b<τ<r−am2/b−yF5. Thus, the proof is completed.Figure 10
There exists an order-1 periodic solution of system (4) with 1−pyF2>yE2.Figure 11
There exists an order-1 periodic solution of system (4) with yF5+τ<yE2.


## 3.3. Existence and Stability of Order-2 Periodic Solution of System (4)

Theorem 8.
Ifτ>r−am2/b and p>1−r−am2/byF3, there exists an order-2 periodic solution in system (4).Proof 6.
Ifτ>r−am2/b and p>1−r−am2/byF3 (see Figure 12), there exists a point A3m2,yA3∈N, satisfying yA3=1−pyF3<yE2. The trajectory of system (4) Γ2 over A3 with x=h2 at point D2, which is pulsed to x=m2 and the phase point, is B2m2,yB2∈N. We know that yE2<yB2=yD2+τ<yF4; the trajectory of system (4) Γ4 over B2 with x=h1 at point C2, which is pulsed to x=m2 and the phase point, is A4m2,yA4∈N, so gA3=yA4−yA3<0.
There exists another pointA1m2,yA1∈N, satisfying 0<yA1≪1. The trajectory of system (4) Γ1 over A1 with x=h2 at point D1, which is pulsed to x=m2 and the phase point, is B1m2,yB1∈N. We know that yB1=yD1+τ>yE2; the trajectory of system (4) Γ3 over B1 with x=h1 at point C1, which is pulsed to x=m2 and the phase point, is A2m2,yA2∈N, so gA1=yA2−yA1>0.
So there must exist a pointAm2,yA∈N, satisfying yA1<yA<yA3, so that gA=0. By Lemma 2, we know that system (4) has an order-2 periodic solution. The proof is completed.Theorem 9.
For system (4), if p>1−r−am2/byF3 and τ>r−am2/b+r−ah2/b, the order-2 periodic solution to it is stable.Proof 7.
According to Lemma3 (see Figure 12), let x=ξt and y=ηt be a periodic solution to system (4) and ξ0=xA=m2 and η0=yA; ξT1=xD=h2 and ηT1=yD; we have ξT1+0=m2=xB, ηT1+0=yD+τ=yB, ξT1+T2=h1=xC, and ηT1+T2=yC. We also have ξT1+T2+0=m2=xA and ηT1+T2+0=1−pyC=yA
(14)Px,y=xr−ax−by,Qx,y=yx−m1m2−x,α1x,y=m2−h1,β1x,y=−py,φ1x,y=x−h1,α2x,y=m2−h2,β2x,y=τ,φ2x,y=x−h2.
Then(15)∂P∂x=r−2ax−by,∂Q∂x=x−m1m2−x,∂α1∂x=∂α1∂y=0,∂β1∂x=0,∂β1∂y=−p,∂φ1∂x=1,∂φ1∂y=0,∂α2∂x=0,∂α2∂y=0,∂β2∂x=0,∂β2∂y=0,∂φ2∂x=1,∂φ2∂y=0,Δ1=P+∂β1/∂y∂φ1/∂x−∂β1/∂x∂φ1/∂y+∂φ1/∂x+Q+∂α1/∂x∂φ/∂y−∂α1/∂y∂φ1/∂x+∂φ1/∂yP∂φ1/∂x+Q∂φ1/∂y=P+ξT1+T2+0,ηT1+T2+01−pPξT1+T2,ηT1+T2=1−pPAPC=m2r−am2−b1−pyC1−ph1r−ah1−byC,Δ2=P+∂β2/∂y∂φ2/∂x−∂β2/∂x∂φ2/∂y+∂φ2/∂x+Q+∂α2/∂x∂φ2/∂y−∂α2/∂y∂φ2/∂x+∂φ2/∂yP∂φ2/∂x+Q∂φ2/∂y=P+ξT1+0,ηT1+0PξT1,ηT1=PBPD=m2r−am2−byD+τh2r−ah2−byD,∫0T1+T2∂P∂x+∂Q∂xdt=∫0T1+T2r−2ax−by+x−m1m2−xdt=∫0T1r−2ax−by+x−m1m2−xdt+∫T1T1+T2r−2ax−by+x−m1m2−xdt=∫m2h21xdx+∫1−pyCyD1ydy+∫m2h11xdx+∫yD+τyC1ydy−∫0T1+T2axdx,μ2=Δ1Δ2exp∫0T1+T2∂P∂x+∂Q∂xdt=m2r−am2−b1−pyC1−ph1r−ah1−byCm2r−am2−byD+τh2r−ah2−byDh2m2yD1−pyCh1m2yCyD+τ=r−am2−b1−pyCr−ah1−byCr−am2−byD+τr−ah2−byDyDyD+τ<r−am2−b1−pyCr−ah1−byCr−am2−byD+τr−ah2−byD.
Obviously,μ2<1, if p>1−r−am2/byFC and τ>r−am2/b+r−ah2/b and conditions of Theorem 8 hold. Therefore, the order-2 periodic solution of system (4) is stable, if p>1−r−am2/byF3 and τ>r−am2/b+r−ah2/b. Thus, the proof is completed.Figure 12
There exists an order-2 periodic solution of system (4) with 1−pyF3<yE2 and τ>yE2.


## 3.4. Existence of Order-k (k=1,2) Periodic Solution of System (3)

Case I.
1−q1h2<m2 (see Figure 13).
For notation simplicity, let the intersection of the phase setN and L1 be F8xF8,yF8=F81−q1h2,yF8. The intersection point of L8 and L is denoted by F7xF7,yF7=F71−q1h2,yF7.Theorem 10.
If0<p<1−byF7/r−ah1 and τ>yF7, there exists an order-1 periodic solution in system (3).Proof 8.
The proof is similar to that of Theorem4 and omitted thereby.Theorem 11.
Ifp>1−yF7/yF3 and τ<yF7−yF5, there exists an order-1 periodic solution in system (3).Proof 9.
The proof is similar to that of Theorem6 and omitted thereby.Theorem 12.
Ifp>1−yF7/yF3 and τ>yF7, there exists an order-2 periodic solution in system (3).Proof 10.
The proof is similar to that of Theorem8 and omitted thereby.Case II.
1−q1h2>m2 (see Figure 14).
For notation simplicity, let the intersection of the phase setN and L9 be F9xF9,yF9=F91−q1h2,yF9. The intersection point of L2 and L is denoted by F10xF10,yF10=F101−q1h2,yF10.Theorem 13.
If0<p<1−yF9/yF2 and τ>yF9, there exists an order-1 periodic solution in system (3).Proof 11.
The proof is similar to that of Theorem4 and omitted thereby.Theorem 14.
Ifp>1−yF9/yF3 and τ<yF9−yF6, there exists an order-1 periodic solution in system (3).Proof 12.
The proof is similar to that of Theorem6 and omitted thereby.Theorem 15.
Ifp>1−yF9/yF3 and τ>yF9, there exists an order-2 periodic solution in system (3).Proof 13.
The proof is similar to that of Theorem8 and omitted thereby.Figure 13
Structure diagram inR2+ of system (3) with 1−q1h2<m2.Figure 14
Structure diagram inR2+ of system (3) with 1−q1h2>m2.


## 4. Numerical Simulations and Discussion

In this section, a specific example is presented to verify the theoretical results obtained in the previous section by considering the change of the control parametersp, τ, and q1. From Figures 15–25, we know that the numerical simulation results are consistent with the theoretical results.Figure 15
There exists an order-1 periodic solution in system (16) with q1=1/9, p=0.4, and τ=2.7.Figure 16
There exists an order-1 periodic solution in system (16) with q1=1/9, p=0.8, and τ=0.6.Figure 17
There exists an order-2 periodic solution in system (16) with q1=1/9, p=0.8, and τ=2.7.Figure 18
There exists an order-1 periodic solution in system (16) with q1=1/6, p=0.8, and τ=4.7.Figure 19
There exists an order-1 periodic solution in system (16) with q1=1/6, p=0.9, and τ=0.7.Figure 20
There exists an order-2 periodic solution in system (16) with q1=1/6, p=0.9, and τ=4.7.Figure 21
There exists an order-2 periodic solution in system (16) with q1=1/6, p=0.9, and τ=1.1.Figure 22
There exists an order-1 periodic solution in system (16) with q1=1/18, p=0.6, and τ=4.7.Figure 23
There exists an order-1 periodic solution in system (16) with q1=1/18, p=0.8, and τ=1.7.Figure 24
There exists an order-2 periodic solution in system (16) with q1=1/18, p=0.8, and τ=4.7.Figure 25
There exists an order-2 periodic solution in system (16) with q1=1/18, p=0.95, and τ=4.7.In system (3), let r=4, a=0.4, b=0.5, m1=3, m2=8, h1=6, h2=9, 0<p=p1δ<1, τ>0, and 0<q1<1−h1/h2; we get the following system:
(16)dxdt=x4−0.4x−0.5y,dydt=yx−38−x,6<x<9,△x=3−9q1,△y=−py,x=6,y>3.2,△x=−9q1,△y=τ,x=9,y<0.8,6<x0<9,y0>0.Obviously (see Figure9), we know that E00,0 and E28,1.6 are saddle points and E13,5.6 and E310,0 are stable node points. Let L7:Wx,y=y+Kx−8K+0.8+τ=0; we have F16,3.2, F36,2K+τ+0.8, F48,0.8+τ, F69,0.8, and x0,y0=7.21,0.9.


### 4.1. Verification for the Case1−q1h2=m2

Case I.
Letq1=1/9, p=0.4, and τ=2.7, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 4, there exists an order-1 periodic solution in system (16), which can be seen in Figure 15. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).Case II.
Letq1=1/9, p=0.8, and τ=0.6, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 6, there exists an order-1 periodic solution in system (16), which can be seen in Figure 16. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8 and 9).Case III.
Letq1=1/9, p=0.8, and τ=2.7, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 8, there exists an order-2 periodic solution in system (16), which can be seen in Figure 17. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).


### 4.2. Verification for the Case1−q1h2<m2

Case I.
Letq1=1/6, p=0.8, and τ=4.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 10, there exists an order-1 periodic solution in system (16), which can be seen in Figure 18. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 7.5).Case II.
Letq1=1/6, p=0.9, and τ=0.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 11, there exists an order-1 periodic solution in system (16), which can be seen in Figure 19. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 7.5 and 8).Case III.
Letq1=1/6, p=0.9, and τ=4.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 20. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).Case IV.
Letq1=1/6, p=0.9, and τ=1.1, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 21. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).


### 4.3. Verification for the Case1−q1h2>m2

Case I.
Letq1=1/18, p=0.6, and τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 13, there exists an order-1 periodic solution in system (16), which can be seen in Figure 22. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8.5).Case II.
Letq1=1/18, p=0.8, and τ=1.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 14, there exists an order-1 periodic solution in system (16), which can be seen in Figure 23. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8.5 and 9).Case III.
Letq1=1/18, p=0.8, and τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 24. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).Case IV.
Letq1=1/18, p=0.95, τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 25. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).From Figures1, 15–25, we know that at the initial point (7.21, 0.9) in the system without pulse control, fish species will be extinct, which is harmful to biodiversity. In order to maintain biodiversity, we introduce the state feedback control and put forward the economic threshold (ET) of ecological control to effectively control the amount of water hyacinth population which can prevent the extinction of fish and can fully employ water hyacinth as feedstuff for continuous development of fishery production, changing the harm into benefit.


## 4.1. Verification for the Case1−q1h2=m2

Case I.
Letq1=1/9, p=0.4, and τ=2.7, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 4, there exists an order-1 periodic solution in system (16), which can be seen in Figure 15. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).Case II.
Letq1=1/9, p=0.8, and τ=0.6, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 6, there exists an order-1 periodic solution in system (16), which can be seen in Figure 16. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8 and 9).Case III.
Letq1=1/9, p=0.8, and τ=2.7, then h1<1−q1h2=m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8,y≥0. According to Theorem 8, there exists an order-2 periodic solution in system (16), which can be seen in Figure 17. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).


## 4.2. Verification for the Case1−q1h2<m2

Case I.
Letq1=1/6, p=0.8, and τ=4.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 10, there exists an order-1 periodic solution in system (16), which can be seen in Figure 18. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 7.5).Case II.
Letq1=1/6, p=0.9, and τ=0.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 11, there exists an order-1 periodic solution in system (16), which can be seen in Figure 19. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 7.5 and 8).Case III.
Letq1=1/6, p=0.9, and τ=4.7, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 20. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).Case IV.
Letq1=1/6, p=0.9, and τ=1.1, then h1<1−q1h2<m2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=7.5,y≥0. According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 21. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).


## 4.3. Verification for the Case1−q1h2>m2

Case I.
Letq1=1/18, p=0.6, and τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 13, there exists an order-1 periodic solution in system (16), which can be seen in Figure 22. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8.5).Case II.
Letq1=1/18, p=0.8, and τ=1.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 14, there exists an order-1 periodic solution in system (16), which can be seen in Figure 23. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8.5 and 9).Case III.
Letq1=1/18, p=0.8, and τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 24. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).Case IV.
Letq1=1/18, p=0.95, τ=4.7, then h1<m2<1−q1h2<h2, and the impulsive set and the phase set are M=M1∪M2=x,y∈R+2∣x=6,y>3.2∪x,y∈R+2∣x=9,y<0.8 and N=x,y∈R+2∣x=8.5,y≥0. According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 25. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).From Figures1, 15–25, we know that at the initial point (7.21, 0.9) in the system without pulse control, fish species will be extinct, which is harmful to biodiversity. In order to maintain biodiversity, we introduce the state feedback control and put forward the economic threshold (ET) of ecological control to effectively control the amount of water hyacinth population which can prevent the extinction of fish and can fully employ water hyacinth as feedstuff for continuous development of fishery production, changing the harm into benefit.


## 5. Conclusions

In the system with pulse control, we gain order-1 periodic solutions and order-2 periodic solution of the system, respectively, verifying the accuracy of Theorems4, 6, 8, and 10–15. At the same time, by using the comprehensive control strategies, such as periodically dropping and harvest, we achieve the diversity of species in water hyacinth ecological system.In the future, from the viewpoint of optimization, we will further consider the costs and profits under domination of the market economy and further improve the model, thus realizing the effective protection of the diversity of species in water hyacinth ecology and obtaining the biggest economic profits.



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*Source: 4543976-2018-11-19.xml*