# Dynamic Analysis of Beddington–DeAngelis Predator-Prey System with Nonlinear Impulse Feedback Control

Authors: Dezhao Li; Huidong Cheng; Yu Liu
Journal: Complexity
 (2019)

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## Abstract

In this paper, a predator-prey system with pesticide dose-responded nonlinear pulse of Beddington–DeAngelis functional response is established. First, we construct the Poincaré map of the impulsive semidynamic system and discuss its main properties including the monotonicity, differentiability, fixed point, and asymptote. Second, we address the existence and globally asymptotic stability of the order-1 periodic solution and the sufficient conditions for the existence of the order-k(k ≥ 2) periodic solution. Thirdly, we give the threshold conditions for the existence and stability of boundary periodic solutions and present the parameter analysis. The results show that the pesticide dosage increases with the extension of the control period and decreases with the increase of the threshold. Besides, the state pulse feedback control can manage the pest population at a certain level and avoid excessive application of pesticides.


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## Body

## 1. Introduction

The predator-prey system plays an important role in the relationship of biological populations, so many predator-prey systems with different functional responses have been studied, such as the Monod type [1–5], the Holling type [6–13], and the Ivlev type [14–18]. Currently, the use of chemicals is more and more widespread in agriculture. Also, when the number of pests reaches a critical value, we can release natural enemies. Therefore, the feedback control of pulse state is proposed [17, 19–22]. The Beddington–DeAngelis functional response was introduced by Beddington [23] and DeAngelis et al. [24]. The Beddington–DeAngelis functional response avoided some of the singular behaviors of the ratio-dependent model at low density [25]. Cantrell and Cosner discussed the following predator-prey system with Beddington–DeAngelis functional response [23, 24, 26]:(1)u′t=ru1−uK−muva+bv+cu,v′t=vεmua+bv+cu−μ,u0=u0>0,v0=v0>0,where r,K,m,a,b,c,ε,andμ are positive constants. ut and vt represent the population density of prey and predator at time t, K is the environmental carrying capacity of the prey, and r is the intrinsic growth rate of prey. Function mu/a+bv+cu indicates the Beddington–DeAngelis functional response, and bv stands for mutual interference between the predators. The constants ε and μ represent the rate of conversion and death rate of predators, respectively.For simplicity, we determine dimensionless system (1) and scale it as follows:(2)t⟶rt,u⟶uK,v⟶bvcK.Then, we get(3)u′t=u1−u−suvu+v+A,v′t=δvuu+v+A−d,u0=u0>0,v0=v0>0,where(4)s=mbr,δ=mεcr,d=cμmε,A=acK.In recent years, many pulse equations have been studied that simulate the ecological processes of populations, and most of these studies are pulse differential equations at the fixed time [1, 27–32]. However, feedback control of time pulse has certain defects, which may reduce crop yield and possibly increase management costs. Therefore, we can choose to spray the pesticide when the quantity of pests reaches a certain threshold instead of spraying the pesticide at a fixed time. This measure avoids the possibility of the explosive growth of the number of pests and is more suitable for pest control. This paper studies the Beddington–DeAngelis system with pulse state feedback control strategies:(5)u′t=u1−u−suvu+v+Av′t=δvuu+v+A−du≠TH,ut+=PDutvt+=QDvt+τu=TH.Capturing or using chemicals on predators and prey may impulsively reduce the density of the predators and prey.PD and QD represent the survival rate of prey and predator populations when a given dose D of insecticides is applied and 0≤PD<1 and 0≤QD<1. We assume that insecticides have different insecticidal rates for these two populations, where PD=e−k1D and QD=e−k2D, τ≥0 is the constant number of natural enemies released [33, 34].Some published articles focus on the property of the successor function and Poincaré map to discuss the existence of order-1 periodic solution and the local stability. Besides, if the proposed model has the first integrals, the existence of order-2 periodic solutions can be discussed [35–40]. However, due to the complexity of such model as the Beddington–DeAngelis system in this paper, the problem of global dynamics such as the global stability of the model and the existence of order-kk≥2 periodic solution has not been well solved. Also, there a few researches on the property of Poincaré map while it is applied. So the main arrangement of this paper is as follows. In Section 2, some preliminaries about the pulse semidynamic system and system (3) are given. In Section 3, we construct the Poincaré map, deduce the expression of Poincaré map function of system (5), and then give some of its properties such as the monotonicity, differentiability, fixed point, and asymptote. In Section 4, we prove the existence and stability of the boundary periodic solution and order-kk≥1 periodic solution of system (5). In Section 5, we conducted a numerical simulation.


## 2. Preliminaries

### 2.1. Preliminaries of Pulse Semidynamic Systems

A pulsed semipowered system with state-dependent feedback control can be expressed as [41, 42](6)dutdt=Pu,vdvtdt=Qu,vu,v∉M,ut+=ut+αu,vvt+=vt+βu,vu,v∈M.Here,u,v∈R2, P, Q, α, and β are continuous functions from R2 to R. Let M⊂R2 be the impulse set of system (6), and for any fu,v∈M, the impulse occurs; the map I is defined as(7)f+=If=u+αu,v,v+βu,v=u+,v+∈R2,where f+ is the impulse point for f. We define N=IM as the phase set of the system and N∩M=∅, X=R2 is the metric space, and R+ is the set of all nonnegative reals; we call X,Π,R as a semidynamic system. For any f∈X, Πf,0=f, ΠΠf,t,s=Πf,t+s, where t,s∈R+ [43]. The set(8)D+f=Πf,tt∈R+,is called the positive orbit of f. Furthermore, for any set M∈X, let(9)M+f=D+f∩M−f.Next, we give the definition of impulsive semidynamic system and order-k periodic solution.Definition 1 (see [44, 45]).
The pulsed semidynamic systemX,Π,M,I consists of the nonempty closed subset M of X, the continuous semidynamic system X,Π, and the continuous function I.
We denote the points of discontinuity ofΠf by fn+ and call fn+ an impulsive point of fn. We define a function Φ from X into the extended positive reals R+∪∞ as follows: let f∈X; if M+f=∅, we set Φf=∞; otherwise M+f≠∅, and we set Φf=s, where Πf,t∉M for 0<t<s but Πf,t∈M.Definition 2 (see [46]).
For trajectoryΠf in X,Π,M,I, if there are nonnegative integers m≥0 and k≥1, k is the minimum integer satisfying fm+=fm+k+ and Tk=∑i=mm+k−1Φfi=∑i=mm+k−1si; then, the period of Πf is T, and there is a period of order-k.Definition 3 (see [42, 47]).
The T-periodic solutionu,v=ξt,ηt of thwe system(10)dudt=Pu,v,dvdt=Qu,v,ifϕu,v≠0,u+=u+αu,v,v+=v+βu,v,ifϕu,v=0,is orbitally asymptotically stable and enjoys the property of asymptotic phase if μ1 satisfies μ1<1, where(11)μ1=∏k=1qΔkexp∫0T∂P∂uξt,ηt+∂Q∂vξt,ηtdt,with(12)Δk=P+∂β/∂v∂ϕ/∂u−∂β/∂u∂ϕ/∂v+∂ϕ/∂u+Q+∂α/∂u∂ϕ/∂v−∂α/∂v∂ϕ/∂u+∂ϕ/∂uP∂ϕ/∂u+Q∂ϕ/∂v,and ϕ is continuously differentiable with respect to u and v. u,v∉M can also be denoted by ϕu,v≠0. P, Q, ∂α/∂u, ∂α/∂v, ∂β/∂u, ∂β/∂v, ∂ϕ/∂u, and ∂ϕ/∂v are calculated at the point ξtk,ηtk, P+=Pξtk+,ηtk+, and Q+=Qξtk+,ηtk+.


### 2.2. Preliminaries of System (3)

We know that the solution of system (3) is positive and bounded for all t. If d≥1+A−1, then the equilibrium point 1,0 is globally asymptotically stable [25].If0<d<1+A−1, system (3) has three equilibrium points, 0,0, 1,0, and u∗,v∗, respectively, where u∗ and v∗ are all positive and satisfy the following conditions:(13)1−u∗−sv∗u∗+v∗+A=0,u∗u∗+v∗+A=d.For completeness, we summarize global results for system (3) in Lemma 1 [25, 48, 49].Lemma 1.
(i)
Ifd>1+A−1, 1,0 is globally asymptotically stable
(ii)
Ifd<1+A−1 and trJu∗,v∗≤0, the u∗,v∗ is globally asymptotically stable
(iii)
Ifd<1+A−1 and trJu∗,v∗>0, there is an exact limit cycle


## 2.1. Preliminaries of Pulse Semidynamic Systems

A pulsed semipowered system with state-dependent feedback control can be expressed as [41, 42](6)dutdt=Pu,vdvtdt=Qu,vu,v∉M,ut+=ut+αu,vvt+=vt+βu,vu,v∈M.Here,u,v∈R2, P, Q, α, and β are continuous functions from R2 to R. Let M⊂R2 be the impulse set of system (6), and for any fu,v∈M, the impulse occurs; the map I is defined as(7)f+=If=u+αu,v,v+βu,v=u+,v+∈R2,where f+ is the impulse point for f. We define N=IM as the phase set of the system and N∩M=∅, X=R2 is the metric space, and R+ is the set of all nonnegative reals; we call X,Π,R as a semidynamic system. For any f∈X, Πf,0=f, ΠΠf,t,s=Πf,t+s, where t,s∈R+ [43]. The set(8)D+f=Πf,tt∈R+,is called the positive orbit of f. Furthermore, for any set M∈X, let(9)M+f=D+f∩M−f.Next, we give the definition of impulsive semidynamic system and order-k periodic solution.Definition 1 (see [44, 45]).
The pulsed semidynamic systemX,Π,M,I consists of the nonempty closed subset M of X, the continuous semidynamic system X,Π, and the continuous function I.
We denote the points of discontinuity ofΠf by fn+ and call fn+ an impulsive point of fn. We define a function Φ from X into the extended positive reals R+∪∞ as follows: let f∈X; if M+f=∅, we set Φf=∞; otherwise M+f≠∅, and we set Φf=s, where Πf,t∉M for 0<t<s but Πf,t∈M.Definition 2 (see [46]).
For trajectoryΠf in X,Π,M,I, if there are nonnegative integers m≥0 and k≥1, k is the minimum integer satisfying fm+=fm+k+ and Tk=∑i=mm+k−1Φfi=∑i=mm+k−1si; then, the period of Πf is T, and there is a period of order-k.Definition 3 (see [42, 47]).
The T-periodic solutionu,v=ξt,ηt of thwe system(10)dudt=Pu,v,dvdt=Qu,v,ifϕu,v≠0,u+=u+αu,v,v+=v+βu,v,ifϕu,v=0,is orbitally asymptotically stable and enjoys the property of asymptotic phase if μ1 satisfies μ1<1, where(11)μ1=∏k=1qΔkexp∫0T∂P∂uξt,ηt+∂Q∂vξt,ηtdt,with(12)Δk=P+∂β/∂v∂ϕ/∂u−∂β/∂u∂ϕ/∂v+∂ϕ/∂u+Q+∂α/∂u∂ϕ/∂v−∂α/∂v∂ϕ/∂u+∂ϕ/∂uP∂ϕ/∂u+Q∂ϕ/∂v,and ϕ is continuously differentiable with respect to u and v. u,v∉M can also be denoted by ϕu,v≠0. P, Q, ∂α/∂u, ∂α/∂v, ∂β/∂u, ∂β/∂v, ∂ϕ/∂u, and ∂ϕ/∂v are calculated at the point ξtk,ηtk, P+=Pξtk+,ηtk+, and Q+=Qξtk+,ηtk+.


## 2.2. Preliminaries of System (3)

We know that the solution of system (3) is positive and bounded for all t. If d≥1+A−1, then the equilibrium point 1,0 is globally asymptotically stable [25].If0<d<1+A−1, system (3) has three equilibrium points, 0,0, 1,0, and u∗,v∗, respectively, where u∗ and v∗ are all positive and satisfy the following conditions:(13)1−u∗−sv∗u∗+v∗+A=0,u∗u∗+v∗+A=d.For completeness, we summarize global results for system (3) in Lemma 1 [25, 48, 49].Lemma 1.
(i)
Ifd>1+A−1, 1,0 is globally asymptotically stable
(ii)
Ifd<1+A−1 and trJu∗,v∗≤0, the u∗,v∗ is globally asymptotically stable
(iii)
Ifd<1+A−1 and trJu∗,v∗>0, there is an exact limit cycle


## 3. Poincaré Map

### 3.1. Domains of the Poincaré Map

In the following parts, we only discuss the case ofu∗,v∗ as the globally asymptotically stable point of system (5).System (5) has two isoclinal lines, which are defined as L1 and L2:(14)L1:v=1−Au−u2+As+u−1;L2:v=1−dud−A.Next, the lines associated with the phase set and impulse set are defined asL3 and L4:(15)L3:u=e−k1DTH;L4:u=TH.In this case, the value range ofTH is 0<TH<u∗ and lines L3 and L4 always intersect with line L1. The intersection point of L1 and L4 presents as GTH,vG, and the intersection point of L1 and L3 presents as He−k1DTH,vH, where vG and vH are, respectively,(16)vG=1−ATH−TH2+ATH+s−1,vH=1−Ae−k1DTH−ek12D2TH2+Ae−k1DTH+s−1.The open set defined inR+2 is as follows:(17)Ω=u,vu>0,v>0,u<TH⊂R+2.The impulse setM is the part of line L4 above the U-axis and below point G:(18)M=u,v∈R2u=TH,0≤v<1−ATH−TH2+ATH+s−1.The continuous functionI is expressed as(19)I:TH,v∈M⟶u+,v+=e−k1DTH,e−k2Dv+τ∈R2,so the phase set N is(20)N=IM=u+,v+u+=e−k1DTH,v+∈σ,where σ=τ,e−k2D1−ATH−TH2+A/TH+s−1+τ. We assume that the initial point u0+,v0+ is always on the L3 in the following sections.


### 3.2. Construction of Poincaré Map

The Poincaré map of system (5) can be defined in different ways. In this work, we choose the L3 to define the Poincaré map.Becauseu∗,v∗ is chosen as the globally asymptotically stable point in system (5), and the value range of TH is defined in the impulse set L4:u=TH as 0<TH<u∗, any trajectory starting from the point Zk+e−k1DTH,vk+ on the phase set must intersect with the impulse set L4:u=TH at the point Zk+1TH,vk+1. From Cauchy–Lipschitz theorem, we know the value of vk+ is only determined by vk+1; in order to discuss fluently, we make vk+1=gvk+. The point Zk+1 on the impulse set L4 is mapped to the point Zk+1+e−k1DTH,vk+1+ on L3 after experiencing an impulse. vk+1+=e−k2Dvk+1+τ is obtained by the impulse function vt+=e−k2Dvt+τ of system (5). Zk+1+ is the initial point of the next impulse function on the phase set.So we can express the Poincaré map of system (5) as(21)vi+1+=e−k2Dgvi++τ=φvi+.We can determine the Poincaré map from the points on the phase set. Next, we infer the expression of Poincaré map functionφ and discuss its properties according to the expression of φ.According to the following formula of system (5),(22)Put,vt=u1−u−suvu+v+A,Qut,vt=δvuu+v+A−d.We can rewrite system (5) as a scalar differential equation on the phase set:(23)dvdu=δvu/u+v+A−du1−u−suv/u+v+A=ρu,v,ve−k1DTH=v0+.For model (23), we only focus on the region(24)Ω1=u,v;u>0,v>0,v<1−Au−u2+As+u−1.The functionρx,y is continuous and differentiable in the region Ω1. Besides, let u0+=e−k1DTH and v0+=J, where J∈N, J<vG, and u0+,v0+∈Ω1, so(25)vu=vu;e−k1DTH,J=vu,J,u∈e−k1DTH,TH.From (23), we get(26)vu,J=J+∫e−k1DTHuρj,vj,Jdj.According to (21) and (26), we can obtain the definition of the Poincaré map φ:(27)φJ=e−k2DvTH,J+τ.We simulate a numerical simulation of the Poincaré map function of model (5) (see Figure 1). The two cases are φvH<vH (Figure 1(a)) and φvH>vH (Figure 1(b)).Figure 1
Poincaré mapφ related to the impulsive point series vk+ with parameters fixed as s=2, A=0.5, δ=1, d=0.25, k1=1, k2=0.2, TH=0.6, and τ=0.3. (a) D=1. (b) D=0.5.
(a)
(b)


### 3.3. The Main Properties of Poincaré Map

Through our analysis of the expression and numerical model of the Poincaré mapφ, and assuming τ>0, the following properties of Poincaré map are given.Theorem 1.
The domain ofφ is 0,+∞, and the range of φ is τ,φvH, where φvH=ve−k1DTH,vH+τ. φ monotonically increases on 0,vH and monotonically decreases on vH,+∞, and as the value of vk+ increases continuously, φ approaches the asymptote φ=τ.Proof.
We first prove the domain ofφ is 0,+∞. Because u∗,v∗ in system (3) is globally asymptotically stable, and because TH<u∗, any initial point from the L3 will reach the impulse set M; so the definition domain of φ is 0,+∞.
Next, we divideL3 into two parts, which are 0,vH and vH,+∞. First of all, we can choose two points Zae−k1DTH,va+ and Zbe−k1DTH,vb+ and assume 0<va+<vb+<vH, then the trajectory of these two points will intersect with L4 at the two points Za+1TH,va+1 and Zb+1TH,vb+1. According to Cauchy–Lipschitz theorem, va+1<vb+1 is true for any va+<vb+, and the formula of φ is(28)φva+=e−k2DTH⋅va+1+τ,φvb+=e−k2DTH⋅vb+1+τ.
So it is easy to find for anyva+ and vb+ on 0,vH and va+<vb+, φva+<φvb+ is always true; therefore, φ monotonically increases on 0,vH.
And then, we prove that theφ is monotonically decreasing on vH,+∞. Similarly, we take two numbers va+ and vb+ on vH,+∞ and assume va+<vb+, then the trajectories of the two initial points Zae−k1DTH,va+ and Zbe−k1DTH,vb+ from L3 will cross the isoclinal line L2 and then intersect L3 at Za′e−k1DTH,va′ and Zb′e−k1DTH,vb′, respectively, where va′>vb′. Then, these two curves will intersect L4 at two points Za+1e−k1DTH,va+1 and Zb+1e−k1DTH,vb+1, respectively. According to Cauchy–Lipschitz theorem, va+1>vb+1 is always true, and the expression of φ is obtained in (27), so for any va+<vb+ on vH,+∞, φva+>φvb+ is always true, and φ is monotonically decreasing on vH,+∞.
Then, we prove that as the value ofvk+ increases, φ tends to be stable and approaches the asymptote φ=τ. We define the closure of Ω1 as(29)Ω1=u,v;u>0,v>0,v<1−Au−u2+As+u−1.
Sinceφ increases monotonically on 0,vH and decreases monotonically on vH,+∞, Ω1 is the invariant set of system (5). Let(30)L=v−1−Au−u2+As+u−1,if(31)Pu,v,Qu,v⋅u2+2s−1u+A−1s+1s+u−12,1L=0≤0.
Here⋅ is the scalar product of two vectors, and then, the vector field will eventually reach the boundary Ω1, so Ω1 is an invariant set, and by calculation, one obtains(32)VuL=0=˙u1−u−suvu+v+A⋅u2+2s−1u+A−1s+1s+u−12+δvuu+v+A−d=δvuu+v+A−d<0.
Sinceφ is monotonically increases on 0,vH and monotonically decreases on vH,+¥, φv0+ is bounded for any v0+∈0,vH and φvH,+∞⊂φ0,vH. From the Cauchy–Lipschitz Theorem, vk+1 is only determined by vk+ and can be expressed by vk+1=σvk+. And any point on the phase set N is always going to be dv/dt<0; hence, limvk+⟶¥σvk+=0. Then, we have(33)limvk+⟶¥φvk+=limvk+⟶¥e−k2Dσvk++τ=τ.
So as the value ofvk+ increases, φ tends to be stable and approaches the asymptote φ=τ.
Sinceφ increases monotonically on 0,vH and decreases monotonically on vH,+¥⊂φ0,vH, φvH,+¥; thus, φ takes the maximum value at vH and takes the minimum value at 0, where φvH=ve−k1DTH,vH+τ and φ0=τ, so the range of value is τ,φvH.Theorem 2.
φ is continuously differentiable.Proof.
Here, we can use the initial conditions of continuity and differentiability theorem which is to take parameters of Cauchy theorem and Lipschitz theorem to determine the continuity and differentiability ofφ; by system (5), we can get that both Pu,v and Qu,v functions are continuously differentiable in the first quadrant; so by Cauchy theorem and Lipschitz theorem with parameters, we can get that the φ is a continuously differentiable function.Theorem 3.
φ always has at least one fixed point if τ>0.Proof.
From Theorem1, we know that φ is monotonically increasing on 0,vH and monotonically decreasing on vH,+∞. Then, we divide it into two cases to discuss the existence of the fixed point for φ.Case I: whenφvH<vH, on the one hand, φ0=τ>0, so φ has at least one number vc on 0,vH such that φvc=vc; on the other hand, since φ is monotonically decreasing on vH,+¥ and φvH<vH, φ has no fixed point on vH,+¥. In conclusion, when φvH<vH, the φ has at least one fixed point.Case II: whenφvH≥vH, because φ monotonically decreases on vH,+¥ and as the value of vk+ increases continuously, φ approaches asymptote φ=τ, so φ has only one point vc on vH,+¥ such that φvc=vc. And the number of fixed points on 0,vH is unknown. So when φvH≥vH, the φ has at least one fixed point.
In conclusion,φ always has at least one fixed point.


## 3.1. Domains of the Poincaré Map

In the following parts, we only discuss the case ofu∗,v∗ as the globally asymptotically stable point of system (5).System (5) has two isoclinal lines, which are defined as L1 and L2:(14)L1:v=1−Au−u2+As+u−1;L2:v=1−dud−A.Next, the lines associated with the phase set and impulse set are defined asL3 and L4:(15)L3:u=e−k1DTH;L4:u=TH.In this case, the value range ofTH is 0<TH<u∗ and lines L3 and L4 always intersect with line L1. The intersection point of L1 and L4 presents as GTH,vG, and the intersection point of L1 and L3 presents as He−k1DTH,vH, where vG and vH are, respectively,(16)vG=1−ATH−TH2+ATH+s−1,vH=1−Ae−k1DTH−ek12D2TH2+Ae−k1DTH+s−1.The open set defined inR+2 is as follows:(17)Ω=u,vu>0,v>0,u<TH⊂R+2.The impulse setM is the part of line L4 above the U-axis and below point G:(18)M=u,v∈R2u=TH,0≤v<1−ATH−TH2+ATH+s−1.The continuous functionI is expressed as(19)I:TH,v∈M⟶u+,v+=e−k1DTH,e−k2Dv+τ∈R2,so the phase set N is(20)N=IM=u+,v+u+=e−k1DTH,v+∈σ,where σ=τ,e−k2D1−ATH−TH2+A/TH+s−1+τ. We assume that the initial point u0+,v0+ is always on the L3 in the following sections.


## 3.2. Construction of Poincaré Map

The Poincaré map of system (5) can be defined in different ways. In this work, we choose the L3 to define the Poincaré map.Becauseu∗,v∗ is chosen as the globally asymptotically stable point in system (5), and the value range of TH is defined in the impulse set L4:u=TH as 0<TH<u∗, any trajectory starting from the point Zk+e−k1DTH,vk+ on the phase set must intersect with the impulse set L4:u=TH at the point Zk+1TH,vk+1. From Cauchy–Lipschitz theorem, we know the value of vk+ is only determined by vk+1; in order to discuss fluently, we make vk+1=gvk+. The point Zk+1 on the impulse set L4 is mapped to the point Zk+1+e−k1DTH,vk+1+ on L3 after experiencing an impulse. vk+1+=e−k2Dvk+1+τ is obtained by the impulse function vt+=e−k2Dvt+τ of system (5). Zk+1+ is the initial point of the next impulse function on the phase set.So we can express the Poincaré map of system (5) as(21)vi+1+=e−k2Dgvi++τ=φvi+.We can determine the Poincaré map from the points on the phase set. Next, we infer the expression of Poincaré map functionφ and discuss its properties according to the expression of φ.According to the following formula of system (5),(22)Put,vt=u1−u−suvu+v+A,Qut,vt=δvuu+v+A−d.We can rewrite system (5) as a scalar differential equation on the phase set:(23)dvdu=δvu/u+v+A−du1−u−suv/u+v+A=ρu,v,ve−k1DTH=v0+.For model (23), we only focus on the region(24)Ω1=u,v;u>0,v>0,v<1−Au−u2+As+u−1.The functionρx,y is continuous and differentiable in the region Ω1. Besides, let u0+=e−k1DTH and v0+=J, where J∈N, J<vG, and u0+,v0+∈Ω1, so(25)vu=vu;e−k1DTH,J=vu,J,u∈e−k1DTH,TH.From (23), we get(26)vu,J=J+∫e−k1DTHuρj,vj,Jdj.According to (21) and (26), we can obtain the definition of the Poincaré map φ:(27)φJ=e−k2DvTH,J+τ.We simulate a numerical simulation of the Poincaré map function of model (5) (see Figure 1). The two cases are φvH<vH (Figure 1(a)) and φvH>vH (Figure 1(b)).Figure 1
Poincaré mapφ related to the impulsive point series vk+ with parameters fixed as s=2, A=0.5, δ=1, d=0.25, k1=1, k2=0.2, TH=0.6, and τ=0.3. (a) D=1. (b) D=0.5.
(a)
(b)


## 3.3. The Main Properties of Poincaré Map

Through our analysis of the expression and numerical model of the Poincaré mapφ, and assuming τ>0, the following properties of Poincaré map are given.Theorem 1.
The domain ofφ is 0,+∞, and the range of φ is τ,φvH, where φvH=ve−k1DTH,vH+τ. φ monotonically increases on 0,vH and monotonically decreases on vH,+∞, and as the value of vk+ increases continuously, φ approaches the asymptote φ=τ.Proof.
We first prove the domain ofφ is 0,+∞. Because u∗,v∗ in system (3) is globally asymptotically stable, and because TH<u∗, any initial point from the L3 will reach the impulse set M; so the definition domain of φ is 0,+∞.
Next, we divideL3 into two parts, which are 0,vH and vH,+∞. First of all, we can choose two points Zae−k1DTH,va+ and Zbe−k1DTH,vb+ and assume 0<va+<vb+<vH, then the trajectory of these two points will intersect with L4 at the two points Za+1TH,va+1 and Zb+1TH,vb+1. According to Cauchy–Lipschitz theorem, va+1<vb+1 is true for any va+<vb+, and the formula of φ is(28)φva+=e−k2DTH⋅va+1+τ,φvb+=e−k2DTH⋅vb+1+τ.
So it is easy to find for anyva+ and vb+ on 0,vH and va+<vb+, φva+<φvb+ is always true; therefore, φ monotonically increases on 0,vH.
And then, we prove that theφ is monotonically decreasing on vH,+∞. Similarly, we take two numbers va+ and vb+ on vH,+∞ and assume va+<vb+, then the trajectories of the two initial points Zae−k1DTH,va+ and Zbe−k1DTH,vb+ from L3 will cross the isoclinal line L2 and then intersect L3 at Za′e−k1DTH,va′ and Zb′e−k1DTH,vb′, respectively, where va′>vb′. Then, these two curves will intersect L4 at two points Za+1e−k1DTH,va+1 and Zb+1e−k1DTH,vb+1, respectively. According to Cauchy–Lipschitz theorem, va+1>vb+1 is always true, and the expression of φ is obtained in (27), so for any va+<vb+ on vH,+∞, φva+>φvb+ is always true, and φ is monotonically decreasing on vH,+∞.
Then, we prove that as the value ofvk+ increases, φ tends to be stable and approaches the asymptote φ=τ. We define the closure of Ω1 as(29)Ω1=u,v;u>0,v>0,v<1−Au−u2+As+u−1.
Sinceφ increases monotonically on 0,vH and decreases monotonically on vH,+∞, Ω1 is the invariant set of system (5). Let(30)L=v−1−Au−u2+As+u−1,if(31)Pu,v,Qu,v⋅u2+2s−1u+A−1s+1s+u−12,1L=0≤0.
Here⋅ is the scalar product of two vectors, and then, the vector field will eventually reach the boundary Ω1, so Ω1 is an invariant set, and by calculation, one obtains(32)VuL=0=˙u1−u−suvu+v+A⋅u2+2s−1u+A−1s+1s+u−12+δvuu+v+A−d=δvuu+v+A−d<0.
Sinceφ is monotonically increases on 0,vH and monotonically decreases on vH,+¥, φv0+ is bounded for any v0+∈0,vH and φvH,+∞⊂φ0,vH. From the Cauchy–Lipschitz Theorem, vk+1 is only determined by vk+ and can be expressed by vk+1=σvk+. And any point on the phase set N is always going to be dv/dt<0; hence, limvk+⟶¥σvk+=0. Then, we have(33)limvk+⟶¥φvk+=limvk+⟶¥e−k2Dσvk++τ=τ.
So as the value ofvk+ increases, φ tends to be stable and approaches the asymptote φ=τ.
Sinceφ increases monotonically on 0,vH and decreases monotonically on vH,+¥⊂φ0,vH, φvH,+¥; thus, φ takes the maximum value at vH and takes the minimum value at 0, where φvH=ve−k1DTH,vH+τ and φ0=τ, so the range of value is τ,φvH.Theorem 2.
φ is continuously differentiable.Proof.
Here, we can use the initial conditions of continuity and differentiability theorem which is to take parameters of Cauchy theorem and Lipschitz theorem to determine the continuity and differentiability ofφ; by system (5), we can get that both Pu,v and Qu,v functions are continuously differentiable in the first quadrant; so by Cauchy theorem and Lipschitz theorem with parameters, we can get that the φ is a continuously differentiable function.Theorem 3.
φ always has at least one fixed point if τ>0.Proof.
From Theorem1, we know that φ is monotonically increasing on 0,vH and monotonically decreasing on vH,+∞. Then, we divide it into two cases to discuss the existence of the fixed point for φ.Case I: whenφvH<vH, on the one hand, φ0=τ>0, so φ has at least one number vc on 0,vH such that φvc=vc; on the other hand, since φ is monotonically decreasing on vH,+¥ and φvH<vH, φ has no fixed point on vH,+¥. In conclusion, when φvH<vH, the φ has at least one fixed point.Case II: whenφvH≥vH, because φ monotonically decreases on vH,+¥ and as the value of vk+ increases continuously, φ approaches asymptote φ=τ, so φ has only one point vc on vH,+¥ such that φvc=vc. And the number of fixed points on 0,vH is unknown. So when φvH≥vH, the φ has at least one fixed point.
In conclusion,φ always has at least one fixed point.


## 4. Study on the Periodic Solutions of System (5)

### 4.1. Boundary Periodic Solutions of System (5)

For system (5), if the predator population becomes extinct and the predator also terminates its release, then system (5) has a boundary periodic solution, which produces the following system:(34)u′t=u1−u,ut<TH,ut+=e−k1Dut,ut=TH.Solving (34) with initial value u0+=e−k1DTH,(35)ut=e−k1DTHet1+e−k1DTHet−e−k1DTH.The trajectory from the initial point will eventually intersect with the straight line of the impulse set over time:(36)TH=e−k1DTHeT1+e−k1DTHeT−e−k1DTH.Solving the equation ofT and D, we get(37)T=ln1−e−k1DTHe−k1D1−TH,D=1k1lnTH+eT1−TH,where T is the period of the boundary periodic solution and D is the insecticide dose required to control the number of pests below the TH. Then, the boundary periodic solution of system (5) with a period of T is(38)ut=THet−k−1TTH−THet−k−1T−e−k1D,vt=0.Theorem 4.
The boundary periodic solutionUTt,0 of system (5) is asymptotically stable if(39)R1=e−k1De−k2D1−e−k1DTH1−THexp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt<1.Proof.
By Definition3, we obtain(40)Pu,v=u1−u−suvu+v+A,Qu,v=δvuu+v+A−d,αu,v=ue−k1D−1,βu,v=ve−k2D−1+τ,ϕu,v=u−TH,uTT,vTT=TH,0,uTT+,vTT+=e−k1DTH,0.
From the above formula, we can get(41)∂P∂u=1−2u−svv+Au+v+A2,∂Q∂v=δuu+v+A−uvu+v+A2−d,∂α∂u=e−k1D−1,∂β∂v=e−k2D−1,∂ϕ∂u=1,∂α∂v=∂β∂u=∂ϕ∂v=0,so(42)Δ1=P+∂β/∂v∂ϕ/∂u−∂β/∂u∂ϕ/∂v+∂ϕ/∂u+Q+∂α/∂u∂ϕ/∂v−∂α/∂v∂ϕ/∂u+∂ϕ/∂uP∂ϕ/∂u+Q∂ϕ/∂v=e−k1De−k2D1−e−k1DTH1−TH.
In addition,(43)exp∫0T∂P∂uuTt,vTt+∂Q∂vuTt,vTtdt=exp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt.
The expression ofμ1 is(44)μ1=Δ1exp∫0T∂P∂uuTt,vTt+∂Q∂vuTt,vTtdt=e−k1De−k2D1−e−k1DTH1−THexp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt.
If condition (39) is true, then μ1<1. It means the periodic solution uTt,0 of the boundary is asymptotically stable.


### 4.2. Existence and Stability of Periodic Solutions of Order-kk≥1 at τ>0

From Theorem3, the Poincaré map function φ of system (5) has at least one fixed point; that is to say, system (5) must have at least one order-1 periodic solution.Theorem 5.
The order-1 periodic solutionξt,ηt is orbitally asymptotically stable if and only if(45)e−k2De−k1D1−e−k1DTH−se−k1De−k2Dη0+τ/e−k1DTH+e−k2Dη0+τ+A1−TH−sη0/TH+η0+A⋅exp∫0TUtdt<1,where(46)Ut=exp∫​0T1−2ξt−δd+δξtξt+ηt+A+sηtηt+A+δξtηtξt+ηt+A2dt.Proof.
We useRTH,η0 and R+e−k1DTH,e−k2Dη0+τ to represent the start point and the endpoint of the order-1 periodic solution, respectively. From Theorem 4, we know that the Floquet multiplier(47)μ1=Δ1exp∫0T∂P∂uξTt,ηTt+∂Q∂vξTt,ηTtdt=e−k2De−k1D1−e−k1DTH−se−k1De−k2Dη0+τ/e−k1DTH+e−k2Dη0+τ+A1−TH−sη0/TH+η0+A⋅exp∫0TUtdt.
If (45) is true, then μ1<1, so the order-1 periodic solution is always orbitally asymptotically stable.Theorem 6.
IfφvH<vH, there is at least one locally asymptotically stable order-1 periodic solution in system (5). Furthermore, if there is only one fixed point on 0,vH, then there is a globally asymptotically stable order-1 periodic solution of system (5).Proof.
IfφvH<vH, Poincaré map φ has at least one fixed point. This also proves that there is at least one order-1 periodic solution in system (5). From Theorem 5, we know that the periodic solution is always asymptotic stable if μ1<1. So if φvH<vH, then there is at least one locally asymptotically stable order-1 periodic solution in system (5).
IfφvH<vH, there is only one fixed point on 0,vH. This proves that there is a unique order-1 periodic solution in system (5). According to Theorem 5, we can conclude that the periodic solution is asymptotic stable.
For any trajectory starting frome−k1DTH,v0+, if v0+∈0,vH, then v0+<φv0+<vH. After n times pulses, φnv0+ monotonically increases, so limn⟶+∞φnv0+=vH.
In contrary, ifv0+>vH, we need to discuss this according to different cases. On the one hand, if φnv0+>vH is always holding, we can conclude that φnv0+ is monotonically decreasing because φnv0+<v0+ and limn⟶+∞φnv0+=vH. On the other hand, φnv0+>vH is not true for all n. We make n0 the smallest which satisfies φn0v0+<vH. Then, there must be a positive integer n1>n0 and φn1v0+ monotonically increases as n1 increases, so limn1⟶+∞φnv0+=vH. Therefore, there is a globally asymptotically stable order-1 periodic solution of system (5).Theorem 7.
IfφvH>vH, φ2vH≥vH, and there are no fixed points on 0,vH of φ; then, system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution.Proof.
If there are no fixed points on0,vH of φ, then there is a positive constant i which makes vi+≤vH and vi+1+≥vH. Based on the definition of the Poincaré map φ, we get vi+1+=φvi+≤φvH and vi+1+∈vH,φvH. Because the φ is decreasing on vH,+∞ for any v1+>vH, after one pulse, there is v2+=φv1+≤φvH. Therefore, vi+1+∈vH,φvH holds for all i≥2. Also, φ2 is monotonically increasing on vH,+∞ so that(48)φvH,φvH=φ2vH,φvH⊂vH,φvH.
Next, the existence of the periodic solution is discussed. First of all, for anyv0+∈vH,φyH and make v1+=φv0+≠v0+ and v2+=φ2v0+≠v0+. If v1+=φv0+=v0+ and v2+=φ2v0+=v0+, then v0+ is the fixed point of φ which proves system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution. So the relations about vH, φvH, v0+ ,v1+, and v2+ are needed to be discussed.i vH≤v2+<v0+<v1+≤φvH (Figure 2(a)). Then, v3+=φv2+>φv0+=v1+ and v4+=φv3+>φv1+=v2+. It can be obtained by mathematical induction that(49)vH≤⋯<v2n+2+<v2n+<⋯<v2+<v0+<v1+<⋯<v2n−1+<v2n+1+<⋯≤φvH.ii vH≤v0+<v2+<v1+≤φvH (Figure 2(b)). Then, v1+=φv0+>φv2+=v3+>v2+=φv1+ and v2+=φv1+<φv3+=v4+<v3+=φv2+<v1+; i.e., it can be obtained by mathematical induction that(50)vH≤v0+<v2+<⋯<v2n+<v2n+2+<⋯<v2n+1+<v2n−1+<⋯<v1+≤φvH.iii vH≤v1+<v2+<v0+≤φvH (Figure 2(c)). In the same way, about case ii, we can obtain(51)vH≤v1+<⋯<v2n−1+<v2n+1+<⋯<v2n+2+<v2n+<⋯<v2+<v0+≤φvH.iv vH≤v1+<v0+<v2+≤φvH (Figure 2(d)). By using the same method as case i, we can obtain(52)vH≤⋯<v2n+1+<v2n−1+<⋯<v1+<v0+<v2+<⋯<v2n+<v2n+2+<⋯≤φvH.
For case (ii),φ2nv0+=v2n+ is monotonically increasing and φ2n+1v0+=v2n+1+ is monotonically decreasing; for case (iii), φ2nv0+=v2n+ is monotonically decreasing and φ2n+1v0+=v2n+1+ is monotonically increasing. It is concluded that for case (ii) and case (iii), there exists either a unique fixed point va such that(53)limn⟶∞v2n+=limn⟶∞v2n+1+=va,va∈vH,φvH,or exists two distinct values va and vb and va≠vb such that(54)limn⟶∞v2n+=va,va∈vH,φvH,limn⟶∞v2n+1+=vb,vb∈vH,φvH.
However, for cases (i) and (iv), only the later case can be true.
These results verify that there exists either an order-1 periodic solution or a periodic solution+ (5).Figure 2
Four cases of existence of periodic solutions in Theorem7.
(a)
(b)
(c)
(d)Theorem 8.
IfφvH>vH, then vm+ satisfies vm+=minv+:φmv+=v+, and if there is no fixed point on 0,vH, when φ2vH≥vm+, then system (5) has an order-3 periodic solution.Proof.
IfφvH>vH, and there is no fixed point on 0,vH, it can be seen from Theorem 1 that there is a unique order-1 period solution in vH,φvH:(55)φu˜=u˜,u˜∈vH,φvH,because Poincaré map φ is continuous on closed intervals 0,u˜ and(56)φ0=τ,φu˜=u˜.
According to the intermediate value theorem, there existsvm+∈0,u˜, and φvm+=uH.
Furthermore,(57)φ3vm+=φ2uH<vm+,φ30>0.
According to the properties of continuous functions on closed intervals, there must be at least one value ofu⟶ to enable(58)φ3u⟶=u⟶.
This means that system (5) has an order-3 periodic solution.
If we replace conditionφ2vH≥vm+ in Theorem 8 with condition φk−1uH<vm+, where φvm+=uH, the order-k periodic solution of system (5) can be obtained by a similar method of Theorem 8.


## 4.1. Boundary Periodic Solutions of System (5)

For system (5), if the predator population becomes extinct and the predator also terminates its release, then system (5) has a boundary periodic solution, which produces the following system:(34)u′t=u1−u,ut<TH,ut+=e−k1Dut,ut=TH.Solving (34) with initial value u0+=e−k1DTH,(35)ut=e−k1DTHet1+e−k1DTHet−e−k1DTH.The trajectory from the initial point will eventually intersect with the straight line of the impulse set over time:(36)TH=e−k1DTHeT1+e−k1DTHeT−e−k1DTH.Solving the equation ofT and D, we get(37)T=ln1−e−k1DTHe−k1D1−TH,D=1k1lnTH+eT1−TH,where T is the period of the boundary periodic solution and D is the insecticide dose required to control the number of pests below the TH. Then, the boundary periodic solution of system (5) with a period of T is(38)ut=THet−k−1TTH−THet−k−1T−e−k1D,vt=0.Theorem 4.
The boundary periodic solutionUTt,0 of system (5) is asymptotically stable if(39)R1=e−k1De−k2D1−e−k1DTH1−THexp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt<1.Proof.
By Definition3, we obtain(40)Pu,v=u1−u−suvu+v+A,Qu,v=δvuu+v+A−d,αu,v=ue−k1D−1,βu,v=ve−k2D−1+τ,ϕu,v=u−TH,uTT,vTT=TH,0,uTT+,vTT+=e−k1DTH,0.
From the above formula, we can get(41)∂P∂u=1−2u−svv+Au+v+A2,∂Q∂v=δuu+v+A−uvu+v+A2−d,∂α∂u=e−k1D−1,∂β∂v=e−k2D−1,∂ϕ∂u=1,∂α∂v=∂β∂u=∂ϕ∂v=0,so(42)Δ1=P+∂β/∂v∂ϕ/∂u−∂β/∂u∂ϕ/∂v+∂ϕ/∂u+Q+∂α/∂u∂ϕ/∂v−∂α/∂v∂ϕ/∂u+∂ϕ/∂uP∂ϕ/∂u+Q∂ϕ/∂v=e−k1De−k2D1−e−k1DTH1−TH.
In addition,(43)exp∫0T∂P∂uuTt,vTt+∂Q∂vuTt,vTtdt=exp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt.
The expression ofμ1 is(44)μ1=Δ1exp∫0T∂P∂uuTt,vTt+∂Q∂vuTt,vTtdt=e−k1De−k2D1−e−k1DTH1−THexp∫0T1−2uTt−δd+δuTtuTt+vTt+A+svTtvTt+A+δuTtvTtuTt+vTt+A2dt.
If condition (39) is true, then μ1<1. It means the periodic solution uTt,0 of the boundary is asymptotically stable.


## 4.2. Existence and Stability of Periodic Solutions of Order-kk≥1 at τ>0

From Theorem3, the Poincaré map function φ of system (5) has at least one fixed point; that is to say, system (5) must have at least one order-1 periodic solution.Theorem 5.
The order-1 periodic solutionξt,ηt is orbitally asymptotically stable if and only if(45)e−k2De−k1D1−e−k1DTH−se−k1De−k2Dη0+τ/e−k1DTH+e−k2Dη0+τ+A1−TH−sη0/TH+η0+A⋅exp∫0TUtdt<1,where(46)Ut=exp∫​0T1−2ξt−δd+δξtξt+ηt+A+sηtηt+A+δξtηtξt+ηt+A2dt.Proof.
We useRTH,η0 and R+e−k1DTH,e−k2Dη0+τ to represent the start point and the endpoint of the order-1 periodic solution, respectively. From Theorem 4, we know that the Floquet multiplier(47)μ1=Δ1exp∫0T∂P∂uξTt,ηTt+∂Q∂vξTt,ηTtdt=e−k2De−k1D1−e−k1DTH−se−k1De−k2Dη0+τ/e−k1DTH+e−k2Dη0+τ+A1−TH−sη0/TH+η0+A⋅exp∫0TUtdt.
If (45) is true, then μ1<1, so the order-1 periodic solution is always orbitally asymptotically stable.Theorem 6.
IfφvH<vH, there is at least one locally asymptotically stable order-1 periodic solution in system (5). Furthermore, if there is only one fixed point on 0,vH, then there is a globally asymptotically stable order-1 periodic solution of system (5).Proof.
IfφvH<vH, Poincaré map φ has at least one fixed point. This also proves that there is at least one order-1 periodic solution in system (5). From Theorem 5, we know that the periodic solution is always asymptotic stable if μ1<1. So if φvH<vH, then there is at least one locally asymptotically stable order-1 periodic solution in system (5).
IfφvH<vH, there is only one fixed point on 0,vH. This proves that there is a unique order-1 periodic solution in system (5). According to Theorem 5, we can conclude that the periodic solution is asymptotic stable.
For any trajectory starting frome−k1DTH,v0+, if v0+∈0,vH, then v0+<φv0+<vH. After n times pulses, φnv0+ monotonically increases, so limn⟶+∞φnv0+=vH.
In contrary, ifv0+>vH, we need to discuss this according to different cases. On the one hand, if φnv0+>vH is always holding, we can conclude that φnv0+ is monotonically decreasing because φnv0+<v0+ and limn⟶+∞φnv0+=vH. On the other hand, φnv0+>vH is not true for all n. We make n0 the smallest which satisfies φn0v0+<vH. Then, there must be a positive integer n1>n0 and φn1v0+ monotonically increases as n1 increases, so limn1⟶+∞φnv0+=vH. Therefore, there is a globally asymptotically stable order-1 periodic solution of system (5).Theorem 7.
IfφvH>vH, φ2vH≥vH, and there are no fixed points on 0,vH of φ; then, system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution.Proof.
If there are no fixed points on0,vH of φ, then there is a positive constant i which makes vi+≤vH and vi+1+≥vH. Based on the definition of the Poincaré map φ, we get vi+1+=φvi+≤φvH and vi+1+∈vH,φvH. Because the φ is decreasing on vH,+∞ for any v1+>vH, after one pulse, there is v2+=φv1+≤φvH. Therefore, vi+1+∈vH,φvH holds for all i≥2. Also, φ2 is monotonically increasing on vH,+∞ so that(48)φvH,φvH=φ2vH,φvH⊂vH,φvH.
Next, the existence of the periodic solution is discussed. First of all, for anyv0+∈vH,φyH and make v1+=φv0+≠v0+ and v2+=φ2v0+≠v0+. If v1+=φv0+=v0+ and v2+=φ2v0+=v0+, then v0+ is the fixed point of φ which proves system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution. So the relations about vH, φvH, v0+ ,v1+, and v2+ are needed to be discussed.i vH≤v2+<v0+<v1+≤φvH (Figure 2(a)). Then, v3+=φv2+>φv0+=v1+ and v4+=φv3+>φv1+=v2+. It can be obtained by mathematical induction that(49)vH≤⋯<v2n+2+<v2n+<⋯<v2+<v0+<v1+<⋯<v2n−1+<v2n+1+<⋯≤φvH.ii vH≤v0+<v2+<v1+≤φvH (Figure 2(b)). Then, v1+=φv0+>φv2+=v3+>v2+=φv1+ and v2+=φv1+<φv3+=v4+<v3+=φv2+<v1+; i.e., it can be obtained by mathematical induction that(50)vH≤v0+<v2+<⋯<v2n+<v2n+2+<⋯<v2n+1+<v2n−1+<⋯<v1+≤φvH.iii vH≤v1+<v2+<v0+≤φvH (Figure 2(c)). In the same way, about case ii, we can obtain(51)vH≤v1+<⋯<v2n−1+<v2n+1+<⋯<v2n+2+<v2n+<⋯<v2+<v0+≤φvH.iv vH≤v1+<v0+<v2+≤φvH (Figure 2(d)). By using the same method as case i, we can obtain(52)vH≤⋯<v2n+1+<v2n−1+<⋯<v1+<v0+<v2+<⋯<v2n+<v2n+2+<⋯≤φvH.
For case (ii),φ2nv0+=v2n+ is monotonically increasing and φ2n+1v0+=v2n+1+ is monotonically decreasing; for case (iii), φ2nv0+=v2n+ is monotonically decreasing and φ2n+1v0+=v2n+1+ is monotonically increasing. It is concluded that for case (ii) and case (iii), there exists either a unique fixed point va such that(53)limn⟶∞v2n+=limn⟶∞v2n+1+=va,va∈vH,φvH,or exists two distinct values va and vb and va≠vb such that(54)limn⟶∞v2n+=va,va∈vH,φvH,limn⟶∞v2n+1+=vb,vb∈vH,φvH.
However, for cases (i) and (iv), only the later case can be true.
These results verify that there exists either an order-1 periodic solution or a periodic solution+ (5).Figure 2
Four cases of existence of periodic solutions in Theorem7.
(a)
(b)
(c)
(d)Theorem 8.
IfφvH>vH, then vm+ satisfies vm+=minv+:φmv+=v+, and if there is no fixed point on 0,vH, when φ2vH≥vm+, then system (5) has an order-3 periodic solution.Proof.
IfφvH>vH, and there is no fixed point on 0,vH, it can be seen from Theorem 1 that there is a unique order-1 period solution in vH,φvH:(55)φu˜=u˜,u˜∈vH,φvH,because Poincaré map φ is continuous on closed intervals 0,u˜ and(56)φ0=τ,φu˜=u˜.
According to the intermediate value theorem, there existsvm+∈0,u˜, and φvm+=uH.
Furthermore,(57)φ3vm+=φ2uH<vm+,φ30>0.
According to the properties of continuous functions on closed intervals, there must be at least one value ofu⟶ to enable(58)φ3u⟶=u⟶.
This means that system (5) has an order-3 periodic solution.
If we replace conditionφ2vH≥vm+ in Theorem 8 with condition φk−1uH<vm+, where φvm+=uH, the order-k periodic solution of system (5) can be obtained by a similar method of Theorem 8.


## 5. Numerical Simulation

In the state impulse feedback control, we assign appropriate thresholds forTH and D (see Figures 3 and 4). In Figures 3 and 4, the red line shows the trajectory of the system without the impulse, and the green line shows the trajectory of the system with the impulse; this suggests that populations of predators and pests can be kept within a stable range.Figure 3
State pulse feedback control strategy of system (5): s=2, A=0.5, δ=1, d=0.25, k1=1, k2=0.2, TH=0.6, τ=0.3, and D=0.5.
(a)
(b)
(c)Figure 4
State pulse feedback control strategy of system (5): s=2, A=0.5, δ=1, d=0.25, k1=1, k2=0.2, TH=0.6, τ=0.3, and D=1.
(a)
(b)
(c)It can be seen from Figure5 that different initial points will eventually converge to the same order-1 periodic solution and tend to be stable; this indicates the global asymptotic stability of the order-1 periodic solution.Figure 5
State pulse feedback control strategy of system (5): s=2, A=0.5, δ=1, d=0.25, k1=1, k2=0.2, TH=0.6, τ=0.3, and D=1, and with three different initial points.The above numerical simulation also shows that the number of pests can be controlled in the state pulse feedback control, which verifies the feasibility of state pulse feedback control.In Section4.1, when the predators disappear and the pests reach TH, we obtain the expression of the boundary period solution and the expression of the pesticide dose. Next, we discuss which key factors can affect the pesticide dose D. We gave some reasonable parameters, as shown in Figure 6. The results show that as e−k1DTH decreases, dose D must also be increased (Figure 6(a)). Furthermore, as the T of chemical control increases, the dose D increases (Figure 6(b)). Biologically, we need to consider both the threshold TH and the period T in the process of pest control.Figure 6
Effects of parameterse−k1DTH and T on the D of chemical control: (a) T=20, k1=1.6; (b) TH=0.22, k1=1.6.
(a)
(b)According to condition (39), we can judge whether the chemical control can stabilize the boundary periodic solution alone. R1<1 means that chemical control by D dose alone can control the pest population below the TH, and vice versa. Therefore, how much the dose D and the threshold TH can affect R1 has drawn our attention. For these, we have carried out numerical simulations, as shown in Figure 7(a). The results show that when a single chemical control method is used, high dose D can control the pest population. In addition, as shown in Figure 7(b), for a relatively small TH, we have R1<1, and once TH is greater than a certain value, R1>1. The results show that under the fixed parameter values, the smaller the TH value, the better the prevention and control of pests. In the process of pesticide management, as long as we choose a reasonable threshold TH under pulse state feedback control, we can avoid excessive use of pesticides and reduce some negative effects of pesticides.Figure 7
Effects of control parameters on the threshold conditionR1 and k1=−1.6, k2=−1.8, d=0.25, and A=0.5: (a) TH=0.22; (b) D=0.5.
(a)
(b)


## 6. Conclusion

Compared with previous studies on state-dependent feedback control, we mainly do the following work: the global dynamics of complex models are studied according to the Poincaré map, and the main properties of Poincaré map are studied to prove the existence of fixed points and the existence of order-kk≥1 periodic solutions. Besides, we study the effect of pesticide dose on single chemical control or chemical control combined with biological control. The results show that the pest population density can not only be controlled below the threshold under the state pulse feedback control but also avoid excessive application of pesticides and reduce some negative effects of pesticides.



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*Source: 5308014-2019-11-22.xml*