# The Impulsive Model with Pest Density and Its Change Rate Dependent Feedback Control

Authors: Ihsan Ullah Khan; Sanyi Tang
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

The idea of action threshold depends on the pest density and its change rate is more general and furthermore can produce new modelling techniques related to integrated pest management (IPM) as compared with those that appeared in earlier studies, which definitely bring challenges to analytical analysis and generate new ideas to the state control measures. Keeping this in mind, using the strategies of IPM, we develop a prey-predator system with action threshold depending on the pest density and its change rate, and study its dynamical behavior. We develop new criteria guaranteeing the existence, uniqueness, local and global stability of order-1 periodic solutions. Applying the properties of LambertW function, the Poincaré map is portrayed for the exact phase set, which is helpful to provide the sufficient conditions for the existence and stability of the interior order-1 periodic solutions and boundary order-1 periodic solution, also confirmed by numerical simulations. It is studied in detail that how and under what conditions the fixed point of Poincaré map and its stability are affected by the newly introduced action threshold. The analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.


---

## Body

## 1. Introduction

The risk of pests to agricultural productions may be an enormous issue over the world, which makes pest control being a motivating topic and attracts great attention to the development of effective pest management strategies. Pests will cause vital crop yield declines, even colossal failure. Additionally, they will downsize the standard of farm items. Therefore, countries around the world have established special organizations to review the management procedure of agricultural pests [1–7].Integrated pest management (IPM) is a useful methodology in prevailing pests that have been demonstrated to be more practical than the classic strategies both experimentally [8–10] and theoretically [11, 12]. It is a procedure that is used to solve pest problems while minimizing threats to individuals and the environment. IPM can be utilized to deal with all sorts of pests anywhere in rural, urban, and natural areas or wild land. IPM is an ecosystem-based approach that concentrates on long-term prevention of pests or their damage through a combination of strategies, such as biological control, adjustment of social practices, living space control, and utilization of safe assortments. The objective of IPM is not to eliminate pests, rather to manage the amount of the pests below an associated economic threshold (ET) and ensure ecosystem up to maximum level.Recently, many researchers have proposed impulsive differential equations to examine the dynamics of pest control models [13–18]. Impulsive equations have been brought into population dynamics in relation to impulsive vaccination, chemotherapeutic handling of disease, population ecology, and impulsive birth. Especially, some impulsive differential equations have been presented effectively in population dynamics (agriculture or fishing) and epidemic dynamics. Numerous recent articles have mathematically exhibited a variety of IPM tactics using impulsive differential equations, for example, stage structure in the predator species and periodically changing environmental conditions [19]. Relative models also have been studied in [20].Most of the researchers considered systems with impulses at fixed moments [21–29]. The shortcomings of this kind of systems are that they did not pay enough attention to the management cost and the growth rules of the pest. Impulsive differential equations with impulses happening at fixed time emerge in the modelling of real-world phenomena in which the state of the inspected procedure fluctuates at fixed moments of time. In literature [30], the author extended a model with linear impulsive control tactics to a model with nonlinear impulsive control measures, which revealed further precise conditions for pest control. Wang et al. [31] discussed the threshold condition which guarantee the existence and stability criteria for the pest-free periodic solution. In addition, the complex dynamics for system is discussed when the forward and backward bifurcations could happen once the pest-free periodic solution becomes unstable.State-dependent feedback control approach is generally expressed by an impulsive semi dynamical system, and they can be portrayed in comprehensive terms in real biological problems. For example, control tactics (i.e., pesticide application, harvesting, treatment, etc.) are applied only when a particular species size ranges an earlier known threshold density. Specifically, in [32–35] an excellent example in the series of models encouraged by IPM has been framed and examined. In [36–38], IPM has been exhibited by experiments, and it is demonstrated that IPM is more effective than classical methods.In all the previous literatures, researchers projected models either with a single economic threshold or multiple thresholds [39–45]. There are few drawbacks to this sort of thresholds. However, there are two reasonable circumstances: one is that the number of the pest population is comparatively large, but its change rate is quite small; the other is that the number of population is small, but its change rate is significantly high. The latter case is more obvious at the initial stage of the occurrence of the pest. To overcome these drawbacks, we planned to take the model with action threshold depending on the pest density and its change rate (so-called ratio-dependent AT), and investigate its global dynamics.The paper is ordered as follows: In Section2, the commonly used generalized prey-predator model is proposed and the new ratio-dependent nonlinear action threshold is introduced. In Section 3, the exact impulsive and phase sets are determined for all existing cases. In view of the impulsive and phase sets, the Poincaré map is constructed in Section 4. In Section 5.1, some important relations and lemmas are provided that are very important for the next sections. The boundary order-1 periodic solution is given in Section 5.2. In Section 6, the global properties of system constructed in Section 2 are discussed, including the existence, local and global stability of order-1 periodic solution, and the effect of weighted parameters on the fixed point of Poincaré map. In the same section, the effect of weighted parameters on the different cases is also discussed. To sum up the whole work, a detailed conclusion is given in Section 7.


## 2. Construction of Model and Main Properties

In view of the reasons specified above, we consider the commonly used prey-predator system with pest density and its change rate dependent feedback control, i.e., the action threshold depends not only on pest density but also on its change rate, which can be modeled by(1)
d
x
t
d
t
=
a
x
t
1
−
x
t
K
−
b
x
t
−
q
x
t
y
t
,
d
y
t
d
t
=
λ
x
t
y
t
1
+
w
x
t
−
r
x
t
y
t
−
δ
y
t
,
α
x
t
+
β
d
x
t
d
t
<
A
T
,
x
t
+
=
1
−
p
x
t
,
y
t
+
=
y
t
+
τ
,
α
x
t
+
β
d
x
t
d
t
=
A
T
,whereα
,
β, and A
T are all positive constants with α
+
β
=
1. x
t and y
t respectively, represent the quantities of prey and predator. a denotes the intrinsic growth rate of the pest population and K demonstrates the carrying capacity. The pest population dies at the rate of b
x
t and is predated by the predator population at a rate q
x
t
y
t. The quantity λ
x
t
y
t
/
1
+
w
x
t, which is actually a saturating function of the present quantity of pest, is the expand rate of predator response. The prey population breakdowns the predator response at a rate r
x
t
y
t, and δ
y
t represents the decay rate of the predator in the absence of prey. The quantities 1
−  p
x
t and y
t
+
τ are known as the controlling quantities; whenever the pest population touches the action threshold, the management activities are adapted and the quantities of prey and predator are adjusted according to the controlling actions 1
−  p
x
t and y
t
+
τ respectively. Therefore p denotes the instant killing rate and τ represents the releasing constant.If the value of carrying capacityK
→
+
∞, then for c
=
a
−
b model (1) is reduced to the following form(2)
d
x
t
d
t
=
c
x
t
−
q
x
t
y
t
,
d
y
t
d
t
=
λ
x
t
y
t
1
+
w
x
t
−
r
x
t
y
t
−
δ
y
t
,
α
x
t
+
β
d
x
t
d
t
<
A
T
,
x
t
+
=
1
−
p
x
t
,
y
t
+
=
y
t
+
τ
,
α
x
t
+
β
d
x
t
d
t
=
A
T
.The quantitiesα and β are dependent weighted parameters. It is interesting to note that if the second weighted parameter β disappears, the ratio-dependent A
T will transform into ET [41–45]. Therefore, the ET is an exceptional case of ratio-dependent A
T for β
=
0. From ratio-dependent A
T and the first equation of model (2), it follows that y
=
α
+
c  β
x
−
A  T
/
q
β  x with(3)
lim
x
→
+
∞
α
+
c
β
x
−
A
T
q
β
x
=
α
+
c
β
q
β
.If the weighted parameterα vanishes, the ratio-dependent AT transformed into y
=
c
x
−
A  T
/
q  x. It is obvious that if again pest population x tends to infinity, the predator population y is bounded and approaches its maximum value c
/
q. By applying the controlling quantities on y
=
α
+
c  β
x
−
A  T
/
q
β  x, we get another curve y
+
=
α
+
c  β
x
+
−
A
T
1
−  p
/
q
β
x
+
+
τ. For β
=
0, the curve transforms into the vertical straight line x
+
=
1
−  p
A
T.For convenience, we denotey
=
α
+
c  β
x
−
A  T
/
q
β  x and y
+
=
α
+
c  β
x
+
−
A
T
1
−  p
/
q
β
x
+
+
τ by Γ
I
c and Γ
P
c respectively, as shown in Figure 1. A  T
/
α
+
c  β is the initial value which curve Γ
I
c attains at y
=
0. At this point, the vertical coordinate with Γ
P
c takes the value p
α
+
c  β
/
q  β
+
τ. If β approaches one, then x approaches A  T
/
c and hence in this case, the vertical coordinate with Γ
P
c attains the value p
c
/  q
+
τ. It is an essential assumption that the initial value x
0
+ must satisfy α
x
0
+
+
β
d
x
0
+
/
d  t
<
A
T.Figure 1
Illustration diagrams of the impulsive and phase sets for CasesB
1
,
B
2 and B
3. (a) A
T
/
α
≤
x
2
∗, (b) x
2
∗
<
A
T
/  α
<
x
1
∗, (c) A
T
/
α
≥
x
1
∗. Γ
1 is a trajectory tangent to curve Γ
P
c at point S
=
x
S
,
y
S, Γ
2 is a trajectory tangent to curve Γ
I
c at point T
=
x
T
,
y
T, Γ
3 is homoclinic trajectory which touches curve Γ
P
c at two points x
P
1
,
y
P
1 and x
P
2
,
y
P
2, and its upper and lower right branches touch curve Γ
I
c at points Q
1
=
x
Q
1
,
y
Q
1 and Q
2
=
x
Q
2
,
y
Q
2
, respectively.
(a)
(b)
(c)Our main objective is to discuss the global dynamics of model (2). We will see how the global dynamics are affected if the threshold is not a straight line but complex curve. For the first two equations (i.e., the ODE system without control measures), there always exist trivial equilibrium 0
,
0 and two interior equilibria(4)
E
1
=
x
1
∗
,
c
q
and
E
2
=
x
2
∗
,
c
q
,where(5)
x
1
∗
=
λ
−
r
−
δ
ω
+
λ
−
r
−
δ
ω
2
−
4
r
ω
δ
2
r
ωand(6)
x
2
∗
=
λ
−
r
−
δ
ω
−
λ
−
r
−
δ
ω
2
−
4
r
ω
δ
2
r
ωprovided thatλ
−
r
−
δ
ω
>
0 and λ
−
r
−
δ
ω
>
2
r
δ
ω. If λ
−
r
−
δ
ω
=
2
r
δ
ω, then the two roots will coincide with each other. It is also clear that E
2 is the centre and E
1 is a saddle point.Recently, Tang et al. [46] presented the following prey-predator model with state-dependent feedback control which is the special case of model (1) for α
=
1 and β
=
0
(7)
d
x
t
d
t
=
a
x
t
1
−
x
t
K
−
b
x
t
−
q
x
t
y
t
,
d
y
t
d
t
=
λ
x
t
y
t
1
+
w
x
t
−
r
x
t
y
t
−
δ
y
t
,
x
t
<
A
T
,
x
t
+
=
1
−
p
x
t
,
y
t
+
=
y
t
+
τ
,
x
t
=
A
T
,where AT signifies the economic threshold level, i.e., action threshold transforms into ET.The special case of model (2) for ω
=
0 and λ
−
r
=
d is(8)
d
x
t
d
t
=
c
x
t
−
q
x
t
y
t
,
d
y
t
d
t
=
d
x
t
y
t
−
δ
y
t
,
α
x
t
+
β
d
x
t
d
t
<
A
T
,
x
t
+
=
1
−
p
x
t
,
y
t
+
=
y
t
+
τ
,
α
x
t
+
β
d
x
t
d
t
=
A
T
,which has been considered in [47]. We will see that the results associated to model (8) can be easily obtained based on the results for model (2).


## 3. Impulsive and Phase Sets

In this section, we will find out the exact impulsive and phase sets for the existing cases. The foremost and necessary part is to search out the segment that is free from impulsive effect, i.e., the solution starting fromΓ
P
c cannot reach to curve Γ
I
c for maximum impulsive set. Based on the positions of equilibria E
1, E
2 and curve Γ
I
c, we take the following three cases:(9)
B
1
A
T
α
≤
x
2
∗
;
B
2
x
2
∗
<
A
T
α
<
x
1
∗
and
B
3
A
T
α
≥
x
1
∗
.


### 3.1. Impulsive Set

In CaseB
1, trajectory Γ
1 is tangent to curve Γ
P
c at point S
=
x
S
,
y
S with y
S
≥
c
/
q. If we represent point x
Q
2
,
y
Q
2 by x
i
m
1
,
y
i
m
1, then the domain of the impulsive set becomes as:(10)
M
1
=
x
,
y
∈
R
+
2
A
T
α
+
c
β
≤
x
≤
x
i
m
1
,
0
≤
y
≤
y
i
m
1
.It is obvious from the domain of impulsive setM
1 that in this case, no solution originating from the phase set will reach the interval y
i
m
1
,
c
/  q. In the following lemma, we find out the exact value of y
i
m
1 which depends on the corresponding horizontal coordinate.Lemma 1.
For CaseB
1, the impulsive set is defined as M
1. The maximum vertical coordinate of M
1 is y
i
m
1, where y
i
m
1
=
−
c
/
q
W
−
q
/
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c provided that A
Q
2
1
≤
0.Proof.
Let a solutionΓ
1 is tangent to curve Γ
P
c at point x
S
,
y
S, and it touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2. Then these points must satisfy the following equation(11)
λ
ω
ln
1
+
ω
x
Q
2
1
+
ω
x
S
−
r
x
Q
2
−
x
S
−
δ
ln
x
Q
2
x
S
=
c
ln
y
Q
2
y
S
−
q
y
Q
2
−
y
S
.
Solving this equation fory
Q
2, we get(12)
−
q
c
y
Q
2
e
−
q
/
c
y
Q
2
=
−
q
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c
,
whereA
Q
2
1
=
λ
/  ω
l
n
1
+
ω
x
Q
2
/
1
+
ω
x
S
−
r
x
Q
2
−
x
S
−
δ
l
n
x
Q
2
/
x
S. We can solve the above equation with the help of Lambert W function. Obviously, the above equation will give us two solutions, but only the minimum value lies on curve Γ
I
c as well as Γ
1. If we denote it by y
i
m
1, we obtain(13)
y
i
m
1
=
−
c
q
W
−
q
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c
,
which is well defined due toA
Q
2
1
≤
0.
From Figure1(b), it can be seen that for Case B
2, Γ
2 is tangent to curve Γ
I
c at point T
=
x
T
,
y
T, where y
T
≤
c
/  q. Then based on the positions of equilibria E
1, E
2 and curve Γ
I
c, we discuss the maximum impulsive set for this case as follows:(14)
M
2
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
T
,
0
≤
y
≤
y
T
.
The tangent point of the closed trajectory with curveΓ
I
c varies with the changing estimations of weighted parameters α and β. From the domain of impulsive set M
2, it is obvious that if y
T
<
c
/
q then the interval y
T
,
c
/  q cannot be used for any solution originating from the respective phase set.
Now we discuss the impulsive set for the CaseB
3. This case is more crucial than the previous cases. In this case, homoclinic trajectory exists. This homoclinic trajectory Γ
3 touches curve Γ
P
c at points x
P
1
,
y
P
1 and x
P
2
,
y
P
2, and its lower right branch touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2 (as shown in Figure 1(c)). This is actually the maximum impulsive point for Case B
3. Before finding out the exact value of vertical coordinate y
Q
2, we first provide some necessary quantities which are not only helpful for finding the maximum vertical coordinate of the impulsive set M
3, but also assume a significant role in finding the fixed point of the Poincaré map P
y
i
+. These quantities are listed as follows:(15)
A
P
1
3
=
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
x
P
1
,
(16)
A
1
=
λ
ω
ln
1
+
ω
x
Q
2
1
+
ω
x
P
1
−
r
x
Q
2
−
x
P
1
−
δ
ln
x
Q
2
x
P
1
,
(17)
A
Q
2
3
=
A
P
1
3
−
A
1
=
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
Q
2
−
r
x
1
∗
−
x
Q
2
−
δ
ln
x
1
∗
x
Q
2
.
Replacingx
P
1 by x
P
2 in equations (15) and (16) and denoting the resultant equations by A
P
2
3 and A
2 respectively, then A
Q
2
3 = A
P
2
3
−
A
2. If we denote x
Q
2
,
y
Q
2 by x
i
m
3
,
y
i
m
3, then we find the exact value of y
i
m
3 which depends on the respective horizontal coordinate.Lemma 2.
For CaseB
3, the impulsive set is defined as M
3. The maximum vertical coordinate for this is y
i
m
3, where y
i
m
3
=
−
c
/
q
W
−
e
−
1
−
A
Q
2
3
/
c provided that A
Q
2
3
≥
0.Proof.
In this case, the lower right branch of the homoclinic trajectoryΓ
3 touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2. Combining point Q
2
=
x
Q
2
,
y
Q
2 with E
1
=
x
1
∗
,
c
/  q must satisfy the following relation:(18)
c
ln
c
q
−
c
−
λ
ω
ln
1
+
ω
x
1
∗
+
r
x
1
∗
+
δ
ln
x
1
∗
=
c
ln
y
Q
2
−
q
y
Q
2
−
λ
ω
ln
1
+
ω
x
Q
2
+
r
x
Q
2
+
δ
ln
x
Q
2
,
which can be simplified as(19)
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
Q
2
−
r
x
1
∗
−
x
Q
2
−
δ
ln
x
1
∗
x
Q
2
=
c
ln
c
/
q
y
Q
2
−
q
c
q
−
y
Q
2
.
Solving the above equation fory
Q
2, we get(20)
−
q
c
y
Q
2
e
−
q
/
c
y
Q
2
=
−
e
−
1
−
A
Q
2
3
/
c
.
Following the same way as in Lemma1, applying the properties of Lambert W function, we get two solutions. From Figure 1(c) it is clear that only the minimum value lies both on curve Γ
3 and Γ
I
c. If we denote it by y
i
m
3, then we get(21)
y
i
m
3
=
−
c
q
W
−
e
−
1
−
A
Q
2
3
/
c
,
which is well defined due toA
Q
2
3
≥
0. If we represent the impulsive set by M
3, then it can be expressed as(22)
M
3
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
i
m
3
,
0
≤
y
≤
y
i
m
3
.


### 3.2. Phase Set

In this subsection, we aim to discuss the phase sets for all the existing cases expressed above. The most essential and tough task in the process of discussing phase sets is to find out the segment, which is free from the impulsive effect. To find the exact domain of phase sets, we provide the following intervals:(23)
X
D
1
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
i
m
1
,
Y
D
1
=
τ
,
y
i
m
1
+
τ
,
X
D
2
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
T
,
(24)
Y
D
2
=
τ
,
y
T
+
τ
,
X
D
3
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
i
m
3
,
and
Y
D
3
=
τ
,
y
i
m
3
+
τ
.For CaseB
1, trajectory Γ
1 is tangent to curve Γ
P
c at point S
=
x
S
,
y
S. Thus, the corresponding phase set to the impulsive set M
1 can be expressed as:(25)
N
1
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
D
1
,
y
+
∈
Y
D
1For CaseB
2, the closed trajectory is tangent to curve Γ
I
c at point T
=
x
T
,
y
T, where y
T
≤
c
/  q. We indicate the intersection point of the closed trajectory Γ
2 with line y
=
c
/
q (denoted by L
1) as E
3
=
x
3
,
y
3. If we denote x
P
2
,
y
P
2 by x
m
i
n
2
,
y
m
i
n
2 and x
P
1
,
y
P
1 by x
m
a
x
2
,
y
m
a
x
2, then the phase set corresponds to the impulsive set M
2 can be expressed as follows:(26)
N
2
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
2
,
y
+
∈
Y
P
s
2with(27)
X
P
s
2
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
2
∪
x
max
2
,
+
∞
∩
X
D
2and(28)
Y
P
s
2
=
0
,
y
min
2
∪
y
max
2
,
α
+
c
β
q
β
+
τ
∩
Y
D
2
.From the phase setN
2, it is clear that the solution initiating from the interval y
m
i
n
2
,
y
m
a
x
2 will be free from impulsive effect. In the following lemma, based on the respective horizontal coordinates, the exact values of y
m
a
x
2 and y
m
i
n
2 are given.Lemma 3.
For CaseB
2, the impulsive set is defined as M
2. In this case, any solution initiating from y
min
2
,
y
max
2 will be free from impulse effect, where(29)
y
max
2
=
−
c
q
W
−
1
,
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
a
n
d
y
min
2
=
−
c
q
W
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
2
2
/
c
provided thatA
P
1
2
,
A
P
2
2
≥
0.Proof.
Suppose that the closed trajectoryΓ
2 originates from x
P
1
,
y
P
1, and tangent to curve Γ
I
c at point x
T
,
y
T. Then, these points must satisfy the relation:(30)
λ
ω
ln
1
+
ω
x
T
1
+
ω
x
P
1
−
r
x
T
−
x
P
1
−
δ
ln
x
T
x
P
1
=
c
ln
y
T
y
P
1
−
q
y
T
−
y
P
1
.
Rearranging this equation fory
P
1, we get(31)
−
q
c
y
P
1
e
−
q
/
c
y
P
1
=
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
,
whereA
P
1
2
=
λ
/
ω
ln
1
+
ω
x
T
/
1
+
ω
x
P
1
−
r
x
T
−
x
P
1
−
δ
ln
x
T
/
x
P
1. The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by y
m
a
x
2, we get(32)
y
max
2
=
−
c
q
W
−
1
,
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
.
The value ofy
P
2, denoted by y
m
i
n
2 can also be found by using the same method as above, i.e.,(33)
y
min
2
=
−
c
q
W
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
2
2
/
c
withA
P
2
2
=
λ
/
ω
ln
1
+
ω
x
T
/
1
+
ω
x
P
2
−
r
x
T
−
x
P
2
−
δ
ln
x
T
/
x
P
2.
If weighted parameterβ
=
0, i.e., the threshold level only depends on the pest density then the closed trajectory becomes tangent at y
=
c
/
q. In this case, y
m
i
n
2 and y
m
a
x
2 become as(34)
y
max
2
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
2
/
c
,
and
y
min
2
=
−
c
q
W
−
e
−
1
−
A
P
2
/
c
.
withA
P
1
2
=
A
P
2
2
=
A
P
2.
For CaseB
3, let us denote the intersection of the homoclinic trajectory Γ
3 with line y
=
c
/
q (denoted by L
1) as E
4
=
x
4
,
y
4. Trajectory Γ
3 touches curve Γ
P
c at upper point P
1
=
x
P
1
,
y
P
1 and lower point P
2
=
x
P
2
,
y
P
2, denoted by x
m
a
x
3
,
y
m
a
x
3 and x
m
i
n
3
,
y
m
i
n
3 respectively. In the following lemma, we find the exact values of y
m
a
x
3 and y
m
i
n
3.Lemma 4.
For CaseB
3, the impulsive set is defined as M
3. In this case any solution initiating from y
min
3
,
y
max
3 will be free from impulse effect, where(35)
y
max
3
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
1
3
/
c
a
n
d
y
min
3
=
−
c
q
W
−
e
−
1
−
A
P
2
3
/
c
provided thatA
P
1
3
,
A
P
2
3
≥
0.Proof.
Suppose that the homoclinic trajectoryΓ
3 touches curve Γ
P
c at upper point P
1
=
x
P
1
,
y
P
1. Combining point P
1
=
x
P
1
,
y
P
1 with E
1
=
x
1
∗
,
c
/  q must satisfy the following relation:(36)
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
x
P
1
=
c
ln
c
/
q
y
P
1
−
q
c
q
−
y
P
1
.
Arranging this equation fory
P
1, we get(37)
−
q
c
y
P
1
e
−
q
/
c
y
P
1
=
−
e
−
1
−
A
P
1
3
/
c
,
whereA
P
1
3
=
λ
/
ω
ln
1
+
ω
x
1
∗
/
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
/
x
P
1. The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by y
m
a
x
3, we get(38)
y
max
3
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
1
3
/
c
.
The value ofy
P
2, denoted by y
m
i
n
3 can also be found by following the same way as above, i.e.,(39)
y
min
3
=
−
c
q
W
−
e
−
1
−
A
P
2
3
/
c
withA
P
2
3
=
λ
/
ω
ln
1
+
ω
x
1
∗
/
1
+
ω
x
P
2
−
r
x
1
∗
−
x
P
2
−
δ
ln
x
1
∗
/
x
P
2.
If we represent the phase set for CaseB
3 by N
3, then it can be expressed as(40)
N
3
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
3
,
y
+
∈
Y
P
s
3
with(41)
X
P
s
3
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
3
∪
x
max
3
,
+
∞
∩
X
D
3
and(42)
Y
P
s
3
=
0
,
y
min
3
∪
y
max
3
,
α
+
c
β
q
β
+
τ
∩
Y
D
3
.IfΓ
P
c  lies on the left and does not touch Γ
2 or Γ
3 then the impulsive and phase sets will be transformed into M
1 and N
1 respectively.


### 3.3. The Impulsive and Phase Sets for Model (8)

In view of the model (2) and based on the locations of curve Γ
I
c and stable center δ
/
d
,
c
/  q, the following two cases can be taken for model (8)(43)
B
1
0
A
T
α
≤
δ
d
and
B
2
0
δ
d
<
A
T
α
.In the first case, trajectoryΓ
1
0 is tangent to curve Γ
P
c at point S
0
=
x
S
0
,
y
S
0 with y
S
0
≥
c
/
q (as shown in Figure 2(a)). If we represent point x
Q
2
,
y
Q
2 by x
i
m
0
,
y
i
m
0, then the impulsive set M
1
0 can be expressed as(44)
M
1
0
=
x
,
y
∈
R
+
2
A
T
α
+
c
β
≤
x
≤
x
i
m
0
,
0
≤
y
≤
y
i
m
0
.Figure 2
Illustration diagrams of the impulsive and phase sets for model (8). (a) A
T
/  α
≤
δ
/  d, (b) δ
/  d
<
A
T
/  α. Γ
1
0 is a trajectory tangent to curve Γ
P
c at point S
0
=
x
S
0
,
y
S
0, and Γ
2
0 is a trajectory which is tangent to curve Γ
I
c at point T
0
=
x
T
0
,
y
T
0.
(a)
(b)To discuss the exact domains of the phase set for both cases, we define the intervalsX
D
0
1
=
A
T
1
−
p
/
α
+
c
β
,
1
−
p
x
i
m
0, Y
D
0
1
=
τ
,
y
i
m
0
+
τ, X
D
0
2
=
A
T
1
−
p
/
α
+
c
β
,
1
−
p
x
T
0 and Y
D
0
2
=
τ
,
y
T
0
+
τ. Then, the phase set for Case B
1
0 becomes as:(45)
N
1
0
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
D
0
1
,
y
+
∈
Y
D
0
1In the following lines the analytical value ofy
i
m
0 is given. The proof of the lemma is the same as previous section, so we omit it.Lemma 5.
For CaseB
1
0, the impulsive set is defined as M
1
0. The maximum vertical coordinate for this is y
i
m
0, where y
i
m
0
=
−
c
/
q
W
−
q
/
c
y
S
0
e
−
q
/
c
y
S
0
+
A
Q
2
/
c provided that A
Q
2
=
δ
ln
x
S
0
−
ln
x
Q
2
+
d
x
Q
2
−
x
S
0
≤
0.For CaseB
2
0, we denote the intersection point of the closed trajectory Γ
2
0 with line y
=
c
/
q (denoted by L
1) as E
1
=
x
E
1
,
y
E
1. In this case, the closed trajectory is tangent to curve Γ
I
c at point T
0
=
x
T
0
,
y
T
0, where y
T
0
≤
c
/
q. If we denote x
P
2
,
y
P
2 by x
m
i
n
0
,
y
m
i
n
0 and x
P
1
,
y
P
1 by x
m
a
x
0
,
y
m
a
x
0, then based on the positions of curves Γ
I
c and Γ
P
c, we discuss the impulsive and phase sets as follows:(46)
M
2
0
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
T
0
,
0
≤
y
≤
y
T
0
,
(47)
N
2
0
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
02
,
y
+
∈
Y
P
s
02with(48)
X
P
s
02
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
0
∪
x
max
0
,
+
∞
∩
X
D
0
2and(49)
Y
P
s
02
=
0
,
y
min
0
∪
y
max
0
,
α
+
c
β
q
β
+
τ
∩
Y
D
0
2
.From the phase set, it is clear that the solution initiating from the intervaly
m
i
n
0
,
y
m
a
x
0 will be free from impulsive effect.Lemma 6.
For CaseB
2
0, the impulsive set is defined as M
2
0. In this case any solution initiating from y
m
i
n
0
,
y
m
a
x
0 will be free from impulse effect, where(50)
y
max
0
=
−
c
q
W
−
1
,
−
q
c
y
T
0
e
−
q
/
c
y
T
0
−
A
P
1
/
c
a
n
d
y
min
0
=
−
c
q
W
−
q
c
y
T
0
e
−
q
/
c
y
T
0
−
A
P
2
/
c
provided that(51)
A
P
1
=
δ
ln
x
P
1
−
ln
x
T
0
+
d
x
T
0
−
x
P
1
≥
0
,
A
P
2
=
δ
ln
x
P
2
−
ln
x
T
0
+
d
x
T
0
−
x
P
2
≥
0
.The proof of the Lemma6 can also be shown as previous section, so we also omit it. If the weighted parameter β
=
0, i.e., the threshold level only depends on the pest density then the closed trajectory Γ
2
0 becomes tangent to curve Γ
I
c at y
=
c
/
q. In this case, y
m
i
n
0 and y
m
a
x
0 become as:(52)
y
max
0
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
/
c
,
y
min
0
=
−
c
q
W
−
e
−
1
−
A
P
/
c
.withA
P
1
=
A
P
2
=
A
P.Compared with published work for model (8), we can see that more accurate domains of the impulsive and phase sets have been provided here. From Figure 2(b), it is clear that x
E
1
<
A
T
1
−  p
/
α
+
τ
q  β
<
δ
/  d.In the upcoming discussions, for convenience, we useA
l rather than A
Q
2
1, A
P
1
2, A
P
2
2 or A
1. For Case B
1, A
Q
2
1
≤
0 is equivalent to A
l
≤
0. For Case B
2, A
l
≥
0
⟺
A
P
1
2
,
A
P
2
2
≥
0 and to avoid the complexity, we will focus only on A
l
>
0. Similarly, for Case B
3 it will be more convenient to denote A
1 and A
2 by A
l, A
P
1
3 and A
P
2
3 by A
u, and A
Q
2
3 by A
v.


## 3.1. Impulsive Set

In CaseB
1, trajectory Γ
1 is tangent to curve Γ
P
c at point S
=
x
S
,
y
S with y
S
≥
c
/
q. If we represent point x
Q
2
,
y
Q
2 by x
i
m
1
,
y
i
m
1, then the domain of the impulsive set becomes as:(10)
M
1
=
x
,
y
∈
R
+
2
A
T
α
+
c
β
≤
x
≤
x
i
m
1
,
0
≤
y
≤
y
i
m
1
.It is obvious from the domain of impulsive setM
1 that in this case, no solution originating from the phase set will reach the interval y
i
m
1
,
c
/  q. In the following lemma, we find out the exact value of y
i
m
1 which depends on the corresponding horizontal coordinate.Lemma 1.
For CaseB
1, the impulsive set is defined as M
1. The maximum vertical coordinate of M
1 is y
i
m
1, where y
i
m
1
=
−
c
/
q
W
−
q
/
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c provided that A
Q
2
1
≤
0.Proof.
Let a solutionΓ
1 is tangent to curve Γ
P
c at point x
S
,
y
S, and it touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2. Then these points must satisfy the following equation(11)
λ
ω
ln
1
+
ω
x
Q
2
1
+
ω
x
S
−
r
x
Q
2
−
x
S
−
δ
ln
x
Q
2
x
S
=
c
ln
y
Q
2
y
S
−
q
y
Q
2
−
y
S
.
Solving this equation fory
Q
2, we get(12)
−
q
c
y
Q
2
e
−
q
/
c
y
Q
2
=
−
q
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c
,
whereA
Q
2
1
=
λ
/  ω
l
n
1
+
ω
x
Q
2
/
1
+
ω
x
S
−
r
x
Q
2
−
x
S
−
δ
l
n
x
Q
2
/
x
S. We can solve the above equation with the help of Lambert W function. Obviously, the above equation will give us two solutions, but only the minimum value lies on curve Γ
I
c as well as Γ
1. If we denote it by y
i
m
1, we obtain(13)
y
i
m
1
=
−
c
q
W
−
q
c
y
S
e
−
q
/
c
y
S
+
A
Q
2
1
/
c
,
which is well defined due toA
Q
2
1
≤
0.
From Figure1(b), it can be seen that for Case B
2, Γ
2 is tangent to curve Γ
I
c at point T
=
x
T
,
y
T, where y
T
≤
c
/  q. Then based on the positions of equilibria E
1, E
2 and curve Γ
I
c, we discuss the maximum impulsive set for this case as follows:(14)
M
2
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
T
,
0
≤
y
≤
y
T
.
The tangent point of the closed trajectory with curveΓ
I
c varies with the changing estimations of weighted parameters α and β. From the domain of impulsive set M
2, it is obvious that if y
T
<
c
/
q then the interval y
T
,
c
/  q cannot be used for any solution originating from the respective phase set.
Now we discuss the impulsive set for the CaseB
3. This case is more crucial than the previous cases. In this case, homoclinic trajectory exists. This homoclinic trajectory Γ
3 touches curve Γ
P
c at points x
P
1
,
y
P
1 and x
P
2
,
y
P
2, and its lower right branch touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2 (as shown in Figure 1(c)). This is actually the maximum impulsive point for Case B
3. Before finding out the exact value of vertical coordinate y
Q
2, we first provide some necessary quantities which are not only helpful for finding the maximum vertical coordinate of the impulsive set M
3, but also assume a significant role in finding the fixed point of the Poincaré map P
y
i
+. These quantities are listed as follows:(15)
A
P
1
3
=
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
x
P
1
,
(16)
A
1
=
λ
ω
ln
1
+
ω
x
Q
2
1
+
ω
x
P
1
−
r
x
Q
2
−
x
P
1
−
δ
ln
x
Q
2
x
P
1
,
(17)
A
Q
2
3
=
A
P
1
3
−
A
1
=
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
Q
2
−
r
x
1
∗
−
x
Q
2
−
δ
ln
x
1
∗
x
Q
2
.
Replacingx
P
1 by x
P
2 in equations (15) and (16) and denoting the resultant equations by A
P
2
3 and A
2 respectively, then A
Q
2
3 = A
P
2
3
−
A
2. If we denote x
Q
2
,
y
Q
2 by x
i
m
3
,
y
i
m
3, then we find the exact value of y
i
m
3 which depends on the respective horizontal coordinate.Lemma 2.
For CaseB
3, the impulsive set is defined as M
3. The maximum vertical coordinate for this is y
i
m
3, where y
i
m
3
=
−
c
/
q
W
−
e
−
1
−
A
Q
2
3
/
c provided that A
Q
2
3
≥
0.Proof.
In this case, the lower right branch of the homoclinic trajectoryΓ
3 touches curve Γ
I
c at point Q
2
=
x
Q
2
,
y
Q
2. Combining point Q
2
=
x
Q
2
,
y
Q
2 with E
1
=
x
1
∗
,
c
/  q must satisfy the following relation:(18)
c
ln
c
q
−
c
−
λ
ω
ln
1
+
ω
x
1
∗
+
r
x
1
∗
+
δ
ln
x
1
∗
=
c
ln
y
Q
2
−
q
y
Q
2
−
λ
ω
ln
1
+
ω
x
Q
2
+
r
x
Q
2
+
δ
ln
x
Q
2
,
which can be simplified as(19)
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
Q
2
−
r
x
1
∗
−
x
Q
2
−
δ
ln
x
1
∗
x
Q
2
=
c
ln
c
/
q
y
Q
2
−
q
c
q
−
y
Q
2
.
Solving the above equation fory
Q
2, we get(20)
−
q
c
y
Q
2
e
−
q
/
c
y
Q
2
=
−
e
−
1
−
A
Q
2
3
/
c
.
Following the same way as in Lemma1, applying the properties of Lambert W function, we get two solutions. From Figure 1(c) it is clear that only the minimum value lies both on curve Γ
3 and Γ
I
c. If we denote it by y
i
m
3, then we get(21)
y
i
m
3
=
−
c
q
W
−
e
−
1
−
A
Q
2
3
/
c
,
which is well defined due toA
Q
2
3
≥
0. If we represent the impulsive set by M
3, then it can be expressed as(22)
M
3
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
i
m
3
,
0
≤
y
≤
y
i
m
3
.


## 3.2. Phase Set

In this subsection, we aim to discuss the phase sets for all the existing cases expressed above. The most essential and tough task in the process of discussing phase sets is to find out the segment, which is free from the impulsive effect. To find the exact domain of phase sets, we provide the following intervals:(23)
X
D
1
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
i
m
1
,
Y
D
1
=
τ
,
y
i
m
1
+
τ
,
X
D
2
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
T
,
(24)
Y
D
2
=
τ
,
y
T
+
τ
,
X
D
3
=
A
T
1
−
p
α
+
c
β
,
1
−
p
x
i
m
3
,
and
Y
D
3
=
τ
,
y
i
m
3
+
τ
.For CaseB
1, trajectory Γ
1 is tangent to curve Γ
P
c at point S
=
x
S
,
y
S. Thus, the corresponding phase set to the impulsive set M
1 can be expressed as:(25)
N
1
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
D
1
,
y
+
∈
Y
D
1For CaseB
2, the closed trajectory is tangent to curve Γ
I
c at point T
=
x
T
,
y
T, where y
T
≤
c
/  q. We indicate the intersection point of the closed trajectory Γ
2 with line y
=
c
/
q (denoted by L
1) as E
3
=
x
3
,
y
3. If we denote x
P
2
,
y
P
2 by x
m
i
n
2
,
y
m
i
n
2 and x
P
1
,
y
P
1 by x
m
a
x
2
,
y
m
a
x
2, then the phase set corresponds to the impulsive set M
2 can be expressed as follows:(26)
N
2
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
2
,
y
+
∈
Y
P
s
2with(27)
X
P
s
2
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
2
∪
x
max
2
,
+
∞
∩
X
D
2and(28)
Y
P
s
2
=
0
,
y
min
2
∪
y
max
2
,
α
+
c
β
q
β
+
τ
∩
Y
D
2
.From the phase setN
2, it is clear that the solution initiating from the interval y
m
i
n
2
,
y
m
a
x
2 will be free from impulsive effect. In the following lemma, based on the respective horizontal coordinates, the exact values of y
m
a
x
2 and y
m
i
n
2 are given.Lemma 3.
For CaseB
2, the impulsive set is defined as M
2. In this case, any solution initiating from y
min
2
,
y
max
2 will be free from impulse effect, where(29)
y
max
2
=
−
c
q
W
−
1
,
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
a
n
d
y
min
2
=
−
c
q
W
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
2
2
/
c
provided thatA
P
1
2
,
A
P
2
2
≥
0.Proof.
Suppose that the closed trajectoryΓ
2 originates from x
P
1
,
y
P
1, and tangent to curve Γ
I
c at point x
T
,
y
T. Then, these points must satisfy the relation:(30)
λ
ω
ln
1
+
ω
x
T
1
+
ω
x
P
1
−
r
x
T
−
x
P
1
−
δ
ln
x
T
x
P
1
=
c
ln
y
T
y
P
1
−
q
y
T
−
y
P
1
.
Rearranging this equation fory
P
1, we get(31)
−
q
c
y
P
1
e
−
q
/
c
y
P
1
=
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
,
whereA
P
1
2
=
λ
/
ω
ln
1
+
ω
x
T
/
1
+
ω
x
P
1
−
r
x
T
−
x
P
1
−
δ
ln
x
T
/
x
P
1. The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by y
m
a
x
2, we get(32)
y
max
2
=
−
c
q
W
−
1
,
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
.
The value ofy
P
2, denoted by y
m
i
n
2 can also be found by using the same method as above, i.e.,(33)
y
min
2
=
−
c
q
W
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
2
2
/
c
withA
P
2
2
=
λ
/
ω
ln
1
+
ω
x
T
/
1
+
ω
x
P
2
−
r
x
T
−
x
P
2
−
δ
ln
x
T
/
x
P
2.
If weighted parameterβ
=
0, i.e., the threshold level only depends on the pest density then the closed trajectory becomes tangent at y
=
c
/
q. In this case, y
m
i
n
2 and y
m
a
x
2 become as(34)
y
max
2
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
2
/
c
,
and
y
min
2
=
−
c
q
W
−
e
−
1
−
A
P
2
/
c
.
withA
P
1
2
=
A
P
2
2
=
A
P
2.
For CaseB
3, let us denote the intersection of the homoclinic trajectory Γ
3 with line y
=
c
/
q (denoted by L
1) as E
4
=
x
4
,
y
4. Trajectory Γ
3 touches curve Γ
P
c at upper point P
1
=
x
P
1
,
y
P
1 and lower point P
2
=
x
P
2
,
y
P
2, denoted by x
m
a
x
3
,
y
m
a
x
3 and x
m
i
n
3
,
y
m
i
n
3 respectively. In the following lemma, we find the exact values of y
m
a
x
3 and y
m
i
n
3.Lemma 4.
For CaseB
3, the impulsive set is defined as M
3. In this case any solution initiating from y
min
3
,
y
max
3 will be free from impulse effect, where(35)
y
max
3
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
1
3
/
c
a
n
d
y
min
3
=
−
c
q
W
−
e
−
1
−
A
P
2
3
/
c
provided thatA
P
1
3
,
A
P
2
3
≥
0.Proof.
Suppose that the homoclinic trajectoryΓ
3 touches curve Γ
P
c at upper point P
1
=
x
P
1
,
y
P
1. Combining point P
1
=
x
P
1
,
y
P
1 with E
1
=
x
1
∗
,
c
/  q must satisfy the following relation:(36)
λ
ω
ln
1
+
ω
x
1
∗
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
x
P
1
=
c
ln
c
/
q
y
P
1
−
q
c
q
−
y
P
1
.
Arranging this equation fory
P
1, we get(37)
−
q
c
y
P
1
e
−
q
/
c
y
P
1
=
−
e
−
1
−
A
P
1
3
/
c
,
whereA
P
1
3
=
λ
/
ω
ln
1
+
ω
x
1
∗
/
1
+
ω
x
P
1
−
r
x
1
∗
−
x
P
1
−
δ
ln
x
1
∗
/
x
P
1. The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by y
m
a
x
3, we get(38)
y
max
3
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
1
3
/
c
.
The value ofy
P
2, denoted by y
m
i
n
3 can also be found by following the same way as above, i.e.,(39)
y
min
3
=
−
c
q
W
−
e
−
1
−
A
P
2
3
/
c
withA
P
2
3
=
λ
/
ω
ln
1
+
ω
x
1
∗
/
1
+
ω
x
P
2
−
r
x
1
∗
−
x
P
2
−
δ
ln
x
1
∗
/
x
P
2.
If we represent the phase set for CaseB
3 by N
3, then it can be expressed as(40)
N
3
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
3
,
y
+
∈
Y
P
s
3
with(41)
X
P
s
3
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
3
∪
x
max
3
,
+
∞
∩
X
D
3
and(42)
Y
P
s
3
=
0
,
y
min
3
∪
y
max
3
,
α
+
c
β
q
β
+
τ
∩
Y
D
3
.IfΓ
P
c  lies on the left and does not touch Γ
2 or Γ
3 then the impulsive and phase sets will be transformed into M
1 and N
1 respectively.


## 3.3. The Impulsive and Phase Sets for Model (8)

In view of the model (2) and based on the locations of curve Γ
I
c and stable center δ
/
d
,
c
/  q, the following two cases can be taken for model (8)(43)
B
1
0
A
T
α
≤
δ
d
and
B
2
0
δ
d
<
A
T
α
.In the first case, trajectoryΓ
1
0 is tangent to curve Γ
P
c at point S
0
=
x
S
0
,
y
S
0 with y
S
0
≥
c
/
q (as shown in Figure 2(a)). If we represent point x
Q
2
,
y
Q
2 by x
i
m
0
,
y
i
m
0, then the impulsive set M
1
0 can be expressed as(44)
M
1
0
=
x
,
y
∈
R
+
2
A
T
α
+
c
β
≤
x
≤
x
i
m
0
,
0
≤
y
≤
y
i
m
0
.Figure 2
Illustration diagrams of the impulsive and phase sets for model (8). (a) A
T
/  α
≤
δ
/  d, (b) δ
/  d
<
A
T
/  α. Γ
1
0 is a trajectory tangent to curve Γ
P
c at point S
0
=
x
S
0
,
y
S
0, and Γ
2
0 is a trajectory which is tangent to curve Γ
I
c at point T
0
=
x
T
0
,
y
T
0.
(a)
(b)To discuss the exact domains of the phase set for both cases, we define the intervalsX
D
0
1
=
A
T
1
−
p
/
α
+
c
β
,
1
−
p
x
i
m
0, Y
D
0
1
=
τ
,
y
i
m
0
+
τ, X
D
0
2
=
A
T
1
−
p
/
α
+
c
β
,
1
−
p
x
T
0 and Y
D
0
2
=
τ
,
y
T
0
+
τ. Then, the phase set for Case B
1
0 becomes as:(45)
N
1
0
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
D
0
1
,
y
+
∈
Y
D
0
1In the following lines the analytical value ofy
i
m
0 is given. The proof of the lemma is the same as previous section, so we omit it.Lemma 5.
For CaseB
1
0, the impulsive set is defined as M
1
0. The maximum vertical coordinate for this is y
i
m
0, where y
i
m
0
=
−
c
/
q
W
−
q
/
c
y
S
0
e
−
q
/
c
y
S
0
+
A
Q
2
/
c provided that A
Q
2
=
δ
ln
x
S
0
−
ln
x
Q
2
+
d
x
Q
2
−
x
S
0
≤
0.For CaseB
2
0, we denote the intersection point of the closed trajectory Γ
2
0 with line y
=
c
/
q (denoted by L
1) as E
1
=
x
E
1
,
y
E
1. In this case, the closed trajectory is tangent to curve Γ
I
c at point T
0
=
x
T
0
,
y
T
0, where y
T
0
≤
c
/
q. If we denote x
P
2
,
y
P
2 by x
m
i
n
0
,
y
m
i
n
0 and x
P
1
,
y
P
1 by x
m
a
x
0
,
y
m
a
x
0, then based on the positions of curves Γ
I
c and Γ
P
c, we discuss the impulsive and phase sets as follows:(46)
M
2
0
=
x
,
y
∈
R
2
A
T
α
+
c
β
≤
x
≤
x
T
0
,
0
≤
y
≤
y
T
0
,
(47)
N
2
0
=
x
+
,
y
+
∈
R
+
2
x
+
∈
X
P
s
02
,
y
+
∈
Y
P
s
02with(48)
X
P
s
02
=
A
T
1
−
p
α
+
c
β
+
τ
q
β
,
x
min
0
∪
x
max
0
,
+
∞
∩
X
D
0
2and(49)
Y
P
s
02
=
0
,
y
min
0
∪
y
max
0
,
α
+
c
β
q
β
+
τ
∩
Y
D
0
2
.From the phase set, it is clear that the solution initiating from the intervaly
m
i
n
0
,
y
m
a
x
0 will be free from impulsive effect.Lemma 6.
For CaseB
2
0, the impulsive set is defined as M
2
0. In this case any solution initiating from y
m
i
n
0
,
y
m
a
x
0 will be free from impulse effect, where(50)
y
max
0
=
−
c
q
W
−
1
,
−
q
c
y
T
0
e
−
q
/
c
y
T
0
−
A
P
1
/
c
a
n
d
y
min
0
=
−
c
q
W
−
q
c
y
T
0
e
−
q
/
c
y
T
0
−
A
P
2
/
c
provided that(51)
A
P
1
=
δ
ln
x
P
1
−
ln
x
T
0
+
d
x
T
0
−
x
P
1
≥
0
,
A
P
2
=
δ
ln
x
P
2
−
ln
x
T
0
+
d
x
T
0
−
x
P
2
≥
0
.The proof of the Lemma6 can also be shown as previous section, so we also omit it. If the weighted parameter β
=
0, i.e., the threshold level only depends on the pest density then the closed trajectory Γ
2
0 becomes tangent to curve Γ
I
c at y
=
c
/
q. In this case, y
m
i
n
0 and y
m
a
x
0 become as:(52)
y
max
0
=
−
c
q
W
−
1
,
−
e
−
1
−
A
P
/
c
,
y
min
0
=
−
c
q
W
−
e
−
1
−
A
P
/
c
.withA
P
1
=
A
P
2
=
A
P.Compared with published work for model (8), we can see that more accurate domains of the impulsive and phase sets have been provided here. From Figure 2(b), it is clear that x
E
1
<
A
T
1
−  p
/
α
+
τ
q  β
<
δ
/  d.In the upcoming discussions, for convenience, we useA
l rather than A
Q
2
1, A
P
1
2, A
P
2
2 or A
1. For Case B
1, A
Q
2
1
≤
0 is equivalent to A
l
≤
0. For Case B
2, A
l
≥
0
⟺
A
P
1
2
,
A
P
2
2
≥
0 and to avoid the complexity, we will focus only on A
l
>
0. Similarly, for Case B
3 it will be more convenient to denote A
1 and A
2 by A
l, A
P
1
3 and A
P
2
3 by A
u, and A
Q
2
3 by A
v.


## 4. Formation of Poincaré Map

Theorem 1.
The Poincaré map for the impulsive points of model (2) can be defined as follows:(53)
B
1
:
y
i
+
1
+
=
P
y
i
+
,
y
i
+
∈
Y
D
1
i
f
A
l
≤
0
,
(54)
B
2
:
y
i
+
1
+
=
P
y
i
+
,
y
i
+
∈
Y
P
s
2
i
f
A
l
>
0
,
(55)
B
3
:
y
i
+
1
+
=
P
y
i
+
,
y
i
+
∈
Y
P
s
3
i
f
A
u
≥
A
l
<
0
o
r
A
u
>
0
≥
A
l
,
where(56)
y
i
+
1
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
+
τ
=
P
y
i
+
.Proof.
Assume that a trajectory originate fromx
0
+
,
y
0
+ and repeats the pulse action k times, which can be finite or infinite. Let p
0
+
=
x
i
+
,
y
i
+
∈
Γ
P
c and p
1
=
x
i
+
1
,
y
i
+
1
∈
Γ
I
c be two points of the same trajectory. Then for these points, the following relation can easily be obtained:(57)
λ
ω
ln
1
+
ω
x
i
+
1
1
+
ω
x
i
+
−
r
x
i
+
1
−
x
i
+
−
δ
ln
x
i
+
1
x
i
+
=
c
ln
y
i
+
1
y
i
+
−
q
y
i
+
1
−
y
i
+
.
Applying the properties of LambertW function and solving the above equation for y
i
+
1, we get(58)
y
i
+
1
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
,
where(59)
A
l
=
λ
ω
ln
1
+
ω
x
i
+
1
1
+
ω
x
i
+
−
r
x
i
+
1
−
x
i
+
−
δ
ln
x
i
+
1
x
i
+
.
From (58), we get(60)
y
i
+
1
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
+
τ
=
P
y
i
+
.
Case
B
1: If A
l
≤
0, then for y
i
+
≥
0, equations (58) and (60) are well defined. Actually, if we define g
y
=
−
q
/  c
y
e
x
p
−
q
/  c  y then
(61)
g
′
y
=
q
2
c
2
exp
−
q
c
y
y
−
c
q
,
and it can easily be shown thatg
y achieves its minimum value −
e
−
1 at point y
=
c
/
q. Therefore, −
q
/
c
y
exp
−
q
/
c
y
exp
A
l
/
c
∈
−
e
−
1
,
0 for all A
l
≤
0 and y
>
0. This shows that the Poincaré map is defined by (53) for Case B
1.
Now in order to demonstrate the exact domains of the Poincaré map for CasesB
2 and B
3, the most important part is to find the section of the phase set that is free from impulsive effect, i.e., the solution originating from p
0
+
∈
Γ
P
c cannot reach to point p
1
∈
Γ
I
c.Case
B
2: From Lemma 3, it is obvious that if the starting point p
0
+
=
x
i
+
,
y
i
+ lies inside of the closed trajectory Γ
2, then trajectory cannot reach to curve Γ
I
c. This indicates that points p
0
+ and p
1 cannot lie in the same trajectory, as shown in Figure 1(b). From Lemma 3, it also follows that in this case, we have A
l
>
0 and we need p
0
+
∈
N
2.
IfA
l
>
0, then −
q
/  c
y
e
x
p
−
q
/  c  y
e
x
p
A
l
/  c
≥
−
e
−
1. From this, we get(62)
q
c
y
exp
−
q
c
y
≤
exp
−
1
−
A
l
c
.
The solution givesy
∈
0
,
y
min
2
∪
y
max
2
,
α
+
c
β
/
q
β
+
τ, and from Lemma 3 we know that(63)
y
min
2
=
−
c
q
W
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
2
2
/
c
and
y
max
2
=
−
c
q
W
−
1
,
−
q
c
y
T
e
−
q
/
c
y
T
−
A
P
1
2
/
c
,
The CaseB
3 can be attained directly from the domains of the Poincaré map and applying the proof of Lemma 4. For this case A
u
≥
0, regardless of A
l
>
0 or A
l
≤
0 (as shown in Figure 3), the Poincaré map is given by the case (55). This completes the proof.Figure 3
Fixed point of Poincaré map for CaseC
3. (a) A
l
=
0.2802, τ
=
1.6, τ
3
=
1.2055, y
∗
=
2.5330, A
T
=
12. (b) A
l
=
−
0.2281, τ
=
0.12, τ
4
=
0.1801, y
∗
=
0.6798, A
T
=
20. All other parameter values are fixed as: α
=
0.95, β
=
0.05, c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8.
(a)
(b)Difference equation (56) which explains the Poincaré map reveals the relations between the impulsive points y
i
+ and y
i
+
1
+, so the existence and stability of fixed point of equation (56) indicate the existence and stability of order-1 periodic solution of system (2). Therefore, we conclude that the properties of the Poincaré map play an essential role in exploring the impulsive semi-dynamical system.Corollary 1.
The Poincaré map for model (8) can be defined as:(64)
y
i
+
1
+
=
P
y
i
+
,
y
i
+
∈
Y
D
0
1
if
A
Q
2
≤
0
,
P
y
i
+
,
y
i
+
∈
Y
P
s
02
if
A
P
1
,
A
P
2
>
0
.Following the same way as in Theorem1, we can show that the Poincaré map for model (8) is true. For convenience, we use A
l
0 rather than A
Q
2 or A
P
1 or A
P
2.


## 5. Characterization of Periodic Solution forτ
=
0

In this section, we will focus on the boundary order-1 periodic solution for system (2). To prove this, we first provide some significant relations and lemmas in the following subsection.


### 5.1. Some Important Relations and Notations

In view of the domains of the Poincaré mapP
y
i
+ characterized in Section 4 or the signs of A
l and A
u, we modify the Cases B
1–B
3 as:(65)
C
1
A
l
≤
0
;
C
2
A
l
<
0
and
C
3
i
A
u
≥
A
l
<
0
,
i
i
A
u
≥
0
,
A
l
≤
0
.This shows that the sign ofA
l is crucial for coming analysis. So, while choosing the parameters, we should be very careful. If we change the value of weighted parameters, the sign of A
l not necessarily remains the same.The fixed point of the Poincaré map can be found directly from the analytical formula of the Poincaré map derived in Section4. To do this, let(66)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
+
τ
,i.e.,(67)
−
q
c
y
∗
−
τ
=
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
.By applying the properties of LambertW function, we get(68)
−
q
c
y
∗
−
τ
exp
−
q
c
y
∗
−
τ
=
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
,This demonstrates that there exists a unique fixed point(69)
y
∗
=
τ
1
−
exp
A
l
/
c
−
q
/
c
τor(70)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
.Whenever weighted parameterβ turned out to be zero, the closed trajectory becomes tangent at A
T
,
c
/  q with extreme vertical coordinate. In this case, if the fixed point exists then it must belong to the basic phase set τ
,
c
/  q
+  τ. We will demonstrate that under what condition the fixed point of the Poincaré map belongs to the maximum phase set τ
,
c
/  q
+  τ. We have the following two positions for A
l, i.e., (i) A
l
≤
0 and (ii) A
l
>
0. If A
l
≤
0, then it can easily be shown that y
∗
>
τ and furthermore, the inequality(71)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
c
q
+
τholds true. This shows that ifA
l
≤
0, then y
∗
∈
τ
,
c
/  q
+  τ. If A
l
>
0, then the fixed point exists provided that e
x
p
q
/  c
τ
−
A
l
/  c
>
1. This also ensures that y
∗ is positive and greater than τ. We take(72)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
c
q
+
τ
,which is equivalent to(73)
exp
q
c
τ
−
A
l
c
≥
q
c
τ
+
1
.The following inequality can easily be obtained after simple rearrangement(74)
−
q
c
c
q
+
τ
exp
−
q
c
c
q
+
τ
≥
−
exp
−
1
−
A
l
c
.Solving inequality (74) with respect to c
/  q
+
τ gives either c
/  q
+
τ
≤
y
m
i
n
2 or c
/  q
+
τ
≥
y
m
a
x
2. The first inequality is impossible due to y
m
i
n
2
≤
c
/  q. This shows that c
/  q
+
τ
≥
y
m
a
x
2, and hence y
∗
≤
c
/  q
+
τ when 0
<
A
l
<
q
τ.If weighted parameterβ
>
0, the tangency point of the closed trajectory moves to some other point having vertical coordinate less than c
/
q, and in this case, the basic impulsive set can be written as τ
,
c
/  q
+  τ or τ
,
y
T
+  τ, where y
T
<
c
/
q. This shows that y
∗
<
c
/  q
+
τ or y
∗
≤
y
T
+
τ when 0
<
A
l
<
q
τ.In the following discussion, we give some important relations related toy
i
m
1, y
i
m
3, y
m
a
x
2, y
m
i
n
3 and y
m
a
x
3.Lemma 7.
If0
<
A
l
<
q
τ and β
=
0, then y
∗ attains its minimal value y
min
∗
=
y
max
2 at τ
1
=
y
max
2
−
c
/
q. If β
>
0, then y
∗ attains its minimal value y
min
∗
=
y
max
2 at τ
1
=
y
max
2
−
y
T where y
T
<
c
/
q.Proof.
Taking the derivative ofy
∗ with respect to τ, we get(75)
d
y
∗
d
τ
=
exp
q
/
c
τ
−
A
l
/
c
c
exp
q
/
c
τ
−
A
l
/
c
−
c
−
q
τ
c
exp
q
/
c
τ
−
A
l
/
c
−
1
2
.
Letd
y
∗
/
d
τ
=
0, then the above equality becomes(76)
c
exp
q
c
τ
−
A
l
c
−
c
−
q
τ
=
0
.
From (76), we get(77)
−
1
−
q
τ
c
exp
−
1
−
q
τ
c
=
−
exp
−
1
−
A
l
c
.
We can solve the above equation with the help of LambertW function. It will give us two solutions; however, only the larger solution is positive. The necessary condition for the positivity is A
l
<
q
τ. If we denote the positive root by τ
1, we get(78)
τ
1
=
−
c
q
−
c
q
W
−
1
,
−
e
−
1
−
A
l
/
c
=
y
max
2
−
c
q
.
It is obvious thaty
∗ attains its minimal value at τ
1, and as τ
→
A
l
/
q, y
∗
→
∞. By simple calculations, it can be shown that exp
q
/
c
τ
1
−
A
l
/
c
=
−
W
−
1
,
−
e
−
1
−
A
l
/
c. Finally, by putting this in (70) we get(79)
y
min
∗
=
τ
1
W
−
1
,
−
e
−
1
−
A
l
/
c
1
+
W
−
1
,
−
e
−
1
−
A
l
/
c
=
−
c
q
W
−
1
,
−
e
−
1
−
A
l
/
c
=
y
max
2
.
Forβ
=
0, if we take c
/  q
+
τ
−
y
∗
=
0 then substituting the value of y
∗ and using the equality (76) it can easily be shown that c
/
q
+
τ and y
∗ intersect each other at τ
=
τ
1. For Case C
2, if β
>
0 then it can be seen that y
∗ attains its minimum value at τ
=
τ
1, where τ
1
=
y
m
a
x
2
−
y
T.Lemma 8.
IfA
l
≤
0, then for the fixed point y
∗ the following inequality is satisfied(80)
y
∗
<
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.Proof.
(81)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
τ
exp
q
/
c
τ
exp
q
/
c
τ
−
1
<
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.
After simple rearrangement, we get(82)
c
+
c
2
+
q
2
τ
2
−
q
τ
exp
q
c
τ
−
c
−
q
τ
−
c
2
+
q
2
τ
2
>
0
.
Leth
=
q
/  c
τ, then the above inequality can be rewritten as(83)
e
h
>
1
+
h
+
1
+
h
2
1
−
h
+
1
+
h
2
=
h
+
1
+
h
2
.
To complete the proof, it is enough to show thate
h
−
h
−
1
+
h
2
>
0.
Consider the functionF
h
=
e
h
−
h
−
1
+
h
2, and we have(84)
F
h
=
e
h
−
h
−
1
+
h
2
>
1
+
h
+
h
2
2
−
h
+
1
+
h
2
=
1
+
h
2
2
−
1
+
h
2
>
0
.
Hence, ifA
l
≤
0 then the inequality (80) is true.Lemma 9.
Positive solutions of the following equations exist(85)
i
y
∗
−
y
max
3
=
0
;
i
i
y
∗
−
y
min
3
=
0
.Proof.
iLet y
∗
−
y
m
a
x
3
=
0, then substituting the value of y
∗ and after simple rearrangement we get(86)
q
c
τ
−
y
max
3
exp
q
c
τ
−
y
max
3
=
−
q
c
y
max
3
exp
−
q
c
y
max
3
+
A
l
c
.
By putting the value ofy
m
a
x
3, we get(87)
−
q
c
y
max
3
exp
−
q
c
y
max
3
+
A
l
c
=
−
e
−
1
e
−
A
u
−
A
l
/
c
=
−
e
−
1
−
A
v
/
c
i.e.,(88)
q
c
τ
−
y
max
3
exp
q
c
τ
−
y
max
3
=
−
e
−
1
−
A
v
/
c
.
The above equation can be solved with the help of LambertW function and yields two solutions(89)
τ
2
=
y
max
3
+
c
q
W
−
1
,
−
e
−
1
−
A
v
/
c
and
τ
3
=
y
max
3
+
c
q
W
−
e
−
1
−
A
v
/
c
.
SinceA
v
≥
0, which implies A
u
≥
A
l
>
0 or A
u
>
0
≥
A
l. This shows that the above solutions are well defined. If A
l
≤
0, then the small root τ
2 disappeared and subsequently in this case, we only get the root τ
3 which can also be written as τ
3
=
y
m
a
x
3
−
y
i
m
3.
Following the same way, it can easily be shown that ifA
l
≤
0 then the unique positive solution exists for the equation i  i is τ
4
=
y
min
3
+
c
/
q
W
−
e
−
1
−
A
v
/
c. This solution can also be expressed as τ
4
=
y
min
3
−
y
i
m
3.


### 5.2. Existence and Stability of Boundary Order-1 Periodic Solution

This subsection focuses on the existence and stability of fixed point of Poincaré mapP
y
i
+ for special case τ
=
0. The analytical formula for Poincaré map P
y
i
+ is already formed in Section 4. Let y
∗ be the fixed point of Poincaré map P
y
i
+, then we have y
∗
=
P
y
∗, i.e.,(90)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
.To determine the fixed point, we take the following two cases forA
l, (1) A
l
=
0 and (2) A
l
≠
0. For the first case, equation (90) reduced into(91)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
,by using the properties of LambertW function, the equation (91) can be expressed as −
q
/  c
y
∗
e
−
q
/
c
y
∗
=
−
q
/  c
y
∗
e
−
q
/
c
y
∗. This shows that y
∗ is the fixed point of Poincaré map P
y
i
+.For the second case, we get−
q
/  c
y
∗
e
−
q
/
c
y
∗
=
−
q
/  c
y
∗
e
−
q
/
c
y
∗
+
A
l
/
c. This equality is satisfied if and only if y
∗
=
0. Hence, the only fixed point exists for the Poincaré map is y
∗
=
0.In the above discussion, we have demonstrated that the fixed point of Poincaré map exists. We will now inspect that under what condition the boundary order-1 periodic solution is globally stable. To prove it, we first recall a lemma from [48, 49].Lemma 10.
TheT-periodic solution x
,  y
=
ζ
t
,
ξ
t of system(92)
d
x
d
t
=
L
x
,
y
,
d
y
d
t
=
M
x
,
y
,
i
f
φ
x
,
y
≠
0
,
x
+
=
x
+
α
1
x
,
y
,
y
+
=
y
+
β
1
x
,
y
,
i
f
φ
x
,
y
=
0
is orbitally asymptotically stable if the Floquet multiplierμ
2 satisfies the condition |
μ
2
|
<
1, where(93)
μ
2
=
∏
j
=
1
k
Δ
j
exp
∫
0
T
∂
L
∂
x
ζ
t
,
ξ
t
+
∂
M
∂
y
ζ
t
,
ξ
t
d
t
with(94)
Δ
j
=
L
+
∂
β
1
/
∂
y
∂
φ
/
∂
x
−
∂
β
1
/
∂
x
∂
φ
/
∂
y
+
∂
φ
/
∂
x
+
M
+
∂
α
1
/
∂
x
∂
φ
/
∂
y
−
∂
α
1
/
∂
y
∂
φ
/
∂
x
+
∂
φ
/
∂
y
L
∂
φ
/
∂
x
+
M
∂
φ
/
∂
y
,
andφ is continuously differentiable with respect to x
,
y. L
,
M
,
∂
α
1
/
∂
x
,
∂
α
1
/
∂
y
,
∂
β
1
/
∂
x
,
∂
β
1
/
∂
y
,
∂
φ
/
∂
x, and ∂
φ
/
∂
y are calculated at point ζ
t
j
,
ξ
t
j, L
+
=
L
ζ
t
j
+
,
ξ
t
j
+ and M
+
=
M
ζ
t
j
+
,
ξ
t
j
+, and t
j (j
,
k
∈
N, N is the set of nonnegative integers) is the time of the j-th jump.Theorem 2.
Letτ
=
0 and A
l
=
λ
/
ω
ln
1
+
ω
A
T
/
α
+
c
β
/
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
A
T
/
α
+
c
β
p
+
δ
ln
1
−
p
≠
0. Then boundary order-1 periodic solution x
T
t
,  0 of system (2) is globally asymptotically stable for Case C
1, and locally asymptotically stable for Case C
3
i  i. For Cases C
2 and C
3
i, the boundary order-1 periodic solution is unstable.Proof.
In the following lines, we first prove that the boundary order-1 periodic solutionx
T
t
,  0 exists for system (2) when y
t
=
0 if and only if τ
=
0. For y
t
=
0, the system (2) transformed into the following subsystem:(95)
d
x
t
d
t
=
c
x
t
,
x
t
<
A
T
α
+
c
β
,
x
t
+
=
1
−
p
x
t
,
x
t
=
A
T
α
+
c
β
.
Combining the first equation of subsystem (95) with initial condition x
0
+
=
1
−  p
A
T
/
α
+
c  β gives us the solution(96)
x
t
=
1
−
p
A
T
α
+
c
β
exp
c
t
.
TakingA
T
/
α
+
c  β
=
1
−  p
A
T
/
α
+
c  β
e
x
p
c  T and evaluating it for T, we get T
=
−
1
/
c
l
n
1
−  p. This confirms that T-periodic boundary order-1 solution exists for system (2) as follows:(97)
x
T
t
,
0
=
1
−
p
A
T
α
+
c
β
exp
c
t
,
0
.
Next, we present the global attractivity of the boundary order-1 periodic solutionx
T
t
,  0 for Case C
1. Let p
0
+
=
A
T
1
−
p
/
α
+
c
β
,
y
i
+
∈
L
3 and p
1
=
A
T
/
α
+
c
β
,
y
i
+
1
∈
L
2 be points of the same trajectory, then for these points the following relation is satisfied(98)
A
l
=
λ
ω
ln
1
+
ω
A
T
/
α
+
c
β
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
A
T
α
+
c
β
p
+
δ
ln
1
−
p
=
c
ln
y
i
+
1
y
i
+
−
q
y
i
+
1
−
y
i
+
.
SinceA
l
≠
0, so y
i
+
1
≠
y
i
+. Let us define a function f
y
=
c
l
n
y
−
q
y, then f
′
y
=
c
/  y
−
q. This shows that f
y is monotonically increasing for all those values y of the domain such that y
<
c
/
q.
IfA
l
<
0, then c
l
n
y
i
+
1
/
y
i
+
−
q
y
i
+
1
−
y
i
+
<
0. Since τ
=
0, so the inequality can be rewritten as c
l
n
y
i
+
1
/
y
i
−
q
y
i
+
1
−
y
i
<
0. From this, it is clear that y
i
+
1
<
y
i. Hence, if A
l
≤
0 then the impulsive sequence y
k
+
k
=
0
∞ is monotonically decreasing and satisfies l
i
m
k
→
∞
y
k
+
=
y
∗. This confirms that boundary order-1 periodic solution for Case C
1 is globally attractive. Following the same way, it can easily be shown that if A
l
>
0 then y
i
+
1
>
y
i. Consequently, for Cases C
2 and C
3
i the sequence y
k
+ will be free from impulsive effect after the finite time pulse actions, as shown in Figure 4(a).
Now, we demonstrate that boundary order-1 periodic solution is asymptotically stable. To do this, we employ Lemma10 and denote
Method 1:(99)
L
x
,
y
=
c
−
q
y
x
,
M
x
,
y
=
λ
x
1
+
ω
x
−
r
x
−
δ
y
,
α
1
x
,
y
=
−
p
x
,
β
1
x
,
y
=
τ
,
φ
x
,
y
=
α
+
c
β
x
−
q
β
x
y
−
A
T
,
x
T
T
,
y
T
T
=
A
T
α
+
c
β
,
0
,
x
T
T
+
,
y
T
T
+
=
1
−
p
A
T
α
+
c
β
,
0
.
From above, we can easily calculate:(100)
∂
L
∂
x
=
c
−
q
y
,
∂
M
∂
y
=
λ
x
1
+
ω
x
−
r
x
−
δ
,
∂
α
1
∂
x
=
−
p
,
∂
φ
∂
x
=
α
+
c
β
−
q
β
y
,
∂
φ
∂
y
=
−
q
β
x
,
∂
α
1
∂
y
=
∂
β
1
∂
x
=
∂
β
1
∂
y
=
0
,
and(101)
Δ
1
=
L
+
∂
β
1
/
∂
y
∂
φ
/
∂
x
−
∂
β
1
/
∂
x
∂
φ
/
∂
y
+
∂
φ
/
∂
x
+
M
+
∂
α
1
/
∂
x
∂
φ
/
∂
y
−
∂
α
1
/
∂
y
∂
φ
/
∂
x
+
∂
φ
/
∂
y
L
∂
φ
/
∂
x
+
M
∂
φ
/
∂
y
=
L
+
x
T
T
+
,
y
T
T
+
α
+
c
β
−
q
β
y
+
M
+
x
T
T
+
,
y
T
T
+
p
q
β
x
−
q
β
x
L
x
T
T
,
y
T
T
α
+
c
β
−
q
β
y
−
M
x
T
T
,
y
T
T
q
β
x
=
1
−
p
.
In addition, we also have(102)
exp
∫
0
T
∂
L
∂
x
x
T
t
,
y
T
t
+
∂
M
∂
y
x
T
t
,
y
T
t
d
t
=
exp
∫
0
T
c
+
λ
1
−
p
A
T
/
α
+
c
β
exp
c
t
1
+
ω
1
−
p
A
T
/
α
+
c
β
exp
c
t
−
r
1
−
p
A
T
α
+
c
β
exp
c
t
−
δ
d
t
=
exp
ln
1
1
−
p
+
λ
c
ω
ln
1
+
ω
A
T
/
α
+
c
β
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
c
A
T
α
+
c
β
p
+
δ
c
ln
1
−
p
=
exp
ln
1
1
−
p
+
A
l
c
.
The Floquet multiplierμ
2 can be found as:(103)
μ
2
=
Δ
1
exp
∫
0
T
∂
L
∂
x
x
T
t
,
y
T
t
+
∂
M
∂
y
x
T
t
,
y
T
t
d
t
=
1
−
p
exp
ln
1
1
−
p
+
A
l
c
=
exp
A
l
c
.
IfA
l
<
0 and τ
=
0, then from last equation we can see that |
μ
2
|
<
1. This shows that the boundary order-1 periodic solution x
T
t
,  0 of system (2) is orbitally asymptotically stable for Case C
1. From the domain of phase set, it is also obvious that the boundary order-1 periodic solution is locally asymptotically stability for Case C
3
i  i. For Cases C
2 and C
3
i, the sequence y
k
+ of impulsive points is strictly increasing, and it will be free from impulsive effect after a finite number of pulse actions.
In order to inspect the results shown in Theorem2 and the effects of A
l, we fixed all the parametric values as those appeared in Figure 4. The numerical calculation in Figures 4(a)–4(c) verifies that if A
l
≥
0, then the boundary order-1 periodic solution is unstable while Figures 4(d)–4(f) confirms that if A
l
<
0, then it is stable.
Method 2: The local stability of the boundary order-1 periodic solution can also be obtained from the definition of Poincaré map obtained in (56). Let τ
=
0, then we have(104)
P
y
i
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
.
Taking the derivative of equation (104), we get(105)
d
P
y
i
+
d
y
i
+
y
i
+
=
y
∗
=
d
d
y
i
+
y
i
+
=
y
∗
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
=
−
c
/
q
W
−
q
/
c
y
∗
exp
−
q
/
c
y
∗
+
A
l
/
c
1
+
W
−
q
/
c
y
∗
exp
−
q
/
c
y
∗
+
A
l
/
c
1
y
∗
−
q
c
=
h
y
∗
.
The boundary order-1 periodic solution is stable if and only if the absolute value ofh
y
∗ is less than one. By taking limit of h
y
∗, we get(106)
lim
y
∗
→
0
h
y
∗
=
e
A
l
/
c
.
This limit shows that ifA
l
<
0, then h
y
∗
<
1 as y
∗
→
0, thus the boundary order-1 periodic solution is asymptotically stable. Hence, from all the above outcomes, it can be concluded that boundary order-1 periodic solution x
T
t
,
0 is globally asymptotically stable. This completes the proof.Figure 4
(a–c) Unstable boundary order-1 periodic solution withL
2
=
A
T
/
α
+
c  β
=
3, A
T
=
3.24 and A
l
=
0.02643; (d–f) stable boundary order-1 periodic solution with L
2
=
2, A
T
=
2.16 and A
l
=
−
0.106676. All other parameter values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, α
=
0.9, β
=
0.1, p
=
0.8, τ
=
0.
(a)
(b)
(c)
(d)
(e)
(f)


## 5.1. Some Important Relations and Notations

In view of the domains of the Poincaré mapP
y
i
+ characterized in Section 4 or the signs of A
l and A
u, we modify the Cases B
1–B
3 as:(65)
C
1
A
l
≤
0
;
C
2
A
l
<
0
and
C
3
i
A
u
≥
A
l
<
0
,
i
i
A
u
≥
0
,
A
l
≤
0
.This shows that the sign ofA
l is crucial for coming analysis. So, while choosing the parameters, we should be very careful. If we change the value of weighted parameters, the sign of A
l not necessarily remains the same.The fixed point of the Poincaré map can be found directly from the analytical formula of the Poincaré map derived in Section4. To do this, let(66)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
+
τ
,i.e.,(67)
−
q
c
y
∗
−
τ
=
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
.By applying the properties of LambertW function, we get(68)
−
q
c
y
∗
−
τ
exp
−
q
c
y
∗
−
τ
=
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
,This demonstrates that there exists a unique fixed point(69)
y
∗
=
τ
1
−
exp
A
l
/
c
−
q
/
c
τor(70)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
.Whenever weighted parameterβ turned out to be zero, the closed trajectory becomes tangent at A
T
,
c
/  q with extreme vertical coordinate. In this case, if the fixed point exists then it must belong to the basic phase set τ
,
c
/  q
+  τ. We will demonstrate that under what condition the fixed point of the Poincaré map belongs to the maximum phase set τ
,
c
/  q
+  τ. We have the following two positions for A
l, i.e., (i) A
l
≤
0 and (ii) A
l
>
0. If A
l
≤
0, then it can easily be shown that y
∗
>
τ and furthermore, the inequality(71)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
c
q
+
τholds true. This shows that ifA
l
≤
0, then y
∗
∈
τ
,
c
/  q
+  τ. If A
l
>
0, then the fixed point exists provided that e
x
p
q
/  c
τ
−
A
l
/  c
>
1. This also ensures that y
∗ is positive and greater than τ. We take(72)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
c
q
+
τ
,which is equivalent to(73)
exp
q
c
τ
−
A
l
c
≥
q
c
τ
+
1
.The following inequality can easily be obtained after simple rearrangement(74)
−
q
c
c
q
+
τ
exp
−
q
c
c
q
+
τ
≥
−
exp
−
1
−
A
l
c
.Solving inequality (74) with respect to c
/  q
+
τ gives either c
/  q
+
τ
≤
y
m
i
n
2 or c
/  q
+
τ
≥
y
m
a
x
2. The first inequality is impossible due to y
m
i
n
2
≤
c
/  q. This shows that c
/  q
+
τ
≥
y
m
a
x
2, and hence y
∗
≤
c
/  q
+
τ when 0
<
A
l
<
q
τ.If weighted parameterβ
>
0, the tangency point of the closed trajectory moves to some other point having vertical coordinate less than c
/
q, and in this case, the basic impulsive set can be written as τ
,
c
/  q
+  τ or τ
,
y
T
+  τ, where y
T
<
c
/
q. This shows that y
∗
<
c
/  q
+
τ or y
∗
≤
y
T
+
τ when 0
<
A
l
<
q
τ.In the following discussion, we give some important relations related toy
i
m
1, y
i
m
3, y
m
a
x
2, y
m
i
n
3 and y
m
a
x
3.Lemma 7.
If0
<
A
l
<
q
τ and β
=
0, then y
∗ attains its minimal value y
min
∗
=
y
max
2 at τ
1
=
y
max
2
−
c
/
q. If β
>
0, then y
∗ attains its minimal value y
min
∗
=
y
max
2 at τ
1
=
y
max
2
−
y
T where y
T
<
c
/
q.Proof.
Taking the derivative ofy
∗ with respect to τ, we get(75)
d
y
∗
d
τ
=
exp
q
/
c
τ
−
A
l
/
c
c
exp
q
/
c
τ
−
A
l
/
c
−
c
−
q
τ
c
exp
q
/
c
τ
−
A
l
/
c
−
1
2
.
Letd
y
∗
/
d
τ
=
0, then the above equality becomes(76)
c
exp
q
c
τ
−
A
l
c
−
c
−
q
τ
=
0
.
From (76), we get(77)
−
1
−
q
τ
c
exp
−
1
−
q
τ
c
=
−
exp
−
1
−
A
l
c
.
We can solve the above equation with the help of LambertW function. It will give us two solutions; however, only the larger solution is positive. The necessary condition for the positivity is A
l
<
q
τ. If we denote the positive root by τ
1, we get(78)
τ
1
=
−
c
q
−
c
q
W
−
1
,
−
e
−
1
−
A
l
/
c
=
y
max
2
−
c
q
.
It is obvious thaty
∗ attains its minimal value at τ
1, and as τ
→
A
l
/
q, y
∗
→
∞. By simple calculations, it can be shown that exp
q
/
c
τ
1
−
A
l
/
c
=
−
W
−
1
,
−
e
−
1
−
A
l
/
c. Finally, by putting this in (70) we get(79)
y
min
∗
=
τ
1
W
−
1
,
−
e
−
1
−
A
l
/
c
1
+
W
−
1
,
−
e
−
1
−
A
l
/
c
=
−
c
q
W
−
1
,
−
e
−
1
−
A
l
/
c
=
y
max
2
.
Forβ
=
0, if we take c
/  q
+
τ
−
y
∗
=
0 then substituting the value of y
∗ and using the equality (76) it can easily be shown that c
/
q
+
τ and y
∗ intersect each other at τ
=
τ
1. For Case C
2, if β
>
0 then it can be seen that y
∗ attains its minimum value at τ
=
τ
1, where τ
1
=
y
m
a
x
2
−
y
T.Lemma 8.
IfA
l
≤
0, then for the fixed point y
∗ the following inequality is satisfied(80)
y
∗
<
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.Proof.
(81)
y
∗
=
τ
exp
q
/
c
τ
−
A
l
/
c
exp
q
/
c
τ
−
A
l
/
c
−
1
≤
τ
exp
q
/
c
τ
exp
q
/
c
τ
−
1
<
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.
After simple rearrangement, we get(82)
c
+
c
2
+
q
2
τ
2
−
q
τ
exp
q
c
τ
−
c
−
q
τ
−
c
2
+
q
2
τ
2
>
0
.
Leth
=
q
/  c
τ, then the above inequality can be rewritten as(83)
e
h
>
1
+
h
+
1
+
h
2
1
−
h
+
1
+
h
2
=
h
+
1
+
h
2
.
To complete the proof, it is enough to show thate
h
−
h
−
1
+
h
2
>
0.
Consider the functionF
h
=
e
h
−
h
−
1
+
h
2, and we have(84)
F
h
=
e
h
−
h
−
1
+
h
2
>
1
+
h
+
h
2
2
−
h
+
1
+
h
2
=
1
+
h
2
2
−
1
+
h
2
>
0
.
Hence, ifA
l
≤
0 then the inequality (80) is true.Lemma 9.
Positive solutions of the following equations exist(85)
i
y
∗
−
y
max
3
=
0
;
i
i
y
∗
−
y
min
3
=
0
.Proof.
iLet y
∗
−
y
m
a
x
3
=
0, then substituting the value of y
∗ and after simple rearrangement we get(86)
q
c
τ
−
y
max
3
exp
q
c
τ
−
y
max
3
=
−
q
c
y
max
3
exp
−
q
c
y
max
3
+
A
l
c
.
By putting the value ofy
m
a
x
3, we get(87)
−
q
c
y
max
3
exp
−
q
c
y
max
3
+
A
l
c
=
−
e
−
1
e
−
A
u
−
A
l
/
c
=
−
e
−
1
−
A
v
/
c
i.e.,(88)
q
c
τ
−
y
max
3
exp
q
c
τ
−
y
max
3
=
−
e
−
1
−
A
v
/
c
.
The above equation can be solved with the help of LambertW function and yields two solutions(89)
τ
2
=
y
max
3
+
c
q
W
−
1
,
−
e
−
1
−
A
v
/
c
and
τ
3
=
y
max
3
+
c
q
W
−
e
−
1
−
A
v
/
c
.
SinceA
v
≥
0, which implies A
u
≥
A
l
>
0 or A
u
>
0
≥
A
l. This shows that the above solutions are well defined. If A
l
≤
0, then the small root τ
2 disappeared and subsequently in this case, we only get the root τ
3 which can also be written as τ
3
=
y
m
a
x
3
−
y
i
m
3.
Following the same way, it can easily be shown that ifA
l
≤
0 then the unique positive solution exists for the equation i  i is τ
4
=
y
min
3
+
c
/
q
W
−
e
−
1
−
A
v
/
c. This solution can also be expressed as τ
4
=
y
min
3
−
y
i
m
3.


## 5.2. Existence and Stability of Boundary Order-1 Periodic Solution

This subsection focuses on the existence and stability of fixed point of Poincaré mapP
y
i
+ for special case τ
=
0. The analytical formula for Poincaré map P
y
i
+ is already formed in Section 4. Let y
∗ be the fixed point of Poincaré map P
y
i
+, then we have y
∗
=
P
y
∗, i.e.,(90)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
+
A
l
c
.To determine the fixed point, we take the following two cases forA
l, (1) A
l
=
0 and (2) A
l
≠
0. For the first case, equation (90) reduced into(91)
y
∗
=
−
c
q
W
−
q
c
y
∗
exp
−
q
c
y
∗
,by using the properties of LambertW function, the equation (91) can be expressed as −
q
/  c
y
∗
e
−
q
/
c
y
∗
=
−
q
/  c
y
∗
e
−
q
/
c
y
∗. This shows that y
∗ is the fixed point of Poincaré map P
y
i
+.For the second case, we get−
q
/  c
y
∗
e
−
q
/
c
y
∗
=
−
q
/  c
y
∗
e
−
q
/
c
y
∗
+
A
l
/
c. This equality is satisfied if and only if y
∗
=
0. Hence, the only fixed point exists for the Poincaré map is y
∗
=
0.In the above discussion, we have demonstrated that the fixed point of Poincaré map exists. We will now inspect that under what condition the boundary order-1 periodic solution is globally stable. To prove it, we first recall a lemma from [48, 49].Lemma 10.
TheT-periodic solution x
,  y
=
ζ
t
,
ξ
t of system(92)
d
x
d
t
=
L
x
,
y
,
d
y
d
t
=
M
x
,
y
,
i
f
φ
x
,
y
≠
0
,
x
+
=
x
+
α
1
x
,
y
,
y
+
=
y
+
β
1
x
,
y
,
i
f
φ
x
,
y
=
0
is orbitally asymptotically stable if the Floquet multiplierμ
2 satisfies the condition |
μ
2
|
<
1, where(93)
μ
2
=
∏
j
=
1
k
Δ
j
exp
∫
0
T
∂
L
∂
x
ζ
t
,
ξ
t
+
∂
M
∂
y
ζ
t
,
ξ
t
d
t
with(94)
Δ
j
=
L
+
∂
β
1
/
∂
y
∂
φ
/
∂
x
−
∂
β
1
/
∂
x
∂
φ
/
∂
y
+
∂
φ
/
∂
x
+
M
+
∂
α
1
/
∂
x
∂
φ
/
∂
y
−
∂
α
1
/
∂
y
∂
φ
/
∂
x
+
∂
φ
/
∂
y
L
∂
φ
/
∂
x
+
M
∂
φ
/
∂
y
,
andφ is continuously differentiable with respect to x
,
y. L
,
M
,
∂
α
1
/
∂
x
,
∂
α
1
/
∂
y
,
∂
β
1
/
∂
x
,
∂
β
1
/
∂
y
,
∂
φ
/
∂
x, and ∂
φ
/
∂
y are calculated at point ζ
t
j
,
ξ
t
j, L
+
=
L
ζ
t
j
+
,
ξ
t
j
+ and M
+
=
M
ζ
t
j
+
,
ξ
t
j
+, and t
j (j
,
k
∈
N, N is the set of nonnegative integers) is the time of the j-th jump.Theorem 2.
Letτ
=
0 and A
l
=
λ
/
ω
ln
1
+
ω
A
T
/
α
+
c
β
/
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
A
T
/
α
+
c
β
p
+
δ
ln
1
−
p
≠
0. Then boundary order-1 periodic solution x
T
t
,  0 of system (2) is globally asymptotically stable for Case C
1, and locally asymptotically stable for Case C
3
i  i. For Cases C
2 and C
3
i, the boundary order-1 periodic solution is unstable.Proof.
In the following lines, we first prove that the boundary order-1 periodic solutionx
T
t
,  0 exists for system (2) when y
t
=
0 if and only if τ
=
0. For y
t
=
0, the system (2) transformed into the following subsystem:(95)
d
x
t
d
t
=
c
x
t
,
x
t
<
A
T
α
+
c
β
,
x
t
+
=
1
−
p
x
t
,
x
t
=
A
T
α
+
c
β
.
Combining the first equation of subsystem (95) with initial condition x
0
+
=
1
−  p
A
T
/
α
+
c  β gives us the solution(96)
x
t
=
1
−
p
A
T
α
+
c
β
exp
c
t
.
TakingA
T
/
α
+
c  β
=
1
−  p
A
T
/
α
+
c  β
e
x
p
c  T and evaluating it for T, we get T
=
−
1
/
c
l
n
1
−  p. This confirms that T-periodic boundary order-1 solution exists for system (2) as follows:(97)
x
T
t
,
0
=
1
−
p
A
T
α
+
c
β
exp
c
t
,
0
.
Next, we present the global attractivity of the boundary order-1 periodic solutionx
T
t
,  0 for Case C
1. Let p
0
+
=
A
T
1
−
p
/
α
+
c
β
,
y
i
+
∈
L
3 and p
1
=
A
T
/
α
+
c
β
,
y
i
+
1
∈
L
2 be points of the same trajectory, then for these points the following relation is satisfied(98)
A
l
=
λ
ω
ln
1
+
ω
A
T
/
α
+
c
β
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
A
T
α
+
c
β
p
+
δ
ln
1
−
p
=
c
ln
y
i
+
1
y
i
+
−
q
y
i
+
1
−
y
i
+
.
SinceA
l
≠
0, so y
i
+
1
≠
y
i
+. Let us define a function f
y
=
c
l
n
y
−
q
y, then f
′
y
=
c
/  y
−
q. This shows that f
y is monotonically increasing for all those values y of the domain such that y
<
c
/
q.
IfA
l
<
0, then c
l
n
y
i
+
1
/
y
i
+
−
q
y
i
+
1
−
y
i
+
<
0. Since τ
=
0, so the inequality can be rewritten as c
l
n
y
i
+
1
/
y
i
−
q
y
i
+
1
−
y
i
<
0. From this, it is clear that y
i
+
1
<
y
i. Hence, if A
l
≤
0 then the impulsive sequence y
k
+
k
=
0
∞ is monotonically decreasing and satisfies l
i
m
k
→
∞
y
k
+
=
y
∗. This confirms that boundary order-1 periodic solution for Case C
1 is globally attractive. Following the same way, it can easily be shown that if A
l
>
0 then y
i
+
1
>
y
i. Consequently, for Cases C
2 and C
3
i the sequence y
k
+ will be free from impulsive effect after the finite time pulse actions, as shown in Figure 4(a).
Now, we demonstrate that boundary order-1 periodic solution is asymptotically stable. To do this, we employ Lemma10 and denote
Method 1:(99)
L
x
,
y
=
c
−
q
y
x
,
M
x
,
y
=
λ
x
1
+
ω
x
−
r
x
−
δ
y
,
α
1
x
,
y
=
−
p
x
,
β
1
x
,
y
=
τ
,
φ
x
,
y
=
α
+
c
β
x
−
q
β
x
y
−
A
T
,
x
T
T
,
y
T
T
=
A
T
α
+
c
β
,
0
,
x
T
T
+
,
y
T
T
+
=
1
−
p
A
T
α
+
c
β
,
0
.
From above, we can easily calculate:(100)
∂
L
∂
x
=
c
−
q
y
,
∂
M
∂
y
=
λ
x
1
+
ω
x
−
r
x
−
δ
,
∂
α
1
∂
x
=
−
p
,
∂
φ
∂
x
=
α
+
c
β
−
q
β
y
,
∂
φ
∂
y
=
−
q
β
x
,
∂
α
1
∂
y
=
∂
β
1
∂
x
=
∂
β
1
∂
y
=
0
,
and(101)
Δ
1
=
L
+
∂
β
1
/
∂
y
∂
φ
/
∂
x
−
∂
β
1
/
∂
x
∂
φ
/
∂
y
+
∂
φ
/
∂
x
+
M
+
∂
α
1
/
∂
x
∂
φ
/
∂
y
−
∂
α
1
/
∂
y
∂
φ
/
∂
x
+
∂
φ
/
∂
y
L
∂
φ
/
∂
x
+
M
∂
φ
/
∂
y
=
L
+
x
T
T
+
,
y
T
T
+
α
+
c
β
−
q
β
y
+
M
+
x
T
T
+
,
y
T
T
+
p
q
β
x
−
q
β
x
L
x
T
T
,
y
T
T
α
+
c
β
−
q
β
y
−
M
x
T
T
,
y
T
T
q
β
x
=
1
−
p
.
In addition, we also have(102)
exp
∫
0
T
∂
L
∂
x
x
T
t
,
y
T
t
+
∂
M
∂
y
x
T
t
,
y
T
t
d
t
=
exp
∫
0
T
c
+
λ
1
−
p
A
T
/
α
+
c
β
exp
c
t
1
+
ω
1
−
p
A
T
/
α
+
c
β
exp
c
t
−
r
1
−
p
A
T
α
+
c
β
exp
c
t
−
δ
d
t
=
exp
ln
1
1
−
p
+
λ
c
ω
ln
1
+
ω
A
T
/
α
+
c
β
1
+
ω
A
T
/
α
+
c
β
1
−
p
−
r
c
A
T
α
+
c
β
p
+
δ
c
ln
1
−
p
=
exp
ln
1
1
−
p
+
A
l
c
.
The Floquet multiplierμ
2 can be found as:(103)
μ
2
=
Δ
1
exp
∫
0
T
∂
L
∂
x
x
T
t
,
y
T
t
+
∂
M
∂
y
x
T
t
,
y
T
t
d
t
=
1
−
p
exp
ln
1
1
−
p
+
A
l
c
=
exp
A
l
c
.
IfA
l
<
0 and τ
=
0, then from last equation we can see that |
μ
2
|
<
1. This shows that the boundary order-1 periodic solution x
T
t
,  0 of system (2) is orbitally asymptotically stable for Case C
1. From the domain of phase set, it is also obvious that the boundary order-1 periodic solution is locally asymptotically stability for Case C
3
i  i. For Cases C
2 and C
3
i, the sequence y
k
+ of impulsive points is strictly increasing, and it will be free from impulsive effect after a finite number of pulse actions.
In order to inspect the results shown in Theorem2 and the effects of A
l, we fixed all the parametric values as those appeared in Figure 4. The numerical calculation in Figures 4(a)–4(c) verifies that if A
l
≥
0, then the boundary order-1 periodic solution is unstable while Figures 4(d)–4(f) confirms that if A
l
<
0, then it is stable.
Method 2: The local stability of the boundary order-1 periodic solution can also be obtained from the definition of Poincaré map obtained in (56). Let τ
=
0, then we have(104)
P
y
i
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
.
Taking the derivative of equation (104), we get(105)
d
P
y
i
+
d
y
i
+
y
i
+
=
y
∗
=
d
d
y
i
+
y
i
+
=
y
∗
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
=
−
c
/
q
W
−
q
/
c
y
∗
exp
−
q
/
c
y
∗
+
A
l
/
c
1
+
W
−
q
/
c
y
∗
exp
−
q
/
c
y
∗
+
A
l
/
c
1
y
∗
−
q
c
=
h
y
∗
.
The boundary order-1 periodic solution is stable if and only if the absolute value ofh
y
∗ is less than one. By taking limit of h
y
∗, we get(106)
lim
y
∗
→
0
h
y
∗
=
e
A
l
/
c
.
This limit shows that ifA
l
<
0, then h
y
∗
<
1 as y
∗
→
0, thus the boundary order-1 periodic solution is asymptotically stable. Hence, from all the above outcomes, it can be concluded that boundary order-1 periodic solution x
T
t
,
0 is globally asymptotically stable. This completes the proof.Figure 4
(a–c) Unstable boundary order-1 periodic solution withL
2
=
A
T
/
α
+
c  β
=
3, A
T
=
3.24 and A
l
=
0.02643; (d–f) stable boundary order-1 periodic solution with L
2
=
2, A
T
=
2.16 and A
l
=
−
0.106676. All other parameter values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, α
=
0.9, β
=
0.1, p
=
0.8, τ
=
0.
(a)
(b)
(c)
(d)
(e)
(f)


## 6. Characterization of Periodic Solution forτ
>
0

In this section, we aim to give the detailed conditions for the existence and stability of the fixed point of Poincaré mapP
y
i
+. From above discussion, it is obvious that the impulsive and phase sets are complex curves that rely upon the weighted parameters α and β. So, we will perceive how these parameters differ our results from literature [42]. We first find out the conditions that ensure the existence of order-1 periodic solution for system (2).


### 6.1. Existence of Order-1 Periodic Solution

In order to achieve the target, we first give an important lemma which will be used in the upcoming results.Lemma 11.
IfA
l
>
0 and τ
>
0, then the following inequality is satisfied for the Poincaré map P
y
i
+
(107)
P
y
i
+
>
y
i
+
,
f
o
r
a
l
l
y
i
+
∈
0
,
y
P
2
.Proof.
Let a trajectory originate fromp
0
+
=
x
i
+
,
y
i
+ and it touches curve Γ
I
c at point p
1
=
x
i
+
1
,
y
i
+
1. Here, we assume that y
i
+
,
y
i
+
1
<
c
/
q, then these points must satisfy the relation:(108)
λ
ω
ln
1
+
ω
x
i
+
1
1
+
ω
x
i
+
−
r
x
i
+
1
−
x
i
+
−
δ
ln
x
i
+
1
x
i
+
=
c
ln
y
i
+
1
y
i
+
−
q
y
i
+
1
−
y
i
+
.
From (107), we get(109)
−
q
c
y
i
+
1
e
−
q
/
c
y
i
+
1
=
−
q
c
y
i
+
e
−
q
/
c
y
i
+
+
A
l
/
c
.
IfA
l
>
0, then we get the inequality(110)
−
q
c
y
i
+
1
e
−
q
/
c
y
i
+
1
<
−
q
c
y
i
+
e
−
q
/
c
y
i
+
.
Let us definef
y
=
−
y
exp
−
y, then f
′
y
>
0 if y
>
1 and f
′
y
<
0 if y
∈
0
,
1 . The inequality y
i
+
1
≥
y
i
+ is satisfied for all q
/  c
y
i
+
,
q
/  c
y
i
+
1
∈
0
,
1. We also know that y
i
+
1
+
=
y
i
+
1
+
τ and P
y
i
+
=
y
i
+
1
+. From the above discussions, we conclude that P
y
i
+
>
y
i
+ for all y
i
+
∈
0
,
y
P
2.Theorem 3.
For CaseC
1, the fixed point of Poincaré map P
y
i
+ exists and therefore periodic solution of order-1 exists for system (2), as shown in Figure 5.Proof.
Let a trajectoryΓ
1 is tangent at point x
S
,
y
S and it touches curve Γ
I
c at point x
Q
2
,
y
Q
2, such that y
Q
2
<
c
/
q. If P
y
S
=
y
Q
2
+, then clearly curve  forms an order-1 periodic solution. If P
y
S
≠
y
S, then either P
y
S
>
y
S or P
y
S
<
y
S.
IfP
y
S
>
y
S, then trajectory originating from point P
y
S
=
y
Q
2
+ will touch curve Γ
I
c at point y
Q
3. After one time pulse action, it will be adjusted at point P
y
Q
2
+
=
y
Q
3
+. Hence, we get the inequality(111)
P
y
Q
2
+
<
y
Q
2
+
.
For the second case, the following inequality is satisfied(112)
P
y
S
<
y
S
.
We also know that for the lowest impulsive pointτ, the following inequality is always satisfied(113)
P
τ
>
τ
.
From (111) and (113), due to continuity of the Poincaré map we see that the fixed point exists in the interval τ
,
y
Q
2
+ and from (112) and (114), we can see that the fixed point for Poincaré map exists in the interval τ
,
y
S.Figure 5
Fixed point of Poincaré map for CaseC
1. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.2, A
T
=
1. (a) β
=
0.1, y
∗
=
1.8784, (b) β
=
0.1, (c) β
=
0.2, y
∗
=
1.8789, (d) β
=
0.2.
(a)
(b)
(c)
(d)Theorem 4.
For CaseC
2, if P
y
P
1
>
y
P
1 then the fixed point of the Poincaré map P
y
i
+ exists above the point y
P
1 and therefore periodic solution of order-1 exists for system (2).Proof.
For CaseC
2, there exists a trajectory which touches curve Γ
P
c at two points x
P
1
,
y
P
1 and x
P
2
,
y
P
2 and tangents to curve Γ
I
c at point T
=
x
T
,
y
T. If P
y
P
1
=
y
T
+
=
y
P
1, then curve  forms an order-1 periodic solution for system (2). If P
y
P
1
≠
y
P
1, then either P
y
P
1
>
y
P
1 or P
y
P
1
<
y
P
1.
IfP
y
P
1
>
y
P
1, then trajectory originating from point P
y
P
1
=
y
T
+ touches curve Γ
I
c having vertical coordinate y
T
1 which is less than y
T, i.e., y
T
1
<
y
T. Then after one time pulse action, we get y
T
1
+. It can easily be seen that y
T
1
+
<
y
T
+. Hence, we get the inequality(114)
P
y
T
+
<
y
T
+
.
For this case, we already know that(115)
P
y
P
1
>
y
P
1
.
Combining the inequalities (114) and (115), it is obvious that the fixed point of the Poincaré map exists in the interval y
P
1
,
y
T
+ and hence periodic solution of order-1 exists for system (2). This completes the proof.
IfP
y
P
1
<
y
P
1, then any trajectory initiating from Γ
P
c will touch curve Γ
I
c and after one time pulse action, it will move to the interval τ
,
y
T
+. If τ
>
y
P
2, then trajectory starting from y
+ will directly move inside the closed trajectory after one time pulse action and it will be free from additional impulsive effect. If τ
≤
y
P
2, then using the inequality (107) any trajectory originating from y
+ with y
+
∈
τ
,
y
P
2 will reach at curve Γ
I
c and after a limited number of pulse actions it will finally enter into the closed trajectory Γ
2, and there will be no more pulse actions on it. Thus, for this case, no fixed point of the Poincaré map exists and, therefore, no periodic solution of order-1 exists for system (2).Theorem 5.
For CaseC
3
i, if τ
>
τ
3 then the fixed point of the Poincaré map P
y
i
+ exists above the point y
P
1. For Case C
3
i  i, if τ
<
τ
4
τ
>
τ
3 then the fixed point exists below the point y
P
2 (above y
P
1) and therefore periodic solution of order-1 exists for system (2), as shown in Figure 3.Proof.
For CaseC
3
i, the homoclinic trajectory Γ
3 touches with curve Γ
P
c at two points x
P
1
,
y
P
1 and x
P
2
,
y
P
2 and its right lower branch touches curve Γ
I
c at point x
Q
2
,
y
Q
2. It is obvious that y
Q
2
<
c
/
q. If τ
=
τ
3, then we can see that y
∗, y
Q
2
+ and y
P
1 meet each other. If τ
>
τ
3, then from the relation i of Lemma 9, we conclude that the fixed point lies above point y
P
1.
Ifτ
<
τ
4, then following the relation i  i of Lemma 9, we conclude that the fixed point must be less than y
P
2 and hence belongs to the interval 0
,
y
P
2. If τ
>
τ
3, then following the same way as in Case C
3
i, it can be shown that the fixed point lies above point y
P
1. Hence, an order-1 periodic solution exists for system (2).Corollary 2.
For CaseC
3
i  i, if τ
4
<
τ
<
τ
3 then any trajectory originating from the x
0
+
,
y
0
+, will move inside the trajectory Γ
3 after one time pulse action and there will be no more pulse action on it.Corollary 3.
For CaseC
1
0, if τ
>
0 then for model (8) the fixed point of the Poincaré map exists in the interval τ
,
y
i
m
1
+  τ. For Case C
2
0, if P
y
P
1
>
y
P
1 then the fixed point exists in the interval y
max
0
,
y
T
0
+  τ.Based on all the information given above, we give the exact domains of the fixed points of the Poincaré map in terms ofy
T, y
m
a
x
2, y
m
i
n
3, y
m
a
x
3 ,y
i
m
1, y
i
m
3 in Table 1:Table 1
Exact domains of the fixed point of Poincaré mapP
y
i
+ for system (2).
Cases
Condition
τ
Interval ofy
∗
C
1
A
l
≤
0
τ
>
0
τ
,
y
i
m
1
+  τ
C
2
A
l
>
0
τ
≥
τ
1
y
m
a
x
2
,
y
T
+  τ
C
3
i
A
u
≥
A
l
>
0
τ
>
τ
3
y
m
a
x
3
,
y
i
m
3
+  τ
i  i
A
u
≥
0
,
A
l
≤
0
0
<
τ
<
τ
4
0
,
y
m
i
n
3
i
i  i
A
u
≥
0
,
A
l
≤
0
τ
>
τ
3
y
m
a
x
3
,
y
i
m
3
+  τFigure 6
Effect of weighted parameters on the fixed point of Poincaré map for CaseC
2. (a) β
=
0, (b) β
=
0.1, (c) β
=
0.2, (d) β
=
0.4, (e) β
=
0.6, and (f) β
=
0.9. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.2, A
T
=
6.
(a)
(b)
(c)
(d)
(e)
(f)


### 6.2. Effect of Weighted Parameters on the Dynamic Behavior of System (2)

It is already discussed in detail that under what conditions the fixed point of Poincaré map exists for all existing cases. In the present paper, we have proposed a system with ratio-dependent AT, i.e., instead of vertical straight lines we have complex curves depends on the weighted parametersα and β. These complex curves change their position with a little increase or decrease in the weighted parameters.Figures5–8 reveal the detailed description of the behavior of fixed point y
∗. The fixed point of Poincaré map is affected by the weighted parameters α and β. From Figure 5, we can see that the fixed point of Poincaré map is increasing monotonically for Case C
1. For this case, the minimum fixed point we get whenever weighted parameter β
=
0, i.e., the threshold is only pest density dependent and as β increases the fixed point also increases monotonically. The numerical simulation shows that the fixed point of Poincaré map either increases or decreases monotonically.Figure 7
Effect of weighted parameters on CaseC
3. (a) β
=
0.1, A
l
=
0.0088, (b) β
=
0.3, A
l
=
−
0.0279. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.6, A
T
=
17.
(a)
(b)Figure 8
Effect of weighted parameters on the fixed point of Poincaré map for CaseC
3
i. (a) β
=
0, (b) β
=
0.2, (c) β
=
0.4, (d) β
=
0.6. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.3, A
T
=
12.
(a)
(b)
(c)
(d)The diagrams appearing in Figures6 and 8 describe that the Poincaré map is quite complex in these cases. The fixed points of Poincaré map change as the weighted parameters vary. The diagrams shown in Figure 6 is more complex and amazing. For Case C
2, once the weighted parameter β increases, the fixed point of Poincaré map also increases monotonically, and the fixed point of Poincaré map starts decreasing dramatically as β reaches 0.6. Specifically, it is interesting to note that once β becomes greater than 0.1, the fixed point disappears and it reappears when β crosses the value 0.3. At 0.4, it converts into Case C
3 and when β approaches 1 and α is small enough, the fixed point of Poincaré map again disappears, for example, Figure 6(f).From Figure7, we can also see that how the value of A
l varies only by a small change in the weighted parameters. For this case, if the weighted parameter β increases the fixed point of Poincaré map decreases monotonically, and there does not exist any fixed point once β becomes greater than 0.2, for example, Figures 8(c) and 8(d).


### 6.3. Local and Global Stability of Order-1 Periodic Solution

In this subsection, in the light of the above results, we will examine the local and global stability of the fixed point of Poincaré mapP
y
i
+. To show these results, we suppose that τ
>
0 and y
∗ exists.Theorem 6.
IfA
l
≤
0, then for model (2) the fixed point of Poincaré map is locally stable. If A
l
>
0, then the fixed point of Poincaré map is locally stable provided that y
∗
<
c
+
q
τ
+
c
2
+
q
2
τ
2
/
2
q.Proof.
Lety
∗ be the fixed point of Poincaré map, and let g
y
=
−
q
/
c
y
exp
−
q
/
c
y
+
A
l
/
c then(116)
g
′
y
=
−
q
c
exp
−
q
c
y
+
A
l
c
1
−
q
c
y
.
Using the properties of LambertW function, we get(117)
d
P
y
i
+
d
y
i
+
y
i
+
=
y
∗
=
d
d
y
i
+
y
i
+
=
y
∗
−
c
q
W
g
y
=
−
c
q
W
g
y
∗
1
+
W
g
y
∗
1
y
∗
−
q
c
=
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
=
h
y
∗
.
It can be seen that ify
∗
=
c
/  q
+
τ then h
y
∗
=
−
∞, and hence y
∗ is unstable. Thus, we will only consider the interval τ
,
c
/  q
+  τ. The fixed point is locally stable if h
y
∗
<
1, which is equivalent to(118)
−
1
<
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
<
1
.
The right hand side inequality of (118) is obvious, so we only need to show the left hand side inequality, i.e.,(119)
−
1
<
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
.
By simple calculations, we get(120)
q
y
∗
2
−
c
+
q
τ
y
∗
+
c
τ
2
<
0
,
and solving it with respect toy
∗, we get(121)
y
1
∗
=
c
+
q
τ
−
c
2
+
q
2
τ
2
2
q
and
y
2
∗
=
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.
It is obvious thaty
1
∗
<
y
∗
<
y
2
∗, and also we can easily show that y
1
∗
<
τ
<
y
2
∗
<
c
/  q
+
τ. This shows that if τ
<
y
∗
<
y
2
∗, then for A
l
>
0 the fixed point is locally stable. For case A
l
≤
0, we already proved in Lemma 8 that y
∗
<
y
2
∗. This completes the proof.Corollary 4.
IfA
l
>
0 and y
∗
>
y
2
∗, then for model (2) the fixed point of Poincaré map is unstable.Corollary 5.
IfA
l
0
≤
0, then for model (8) the fixed point of Poincaré map is locally stable. If A
l
0
>
0 and y
∗
<
y
2
∗, then the fixed point is locally stable, and it is unstable if y
∗
>
y
2
∗.For the global stability of the fixed point, based on the domains of the Poincaré map and Figure5, we only focus on the Case C
1 for τ
>
0 and have the following main result.Theorem 7.
Suppose that for CaseC
1, the fixed point y
∗ of the Poincaré map P
y
i
+ exists. Moreover, if(1)P
y
S
<
y
S, then it is globally stable.
(2)P
y
S
>
y
S, then it is globally stable given that P
2
y
+
>
y
+ for y
+
∈
y
S
,
y
∗.Proof.
For CaseC
1, the existence of the fixed point is already discussed in Theorem 3. Here, we first prove that this fixed point is unique and later we check its global stability. From equation (56), we know that(122)
y
i
+
1
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
+
τ
.
For the present case, we also know thatA
l
≤
0. If y
S
>
y
0
+
>
y
1
+
>
τ holds true, then from the monotonicity properties of Lambert W function and g
y, the following relation must be fulfilled by the impulsive point sequence y
k
+
k
=
0
∞:(123)
y
S
>
y
0
+
>
y
1
+
>
⋯
>
y
k
−
1
+
>
y
k
+
>
y
k
+
1
+
>
τ
.
The domain of the Poincaré map justified that ifA
l
≤
0, then the above relation must be fulfilled by all the impulsive points. From which we can state that the impulsive point sequence is monotonically decreasing and it will converge to the unique constant y
∗
∈
τ
,
y
0
+, i.e., l
i
m
k
→
∞
y
k
+
=
y
∗. The above discussion demonstrates that the fixed point is unique. The global stability of the fixed point can be presented as follows.(1) IfP
y
S
<
y
S, and let y
i
+
∈
0
,
y
∗ then y
i
+
<
P
y
i
+
<
y
∗. This shows that P
k
y
i
+ for k
≥
1, is monotonically increasing and l
i
m
k
→
∞
P
k
y
i
+
=
y
∗. Again let y
i
+
∈
y
∗
,
α
+
c  β
/
q  β
+  τ, then there are two possibilities: i for all k, we have P
k
y
i
+
>
y
∗. We know that in this interval P
y
i
+
<
y
i
+, so P
k
y
i
+ is monotonically decreasing and as a result, we can write l
i
m
k
→
∞
P
k
y
i
+
=
y
∗; i  i let P
k
y
i
+
>
y
∗ be not valid for all k, and let l
1 be the smallest positive integer such that P
l
1
y
i
+
<
y
∗. Then, by using the same method for y
i
+
∈
0
,
y
∗, if k is increasing then P
l
1
+
k
y
i
+ is also monotonically increasing and l
i
m
k
→
∞
P
l
1
+
k
y
i
+
=
y
∗. This shows that the result given in Case (1) is true.
(2) IfP
y
S
>
y
S, we take the following three intervals: a
y
i
+
∈
y
S
,
y
∗
,
b
y
i
+
∈
0
,
y
S
,
c
y
i
+
∈
y
∗
,
α
+
c  β
/
q
β
+  τ For interval (a), the Poincaré map P
y
i
+ is monotonically decreasing and hence for all y
i
+
∈
y
S
,
y
∗ we have P
y
S
≥
P
y
i
+
>
y
∗. Also applying the second condition P
2
y
i
+
>
y
i
+, we get y
i
+
<
P
2
y
i
+
<
y
∗ for all y
i
+
∈
y
S
,
y
∗. By induction, it is concluded that P
2
k
−  1
y
i
+
<
P
2
k
y
i
+
<
y
∗ for all k
≥
1. This shows that P
2
k
y
i
+ is monotonically increasing with l
i
m
k
→
∞
P
2
k
y
i
+
=
y
∗.
For interval (b), the Poincaré mapP
y
i
+ is monotonically increasing, so there must exist l
2
≥
1 such that either P
l
2
y
i
+
>
y
∗ or P
l
2
y
i
+
∈
y
S
,
y
∗. For the case, when P
l
2
y
i
+
>
y
∗ there must exist y
⌣
∈
y
S
,
y
∗ such that P
l
2
y
i
+
=
P
y
⌣, and hence l
i
m
k
→
∞
P
l
2
+
2
k
y
i
+
=
y
∗ monotonically. If P
l
2
y
i
+
∈
y
S
,
y
∗ then following the same way as in interval (a), we get l
i
m
k
→
∞
P
l
2
+
2
k
y
i
+
=
y
∗.
For interval (c), there exists a positive integerl
3 such that either P
l
3
y
i
+
∈
0
,
y
S or y
S
,
y
∗. Hence, in the light of above intervals (a) and (b), it can be shown that the result given in Case (2) is true. This completes the proof of Theorem 7.Corollary 6.
For CaseC
3
i  i, if 0
<
τ
<
τ
4, then there exists a unique fixed point for system (2) which is globally stable.


## 6.1. Existence of Order-1 Periodic Solution

In order to achieve the target, we first give an important lemma which will be used in the upcoming results.Lemma 11.
IfA
l
>
0 and τ
>
0, then the following inequality is satisfied for the Poincaré map P
y
i
+
(107)
P
y
i
+
>
y
i
+
,
f
o
r
a
l
l
y
i
+
∈
0
,
y
P
2
.Proof.
Let a trajectory originate fromp
0
+
=
x
i
+
,
y
i
+ and it touches curve Γ
I
c at point p
1
=
x
i
+
1
,
y
i
+
1. Here, we assume that y
i
+
,
y
i
+
1
<
c
/
q, then these points must satisfy the relation:(108)
λ
ω
ln
1
+
ω
x
i
+
1
1
+
ω
x
i
+
−
r
x
i
+
1
−
x
i
+
−
δ
ln
x
i
+
1
x
i
+
=
c
ln
y
i
+
1
y
i
+
−
q
y
i
+
1
−
y
i
+
.
From (107), we get(109)
−
q
c
y
i
+
1
e
−
q
/
c
y
i
+
1
=
−
q
c
y
i
+
e
−
q
/
c
y
i
+
+
A
l
/
c
.
IfA
l
>
0, then we get the inequality(110)
−
q
c
y
i
+
1
e
−
q
/
c
y
i
+
1
<
−
q
c
y
i
+
e
−
q
/
c
y
i
+
.
Let us definef
y
=
−
y
exp
−
y, then f
′
y
>
0 if y
>
1 and f
′
y
<
0 if y
∈
0
,
1 . The inequality y
i
+
1
≥
y
i
+ is satisfied for all q
/  c
y
i
+
,
q
/  c
y
i
+
1
∈
0
,
1. We also know that y
i
+
1
+
=
y
i
+
1
+
τ and P
y
i
+
=
y
i
+
1
+. From the above discussions, we conclude that P
y
i
+
>
y
i
+ for all y
i
+
∈
0
,
y
P
2.Theorem 3.
For CaseC
1, the fixed point of Poincaré map P
y
i
+ exists and therefore periodic solution of order-1 exists for system (2), as shown in Figure 5.Proof.
Let a trajectoryΓ
1 is tangent at point x
S
,
y
S and it touches curve Γ
I
c at point x
Q
2
,
y
Q
2, such that y
Q
2
<
c
/
q. If P
y
S
=
y
Q
2
+, then clearly curve  forms an order-1 periodic solution. If P
y
S
≠
y
S, then either P
y
S
>
y
S or P
y
S
<
y
S.
IfP
y
S
>
y
S, then trajectory originating from point P
y
S
=
y
Q
2
+ will touch curve Γ
I
c at point y
Q
3. After one time pulse action, it will be adjusted at point P
y
Q
2
+
=
y
Q
3
+. Hence, we get the inequality(111)
P
y
Q
2
+
<
y
Q
2
+
.
For the second case, the following inequality is satisfied(112)
P
y
S
<
y
S
.
We also know that for the lowest impulsive pointτ, the following inequality is always satisfied(113)
P
τ
>
τ
.
From (111) and (113), due to continuity of the Poincaré map we see that the fixed point exists in the interval τ
,
y
Q
2
+ and from (112) and (114), we can see that the fixed point for Poincaré map exists in the interval τ
,
y
S.Figure 5
Fixed point of Poincaré map for CaseC
1. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.2, A
T
=
1. (a) β
=
0.1, y
∗
=
1.8784, (b) β
=
0.1, (c) β
=
0.2, y
∗
=
1.8789, (d) β
=
0.2.
(a)
(b)
(c)
(d)Theorem 4.
For CaseC
2, if P
y
P
1
>
y
P
1 then the fixed point of the Poincaré map P
y
i
+ exists above the point y
P
1 and therefore periodic solution of order-1 exists for system (2).Proof.
For CaseC
2, there exists a trajectory which touches curve Γ
P
c at two points x
P
1
,
y
P
1 and x
P
2
,
y
P
2 and tangents to curve Γ
I
c at point T
=
x
T
,
y
T. If P
y
P
1
=
y
T
+
=
y
P
1, then curve  forms an order-1 periodic solution for system (2). If P
y
P
1
≠
y
P
1, then either P
y
P
1
>
y
P
1 or P
y
P
1
<
y
P
1.
IfP
y
P
1
>
y
P
1, then trajectory originating from point P
y
P
1
=
y
T
+ touches curve Γ
I
c having vertical coordinate y
T
1 which is less than y
T, i.e., y
T
1
<
y
T. Then after one time pulse action, we get y
T
1
+. It can easily be seen that y
T
1
+
<
y
T
+. Hence, we get the inequality(114)
P
y
T
+
<
y
T
+
.
For this case, we already know that(115)
P
y
P
1
>
y
P
1
.
Combining the inequalities (114) and (115), it is obvious that the fixed point of the Poincaré map exists in the interval y
P
1
,
y
T
+ and hence periodic solution of order-1 exists for system (2). This completes the proof.
IfP
y
P
1
<
y
P
1, then any trajectory initiating from Γ
P
c will touch curve Γ
I
c and after one time pulse action, it will move to the interval τ
,
y
T
+. If τ
>
y
P
2, then trajectory starting from y
+ will directly move inside the closed trajectory after one time pulse action and it will be free from additional impulsive effect. If τ
≤
y
P
2, then using the inequality (107) any trajectory originating from y
+ with y
+
∈
τ
,
y
P
2 will reach at curve Γ
I
c and after a limited number of pulse actions it will finally enter into the closed trajectory Γ
2, and there will be no more pulse actions on it. Thus, for this case, no fixed point of the Poincaré map exists and, therefore, no periodic solution of order-1 exists for system (2).Theorem 5.
For CaseC
3
i, if τ
>
τ
3 then the fixed point of the Poincaré map P
y
i
+ exists above the point y
P
1. For Case C
3
i  i, if τ
<
τ
4
τ
>
τ
3 then the fixed point exists below the point y
P
2 (above y
P
1) and therefore periodic solution of order-1 exists for system (2), as shown in Figure 3.Proof.
For CaseC
3
i, the homoclinic trajectory Γ
3 touches with curve Γ
P
c at two points x
P
1
,
y
P
1 and x
P
2
,
y
P
2 and its right lower branch touches curve Γ
I
c at point x
Q
2
,
y
Q
2. It is obvious that y
Q
2
<
c
/
q. If τ
=
τ
3, then we can see that y
∗, y
Q
2
+ and y
P
1 meet each other. If τ
>
τ
3, then from the relation i of Lemma 9, we conclude that the fixed point lies above point y
P
1.
Ifτ
<
τ
4, then following the relation i  i of Lemma 9, we conclude that the fixed point must be less than y
P
2 and hence belongs to the interval 0
,
y
P
2. If τ
>
τ
3, then following the same way as in Case C
3
i, it can be shown that the fixed point lies above point y
P
1. Hence, an order-1 periodic solution exists for system (2).Corollary 2.
For CaseC
3
i  i, if τ
4
<
τ
<
τ
3 then any trajectory originating from the x
0
+
,
y
0
+, will move inside the trajectory Γ
3 after one time pulse action and there will be no more pulse action on it.Corollary 3.
For CaseC
1
0, if τ
>
0 then for model (8) the fixed point of the Poincaré map exists in the interval τ
,
y
i
m
1
+  τ. For Case C
2
0, if P
y
P
1
>
y
P
1 then the fixed point exists in the interval y
max
0
,
y
T
0
+  τ.Based on all the information given above, we give the exact domains of the fixed points of the Poincaré map in terms ofy
T, y
m
a
x
2, y
m
i
n
3, y
m
a
x
3 ,y
i
m
1, y
i
m
3 in Table 1:Table 1
Exact domains of the fixed point of Poincaré mapP
y
i
+ for system (2).
Cases
Condition
τ
Interval ofy
∗
C
1
A
l
≤
0
τ
>
0
τ
,
y
i
m
1
+  τ
C
2
A
l
>
0
τ
≥
τ
1
y
m
a
x
2
,
y
T
+  τ
C
3
i
A
u
≥
A
l
>
0
τ
>
τ
3
y
m
a
x
3
,
y
i
m
3
+  τ
i  i
A
u
≥
0
,
A
l
≤
0
0
<
τ
<
τ
4
0
,
y
m
i
n
3
i
i  i
A
u
≥
0
,
A
l
≤
0
τ
>
τ
3
y
m
a
x
3
,
y
i
m
3
+  τFigure 6
Effect of weighted parameters on the fixed point of Poincaré map for CaseC
2. (a) β
=
0, (b) β
=
0.1, (c) β
=
0.2, (d) β
=
0.4, (e) β
=
0.6, and (f) β
=
0.9. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.2, A
T
=
6.
(a)
(b)
(c)
(d)
(e)
(f)


## 6.2. Effect of Weighted Parameters on the Dynamic Behavior of System (2)

It is already discussed in detail that under what conditions the fixed point of Poincaré map exists for all existing cases. In the present paper, we have proposed a system with ratio-dependent AT, i.e., instead of vertical straight lines we have complex curves depends on the weighted parametersα and β. These complex curves change their position with a little increase or decrease in the weighted parameters.Figures5–8 reveal the detailed description of the behavior of fixed point y
∗. The fixed point of Poincaré map is affected by the weighted parameters α and β. From Figure 5, we can see that the fixed point of Poincaré map is increasing monotonically for Case C
1. For this case, the minimum fixed point we get whenever weighted parameter β
=
0, i.e., the threshold is only pest density dependent and as β increases the fixed point also increases monotonically. The numerical simulation shows that the fixed point of Poincaré map either increases or decreases monotonically.Figure 7
Effect of weighted parameters on CaseC
3. (a) β
=
0.1, A
l
=
0.0088, (b) β
=
0.3, A
l
=
−
0.0279. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.6, A
T
=
17.
(a)
(b)Figure 8
Effect of weighted parameters on the fixed point of Poincaré map for CaseC
3
i. (a) β
=
0, (b) β
=
0.2, (c) β
=
0.4, (d) β
=
0.6. All other parametric values are fixed as: c
=
1.8, q
=
1.3, λ
=
0.52, ω
=
0.1, r
=
0.23, δ
=
0.3, p
=
0.8, τ
=
1.3, A
T
=
12.
(a)
(b)
(c)
(d)The diagrams appearing in Figures6 and 8 describe that the Poincaré map is quite complex in these cases. The fixed points of Poincaré map change as the weighted parameters vary. The diagrams shown in Figure 6 is more complex and amazing. For Case C
2, once the weighted parameter β increases, the fixed point of Poincaré map also increases monotonically, and the fixed point of Poincaré map starts decreasing dramatically as β reaches 0.6. Specifically, it is interesting to note that once β becomes greater than 0.1, the fixed point disappears and it reappears when β crosses the value 0.3. At 0.4, it converts into Case C
3 and when β approaches 1 and α is small enough, the fixed point of Poincaré map again disappears, for example, Figure 6(f).From Figure7, we can also see that how the value of A
l varies only by a small change in the weighted parameters. For this case, if the weighted parameter β increases the fixed point of Poincaré map decreases monotonically, and there does not exist any fixed point once β becomes greater than 0.2, for example, Figures 8(c) and 8(d).


## 6.3. Local and Global Stability of Order-1 Periodic Solution

In this subsection, in the light of the above results, we will examine the local and global stability of the fixed point of Poincaré mapP
y
i
+. To show these results, we suppose that τ
>
0 and y
∗ exists.Theorem 6.
IfA
l
≤
0, then for model (2) the fixed point of Poincaré map is locally stable. If A
l
>
0, then the fixed point of Poincaré map is locally stable provided that y
∗
<
c
+
q
τ
+
c
2
+
q
2
τ
2
/
2
q.Proof.
Lety
∗ be the fixed point of Poincaré map, and let g
y
=
−
q
/
c
y
exp
−
q
/
c
y
+
A
l
/
c then(116)
g
′
y
=
−
q
c
exp
−
q
c
y
+
A
l
c
1
−
q
c
y
.
Using the properties of LambertW function, we get(117)
d
P
y
i
+
d
y
i
+
y
i
+
=
y
∗
=
d
d
y
i
+
y
i
+
=
y
∗
−
c
q
W
g
y
=
−
c
q
W
g
y
∗
1
+
W
g
y
∗
1
y
∗
−
q
c
=
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
=
h
y
∗
.
It can be seen that ify
∗
=
c
/  q
+
τ then h
y
∗
=
−
∞, and hence y
∗ is unstable. Thus, we will only consider the interval τ
,
c
/  q
+  τ. The fixed point is locally stable if h
y
∗
<
1, which is equivalent to(118)
−
1
<
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
<
1
.
The right hand side inequality of (118) is obvious, so we only need to show the left hand side inequality, i.e.,(119)
−
1
<
y
∗
−
τ
c
−
q
y
∗
y
∗
c
−
q
y
∗
−
τ
.
By simple calculations, we get(120)
q
y
∗
2
−
c
+
q
τ
y
∗
+
c
τ
2
<
0
,
and solving it with respect toy
∗, we get(121)
y
1
∗
=
c
+
q
τ
−
c
2
+
q
2
τ
2
2
q
and
y
2
∗
=
c
+
q
τ
+
c
2
+
q
2
τ
2
2
q
.
It is obvious thaty
1
∗
<
y
∗
<
y
2
∗, and also we can easily show that y
1
∗
<
τ
<
y
2
∗
<
c
/  q
+
τ. This shows that if τ
<
y
∗
<
y
2
∗, then for A
l
>
0 the fixed point is locally stable. For case A
l
≤
0, we already proved in Lemma 8 that y
∗
<
y
2
∗. This completes the proof.Corollary 4.
IfA
l
>
0 and y
∗
>
y
2
∗, then for model (2) the fixed point of Poincaré map is unstable.Corollary 5.
IfA
l
0
≤
0, then for model (8) the fixed point of Poincaré map is locally stable. If A
l
0
>
0 and y
∗
<
y
2
∗, then the fixed point is locally stable, and it is unstable if y
∗
>
y
2
∗.For the global stability of the fixed point, based on the domains of the Poincaré map and Figure5, we only focus on the Case C
1 for τ
>
0 and have the following main result.Theorem 7.
Suppose that for CaseC
1, the fixed point y
∗ of the Poincaré map P
y
i
+ exists. Moreover, if(1)P
y
S
<
y
S, then it is globally stable.
(2)P
y
S
>
y
S, then it is globally stable given that P
2
y
+
>
y
+ for y
+
∈
y
S
,
y
∗.Proof.
For CaseC
1, the existence of the fixed point is already discussed in Theorem 3. Here, we first prove that this fixed point is unique and later we check its global stability. From equation (56), we know that(122)
y
i
+
1
+
=
−
c
q
W
−
q
c
y
i
+
exp
−
q
c
y
i
+
+
A
l
c
+
τ
.
For the present case, we also know thatA
l
≤
0. If y
S
>
y
0
+
>
y
1
+
>
τ holds true, then from the monotonicity properties of Lambert W function and g
y, the following relation must be fulfilled by the impulsive point sequence y
k
+
k
=
0
∞:(123)
y
S
>
y
0
+
>
y
1
+
>
⋯
>
y
k
−
1
+
>
y
k
+
>
y
k
+
1
+
>
τ
.
The domain of the Poincaré map justified that ifA
l
≤
0, then the above relation must be fulfilled by all the impulsive points. From which we can state that the impulsive point sequence is monotonically decreasing and it will converge to the unique constant y
∗
∈
τ
,
y
0
+, i.e., l
i
m
k
→
∞
y
k
+
=
y
∗. The above discussion demonstrates that the fixed point is unique. The global stability of the fixed point can be presented as follows.(1) IfP
y
S
<
y
S, and let y
i
+
∈
0
,
y
∗ then y
i
+
<
P
y
i
+
<
y
∗. This shows that P
k
y
i
+ for k
≥
1, is monotonically increasing and l
i
m
k
→
∞
P
k
y
i
+
=
y
∗. Again let y
i
+
∈
y
∗
,
α
+
c  β
/
q  β
+  τ, then there are two possibilities: i for all k, we have P
k
y
i
+
>
y
∗. We know that in this interval P
y
i
+
<
y
i
+, so P
k
y
i
+ is monotonically decreasing and as a result, we can write l
i
m
k
→
∞
P
k
y
i
+
=
y
∗; i  i let P
k
y
i
+
>
y
∗ be not valid for all k, and let l
1 be the smallest positive integer such that P
l
1
y
i
+
<
y
∗. Then, by using the same method for y
i
+
∈
0
,
y
∗, if k is increasing then P
l
1
+
k
y
i
+ is also monotonically increasing and l
i
m
k
→
∞
P
l
1
+
k
y
i
+
=
y
∗. This shows that the result given in Case (1) is true.
(2) IfP
y
S
>
y
S, we take the following three intervals: a
y
i
+
∈
y
S
,
y
∗
,
b
y
i
+
∈
0
,
y
S
,
c
y
i
+
∈
y
∗
,
α
+
c  β
/
q
β
+  τ For interval (a), the Poincaré map P
y
i
+ is monotonically decreasing and hence for all y
i
+
∈
y
S
,
y
∗ we have P
y
S
≥
P
y
i
+
>
y
∗. Also applying the second condition P
2
y
i
+
>
y
i
+, we get y
i
+
<
P
2
y
i
+
<
y
∗ for all y
i
+
∈
y
S
,
y
∗. By induction, it is concluded that P
2
k
−  1
y
i
+
<
P
2
k
y
i
+
<
y
∗ for all k
≥
1. This shows that P
2
k
y
i
+ is monotonically increasing with l
i
m
k
→
∞
P
2
k
y
i
+
=
y
∗.
For interval (b), the Poincaré mapP
y
i
+ is monotonically increasing, so there must exist l
2
≥
1 such that either P
l
2
y
i
+
>
y
∗ or P
l
2
y
i
+
∈
y
S
,
y
∗. For the case, when P
l
2
y
i
+
>
y
∗ there must exist y
⌣
∈
y
S
,
y
∗ such that P
l
2
y
i
+
=
P
y
⌣, and hence l
i
m
k
→
∞
P
l
2
+
2
k
y
i
+
=
y
∗ monotonically. If P
l
2
y
i
+
∈
y
S
,
y
∗ then following the same way as in interval (a), we get l
i
m
k
→
∞
P
l
2
+
2
k
y
i
+
=
y
∗.
For interval (c), there exists a positive integerl
3 such that either P
l
3
y
i
+
∈
0
,
y
S or y
S
,
y
∗. Hence, in the light of above intervals (a) and (b), it can be shown that the result given in Case (2) is true. This completes the proof of Theorem 7.Corollary 6.
For CaseC
3
i  i, if 0
<
τ
<
τ
4, then there exists a unique fixed point for system (2) which is globally stable.


## 7. Conclusion

Mathematical ecology is one of the basic elements of IPM process. It is the study of populations that interact, the way they affect the growth rates of each other. The Lotka–Volterra model is a very special case of such an interaction, in which there are two species, one of which is a prey and another one is a predator. Prey-predator models have received a high concentration of scholars due to their prosperous dynamic behavior. Prey and predator can impact each other’s development, and such pairs exist throughout nature. It represents one of the primary models in mathematical ecology. Another fundamental concept of IPM process is that of using sound ET. It is the practical rule used to determine when to take management action.In this paper, concerning IPM system, we have proposed and examined a commonly used prey-predator impulsive dynamical model with action threshold which depends on pest density and its change rate, which implies that the threshold is not only pest density dependent but also depends on the density of natural enemy. The threshold contains two weighted quantitiesα and β. Once the weighted parameter β vanishes, i.e., β
=
0 the action threshold just relies upon the density of pest population. Then the action threshold will be transformed into ET, which has been extensively demonstrated and explored in past writings [39–45].The reason for choosing ratio-dependent AT is the presence of some practical issues in the previous used models during investigation on this topic. Firstly, for a comparatively large number of the pest population, its change rate is quite small. The second reason is that the number of population is small, but its change rate is significantly high which is more clear at the initial stage of the occurrence of the pest. Therefore, in order to overcome those shortcomings, we intended to take the model with action threshold depending on the pest density and its changing rate, which will result in complex curves for impulsive and phase sets.Comparing with the main outcomes acquired for the prey-predator model in [46], we conclude that the ratio-dependent AT can altogether impact the dynamics of system proposed here including Poincaré map and fixed point, which is very useful for structuring proper pest control measures. The complex and rich dynamics occur when model (2) does not exist the fixed point of Poincaré map. Moreover, the increasing and extensive uses of systems with ratio-dependent AT as control measures in a wide variety of fields require much more advanced and new qualitative techniques to explore their whole dynamics and reveal the important biological implications. This is an enormous task for analyzing the system with ratio-dependent AT, and new methodologies need to be established.Applying the Lambert W function function and its properties, the exact impulsive and phase sets were found. Based on these, the Poincaré map is shaped for the exact phase set. The conditions for the existence and stability of the boundary order-1 periodic solution are provided. From Figure4, it can be seen that the numerical simulation also agrees with the theoretical outcomes. Sufficient conditions that confirm the order-1 periodic solution and its stability were studied. It is also studied in detail how and under what conditions the fixed point of Poincaré map and its stability are affected by the weighted parameters α and β.Figures5, 6, and 8 demonstrate that the definition domain of the Poincaré map is indeed very complex for system (2). Numerical simulation shows how the shapes of Poincaré map vary with the small changes in the weighted parameters α and β. The fixed point of the Poincaré map, i.e., periodic solution of order-1 is affected by the weighted parameters. If the weighted parameter β increases, for some cases it decreases monotonically and for some cases it increases monotonically. For those cases where the fixed point is increasing, we get its minimum value whenever β
=
0, i.e., the threshold relies upon the pest density, as shown in Figure 5. For those cases where the fixed point is decreasing, we get its maximum value whenever β
=
0, as shown in Figure 8.The new investigative procedures built up in this paper could not easily be applied to other generalized models with state-dependent feedback control [50–53], yet also can assist us in comprehending further the qualitative behavior of the planar impulsive semidynamical system and encourage us to address more extensive issues. Compared with the previous work, we provided the exact domains for impulsive and phase sets. We believe that the idea of action threshold is more general and practical as it depends on pest density and its change rate. It also can generate new significant directions as compared with those introduced in previous studies.We considered the more general pest and natural enemy systems with ratio-dependent AT, and no doubt it is crucial to determine the Poincaré map and analyze the global dynamics. The impulsive and phase sets are complex curves rather than straight lines. The main results of this paper exhibit that the pests can be entirely controlled by applying control action for a predetermined number of times such that the ratio-dependent AT is not exceeded. Numerical simulation additionally illustrates another essential reality that the impulsive and phase sets not just change with the change of the weight parametersα and β, yet also rely upon the interaction between the pest and its natural enemy.



---

*Source: 4561241-2020-02-21.xml*