# A State-Dependent Impulsive Nonlinear System with Ratio-Dependent Action Threshold for Investigating the Pest-Natural Enemy Model

Authors: Ihsan Ullah Khan; Saif Ullah; Ebenezer Bonyah; Basem Al Alwan; Ahmed Alshehri
Journal: Complexity
 (2022)

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## Abstract

Based on the Lotka–Volterra system, a pest-natural enemy model with nonlinear feedback control as well as nonlinear action threshold is introduced. The model characterizes the implementation of comprehensive prevention and control measures when the pest density reaches the nonlinear action threshold level depending on the pest density and its change rate. The mortality rate of the pest is a saturation function that strictly depends on their density while the release of natural enemies is also a nonlinear pulse term depending on the density of real-time natural enemies. The exact impulsive and phase sets are given. The definition and properties of the Poincaré map corresponding to the pulse points on the phase set are provided. We investigate the existence and stability of boundary and interior order-1 periodic solution. The theoretical analysis developed in the present paper combined with nonlinear controlling measures as well as nonlinear action threshold methods and techniques laid the foundation for the establishment and analysis of other state-dependent feedback control models.


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## Body

## 1. Introduction

Pest control [1–6] is not only an ancient problem but also a new challenge faced by the modern world. Various scientific and effective methods [7–13] are needed to comprehensively prevent and control pest outbreaks and reinfestation. The most common early method was chemical control [14, 15], that is, the method of controlling pest by spraying pesticides during pest outbreaks. The main advantages of chemical control are quick effect and convenient use. It can eradicate or maintain the number of pests at a lower level within a short period of time. Therefore, chemical control is still one of the important means to control pest population. Biological control [16–18] is another important control method, which has the advantages of strong effect and long duration, and is also an environmental friendly control method. Maiti et al. [19] used a valuable technique known as sterile insect release method (SIRM) to manage the pest population. The authors discussed the effect of uncertain ecological variations on sterile and fertile insects. Other main methods are physical control and agricultural control. For example, the agricultural control method is a method to reduce or control pests through measures such as crop rotation, intercropping, and reasonable adjustment of cultivation procedures.Each pest control method has its advantages and disadvantages. Due to long-term and high-dose use, pests can easily develop resistance to specific pesticides, resulting in pest control failure and pest reemergance. However, other control strategies cannot effectively reduce the number of pests in a short time because of their slow effectiveness. Therefore, how to effectively and reasonably use multiple methods is the best choice for pest control. Based on this, the Food and Agriculture Organization of the United Nations (FAO) proposed the concept of integrated pest management (IPM) [1, 20, 21] and defined it as follows: “IPM is a pest control system that comprehensively considers the population dynamics of the pest and its related environment and uses all appropriate control techniques and methods that work as closely as possible to maintain levels at which pest populations do not cause economic harm.” Both experimentally [22, 23] and theoretically [24, 25], it has been proved that IPM is more practical than the classic approach. This is one of the most useful methods which minimizes damage to individuals and the environment in addressing pest control.In this perspective, researchers have studied the mathematical problems based on impulsive differential equations in order investigate the dynamics of IPM and compass biped robotic systems. In numerous realistic problems, impulses often occur at state-dependent. Therefore, it is more feasible to apply the procedure of state-dependent feedback control to model real-world issues. Znegui et al. [26] used an impulsive hybrid nonlinear system to construct a passive biped robot model that demonstrates complicated behaviors. In [27], the authors constructed a Poincaré map which was further utilized to examine the existence and stability of order-1 periodic type solution of the problem under consideration. Many new systems on the design of specific analytical expression of the hybrid state-dependent Poincaré were studied in [28, 29]. The authors in [26–29] portrayed an expression of the controlled Poincaré map to discuss the stabilization of passive dynamic walking of the compass-gait biped robot. The compass-gait biped robot is a two-DoF legged mechanical system which is identified by its passive dynamic walking. The one-DoF mechanical systems are also of great importance. Some articles related to one-DoF state-feedback control with respect to different perspectives can be found in [30, 31].The impulsive differential equations are also used proficiently in epidemic dynamics [32] and population dynamics [33–35]. A basic assumption of the above series of studies is that regardless of how huge the number of pests or the growth rate is, as long as the number of pest populations touches economic threshold (ET) [33–35], the IPM strategy can be implemented. However, there are two basic situations of actual pest growth that require high attention: first, the number of pests is comparatively large, and the rate of change is small; second, the population is small, but the rate of change is high. A fundamental problem illustrated by these two situations is that when the pest population is large (such as exceeding ET), the growth rate is small or even negative at this time. In this case, even if the IPM strategy is not implemented, the number of pests may not exceed economic injury level (EIL) [36]. Another situation is that the number of pests is not large, and the rate at which the pest population is growing is very large. In this case, if the control strategy is not implemented in time, it may lead to a large outbreak of pests. Next, in order to establish appropriate and effective integrated controlling strategies, the IPM process needs precise inspection of the pest quantity. The mortality rate should be fluctuated according to the saturating function which relies upon the density of pest, and the releasing quantity of natural enemies should be a function of their density. Therefore, keeping in mind the above factors, a feasible new state-feedback control pest-natural enemy ecosystem with nonlinear controlling measures as well as nonlinear action threshold system is proposed. The corresponding analytical techniques and numerical methods are developed to examine the dynamical aspects of the system under consideration.The main research contents are reflected in the following aspects. We construct a Lotka–Volterra prey-predator model involving both nonlinear feedback and action threshold depending on the density of pest and its change rate. In the model, we use the action threshold instead of the economic threshold to characterize the implementation of control measures, that is, when the number of pests reaches the action threshold depending on the density of pest and its change rate, a comprehensive pest control tactic is applied so that the number of pests does not exceed the nonlinear ratio-dependent AT. On the other hand, the use of nonlinear controlling factors in the feedback control makes the model closer to reality. Properties of the nonlinear ratio-dependent AT are given. Then, the classification is performed according to the positional relationship between the action threshold level and the stable equilibrium point of the corresponding ordinary differential system. By using the definition and properties of Lambert W function, the analytical expression of the Poincaré map is given. Furthermore, by using the analytical properties of Poincaré map, the existence, uniqueness, and stability of the pest-free and interior-order one periodic solution of the pest-natural enemy system are given, and corresponding sufficient conditions are obtained. The main results are confirmed by numerical simulations.


## 2. Model Construction and Main Properties of Action Threshold

### 2.1. Construction of Model

In view of the above objective factors, we propose the following nonlinear state-dependent feedback control model combined with nonlinear ratio-dependent AT:(1)dxtdt=axt−bxtyt,dytdt=cxtyt−dyt,θ1xt+θ2dxtdt<AT,xt+=1−δxtxt+αxt,yt+=yt+υ1+βyt,θ1xt+θ2dxtdt=AT.It can be seen that without pulse control measures, the model is simply based on the classical Lotka–Volterra type problem which is extensively used to describe the relation between the populations of pest and natural enemy shown byxt and yt, respectively. Weighted parameters θ1, θ2, and AT are positive constants, which satisfy θ1+θ2=1. The discontinuous mapping shown in the third and fourth equations in system (1) represents that the implementation of comprehensive control measures depends on the action level, that is, once the pest density reaches action threshold, the densities of pests as well as the natural enemies are immediately updated to 1−δxt/xt+αxt and yt+υ/1+βyt, respectively. α>0 represents the semisaturation constant, δ>0 is defined as the maximum instantaneous killing rate after the use of pesticides, and υ>0 is the maximum natural enemy when executing the control strategy. The amount β>0 is the natural enemy density adjustment parameter. The nonlinear term υ/1+βyt shows a function of yt which decreases monotonically, and the maximum amount of natural enemy release does not exceed. The symbols x0+ with y0+, respectively, represent the initial populations of pests and natural enemies and satisfy x0++y0+<AT. In model (1), there always exist a stable centre E0=d/c,a/b and a saddle point 0,0 which is unstable.The special cases of the above model for different parameters were considered in [37–39]. The biological significance and main properties of the corresponding ODE model can be seen in [37]. In [38], Tian et al. extended the classic pest-natural enemy model with linear state-dependent control measures to a model with nonlinear state-dependent impulsive control tactics. In [39], the authors for the first time introduced and provided the concept of action threshold depending on the density of pest and its rate of change. They used the definition and properties of the Lambert W function to construct the analytical expression of the Poincaré map. Furthermore, by using the analytical properties of Poincaré map, the existence, uniqueness, and stability of the natural enemy free periodic solution and internal periodic solution were discussed in detail. The results explain the significance of nonlinear ratio-dependent AT in integrated pest control and the important guiding role in IPM strategy.


### 2.2. Properties of Action Threshold

The quantitiesθ1 and θ2 are dependent weighted parameters. If θ2=0, then the ratio-dependent AT converts into ET. Therefore, we can say that ET is a special case of ratio-dependent AT for θ2=0. Combining the first equation of ODE model (1) with ratio-dependent AT, we get(2)limx⟶+∞θ1+aθ2x−ATbθ2x=θ1+aθ2bθ2.If we putθ1=0, then the ratio-dependent AT converts into y=ax−AT/bx. In this case, if x⟶+∞, then y is bounded and reaches its highest value a/b. Further, with the utilization of the control actions on y=θ1+aθ2x−AT/bθ2x, we get another curve y+=θ1+aθ2x+−AT1−δxt/xt+α/bθ2x++υ/1+βy. For θ2=0, the curve changes into x+=1−δx/x+αAT showing a vertical straight line. Let PAT=δx/x+α; then, for convenience, we denote the two curves y=θ1+aθ2x−AT/bθ2x and y+=θ1+aθ2x+−AT1−PAT/bθ2x++υ/1+βy by ΓIS and ΓPS, respectively, as shown in Figure 1.Figure 1
Detailed diagrams describing the impulsive along with phase sets where (a)AT/θ1≤d/c and AT/θ1>d/c. In sub-plot (a), Γ1 shows the tangent trajectory to the curve ΓPS and touches the curve ΓIS at xG2,yG2. In sub-plot (b), Γ2 touches the curve ΓPS at two points xK1,yK1 and xK2,yK2, and tangent to the curve ΓIS at point xS,yS.
(a)(b)


## 2.1. Construction of Model

In view of the above objective factors, we propose the following nonlinear state-dependent feedback control model combined with nonlinear ratio-dependent AT:(1)dxtdt=axt−bxtyt,dytdt=cxtyt−dyt,θ1xt+θ2dxtdt<AT,xt+=1−δxtxt+αxt,yt+=yt+υ1+βyt,θ1xt+θ2dxtdt=AT.It can be seen that without pulse control measures, the model is simply based on the classical Lotka–Volterra type problem which is extensively used to describe the relation between the populations of pest and natural enemy shown byxt and yt, respectively. Weighted parameters θ1, θ2, and AT are positive constants, which satisfy θ1+θ2=1. The discontinuous mapping shown in the third and fourth equations in system (1) represents that the implementation of comprehensive control measures depends on the action level, that is, once the pest density reaches action threshold, the densities of pests as well as the natural enemies are immediately updated to 1−δxt/xt+αxt and yt+υ/1+βyt, respectively. α>0 represents the semisaturation constant, δ>0 is defined as the maximum instantaneous killing rate after the use of pesticides, and υ>0 is the maximum natural enemy when executing the control strategy. The amount β>0 is the natural enemy density adjustment parameter. The nonlinear term υ/1+βyt shows a function of yt which decreases monotonically, and the maximum amount of natural enemy release does not exceed. The symbols x0+ with y0+, respectively, represent the initial populations of pests and natural enemies and satisfy x0++y0+<AT. In model (1), there always exist a stable centre E0=d/c,a/b and a saddle point 0,0 which is unstable.The special cases of the above model for different parameters were considered in [37–39]. The biological significance and main properties of the corresponding ODE model can be seen in [37]. In [38], Tian et al. extended the classic pest-natural enemy model with linear state-dependent control measures to a model with nonlinear state-dependent impulsive control tactics. In [39], the authors for the first time introduced and provided the concept of action threshold depending on the density of pest and its rate of change. They used the definition and properties of the Lambert W function to construct the analytical expression of the Poincaré map. Furthermore, by using the analytical properties of Poincaré map, the existence, uniqueness, and stability of the natural enemy free periodic solution and internal periodic solution were discussed in detail. The results explain the significance of nonlinear ratio-dependent AT in integrated pest control and the important guiding role in IPM strategy.


## 2.2. Properties of Action Threshold

The quantitiesθ1 and θ2 are dependent weighted parameters. If θ2=0, then the ratio-dependent AT converts into ET. Therefore, we can say that ET is a special case of ratio-dependent AT for θ2=0. Combining the first equation of ODE model (1) with ratio-dependent AT, we get(2)limx⟶+∞θ1+aθ2x−ATbθ2x=θ1+aθ2bθ2.If we putθ1=0, then the ratio-dependent AT converts into y=ax−AT/bx. In this case, if x⟶+∞, then y is bounded and reaches its highest value a/b. Further, with the utilization of the control actions on y=θ1+aθ2x−AT/bθ2x, we get another curve y+=θ1+aθ2x+−AT1−δxt/xt+α/bθ2x++υ/1+βy. For θ2=0, the curve changes into x+=1−δx/x+αAT showing a vertical straight line. Let PAT=δx/x+α; then, for convenience, we denote the two curves y=θ1+aθ2x−AT/bθ2x and y+=θ1+aθ2x+−AT1−PAT/bθ2x++υ/1+βy by ΓIS and ΓPS, respectively, as shown in Figure 1.Figure 1
Detailed diagrams describing the impulsive along with phase sets where (a)AT/θ1≤d/c and AT/θ1>d/c. In sub-plot (a), Γ1 shows the tangent trajectory to the curve ΓPS and touches the curve ΓIS at xG2,yG2. In sub-plot (b), Γ2 touches the curve ΓPS at two points xK1,yK1 and xK2,yK2, and tangent to the curve ΓIS at point xS,yS.
(a)(b)


## 3. Impulsive and Phase Sets

This section is devoted to present the dynamical aspects of the system (1), and we can use the Poincaré map on the sequence of pulse points which will be formulated later. Let AT/θ1 be the abscissa of the curves ΓIS at y=a/b.Then, we take the following cases based on the equilibriumE0 and curve ΓIS.(3)AATθ1≤dc;Bdc<ATθ1.The necessary and primary component is to examine the section that is not used during the pulse effect process, which means that the trajectory initiating fromΓPS cannot touch the curve ΓIS in the case of maximum impulsive set. In the following part of the paper, we address the definition of impulsive sets.


### 3.1. Impulsive Set

In CaseA, the solution Γ1 is tangent to the curve ΓPS at point T=xT,yT. If we denote the impulsive set by ℳ1, then it can be written as(4)ℳ1=x,y∈ℝ+2|ATθ1+aθ2≤x≤xG2,0≤y≤yG2.Now based on the corresponding horizontal coordinate, we search the exact value ofyG2 in the following lemma. The point yG2 is actually the maximum value of the impulsive set ℳ1 for Case A.Lemma 1.
For CaseA, the maximum impulsive set is defined as ℳ1 with(5)yG2=−abW−bayTe−b/ayT+AG2/aprovided thatAG21≤0.Proof.
LetΓ1 be a trajectory tangent at T=xT,yT, and it touches the curve ΓIS at point G2=xG2,yG2. Then, T and G2 must satisfy the following equation:(6)alnyG2−byG2+dlnxG2−cxG2=alnyT−byT+dlnxT−cxT.
Solving this equation foryG2, we get(7)−bayG2e−b/ayG2=−bayTe−b/ayT+AG2/a,where AG2=dlnxT−lnxQ2+cxG2−xT. The above equation obviously gives two solutions when we solve it by using Lambert W function. The minimum solution can be written as follows:(8)yG2=−abW−bayTe−b/ayT+AG2/a,which is well defined because AG2≤0.
For CaseB, it is clear from Figure 1(b) that at point S=xS,yS, Γ2 is tangent to the curve ΓIS where yS≤a/b. Then, taking into account the locations of equilibrium E0 and the curve ΓIS, we can write the maximum impulsive set for Case B as(9)ℳ2=x,y∈ℝ2|ATθ1+aθ2≤x≤xS,0≤y≤yS.
The above information shows that for this case, the tangent point withΓIS varies due to small changes in θ1 and θ2.
If the weighted parameterθ2 decreases, then the quantity yS approaches its maximum value a/b.


### 3.2. Phase Set

To determine the exact phase set of system (1) under different conditions, we need to know whether the solution from initial point x0+,y0+ reaches the corresponding impulsive set and whether the pulse action occurs or not. To provide the exact domain of phase sets, we first discuss the interval which is free of impulsive effect.Lemma 2.
For CaseB, any solution starting from the phase set with initial point x0+,y0+ (where y0+∈yK2,yK1) will not reach the impulsive set ℳ2, where(10)yK1=−abW−1,−baySe−b/ayS−AK1/aandyK2=−abW−baySe−b/ayS−AK2/a,provided that AK1,AK2≥0.Proof.
Assume that the closed trajectoryΓ1 starts from K1=xK1,yK1 and touches the curve ΓIS at point S=xS,yS. Then, K1 and S must satisfy the following relationship:(11)alnyK1−byK1+dlnxK1−cxK1=alnyS−byS+dlnxS−cxS.
Rearranging this equation foryK1, we get(12)−bayK1e−b/ayK1=−baySe−b/ayS−AK1/a,where AK1=dlnxK1−lnxS+cxS−xK1. The above equation can be easily solved utilizing the Lambert W function approach which clearly will result in two solutions of the problem. The maximum solution can be written as(13)yK1=−abW−1,−baySe−b/ayS−AK1/a.
The value ofyK2 can be found in the similar way as above, i.e.,(14)yK2=−abW−baySe−b/ayS−AK2/a,with AK2=dlnxK2−lnxS+cxS−xK2.
As a result, any solution curve initiating fromx0+,y0+ with y0+∈yK2,yK1 will be free from the effect of impulsive set.
For the case whenθ2=0, the trajectory shown by Γ2 becomes tangent at y=a/b. So, yK2 and yK1 become(15)yK2=−abW−1,−e−1−AK1/a,andyK1=−abW−e−1−AK2/a.
The impulsive function described byyt+=yt+υ/1+βyt satisfies some properties which are very important.
To do this, we indicate(16)Fu=u+υ1+βu,u∈0,ab,and then we get F´u=1−υβ/1+βu2 and F´u=0 at u=υβ−1/β.
AAT/θ1≤d/c. From Lemma 1, we can describe the impulsive set ℳ1 as ℳ1=x,y∈ℝ+2|AT/θ1+aθ2≤x≤xG2,0≤y≤yG2. Further, we can take three subclasses as follows.(i)
υβ−1/β≤0.For this subcase,F´u≥0 for all u∈0,yG2, which shows that υ≤Fu≤yG2+υ/1+βyG2. Then, the corresponding phase set to ℳ1 can be expressed as(17)N11=x+,y+∈ℝ+2|x+∈X11,y+∈Y11,with(18)X11=AT1−PATθ1+aθ2,1−PATxG2,Y11=υ,yG2+υ1+βyG2.(ii)
υβ−1/β≥yG2.For this subcase,F´u≤0 for u∈0,yG2, which denotes that yG2+υ/1+βyG2≤Fu≤υ. Then, the corresponding phase set to ℳ1 is expressed as follows:(19)N12=x+,y+∈ℝ+×ℝ+|x+∈X21,y+∈Y21,with(20)X21=1−PATxG2,AT1−PATθ1+aθ2,Y21=yG2+υ1+βyG2,υ.(iii)
0<υβ−1/β<yG2.
For the present subcase, the impulsive setℳ1 becomes ℳ1=ℳ11∪ℳ12, where(21)ℳ11=x,y∈ℝ+2|x∈X31,y∈Y31,with(22)X31=ATθ1+aθ2,ATβθ1+aθ2β−υβ−1bθ2,Y31=0,υβ−1β, (23)ℳ12=x,y∈ℝ+2|<x∈X41,y∈Y41,with(24)X41=ATβθ1+aθ2β−υβ−1bθ2,xG2,Y41=υβ−1β,yG2.
Hence, the corresponding phase set to the impulsive setℳ1  =  ℳ11∪ℳ12 is N13∪N14, where(25)N13=x+,y+∈ℝ+2|x+∈X51,y+∈Y51,with(26)X51=AT1−PATβθ1+aθ2β−υβ−1bθ2,AT1−PATθ1+aθ2,Y51=2υβ−1β,υ, (27)N14=x+,y+∈ℝ+2|x+∈X61,y+∈Y61,with(28)X61=AT1−PATβθ1+aθ2β−υβ−1bθ2,1−PATxG2,Y61=2υβ−1β,yG2+υ1+βyG2.
Bd/c<AT/θ1. For this case, we express the impulsive set as follows.
ℳ2=x,y∈ℝ+×ℝ+|AT/θ1+aθ2≤x≤xS,0≤y≤yS. In order to give the exact domain of phase sets for Case B, based on Lemma 2, we describe the following sets:(29)XDl=AT1−PATθ1+aθ2+υbθ2,xK2∪xK1,∞,YDl=0,yK2∪yK1,θ1+aθ2bθ2+bθ2υbθ2+βθ1+aθ2.
The following three subcases can be taken based on the definition of the phase set.(i)
υβ−1/θ≤0.For this subcase,F´u≥0 for all values of u belongs to 0,yS. This shows that υ≤Fu≤yS+υ/1+βyS. The corresponding phase set to ℳ2 can be expressed as(30)N21=x+,y+∈ℝ+×ℝ+|x+∈X12,y+∈Y12,with(31)X210=AT1−PATθ1+aθ2,1−PATxS,X12=XDl∩X210,(32)Y210=υ,yS+υ1+βyS,Y12=YDl∩Y210.(ii)
υβ−1/β≥yS.For this subcase,F´u≤0 for u∈0,yS, which denotes that yS+υ/1+βyS≤Fu≤υ. Hence, the phase set corresponding to ℳ2 is given as(33)N22=x+,y+∈ℝ+×ℝ+|x+∈X22,y+∈Y22,with(34)X220=1−PATxS,AT1−PATθ1+aθ2,X22=XDl∩X220,(35)Y220=yS+υ1+βyS,υ,Y22=YDl∩Y220.(iii)
0<υβ−1/β<yS.
If0≤u≤υβ−1/β, then F´u≤0 and 2υβ−1/β≤Fu≤υ. If υθ−1/β<u≤yS, then F´u>0 and 2υβ−1/β<Fu≤yS+υ/1+βyS.
The impulsive setℳ2 is now can be explained in the form ℳ2=ℳ21∪ℳ22, where(36)ℳ21=x,y∈ℝ+2|x∈X32,y∈Y32,with(37)X32=ATθ1+aθ2,ATβθ1+aθ2β−υβ−1bθ2,Y32=0,υβ−1β,and(38)ℳ22=x,y∈ℝ+2|x∈X42,y∈Y42,with(39)X42=ATβθ1+aθ2β−υβ−1bθ2,xS,Y42=υβ−1β,yS.
Hence, the phase set corresponding to the impulsive setℳ2  =  ℳ21∪ℳ22 can be expressed as N23∪N24, where(40)N23=x+,y+∈ℝ+2|x+∈X52,y+∈Y52,with(41)X230=AT1−PATβθ1+aθ2β−υβ−1bθ2,AT1−PATθ1+aθ2,X52=XDl∩X230,Y230=2υβ−1β,υ,Y52=YDl∩Y230, (42)N24=x+,y+∈ℝ+2|x+∈X62,y+∈Y62,with(43)X240=AT1−PATβθ1+aθ2β−υβ−1bθ2,1−PATxS,X62=XDl∩X240, (44)Y240=2υβ−1β,yS+υ1+βyS,Y62=YDl∩Y240.
For CaseA, if AT/θ1≤d/c, then the solution from the phase set does not reach the interval yG2,a/b. It is also important to note that if yT=a/b and AG2=0, then yG2=a/b. For Case B, it can be seen from the vector field of system (1) that if the closed orbit is tangent or does not touch the curve ΓPS, then there must be a trajectory that is tangent to the curve ΓPS at a point xT,yT, and the trajectory intersects the curve ΓIS at lower point G2. This proves that the impulsive set in this case is defined by ℳ1, as shown in Figure 1(b).
If the closed trajectory is tangent toΓIS at point S=xS,yS and intersects the curve ΓPS at two points, then it can be seen that for any solution from the phase set, it is impossible to reach the interval yS,a/b. The above theory shows that nonlinear terms of the controlling measure combined with nonlinear action threshold make impulse system (1) quite complicated, and it is very difficult to analyze each situation in detail.


## 3.1. Impulsive Set

In CaseA, the solution Γ1 is tangent to the curve ΓPS at point T=xT,yT. If we denote the impulsive set by ℳ1, then it can be written as(4)ℳ1=x,y∈ℝ+2|ATθ1+aθ2≤x≤xG2,0≤y≤yG2.Now based on the corresponding horizontal coordinate, we search the exact value ofyG2 in the following lemma. The point yG2 is actually the maximum value of the impulsive set ℳ1 for Case A.Lemma 1.
For CaseA, the maximum impulsive set is defined as ℳ1 with(5)yG2=−abW−bayTe−b/ayT+AG2/aprovided thatAG21≤0.Proof.
LetΓ1 be a trajectory tangent at T=xT,yT, and it touches the curve ΓIS at point G2=xG2,yG2. Then, T and G2 must satisfy the following equation:(6)alnyG2−byG2+dlnxG2−cxG2=alnyT−byT+dlnxT−cxT.
Solving this equation foryG2, we get(7)−bayG2e−b/ayG2=−bayTe−b/ayT+AG2/a,where AG2=dlnxT−lnxQ2+cxG2−xT. The above equation obviously gives two solutions when we solve it by using Lambert W function. The minimum solution can be written as follows:(8)yG2=−abW−bayTe−b/ayT+AG2/a,which is well defined because AG2≤0.
For CaseB, it is clear from Figure 1(b) that at point S=xS,yS, Γ2 is tangent to the curve ΓIS where yS≤a/b. Then, taking into account the locations of equilibrium E0 and the curve ΓIS, we can write the maximum impulsive set for Case B as(9)ℳ2=x,y∈ℝ2|ATθ1+aθ2≤x≤xS,0≤y≤yS.
The above information shows that for this case, the tangent point withΓIS varies due to small changes in θ1 and θ2.
If the weighted parameterθ2 decreases, then the quantity yS approaches its maximum value a/b.


## 3.2. Phase Set

To determine the exact phase set of system (1) under different conditions, we need to know whether the solution from initial point x0+,y0+ reaches the corresponding impulsive set and whether the pulse action occurs or not. To provide the exact domain of phase sets, we first discuss the interval which is free of impulsive effect.Lemma 2.
For CaseB, any solution starting from the phase set with initial point x0+,y0+ (where y0+∈yK2,yK1) will not reach the impulsive set ℳ2, where(10)yK1=−abW−1,−baySe−b/ayS−AK1/aandyK2=−abW−baySe−b/ayS−AK2/a,provided that AK1,AK2≥0.Proof.
Assume that the closed trajectoryΓ1 starts from K1=xK1,yK1 and touches the curve ΓIS at point S=xS,yS. Then, K1 and S must satisfy the following relationship:(11)alnyK1−byK1+dlnxK1−cxK1=alnyS−byS+dlnxS−cxS.
Rearranging this equation foryK1, we get(12)−bayK1e−b/ayK1=−baySe−b/ayS−AK1/a,where AK1=dlnxK1−lnxS+cxS−xK1. The above equation can be easily solved utilizing the Lambert W function approach which clearly will result in two solutions of the problem. The maximum solution can be written as(13)yK1=−abW−1,−baySe−b/ayS−AK1/a.
The value ofyK2 can be found in the similar way as above, i.e.,(14)yK2=−abW−baySe−b/ayS−AK2/a,with AK2=dlnxK2−lnxS+cxS−xK2.
As a result, any solution curve initiating fromx0+,y0+ with y0+∈yK2,yK1 will be free from the effect of impulsive set.
For the case whenθ2=0, the trajectory shown by Γ2 becomes tangent at y=a/b. So, yK2 and yK1 become(15)yK2=−abW−1,−e−1−AK1/a,andyK1=−abW−e−1−AK2/a.
The impulsive function described byyt+=yt+υ/1+βyt satisfies some properties which are very important.
To do this, we indicate(16)Fu=u+υ1+βu,u∈0,ab,and then we get F´u=1−υβ/1+βu2 and F´u=0 at u=υβ−1/β.
AAT/θ1≤d/c. From Lemma 1, we can describe the impulsive set ℳ1 as ℳ1=x,y∈ℝ+2|AT/θ1+aθ2≤x≤xG2,0≤y≤yG2. Further, we can take three subclasses as follows.(i)
υβ−1/β≤0.For this subcase,F´u≥0 for all u∈0,yG2, which shows that υ≤Fu≤yG2+υ/1+βyG2. Then, the corresponding phase set to ℳ1 can be expressed as(17)N11=x+,y+∈ℝ+2|x+∈X11,y+∈Y11,with(18)X11=AT1−PATθ1+aθ2,1−PATxG2,Y11=υ,yG2+υ1+βyG2.(ii)
υβ−1/β≥yG2.For this subcase,F´u≤0 for u∈0,yG2, which denotes that yG2+υ/1+βyG2≤Fu≤υ. Then, the corresponding phase set to ℳ1 is expressed as follows:(19)N12=x+,y+∈ℝ+×ℝ+|x+∈X21,y+∈Y21,with(20)X21=1−PATxG2,AT1−PATθ1+aθ2,Y21=yG2+υ1+βyG2,υ.(iii)
0<υβ−1/β<yG2.
For the present subcase, the impulsive setℳ1 becomes ℳ1=ℳ11∪ℳ12, where(21)ℳ11=x,y∈ℝ+2|x∈X31,y∈Y31,with(22)X31=ATθ1+aθ2,ATβθ1+aθ2β−υβ−1bθ2,Y31=0,υβ−1β, (23)ℳ12=x,y∈ℝ+2|<x∈X41,y∈Y41,with(24)X41=ATβθ1+aθ2β−υβ−1bθ2,xG2,Y41=υβ−1β,yG2.
Hence, the corresponding phase set to the impulsive setℳ1  =  ℳ11∪ℳ12 is N13∪N14, where(25)N13=x+,y+∈ℝ+2|x+∈X51,y+∈Y51,with(26)X51=AT1−PATβθ1+aθ2β−υβ−1bθ2,AT1−PATθ1+aθ2,Y51=2υβ−1β,υ, (27)N14=x+,y+∈ℝ+2|x+∈X61,y+∈Y61,with(28)X61=AT1−PATβθ1+aθ2β−υβ−1bθ2,1−PATxG2,Y61=2υβ−1β,yG2+υ1+βyG2.
Bd/c<AT/θ1. For this case, we express the impulsive set as follows.
ℳ2=x,y∈ℝ+×ℝ+|AT/θ1+aθ2≤x≤xS,0≤y≤yS. In order to give the exact domain of phase sets for Case B, based on Lemma 2, we describe the following sets:(29)XDl=AT1−PATθ1+aθ2+υbθ2,xK2∪xK1,∞,YDl=0,yK2∪yK1,θ1+aθ2bθ2+bθ2υbθ2+βθ1+aθ2.
The following three subcases can be taken based on the definition of the phase set.(i)
υβ−1/θ≤0.For this subcase,F´u≥0 for all values of u belongs to 0,yS. This shows that υ≤Fu≤yS+υ/1+βyS. The corresponding phase set to ℳ2 can be expressed as(30)N21=x+,y+∈ℝ+×ℝ+|x+∈X12,y+∈Y12,with(31)X210=AT1−PATθ1+aθ2,1−PATxS,X12=XDl∩X210,(32)Y210=υ,yS+υ1+βyS,Y12=YDl∩Y210.(ii)
υβ−1/β≥yS.For this subcase,F´u≤0 for u∈0,yS, which denotes that yS+υ/1+βyS≤Fu≤υ. Hence, the phase set corresponding to ℳ2 is given as(33)N22=x+,y+∈ℝ+×ℝ+|x+∈X22,y+∈Y22,with(34)X220=1−PATxS,AT1−PATθ1+aθ2,X22=XDl∩X220,(35)Y220=yS+υ1+βyS,υ,Y22=YDl∩Y220.(iii)
0<υβ−1/β<yS.
If0≤u≤υβ−1/β, then F´u≤0 and 2υβ−1/β≤Fu≤υ. If υθ−1/β<u≤yS, then F´u>0 and 2υβ−1/β<Fu≤yS+υ/1+βyS.
The impulsive setℳ2 is now can be explained in the form ℳ2=ℳ21∪ℳ22, where(36)ℳ21=x,y∈ℝ+2|x∈X32,y∈Y32,with(37)X32=ATθ1+aθ2,ATβθ1+aθ2β−υβ−1bθ2,Y32=0,υβ−1β,and(38)ℳ22=x,y∈ℝ+2|x∈X42,y∈Y42,with(39)X42=ATβθ1+aθ2β−υβ−1bθ2,xS,Y42=υβ−1β,yS.
Hence, the phase set corresponding to the impulsive setℳ2  =  ℳ21∪ℳ22 can be expressed as N23∪N24, where(40)N23=x+,y+∈ℝ+2|x+∈X52,y+∈Y52,with(41)X230=AT1−PATβθ1+aθ2β−υβ−1bθ2,AT1−PATθ1+aθ2,X52=XDl∩X230,Y230=2υβ−1β,υ,Y52=YDl∩Y230, (42)N24=x+,y+∈ℝ+2|x+∈X62,y+∈Y62,with(43)X240=AT1−PATβθ1+aθ2β−υβ−1bθ2,1−PATxS,X62=XDl∩X240, (44)Y240=2υβ−1β,yS+υ1+βyS,Y62=YDl∩Y240.
For CaseA, if AT/θ1≤d/c, then the solution from the phase set does not reach the interval yG2,a/b. It is also important to note that if yT=a/b and AG2=0, then yG2=a/b. For Case B, it can be seen from the vector field of system (1) that if the closed orbit is tangent or does not touch the curve ΓPS, then there must be a trajectory that is tangent to the curve ΓPS at a point xT,yT, and the trajectory intersects the curve ΓIS at lower point G2. This proves that the impulsive set in this case is defined by ℳ1, as shown in Figure 1(b).
If the closed trajectory is tangent toΓIS at point S=xS,yS and intersects the curve ΓPS at two points, then it can be seen that for any solution from the phase set, it is impossible to reach the interval yS,a/b. The above theory shows that nonlinear terms of the controlling measure combined with nonlinear action threshold make impulse system (1) quite complicated, and it is very difficult to analyze each situation in detail.


## 4. Poincaré Map

Poincaré map [40–42] plays a very helpful role in examining the qualitative behavior of a dynamical system, most prominently the asymptotic stability of periodic or almost periodic orbits. Based on the impulse and phase sets discussed above, the following related theorem for Poincaré map can be obtained.Theorem 1.
For the impulsive points of model (1), the Poincaré map for Cases AandB has the following form.(A)
AT/θ1≤d/c:(45)yi+1+=ψyi+,yi+∈Y11,ifυβ−1β≤0,ψyi+,yi+∈Y21,ifυβ−1β≥yG2,ψyi+,yi+∈Y51∪Y61,if0<υβ−1β<yG2.(B)
d/c<AT/θ1:(46)yi+1+=ψyi+,yi+∈Y12,ifυβ−1β≤0,ψyi+,yi+∈Y22,ifυβ−1β≥yS,ψyi+,yi+∈Y52∪Y62,if0<υβ−1β<yS,where(47)ψyi+=−abW−bayi+exp−bayi++Ala+υ1−βa/bW−b/ayi+exp−b/ayi++Al/a.Proof.
Suppose that a trajectory initiating fromx0+,y0+ repeats k (finite or infinite) times pulse action. Let the points of the impulse set be represented by pi=xi,yi, and after the pulse action, the corresponding points of phase set are represented by pi+=xi+,yi+. If p0+=xi+,yi+∈ΓPS and p1=xi+1,yi+1∈ΓIS are on the same trajectory above, then the coordinates of the two points satisfy the following trajectory equation:(48)dlnxi+−dlnxi+1+cxi+1−cxi+=alnyi+1yi+−byi+1−yi+.
Solving the above equation foryi+1, we get(49)yi+1=−abW−bayi+exp−bayi++A1a,where(50)Al=dlnxi+−dlnxi+1+cxi+1−cxi+,and therefore(51)ψyi+=−abW−bayi+exp−bayi++A1a+υ/1−βabW−bayi+exp−bayi++Ala=yi+1+.
From above equation, we can see that the Poincaré map given in (47) depends on both the Lambert W function and the sign of Al.
Case. A. If Al≤0, then for yi+≥0, the above expressions defined in (9) and (10) are well defined. Further, if we define gy=−b/ayexp−b/ay, then it is easy to prove that gy achieved its minimum value −e−1 at y=a/b. Therefore, −b/ayexp−b/ayexpAl/a∈−e−1,0 for all Al≤0 and y>0. This denotes that the Poincaré map defined relative to Case A is (7).
For CaseB, if Al>0, then −b/ayexp−b/ayexpAl/a≥−exp−1. From this, we obtain the following:(52)bayexp−yba≤exp−1+Ala.
This solution further simplifies asy∈0,yK2∪yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, and from Lemma 2 we know that(53)yK2=−abW−baySe−b/ayS−AK2/aandyK1=−abW−1,−baySe−b/ayS−AK1/a.
Hence, in the same way, the Poincaré map domain for all remaining cases provided in Section3 and Table 1 can be found. This finalized the proof.Table 1
The exact impulsive and phase sets for system (1) under Cases (A) and (B).
CasesConditionImpulsive setPhase setAiATθ1≤dcℳ1N11iiN12iiiN13∪N14BiATθ1≤dcℳ2N21iiN22iiiN23∪N24


## 5. Characteristics of Poincaré Map

To discuss the existence as well as the stability for the order-1 periodic solution of problem (1), we first analyze the different characteristics of Poincaré map for the above existing cases. For this, we define an important point G=xG,yG=ATβ/θ1+aθ2β−υβ−1bθ2,υβ−1/β which will be used in the following discussion. If G∈ΓIS, then after one time pulse, the corresponding impulse point can be presented as G+:xG+,yG+=AT1−PATβ/θ1+aθ2β−υβ−1bθ2,2υβ−1/β.Theorem 2.
The Poincaré mapψyi+ for Cases A and B provided in Table 2 satisfies different properties as follows:(A)
AT/θ1≤d/c and Al≤0.(i)
It shows increasing behavior on0,yT and decreasing behavior on yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 for υβ−1/β≤0.(ii)
It is increasing onyT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 and decreasing on 0,yT for υβ−1/β≥yG2.(iii)
It is decreasing on0,yn2 and yT,yn1 and increasing on yn2,yT and yn1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 for 0<υβ−1/β<yG2, where yn2=miny+:ψy+=yG+,yn1=maxy+:ψy+=yG+.(B)
d/c<AT/θ1 and Al>0.(i)
It shows increasing behavior over the closed interval0,yK2 and decreasing behavior on yK1,θ1+aθ2/bθ+bθ2υ/bθ2+βθ1+aθ2 for υβ−1/β≤0.(ii)
It is increasing onyK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 and decreasing on 0,yK2 for υβ−1/β≥yS.(iii)
It is decreasing on0,yN2 and yK1,yN1 and increasing on yN2,yK2 and yN1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 for 0<υβ−1/β<yS, where yN2=miny+:ψy+=yG+, yN1=maxy+:ψy+=yG+.Table 2
The domain of the Poincaré map for CasesA and B.
CasesConditionAlψyi+AiAT/θ1≤d/cAl≤0yi+∈Y11iiyi+∈Y21iiiyi+∈Y51∪Y61BiAT/θ1≤d/cAl>0yi+∈Y12iiyi+∈Y22iiiyi+∈Y52∪Y62Proof.
Assuming thatqi+=xi+,yi+∈ΓPS, the solution initiating from qi+ intersects the curve ΓIS at qi+1=xi+1,yi+1. If qi+ and qi+1 lie in one trajectory, then yi+1 is established by yi+ and can be expressed as yi+1=Fyi+. The corresponding vector field relative of the system given in (1) confirms that the domain of consideration of Poincaré map ψyi+ for Case Ai is defined by 0,yT∪yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. Furthermore, for this case, the corresponding impulsive function F has increasing behavior over the closed interval 0,yT. Therefore, based on the definition of ψyi+, it is increasing on 0,yT and decreasing on yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. The function F is decreasing upon 0,yT in Case Aii, which shows that ψyi+ is decreasing over the interval 0,yT and increasing over the closed interval yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For Case Aiii, F is decreasing over 0,yG and increasing upon yG,yT. Therefore, ψyi+ is decreasing on 0,yN2 and yK1,yN1 and increasing on yN2,yK2 and yN1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2.
By using the same methods as above, we can prove that the monotonicities of the Poincaré map for CasesBi,ii,iii in Theorem 2 are true.Lemma 3.
IfAl>0 and υ>0, then the inequality(54)ψyi+>yi+,for allyi+∈0,yK2,is fulfilled for the corresponding Poincaré map shown by ψyi+.Proof.
Let a solution originate fromp0+=xi+,yi+, and it touches the curve ΓIS at point p1=xi+1,yi+1. We assume that yi+,yi+1<a/b; then,(55)alnyi+1−byi+1+dlnxi+1−cxi+1=alnyi+−byi++dlnxi+−cxi+.
From (55), we get(56)−bayi+1e−b/ayi+1=−bayi+e−b/ayi++Al/a.
IfAl>0, then we get the inequality(57)−bayi+1e−b/ayi+1<−bayi+e−b/ayi+.
Letfy=−yexp−y; then, f′y>0 if y>1 and f′y<0 if y∈0,1. The inequality yi+1>yi+ is satisfied for all b/ayi+,b/ayi+1∈0,1. We also know that yi+1+=yi+1+υ and ψyi+=yi+1+. Hence, we deduce that ψyi+>yi+ for all yi+∈0,yK2.
In light of the above explained properties of Poincaré map, the existence of the fixed point of Poincaré mapψyi+ for υ>0 is discussed in following section.


## 6. Characteristics of Boundary Periodic Solution

In Section4, the formula for Poincaré map ψyi+ has been attained. We will use this formula to study the existence of fixed point, where the fixed point is indicated as y∗, satisfying ψy∗=y∗, such as(58)y∗=−abW×−bay∗exp−bay∗+Ala+υ1−βa/bW−b/ay∗exp−b/ay∗+Al/a.Forυ=0, we get the following equation from above:(59)y∗=−abW−bay∗exp−bay∗+Ala.IfAl=0, the fixed point shown by y∗ of the respective Poincaré map ψyi+ becomes(60)y∗=−abW−bay∗exp−bay∗.This shows that ifυ=0, Al=0, then every point is the fixed point of ψyi+. If υ=0, Al≠0, then y∗ (a fix point) of the ψyi+ fulfils(61)y∗=−abW−bay∗exp−bay∗+Ala.In this case,ψy∗=y∗ holds ⇔y∗=0. Thus, we deduced that y∗=0 is a unique fixed point for system (1).In the following result, we present the conditions of global stability for boundary order-1 periodic solution. To demonstrate it, we first discuss an important lemma [43, 44].Lemma 4.
TheT-periodic solution x,y=ζt,ξt of system(62)dxdt=Cx,y,dydt=Dx,y,ifθx,y≠0,x+=x+εx,y,y+=y+εx,y,ifθx,y=0,.is orbitally asymptotically stable if the Floquet multiplier μ2 satisfies μ2<1, where(63)μ2=∏j=1kΔjexp∫0T∂C∂xζt,ξt+∂D∂yζt,ξtdt,with(64)Δj=C+∂ε/∂y∂θ/∂x−∂ε/∂x∂θ/∂y+∂θ/∂x+D+∂ϵ/∂x∂θ/∂y−∂ϵ/∂y∂θ/∂x+∂θ/∂yC∂θ/∂x+B∂θ/∂y,and θ is continuously differentiable corresponding to both x,y. C,D,∂ϵ/∂x,∂ϵ/∂y,∂ε/∂x,∂ε/∂y,∂θ/∂x and ∂θ/∂y are evaluated at ζtj,ξtj, C+=Cζtj+,ξtj+ and D+=Dζtj+,ξtj+, and tj (j,k∈ℕ, ℕ is the set of nonnegative integers) is the time of the j-th jump.Theorem 3.
IfAl=0 and υ=0, then the fixed point y∗ of Poincaré map ψyi+ is stable in the phase set. If Al<0 and υ=0, then xTt,0 is globally asymptotically stable. If Al>0 and υ=0, then it is unstable.Proof.
Ifυ=0, Al=0, then y∗ in the phase set is a fixed point of the Poincaré map ψyi+. This case confirms the stable solution of the problem but is not asymptotically stable. We first show that when yt=0 if and only if υ=0, and then boundary order-1 periodic solution exists for system (1). For yt=0, system (1) is converted into the subsystem given below:(65)dxtdt=a×xt,xt<ATθ1+aθ2,xt+=xt1−δxtxt+α,xt=ATθ1+aθ2,
The first equation of the subsystem (14), combining with the respective initial condition shown as x0+=1−PATAT/θ1+aθ2, where PAT=δxt/xt+α, gives us the solution(66)xt=1−PATATθ1+aθ2expat.
Taking the equationAT/θ1+aθ2=1−PATAT/θ1+aθ2expaT and evaluating it for T, we get T=−1/aln1−PAT. This shows that T-periodic boundary order-1 solution exists for system (1) as(67)xTt,0=1−PATATθ2+aθ2expat,0.
Next, we show thatxTt,0 is asymptotically stable. For this, we apply Lemma 4 and present the following.
Method1.(68)Cx,y=a−byx,Dx,y=ycx−d,εx,y=−PATx,εx,y=υ1+βy,θx,y=θ1+aθ2x−bθ2xy−AT,xTT,yTT=ATθ1+aθ2,0,xTT+,yTT+=1−PATATθ1+aθ2,0.
From the above, we get(69)∂C∂x=a−by,∂D∂y=cx−d,∂ε∂x=−δx2+2δαxx+α2,∂ε∂x=υβ1+βy2,∂θ∂x=θ1+aθ2−bθ2y,∂θ∂y=−bθ2x,∂ε∂y=∂ε∂y=0,(70)Δ1=C+∂ε/∂y∂θ/∂x−∂ε/∂x∂θ/∂y+∂θ/∂x+D+∂ε/∂x∂θ/∂y−∂ε/∂y∂θ/∂x+∂θ/∂yC∂θ/∂x+D∂θ/∂y=C+xTT+,yTT+θ1+aθ2−bθ2y+D+xTT+,yTT+PATbθ2x−bθ2xCxTT,yTTθ1+aθ2−bθ2y−DxTT,yTTbθ2x=1−PAT.
Based on the above information, the Floquet multiplier denoted byμ2 is defined as(71)μ2=Δ1exp∫0T∂C∂xxTt,yTt+∂D∂yxTt,yTtdt=1−PATexpln11−PAT+Ala=expAla.
IfAl<0 and υ=0, then we get μ2<1. This indicates that for the problem described in (1), the boundary order-1 periodic solution xTt,0 is orbitally stable asymptotically. If Al>0, the sequence yk+ of pulse points is increasing strictly and additionally will be free from more pulse action only after limited time pulse effects.
Method2. The asymptotic stability of boundary order-1 periodic solution can also be discussed directly from Poincaré map portrayed in (47). Let υ=0; then,(72)ψyi+=−abW−bayi+exp−bayi++Ala.
Taking the derivative of (72), we get(73)dψyi+dyi+yi+=y∗=ddyi+yi+=y∗−abW−bayi+exp−bayi++Ala=−a/bW−b/ay∗exp−b/ay∗+Al/a1+W−b/ay∗exp−b/ay∗+Al/a1/y∗−b/a=hy∗.
The boundary order-1 periodic solution is stable⇔hy∗<1. By utilizing the limit of hy∗, we get(74)limy∗⟶0hy∗=eAl/a.
This denotes that ify∗⟶0, then hy∗<1 for Al<0, and hence xTt,0 is asymptotically stable.
In the following, we show the global attractivity of the boundary order-1 periodic solutionxTt,0. Let p0+=AT1−PAT/θ1+aθ2,y1+∈L3 and p1=AT/θ1+aθ2,y2∈L2 be the points of the same trajectory; then,(75)Al=dln1−PAT+cATθ1+aθ2PAT=alny2y1+−by2−y1+.
LetAl≠0; then, from (75), it is clear that y2≠y1+. If fy=alny−by, then f´y=a/y−b. This indicates that if y<a/b, then fy is monotonically increasing.
IfAl<0, then alny2/y1+−by2−y1+<0. Since υ=0, the inequality becomes alny2/y1−by2−y1<0. This shows that y2<y1. Therefore, ifAl≤0, then the impulsive sequence yk+k=0∞ is monotonically decreasing and limk⟶∞yk+=y∗. These kinds of information affirm that the boundary order-1 periodic solution is globally attractive. In the same way as above, we can prove that if Al>0, then y2>y1. Therefore, the sequence yk+ will be free from impulsive effect after finite time pulse actions, as shown in Figure 2(b). Hence, from all the above outcomes, it can be concluded that if Al<0, then the boundary order-1 periodic solution, i.e., xTt,0, is globally asymptotically stable.
The numerical calculation in Figure2(a) shows that if Al<0, then the boundary order-1 periodic solution is stable while Figure 2(b) confirms that if Al≥0, then it is unstable.Figure 2
(a) The stable boundary order-1 periodic solution whereL2=1.4 and Al=−0.050,c=0.50,d=1.20. (b) The unstable boundary order-1 periodic solution with L2=1.20 and Al=0.0030,d=0.20,c=0.20. The rest of the parameter values are fixed with a=1,δ=0.3,b=0.1,α=4,β=1,υ=0.
(a)(b)


## 7. Existence of Order-1 Periodic Solution

In this section, we will discuss and analyze the order-1 periodic solution for system (1) when υ>0.Theorem 4.
For CaseAiorii, the fixed point of Poincaré mapψyi+exists, and therefore an order-1 periodic solution exists for system (1).Proof.
For CaseAi, the trajectory Γ1 is tangent to the curve ΓPS at point xT,yT and intersects the curve ΓIS at lower point G2. If ψyT=yG2+=yT, then the curve TG2^ forms an order-1 periodic solution for system (1).
For CaseAi, if yG2+>yT or yG2+<yT, then the solution originating from the point G2+ touches the curve ΓPS at a point G3=xG3,yG3 with yG3<yG2. The pulse action is applied and the point G3 maps to a point G3+=xG3+,yG3+, and yG3+=FyG3. For Case Ai, F is increasing on 0,yG2. Therefore, yG3+=ψyG2+ satisfies the inequality(76)ψyG2+<yG2+.
The pointψυAT1−PAT/θ1+aθ2,υ being the lowest impulsive point satisfies(77)ψυ>υ.
Inequalities (17) and (18) confirm that a fixed point of the Poincaré map exists, and therefore an order-1 periodic solution exists for system (1).
For CaseAii, F is decreasing on 0,yG2. If yG2+>yT or yG2+<yT, we get(78)ψyG2+>yG2+.
Moreover, the highest impulsive point isψυAT1−PAT/θ1+aθ2,υ, and we get(79)ψυ<υ.
Inequalities (19) and (20) confirm that there exists a fixed point for the Poincaré map, and therefore an order-1 periodic solution exists for system (1). This completes the proof.Theorem 5.
For CaseAiii, the fixed point of Poincaré mapψyi+exists, and therefore an order-1 periodic solution exists for system (1).Proof.
IfyG2+=yT, then the curve TG2^ forms an order-1 periodic solution for the problem given in system (1). If yG2+≠yT, then the following two cases are taken into consideration.(80)1yG2+≥υ,2yG2+<υ.
For Case (1), ifyG2+>yT, then we can write(81)ψyT>yT.
AsG2+ is the lowest impulsive point, it satisfies(82)ψyG2+<yG2+.
Thus, inequalities (21) and (22) confirm that we can find a fixed point of Poincaré map ψyi+.
IfyG2+<yT, then we can write(83)ψyT<yT.
Moreover, ifG+ is the least impulsive point, then it leads to the following:(84)ψyG+≥yG+.
Thus, the above two inequalities (83) and (84) confirm that there exists a fixed point of Poincare map ψyi+.
For Case (2), ifyG2+>yT, then ψyT>yT. On the other hand, if the highest impulsive point is ψυAT1−PAT/θ1+aθ2,υ, then ψυ<υ. The above two inequalities affirm that there exists a fixed point of the Poincaré map ψyi+.
IfyG2+<yT, then ψyT<yT. Moreover, as G+ is the least impulsive point, we get ψyG+≥yG+. It confirms that there exists a fixed point for the map shown by ψyi+, and hence an order-1 periodic solution exists for system (1).Theorem 6.
For CaseBiorii, ifyS+>yK1, then the fixed point of Poincaré mapψyi+exists, and therefore an order-1 periodic solution exists for system (1).Proof.
For CaseBi, we know that there exists a curve Γ2, which is tangent to ΓIS at point S=xS,yS and intersects the curve ΓPS at two points K1 and K2. If yS+=yK1, then the curve K1S^ forms an order-1 periodic solution for the problem stated in (1).
Further, for CaseBi, if yS+>yK1, then the point demoted by S+ lies above the point K1, and we get(85)ψyK1>yK1.
In addition, the solution initiating from the pointS+ meets the curve ΓIS at a point S1 which lies below the point S, i.e., yS1<yS. As F is increasing on 0,yS, we have FyS1<FyS, i.e., yS1+<yS+. All the above results affirm that the Poincaré map for Case Bi satisfies(86)ψyS+<yS+.
Inequalities (25) and (26) confirm that a fixed point in yK1,yT+ will exist. Hence, an order-1 periodic solution exists for problem (1).
IfyS+<yK1, then after a one time impulsive effect, the solution will directly map to the interval υ,yS+. Thus, if yK2≥υ, then according to inequality (1), any trajectory originating from y+ with υ≤y+≤yK2 will intersect the curve ΓIS and experience a limited time of pulse actions and at last enter into Int Γ2 and will be free from more pulse action. If yK2<υ<yS+, then each solution curve of problem (1) will map to the Int Γ2 after a one time impulsive effect. Hence, if yS+<yK1, then a fixed point does not exist.
For CaseBii, if yS+>yK1, then ψyK1>yK1. We also know that the function F is decreasing on 0,yS. So, the solution y+ initiating from 0,yK2∪yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 will map to the interval yS+,υ after a one time impulsive effect. Therefore, the trajectory originating from the point ψυAT1−PAT/θ1+aθ2,υ will satisfy ψυ<υ. From the above inequalities, it follows that the fixed point exists in the interval yK1,υ.Theorem 7.
For CaseBiii, ifyS+>yK1, then the fixed point of Poincaré mapψyi+exists, and therefore an order-1 periodic solution exists for system (1).Proof.
IfyS+=yK1, then for system (1), the curve K1S^ forms an order-1 periodic solution. If yS+≠yK1, then we consider the following two cases.(87)1yS+≥υ,2yS+<υ.
For Case (1), ifyT+>yK1, then ψyK1>yK1. Moreover, according to the exact domain of the Poincaré map ψyi+, the impulsive point S1+ of S+ lies below the point S+, i.e., S1+<S+ for yS+≥υ. Therefore, inequality ψyS+<yS+ is true, which shows that the fixed point exists in the interval yK1,yS+.
IfyS+<yK1, then applying the same techniques as those given in Theorem 6, it can easily be shown there must exist a finite number of pulse effects for any solution of system (1). Furthermore, the solution enters into Int Γ2 and becomes free from more pulse actions.
For Case (2), ifyS+>yK1, then ψyK1>yK1 holds true. We also know that the highest impulsive point is ψυAT1−PAT/θ1+aθ2,υ because yS+<υ. Therefore, we get ψυ<υ, and hence the theorem is true.
IfyS+<yK1, then any trajectory of system (1) tends into Int Γ2 only after finite pulse effects. This completes the proof.


## 8. Stability of Order-1 Periodic Solution

The monotonicities of Poincaré mapψyi+ and existence of its fixed point were discussed in previous sections. Now, based on these, we will discuss the stability of fixed point of Poincaré map ψyi+ for system (1).Theorem 8.
For CaseAi, if the fixed point of Poincaré map ψyi+ is unique and one of the following two conditions is satisfied, then the corresponding fixed point denoted by y∗ is stable globally.(a)
IfψyT<yT.(b)
IfψyT>yTandψ2yi+>yi+foryi+∈yT,y∗.Proof.
From Theorem4, we know that for Case Ai, the fixed point of Poincaré map ψyi+ exists. Let the fixed point y∗ be unique; then, the global stability can be discussed as follows:(a)
IfψyT<yT, then yi+<ψyi+<y∗ for all yi+∈0,y∗. This means that as j increases, ψjyi+ increases monotonically and satisfies limj⟶+∞ψjyi+=y∗. If yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, then we take two cases. (1) If yi+∈y∗,yT, then according to the relation y∗<ψyi+<yi+, ψyi+ decreases monotonically, i.e., y∗<ψjyi+<ψj−1yi+ for all j≥1 and we get limj⟶+∞ψjyi+=y∗. (2) If yi+∈yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, then ψyi+∈0,yT and limj⟶+∞ψ1+jyi+=y∗. Therefore, the conclusion in a is true.(b)
IfψyT>yT, then we take three intervals: (1) yi+∈yT,y∗; (2) yi+∈0,yT; (3) yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For interval (1), since yT≤yi+<y∗ and Poincaré map ψyi+ is monotonically decreasing in this interval, it is easy to get yT≥ψyi+>y∗. At the same time, by using the second condition ψ2yi+>yi+, we get yi+<ψ2yi+<y∗. This means that for all j≥1, ψ2j−1yi+<ψ2jyi+<y∗. This shows that ψ2jyi+ increases monotonically, and limj⟶+∞ψ2jyi+=y∗.
For intervals (2) and (3), using the same method as those in (1), we can prove that there must exist n≥1 such that ψnyi+∈yT,y∗, and hence the fixed point of Poincaré map ψyi+ is globally stable under conditions (2) and (3). This completes the proof.Theorem 9.
For CaseAii, if the fixed point y∗ of Poincaré map ψyi+ is unique and one of the following two conditions is true, then y∗ is globally stable.(a)
IfψyT>yT.(b)
IfψyT<yT and ψ2yi+<yi+ for yi+∈y∗,yT.Proof.
Theorem4 shows that for Case Aii, there exists a fixed point of the map ψyi+. Assuming that the fixed point is unique, we have the following conclusions regarding its stability:(a)
From Theorem2, it is clear that the Poincaré map is monotonically increasing in the interval yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 and monotonically decreasing in the interval 0,yT. If ψyT>yT, then the fixed point satisfies y∗>yT for any yi+∈yT,y∗ and ψj1yi+ increases with the increasing value of j1 such that limj⟶+∞ψj1yi+=y∗ for all yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. ψj2yi+ decreases as j2 increases, and limj⟶+∞ψj2yi+=y∗.For allyi+∈0,yT, there is ψyi+∈yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2; therefore, limj⟶+∞ψ1+j1yi+=y∗ or limj⟶+∞ψ1+j2yi+=y∗. In summary, the only fixed point y∗ is globally stable.(b)
The Poincaré mapψyi+ is monotonically decreasing in the interval 0,yT, and for yi+∈y∗,yT, the condition ψ2yi+<yi+ is satisfied. So, it is easy to get y∗<ψ4yi+<ψ2yi+. By induction, there is a relation y∗<ψ2jyi+<ψ2j−1yi+ for all j≥1. This shows that ψ2jyi+ monotonically decreases with increasing value of j, and limj⟶+∞ψ2jyi+=y∗. In addition, for all yi+∈0,y∗∪yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, there must exist l≥1 such that ψlyi+∈y∗,yT, and hence limj⟶+∞ψl+2jyi+=y∗.Theorem 10.
For CaseAiii, if the fixed point y∗ is unique and one of the following conditions is true, then it is globally stable.(a)
Ifψyni>ynii=1,2.(b)
Ifψyni<ynii=1,2, and ψ2yi+<yi+ for all yi+∈y∗,yn2.(c)
IfψyT>yT, ψyn2>yn2, and ψyn1<yn1, for yi+∈y∗,yn1 when y∗<ψ2yi+<yi+.(d)
IfψyT<yT, ψyn2>yn2, and ψyn1<yn1.Proof.
Theorem5 shows that there exists a fixed point of Poincaré map ψyi+ for Case Aiii. Moreover, if y∗ is unique, then its global stability can be described as follows:(a)
Ifψyni>ynifori=1,2, then we take three intervals: (1) yn1,y∗; (2) y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2; (3) 0,yn1. For all yi+∈yn1,y∗, we get yi+<ψyi+<y∗. The Poincaré map ψyi+ is monotonically increasing in the interval yn1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, and ψyi+<ψ2yi+<y∗. By induction, we get ψj−1yi+<ψjyi+<y∗ for all j≥1, which means that ψjyi+ monotonically increases as j increases, and limj⟶+∞ψjyi+=y∗, yi+∈yn1,y∗.For allyi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, we get y∗<ψyi+<yi+. From the monotonicity of ψyi+, we have y∗<ψ2yi+<ψyi+, which means that ψjyi+ decreases with increasing value of j and limj⟶+∞ψjyi+=y∗ for all yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For all yi+∈0,yn1, it is easy to get ψyi+∈yn1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, and according to the previous conclusion, we get limj⟶+∞ψ1+jyi+=y∗. Therefore, the result in Case a is true.(b)
Ifψyni<ynifori=1,2, then we take two cases: (1) yi+∈y∗,yn2; (2) yi+∈0,y∗∪yn2,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For all yi+∈y∗,yn2 and according to the monotonicity of the Poincaré map, ψyi+ satisfies ψ2yi+<yi+. From this, it is easy to get y∗<ψ4yi+<ψ2yi+. By induction, the inequality y∗<ψ2jyi+<ψ2j−1yi+ for all j≥1 holds, which means that as j increases, the mapping ψ2jyi+ monotonically decreases, and limj⟶+∞ψ2jyi+=y∗ for all yi+∈y∗,yn2. For all yi+∈0,y∗∪yn2,+∞, there exists k≥1, such that ψkyi+∈y∗,yn2. From this, we get limj⟶+∞ψk+2jyi+=y∗ for all yi+∈0,y∗∪yn2,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. All the above conclusions indicate that Case b is true.(c)
We again take two conditions: (1)yi+∈y∗,yn1; (2) yi+∈0,y∗∪yn1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For all yi+∈y∗,yn1, the Poincaré map ψyi+ is monotonically decreasing, and the inequality y∗<ψ2yi+<yi+ is satisfied. We can easily get the relationship y∗<ψ4yi+<ψ2yi+, and by induction, y∗<ψ2jyi+<ψ2j−1yi+ for all j≥1. This means that as j increases, the mapping ψ2jyi+ monotonically decreases, and limj⟶+∞ψ2jyi+=y∗. For all yi+∈0,y∗∪yn1,+∞, there must exist l≥1, such that ψlyi+∈y∗,yn1. Therefore, we get limj⟶+∞ψl+2jyi+=y∗ for all yi+∈0,y∗∪yn1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, which means that Case (c) is true.(d)
If the conditions given in statement are satisfied, we consider two intervals: (1)yi+∈yn2,yT; (2) yi+∈0,yn2∪yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. If yi+∈yn2,yT, then according to the monotonicity of the Poincaré map ψyi+, ψj1yi+ monotonically increases as j1 increases, and limj⟶+∞ψj1yi+=y∗. If yi+∈y∗,yT, then ψj2yi+ monotonically decreases as j2 increases, and limj⟶+∞ψj2yi+=y∗. For all yi+∈0,yn2∪yT,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, it is easy to know that there must exist a positive integer k, such that ψkyi+∈yn2,yT, and at the same time, limj⟶+∞ψk+j1yi+=y∗ or limj⟶+∞ψk+j2yi+=y∗. Hence, the Case (d) is true.Theorem 11.
For CaseBi, if ψyK1>yK1, then the fixed point y∗ of Poincaré map ψyi+ is globally asymptotically stable provided that ψ2yi+>yi+ for all yi+∈yK1,y∗.Proof.
From Theorem6, we know that for Case Bi, a fixed point of Poincaré map ψyi+ exists.
According to the inequality given in Lemma3, ψyi+>yi+ for all yi+∈0,yK2. At the same time, the inequality ψ0=υ>0 is satisfied. So, the fixed point y∗ does not lie in the interval 0,yK2. This shows that the unique fixed point belongs to the interval yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2.
IfyK1≤yi+<y∗, then from the monotonicity of the mapping ψyi+, we get ψyK1≥ψyi+>y∗. By applying the inequality ψ2yi+>yi+ for all yi+∈yK1,y∗, we get yi+<ψ2yi+<y∗. By induction, there exists a relationship ψ2j−1yi+<ψ2jyi+<y∗ for all j≥1. This means that as j increases, ψ2jyi+ increases monotonically, and hence limj⟶+∞ψ2jyi+=y∗.Theorem 12.
For CaseBii, ifψyK1>yK1, then the fixed point of Poincaré mapψyi+is globally stable.Proof.
From Theorem6, there exists a fixed point of Poincaré map ψyi+ for Case Bii. Using the same method as in Theorem 11, there is no fixed point on the interval 0,yK2, and y∗ is located in the interval yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. Moreover, under the uniqueness of y∗, the global stability can be described as follows.
For CaseBii, the Poincaré map ψyi+ is monotonically decreasing in the interval 0,yK2 and monotonically increasing in the interval yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. If yi+∈yK1,y∗, then according to the relationship yi+<ψyi+<y∗, it is obvious that ψjy+ increases monotonically towards y∗ as j increases, i.e., limj⟶+∞ψjyi+=y∗. For all yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, according to the relationship y∗<ψyi+<yi+ and properties of Poincaré map ψyi+, we know that ψjyi+ monotonically decreases with the increasing value of j, and limj⟶+∞ψjyi+=y∗.
Ifyi+∈0,yK2, then there must exist some l≥1 such that ψyi+∈yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, and therefore limj⟶+∞ψ1+jyi+=y∗. Hence, the result in Theorem 12 is correct.Theorem 13.
For CaseBiii, if ψyK1>yK1, then the unique fixed point y∗ of Poincaré map ψyi+ exists. If one of the conditions (a) and (b) given below is true, then y∗ is globally stable.(a)
IfψyNi>yNii=1,2.(b)
IfyK1≤ψyN2=ψyN1≤yN1, and ψ2yi+>yi+ for all yi+∈yK1,y∗.Proof.
For CaseBiii, if ψyK1>yK1, then from Lemma 3 and Theorem 7, we know that Poincaré mapping ψyi+ has at least one fixed point y∗ belonging to the interval yK1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. Under the uniqueness of y∗, the global stability can be demonstrated as follows:(a)
IfψyNi>yNii=1,2, then only y∗ exists in the interval yN1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. From Theorem 2, we can see that Poincaré map ψyi+ is monotonically increasing in the interval yN1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. For any yi+∈yN1,y∗, we get yi+<ψyi+<y∗, which shows that ψjyi+ for j≥1 increases monotonically, and limj⟶+∞ψjyi+=y∗. For any yi+∈y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2, we get the relation y∗<ψyi+<yi+. Therefore, from the monotonicity of ψyi+, ψjyi+ monotonically decreases with increasing value of j, and we get limj⟶+∞ψjyi+=y∗. For all yi+∈0,yK2∪yK1,yN1, it is obvious that there exists an integer l≥0, such that ψlyi+∈yN1,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2. Hence, for all yi+∈0,yK2∪yK1,yN1, we get limj⟶+∞ψl+jyi+=y∗.All these results show that ifψyNi>yNii=1,2, then the unique fixed point y∗ of the mapping ψyi+ is globally stable.(b)
IfyK1≤ψyN2=ψyN1≤yN1, then combined with the inequality ψyK1>yK1 given in the statement, it is clear that there exists only one y∗ in the interval yK1,yN1. The mapping ψyi+ monotonically decreases in the interval yK1,y∗, i.e., for all yi+∈yK1,y∗, we have ψyK1≥ψyi+>y∗. In addition, by applying the condition ψ2yi+>yi+, we get yi+<ψ2yi+<y∗. Hence, we get ψ2j−1yi+<ψ2jyi+<y∗ for j≥1. This shows that ψ2jyi+ monotonically increases with the increasing value of j and limj⟶+∞ψ2jyi+=y∗ for all yi+∈yK1,y∗.
Ifyi+∈0,yK2∪y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2 and yK1≤ψyN2=ψyN1≤yN1, then there must exist l≥1 such that ψlyi+∈yK1,y∗. By using the same way as above, we get limj⟶+∞ψl+2jyi+=y∗ for all yi+∈0,yK2∪y∗,θ1+aθ2/bθ2+bθ2υ/bθ2+βθ1+aθ2.
Therefore, ifyK1≤ψyN2=ψyN1≤yN1 and ψ2yi+>yi+ for all yi+∈yK1,y∗, then the fixed point y∗ is globally stable.


## 9. Conclusions

The IPM strategy is a dynamic management system. From a mathematical perspective, this is actually an optimal control problem under multiple objectives. The IPM approach’s purpose is to monitor the number of pest populations in real time and decide whether to implement a control strategy based on the size of the population. The state-dependent impulsive differential equation [20, 45–47] is needed to truly characterize the IPM strategy and the dynamic evolution of pest-natural system. Moreover, in recent years, researchers have proposed a variety of state-dependent pest-natural enemy feedback control systems.The change rate of pest population plays an important role in state-dependent prey-predator ecological system. There are two fundamental circumstances in the previous studies which require high attention. First, the pest population is comparatively high and the change rate is little; second, the population of pest is small, but the change rate is high. A crucial issue illustrated by these two situations is that when the pest population is large, the growth rate is small or even negative at this time. In this case, even if the IPM strategy is not implemented, the number of pests may not exceed EIL. Another situation is that although the number of pests is not large, the growth rate of the pest population is very large. If the control strategy is not implemented in time, it may lead to a large outbreak of pests. Next, the IPM process needs precise checking of the pest populations, and consequently suitable integrated control strategies can be prepared. The pest killing rate should be a function of their density, whereas the releasing quantity of natural enemies should be a function of their density. Based on this, a feasible new nonlinear state-feedback system with nonlinear ratio-dependent AT is proposed.The use of nonlinear pulse as state-dependent feedback control with nonlinear ratio-dependent AT is more reasonable and closer to reality in a biological sense, but the impulsive model becomes very difficult because of the existence of two population quantities in the control actions. By including the densities of pest and its natural enemy in controlling measures, we can develop the pest control model based on the practical importance according to the growth direction of agriculture and forestry. Corresponding analytical techniques and numerical methods were developed, the dynamic behavior of the system was examined, and the important role of the main conclusions in integrated pest control was given.To avoid the complexity, in this paper, we proposed the simple Lotka–Volterra impulsive mathematical model. Our aim is to reveal how nonlinear pulse control with nonlinear ratio-dependent AT affects the whole dynamics and concentrate on the biological implications. The definition and properties of Poincaré map for phase-concentrated pulse points in various cases are discussed and studied. The existence, uniqueness, and global stability of boundary and interior periodic solutions of order 1 for model (1) are analyzed by using the definition of Poincaré map. In the present paper, some basic techniques were used for the qualitative analysis of nonlinear pulsed model with nonlinear ratio-depended AT, which can be widely used in the study of feedback control systems with critical conditions, such as the blood glucose-insulin regulation system.



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*Source: 7903450-2022-01-18.xml*