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Citation: Quan, Q.; Dai, X.; Jiao, J.
Dynamics of a Predator–Prey Model
with Impulsive Diffusion and
Transient/Nontransient Impulsive
Harvesting. Mathematics 2023,11,
3254. https://doi.org/10.3390/
math11143254
Academic Editor: Patricia J. Y. Wong
Received: 29 June 2023
Revised: 20 July 2023
Accepted: 21 July 2023
Published: 24 July 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Dynamics of a Predator–Prey Model with Impulsive Diffusion
and Transient/Nontransient Impulsive Harvesting
Qi Quan 1, Xiangjun Dai 1,2 and Jianjun Jiao 1,3,*
1School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China;
21030060033@gznu.edu.cn or quanqi2020@126.com (Q.Q.); daiaga0921@126.com (X.D.)
2School of Date Science, Tongren University, Tongren 554300, China; 21030060032@gznu.edu.cn
3
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
*Correspondence: jiaojianjun05@126.com or jiaojianju@mail.gufe.edu.cn
Abstract:
Harvesting is one of the ways for humans to realize economic interests, while unrestricted
harvesting will lead to the extinction of populations. This paper proposes a predator–prey model with
impulsive diffusion and transient/nontransient impulsive harvesting. In this model, we consider
both impulsive harvesting and impulsive diffusion; additionally, predator and prey are harvested
simultaneously. First, we obtain the subsystems of the system in prey extinction and predator
extinction. We obtain the fixed points of the subsystems by the stroboscopic map theories of impulsive
differential equations and analyze their stabilities. Further, we establish the globally asymptotically
stable conditions for the prey/predator-extinction periodic solution and the trivial solution of the
system, and then the sufficient conditions for the permanence of the system are given. We also
perform several numerical simulations to substantiate our results. It is shown that the transient and
nontransient impulsive harvesting have strong impacts on the persistence of the predator–prey model.
Keywords:
impulsive diffusion; transient and non-transient impulsive harvesting; predator–prey
model; permanence
MSC: 34A37; 34D05; 34D23; 34E05; 37M05
1. Introduction
In nature, species cannot exist alone; they always interact with other species, such as in
competition, predator–prey, or reciprocity. As one of them, the predator–prey relationship
is widespread and very important. It is also a main research topic in population dynamics.
In the 1940s, Lotka and Volterra proposed the classic predator–prey system. Afterward,
the classic predator–prey model has been followed and developed in much literature [
1
–
8
],
and the study of the dynamics of the predator–prey model has been observed widely in
applied mathematics. There are many factors, for example, weather, food supply, mating
habits or harvesting, by which the dynamics of the predator–prey population are affected.
In [1], Brauer studied the following system:
(x0=x f (x,y)−F,
y0=yg(x,y),(1)
where prey population
x(t)
is harvested at a constant time rate F, and f(x,y) and g(x,y)
denote the per capita growth rates of prey population
x(t)
and predator population
y(t)
,
respectively. Similar to reference [
1
], the activities of harvesting are usually assumed
to be continuous in formerly published results. Kumar and Kharbanda [
2
] studied a
predator–prey
model with nonlinear harvesting. Lv et al. [
3
] investigated a prey–predator
model with continuous harvesting, and the stability of the model is discussed from both
local and global perspectives. Although it is preferable from the point of view of both
Mathematics 2023,11, 3254. https://doi.org/10.3390/math11143254 https://www.mdpi.com/journal/mathematics
Mathematics 2023,11, 3254 2 of 25
maximizing harvest and sustainability, continuous harvesting is not always realistic, be-
cause the harvesting is seasonal or occurs in regular pulses for most species. In [
4
], a
logistic system with impulsive perturbations was investigated. The specific form of the
model is as follows:
(x0(t) = x(t)(r(t)−a(t)x(t)),t6=tk,
∆x(t) = bkx(t),t=tk,(2)
when
bk<
0, the perturbation means harvesting,
∆x(tk) = x(tk+)−x(tk)
. Recently,
predator–prey models with impulsive harvesting have been intensively researched. Tian
and Gao [
5
] discussed an instantaneous harvest fishery model. Liu et al. [
6
] considered
a predator–prey model in which predator and prey species are harvested independently
with proportion. Wei et al. [
7
] proposed a ratio-dependent prey–predator model with
state-dependent impulsive harvesting. Especially, Jiao [
8
] mentioned that transient and
nontransient pulse harvesting constitute the whole harvesting process in the reality of
biological resource management and presented the following model with impulsive effects:
dx1(t)
dt =−(c1+d1)x1(t),
dx2(t)
dt =c1x1(t)−d2x2(t),
t∈(nτ,(n+l)τ],
∆x1(t) = −u1x1(t),
∆x2(t) = −u2x2(t),)t= (n+l)τ,
dx1(t)
dt =−(c2+d3)x1(t)−E1x1(t),
dx2(t)
dt =c2x1(t)−d4x2(t)−E2x2(t),
t∈((n+l)τ,(n+1)τ],
∆x1(t) = x2(t)(a−bx2(t)),
∆x2(t) = 0, )t= (n+1)τ,
(3)
where the transient impulsive harvesting rate is denoted by
ui(i=
1, 2
)
and the nontransient
impulsive harvesting coefficient is denoted by
Ei(i=
1, 2
)
. The biological significance
of other parameters refer to [
5
]. In [
5
–
8
], and scholars have studied the persistence and
extinction of the investigated predator–prey models. All results show that through proper
pulse control, the population will coexist, and then, the purpose of maintaining the balance
of the ecosystem can be achieved.
The diffusion of populations is very common in nature and affects the dynamics
of the system and the ecological balance. Modern biologists believe that dispersion and
migration become necessary for populations due to seasonal changes, lack of food, breeding,
or avoidance of predators [
9
–
12
]. Paying attention to species living in patches of the
environment, Takeuchi [
13
] considered the following general single-population system
with diffusion:
˙
xi=xigi(xi) +
n
∑
j=1
Dij (xj−xi),x(0)>0, i=1, 2, . . . n, (4)
where
xi
is the population density in patch
i
,
gi(xi)
is the natural growth rate, and
Dij
is the dispersal rate. Initially, researchers assumed that diffusion between patches was
continuous or discrete; however, many species only diffuse over a single period of time in
Mathematics 2023,11, 3254 3 of 25
practice. In [
14
], a model describing the dynamics of single species with impulsive diffusion
was given by
dx1(t)
dt =x1(t)(a1−b1x1(t)),
dx2(t)
dt =x2(t)(a2−b2x2(t)),
t6=nτ,
∆x1(t) = d1(x2(t)−x1(t)),
∆x2(t) = d2(x1(t)−x2(t)),)t=nτ,
(5)
where
di(i=
1, 2
)
is the dispersal rate in the
i
-th patch, and the dispersal behavior of
species occurs every
τ
period. Other examples specific to diffusion models can be seen
in [15–17]
. Cui [
15
] studied a time-varying logistic population growth model with diffusion.
Zhong et al. [
16
] proposed a fishery model with impulsive diffusion; they assumed that the
system consists of two paths connected by diffusion and that the inshore subpopulation is
harvested at fixed moments in time. In [
17
], a predator–prey model assuming diffusion and
harvesting occurring at different fixed times was studied by Jiao et al. They considered the
case of harvesting both prey and predator populations and performed a dynamic analysis
of the model.
Most of the previous research focused only on impulsive harvesting or impulsive
diffusion and carried out unilateral harvesting of predators or prey. There still has been no
investigation of the predator–prey model with transient/nontransient impulsive harvest-
ing considering both pulse harvesting and diffusion in the literature. In addition, pulse
harvesting consists of transient and nontransient impulsive harvesting; predator and prey
may also be harvested at the same time. The transient impulsive harvesting process is
extremely short, which will cause sudden changes in the population. The nontransient
pulse harvesting depends on the current state and will last for a while, which is crucial to
the entire process of system development and cannot be ignored in both theoretical analysis
and practical application.
2. The Model
Higher-order predators such as tigers are able to create territories. They will not
interfere with other areas and only prey in their own territories [
18
–
20
]. In this paper, we
assume predator species are restricted to a single patch, and prey species diffuse between
two patches at a fixed moment of time for foraging, breeding, or avoiding predators. From
the above point of view and considering transient and nontransient impulsive harvesting
exist in populations of both prey and predator, we propose a new predator–prey model
with pulse effects, defined as
dx1(t)
dt =x1(t)(a1−b1x1(t)),
dx2(t)
dt =−d1x2(t)−β1x2(t)y(t),
dy(t)
dt =y(t)(a2−b2y(t)) + k1β1x2(t)y(t),
t∈(nσ,(n+ξ)σ],
∆x1(t) = −m1x1(t),
∆x2(t) = −m2x2(t),
∆y(t) = −m3y(t),
t= (n+ξ)σ,
dx1(t)
dt =x1(t)(a3−b3x1(t)) −h1x1(t),
dx2(t)
dt =−d2x2(t)−h2x2(t)−β2x2(t)y(t),
dy(t)
dt =y(t)(a4−b4y(t)) −h3y(t) + k2β2x2(t)y(t),
t∈((n+ξ)σ,(n+1)σ],
∆x1(t) = d(x2(t)−x1(t)),
∆x2(t) = d(x1(t)−x2(t)),
∆y(t) = 0,
t= (n+1)σ.
(6)
Mathematics 2023,11, 3254 4 of 25
where
x1(t)
is the population density of prey in patch 1.
x2(t)
and
y(t)
are the population
densities of prey and predator in patch 2, respectively. The parameters
a1
,
b1
denote
the intrinsic growth rate and intraspecific competition coefficient of
x1
, respectively, on
(nσ
,
(n+ξ)σ]
.
d1
is the natural death rate of
x2
,
β1
is the prey captured rate by
y
, and
k1
is
the rate of conversion of nutrients into the reproduction rate of
y
, on
(nσ
,
(n+ξ)σ]
.
a2
,
b2
denote the intrinsic growth rate and intraspecific competition coefficient of
y
, respectively,
on
(nσ
,
(n+ξ)σ]
.
m1
,
m2
, and
m3
represent the transient impulsive harvesting rate of
x1
,
x2
, and
y
at time
t= (n+ξ)σ
, respectively.
a3
,
b3
are the intrinsic growth rate and
intraspecific competition coefficient of
x1
, respectively, on
((n+ξ)σ
,
(n+
1
)σ]
.
h1
,
h2
, and
h3
represent the nontransient impulsive harvesting rate of
x1
,
x2
, and
y
, respectively, on
((n+ξ)σ
,
(n+
1
)σ]
.
d2
is the natural death rate of
x2
,
β2
is the prey captured rate by
y
, and
k2
represents the rate of conversion of nutrients into the reproduction rate of
y
on
((n+ξ)σ
,
(n+
1
)σ]
.
a4
,
b4
are the intrinsic growth rate and intraspecific competition
coefficients of
y
, respectively, on
((n+ξ)σ
,
(n+
1
)σ]
. 0
<d<
1 denotes the dispersal
rate of the prey between two patches.
((n+ξ)σ
,
(n+
1
)σ]
is the nontransient impulsive
harvesting interval. The pulse diffusion and impulsive harvesting occur every
σ
period.
All the parameters are assumed to be positive for biological considerations.
3. Some Lemmas
Denote
U(t) = (x1(t),x2(t),y(t))T
as the solution of system (6). It is a piecewise con-
tinuous function
U:R+→R3
+
and continuous on
(nσ
,
(n+ξ)σ]×R3
+
and
((n+ξ)σ
,
(n+
1
)σ]×R3
+
, respectively, where
R+= [
0,
∞)
,
R3
+={(x1
,
x2
,
y):x1≥
0,
x2≥
0,
y≥
0
}
. The global existence and uniqueness of solutions of system
(
6
)
is guaran-
teed by the smoothness properties of
f= ( f1
,
f2
,
f3)
, which denotes the mapping defined
by the right side of system (6)[21].
Lemma 1.
There exists a constant
M0>
0such that
x1(t)≤M0
,
x2(t)≤M0
,
y(t)≤M0
for
each solution (x1(t),x2(t),y(t)) of system (6)with a t large enough.
Proof.
Define
V(t) = x1(t) + kx2(t) + y(t)
, and choose
k=max{k1
,
k2}
,
dL=min{d1
,
d2+
h2}. Then, we have
D+V(t) + dLV(t) = (a1+dL)x1(t)−b1x12(t)−(k−k1)β1x2(t)y(t)−k(d1−dL)x2(t)
+(a2+dL)y(t)−b2y2(t)≤γ1,t∈(nσ,(n+ξ)σ],
V(t+)≤V(t),t= (n+ξ)σ,
D+V(t) + dLV(t) = [(a3−h1) + dL]x1(t)−b3x12(t)−(k−k2)β2x2(t)y(t) + kdLx2(t)
−k(d2+h2)x2(t) + [(a4−h3) + dL]y(t)−b4y2(t)≤γ2,t∈((n+ξ)σ,(n+1)σ],
V(t+)≤(1−d+kd +d
k)V(t),t= (n+1)σ,
(7)
here,
γ1=(a1+dL)2
4b1
+(a2+dL)2
4b2
,
γ2=[(a3−h1) + dL]2
4b3
+[(a4−h3) + dL]2
4b4
. Take
γ=max{γ1,γ2}, when t6= (n+ξ)σ,t6= (n+1)σ, we obtain
D+V(t) + dLV(t)≤γ,
V(t+)≤(1−d+kd +d
k)V(t),t= (n+1)τ.
(8)
With reference to [11], we obtain
V(t)≤V(0+)(1−d+kd +d
k)e−dLt+γ
dL
(1−d+kd +d
k)(1−e−dLt)(9)
Mathematics 2023,11, 3254 5 of 25
→γ
dL
(1−d+kd +d
k)as t→∞.
Hence,
V(t)
is uniformly ultimately bounded. By the definition of
V(t)
, there exists a
constant M0>0 such that x1(t)≤M0,x2(t)≤M0,y(t)≤M0for a tlarge enough.
Considering the subsystem of system (6)with y(t)=0, we have:
dx1(t)
dt =x1(t)(a1−b1x1(t)),
dx2(t)
dt =−d1x2(t),
t∈(nσ,(n+ξ)σ],
∆x1(t) = −m1x1(t),
∆x2(t) = −m2x2(t),)t= (n+ξ)σ,
dx1(t)
dt =x1(t)(a3−b3x1(t)) −h1x1(t),
dx2(t)
dt =−d2x2(t)−h2x2(t),
t∈((n+ξ)σ,(n+1)σ],
∆x1(t) = d(x2(t)−x1(t)),
∆x2(t) = d(x1(t)−x2(t)),
t= (n+1)σ.
(10)
By calculation, we obtain the analytic solution of system (7)between pluses:
x1(t) =
a1ea1(t−nσ)x1(nσ+)
a1+b1(ea1(t−nσ)−1)x1(nσ+),t∈(nσ,(n+ξ)σ],
(a3−h1)e(a3−h1)(t−(n+ξ)σ)x1((n+ξ)σ+)
(a3−h1) + b3(e(a3−h1)(t−(n+ξ)σ)−1)x1((n+ξ)σ+),
t∈((n+ξ)σ,(n+1)σ],
x2(t) = (e−d1(t−nσ)x2(nσ+),t∈(nσ,(n+ξ)σ],
e−(d2+h2)(t−(n+ξ)σ)x2((n+ξ)σ+),t∈((n+ξ)σ,(n+1)σ],
(11)
and the stroboscopic map of system (10):
x1((n+1)σ+) = (1−d)ABx1(nσ+)
B+Cx1(nσ+)+dDx2(nσ+),
x2((n+1)σ+) = dABx1(nσ+)
B+Cx1(nσ+)+ (1−d)Dx2(nσ+),
(12)
here,
A= (
1
−m1)ea1ξσ+(a3−h1)(1−ξ)σ>
0,
B=a1(a3−h1)
,
C=b1(a3−h1)(ea1ξ σ −
1
) +
a1b3(
1
−m1)ea1ξσ (e(a3−h1)(1−ξ)σ−
1
)
, 0
<D= (
1
−m2)e−d1ξσ−(d2+h2)(1−ξ)σ<
1. It is easy
to see that system (12)has two fixed points (0, 0)and (x∗
1,x∗
2), where
x∗
1=B{(1−d)(A+D)−[1+ (1−2d)AD]}
C[1−(1−d)D],
x∗
2=dB{(1−d)(A+D)−[1+ (1−2d)AD]}
C[1−(1−d)D][(1−d)−(1−2d)D],
(13)
with condition (1−d)(A+D)>[1+ (1−2d)AD].
Lemma 2.
(i) If
(
1
−d)(A+D)<[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the fixed point
(0, 0)is locally stable,
(ii) If
(
1
−d)(A+D)>[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the positive fixed point
(x1∗,x2∗)is locally stable.
Proof. Denote (x1n,x2n) = (x1(nσ+),x2(nσ+)).
Mathematics 2023,11, 3254 6 of 25
(i) The linearized equation of (12)around (0, 0)is
x1n+1
x2n+1=M1x1n
x2n, (14)
where
M1= (1−d)A dD
dA (1−d)D!. (15)
Apparently, the near dynamics of the fixed point
(
0, 0
)
are determined by linear
system
(
14
)
. The stability of the fixed point
(
0, 0
)
is determined by the eigenvalues of
M1
less than 1. This is true only if M1satisfies the three Jury conditions [22]:
1−det M1>0,
1+trM1+det M1>0,
1−trM1+det M1>0.
. (16)
By
(
15
)
and Conditions for
(i)
in Lemma 2, it is clear that
trM1= (
1
−d)A+ (
1
−d)D>
0. Hence, 1 +trM1+det M1>0 holds, if 1 −trM1+det M1>0 is true. Calculating
1−detM1=1−[(1−d)2AD −d2AD] = 1−(1−2d)AD >0.
1−trM1+det M1=1−[(1−d)A+ (1−d)D] + [(1−d)2AD −d2AD]
=1+ (1−2d)AD)−(1−d)(A+D)>0.
. (17)
Therefore, the fixed point (0, 0)is locally stable.
(ii) Similarly, we can study the local stability of positive fixed point
(x1∗
,
x2∗)
by
Jury conditions. In the neighborhood of
(x1∗
,
x2∗)
, system
(
12
)
is controlled by the
linearization of x1n+1−x1∗
x2n+1−x2∗=M2x1n−x1∗
x2n−x2∗, (18)
where
M2=
(1−d)AB2
(B+Cx1∗)2dD
dAB2
(B+Cx1∗)2(1−d)D
. (19)
Obviously,
trM2=(1−d)AB2
(B+Cx1∗)2+ (
1
−d)D>
0. Hence, 1
+trM2+det M2>
0
holds, if 1 −tr2M+det M2>0 is true. Calculating
1−detM2=1−[(1−d)2AB2D
(B+Cx1∗)2−d2AB2D
(B+Cx1∗)2]
=1−(1−2d)AD B2
(B+Cx1∗)2>0.
1−trM2+det M2=1−[(1−d)AB2
(B+Cx1∗)2+ (1−d)D]+[(1−d)2AB2D
(B+Cx1∗)2−d2AB2D
(B+Cx1∗)2]
=1−(1−d)D−AB2[(1−d) + (2d−1)D]
(B+Cx1∗)2
=[1−(1−d)D]{(1−d)(A+D)−[1+ (1−2d)AD]}
A[(1−d)−(1−2d)D]>0.
(20)
Therefore, the positive fixed point (x1∗,x2∗)is locally stable.
Mathematics 2023,11, 3254 7 of 25
Lemma 3.
(i) If
(
1
−d)(A+D)<[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the fixed point
(0, 0)is globally asymptotically stable,
(ii) If
(
1
−d)(A+D)>[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the positive fixed point
(x1∗,x2∗)is globally asymptotically stable.
Proof.
In lemma 2, we proved that the two fixed point are locally stable under the corre-
sponding conditions, respectively. Next, we only need to prove the global attractiveness.
According to Theorem 2.2 in reference [
23
], we rewrite system
(
12
)
as a map
T:R2
+→R2
+
:
T1(x1,x2) = (1−d)ABx1
B+Cx1
+dDx2,
T2(x1,x2) = dABx1
B+Cx1
+ (1−d)Dx2.
(21)
For any
(x1
,
x2)>
0, it is obvious that
T:R2
+→R2
+
is continuous, and
C1
in
int(R2
+)
and T(0, 0) = 0. Since
DT(x1,x2) =
∂T1
∂x1
∂T1
∂x2
∂T2
∂x1
∂T2
∂x2
=
(1−d)AB2
(B+Cx1)2dD
dAB2
(B+Cx1)2(1−d)D
, (22)
then DT(0, 0) = M1and limxi→0,xi>0(i=1,2)DT(x1,x2) = DT(0, 0). Moreover,
(a)DT(x1,x2)>0 for (x1,x2)>0,
(b)If 0 <(x1,x2)<(ˆ
x1,ˆ
x2), then DT(ˆ
x1,ˆ
x2)≤DT(x1,x2)(6≡ DT(x1,x2)).
Let
λ∗=ρ(DT(
0, 0
))
; due to
DT(
0, 0
) = M1
, we have
λ∗<
1 for
(
1
−d)(A+D)<
[
1
+ (
1
−
2
d)AD]
, while
λ∗>
1 for
(
1
−d)(A+D)>[
1
+ (
1
−
2
d)AD]
. According
to theorem 2.2 in reference [
23
] and boundedness of solutions, we can see that for any
(x1
,
x2)>
0, if
(
1
−d)(A+D)<[
1
+ (
1
−
2
d)AD]
, then
limn→∞Tn(x1
,
x2) = (
0, 0
)
,
and there is a unique nonzero fixed point
q= (q
1,
q
2
)
of
T(x1
,
x2)
; if
(
1
−d)(A+D)>
[1+ (1−2d)AD], then limn→∞Tn(x1,x2) = (q1, q2).
From the above discussion, we know that
q= (x1∗
,
x2∗)
. Hence, for
(
1
−d)(A+D)>
[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, system
(
12
)
has a unique positive fixed point
(x1∗,x2∗)and it is globally asymptotically stable.
Similarly to Refs. [8,17], we can obtain the next lemma.
Lemma 4.
(i) If
(
1
−d)(A+D)<[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the trivial periodic
solution (0, 0)of system (10)is globally asymptotically stable,
(ii) If
(
1
−d)(A+D)>[
1
+ (
1
−
2
d)AD]
and
(
1
−
2
d)AD <
1, the periodic solution
(
]
x1(t),
]
x2(t)) of system (10)is globally asymptotically stable, where
]
x1(t) =
a1x1∗ea1(t−nσ)
a1+b1x1∗(ea1(t−nσ)−1),t∈(nσ,(n+ξ)σ],
(a3−h1)x1∗∗e(a3−h1)(t−(n+ξ)σ)
(a3−h1) + b3x1∗∗(e(a3−h1)(t−(n+ξ)σ)−1),t∈((n+ξ)σ,(n+1)σ],
]
x2(t) = (x2∗e−d1(t−nσ),t∈(nσ,(n+ξ)σ],
x2∗∗e−(d2+h2)(t−(n+ξ)σ),t∈((n+ξ)σ,(n+1)σ],
(23)
here, x1∗, x2∗(see (13)) and x1∗∗, x2∗∗ are determined as
x1∗∗ =(1−m1)a1ea1ξσx1∗
a1+b1x1∗ea1ξσ −1,
x2∗∗ = (1−m2)e−d1ξσ x2∗.
(24)
Mathematics 2023,11, 3254 8 of 25
Considering another subsystem of system (6)with xi(t) = 0(i=1, 2), we have
dy(t)
dt =y(t)(a2−b2y(t)),t∈(nσ,(n+ξ)σ],
∆y(t) = −m3y(t),t= (n+ξ)σ,
dy(t)
dt =y(t)(a4−b4y(t)) −h3y(t),t∈((n+ξ)σ,(n+1)σ],
∆y(t) = 0, t= (n+1)σ.
(25)
By calculation, we obtain the analytic solution of system (25)between pluses:
y(t) =
a2ea2(t−nσ)z(nσ+)
a2+b2(ea2(t−nσ)−1)z(nσ+),t∈(nσ,(n+ξ)σ],
(a4−h3)e(a4−h3)(t−(n+ξ)σ)z((n+ξ)σ+)
(a4−h3) + b4(e(a4−h3)(t−(n+ξ)σ)−1)z((n+ξ)σ+),
t∈((n+ξ)σ,(n+1)σ],
(26)
and the stroboscopic map of system (25):
y((n+1)σ+) = a2(a4−h3)Azy(nσ+)
a2(a4−h3) + Bzy(nσ+),(27)
where
Az= (1−m3)ea2ξσ+(a4−h3)(1−ξ)σ>0,
Bz=b2(a4−h3)(ea2ξ σ −1) + a2b4(1−m3)ea2ξσ(e(a4−h3)(1−ξ)σ−1).
(28)
Two fixed points of system (27)are obtained as y0and y∗, where
y∗=a2(a4−h3)(Az−1)
Bz
(29)
with condition Az>1.
Lemma 5. (i) If Az<1, the fixed point y0is globally asymptotically stable.
(ii) If Az>1, the positive fixed point y∗is globally asymptotically stable.
Proof. Denote yn=y(nσ+), then (27)can be written as
F(yn) = a2(a4−h3)Azyn
a2(a4−h3) + Bzyn,(30)
then dF(yn)
dyn
=a22(a4−h3)2Az
(a2(a4−h3) + Bzyn)2.(31)
(i) If Az<1, y0is the unique fixed point of (27),
dF(yn)
dynyn=0=a22(a4−h3)2Az
(a2(a4−h3))2=Az<1. (32)
Therefore, if y0is locally stable, then it is globally asymptotically stable.
(ii) If Az>1, y0is unstable, y∗exists, and
dF(yn)
dynyn=y∗=a22(a4−h3)2Az
(a2(a4−h3)) + Bzy∗)2=a22(a4−h3)2
a22(a4−h3)2Az=1
Az
<1. (33)
Mathematics 2023,11, 3254 9 of 25
Therefore, if y∗is locally stable, then it is globally asymptotically stable.
Similarly to Ref. [24], we can obtain the next lemma.
Lemma 6.
(i) If
Az<
1, the trivial periodic solution of system
(
25
)
is globally asymptotically
stable.
(ii) If
Az>
1, the periodic solution
g
y(t)
of system
(
25
)
is globally asymptotically stable, where
g
y(t) =
a2y∗ea2(t−nσ)
a2+b2y∗(ea2(t−nσ)−1),t∈(nσ,(n+ξ)σ],
(a4−h3)y∗∗e(a4−h3)(t−(n+ξ)σ)
(a4−h3) + b4y∗∗(e(a4−h3)(t−(n+ξ)σ)−1),t∈((n+ξ)σ,(n+1)σ],
(34)
and
y∗∗ =(1−m3)a2ea2ξσy∗
a2+b2(ea2ξσ −1)y∗. (35)
4. The Dynamics
Firstly, we study the global asymptotic stability of the boundary periodic solutions
(
]
x1(t),
]
x2(t), 0),(0, 0, g
y(t)) and the trivial solution (0, 0, 0)of system (6).
Theorem 1. If
(1−d)(A+D)>[1+ (1−2d)AD],(36)
and
(1−2d)AD <1, (37)
and
(1−d)(AE +D)<1, (38)
and
ln 1
1−m3
>a2ξσ + (a4−h3)(1−ξ)σ+k1β1(1−e−d1ξ σ )
d1
x2∗
+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗
(39)
hold, the predator-extinction periodic solution
(
]
x1(t)
,
]
x2(t)
, 0
)
of system
(
6
)
is globally asymptoti-
cally stable, where E =e−Rξ σ
02b1
]
x1(s)ds−Rσ
ξσ 2b3
]
x1(s)ds, x2∗and x2∗∗ see (13)and (24).
Proof.
Firstly, define
z1(t) = x1(t)−
]
x1(t)
,
z2(t) = x2(t)−
]
x2(t)
,
z3(t) = y(t)
, we obtain
the following linearly similar system for system (6):
dz1(t)
dt
dz2(t)
dt
dz3(t)
dt
=
a1−2b1
]
x1(t)0 0
0−d1−β1
]
x2(t)
0 0 a2+k1β1
]
x2(t)
z1(t)
z2(t)
z3(t)
,t∈(nσ,(n+ξ)σ], (40)
and
dz1(t)
dt
dz2(t)
dt
dz3(t)
dt
=
(a3−h1)−2b3
]
x1(t)0 0
0−(d2+h2)−β2
]
x2(t)
0 0 (a4−h3) + k2β2
]
x2(t)
z1(t)
z2(t)
z3(t)
, (41)
Mathematics 2023,11, 3254 10 of 25
t∈((n+ξ)σ,(n+1)σ].
For
t∈(nσ
,
(n+ξ)σ]
and
t∈((n+ξ)σ
,
(n+
1
)σ]
, it is easy to obtain the fundamental
solution matrixes:
φ1(t) =
eRt
nσa1−2b1
]
x1(s)ds 0 0
0e−d1(t−nσ)†1
0 0 eRt
nσa2+k1β1
]
x2(s)ds
, (42)
and
φ2(t) =
eRt
(n+ξ)σ(a3−h1)−2b3
]
x1(s)ds 0 0
0e−(d2+h2)(t−(n+ξ)σ)†2
0 0 eRt
(n+ξ)σ(a4−h3)+k2β2
]
x2(s)ds
. (43)
As
†1
,
†2
are not required for the following analysis, its exact form is not necessary to
obtain. The linearization of the fourth, fifth and sixth equations of system (6)is
z1((n+ξ)σ+)
z2((n+ξ)σ+)
z3((n+ξ)σ+)
=
1−m10 0
0 1 −m20
0 0 1 −m3
z1((n+ξ)σ)
z2((n+ξ)σ)
z3((n+ξ)σ)
. (44)
The linearization of the tenth, eleventh and twelfth equations of system (6)is
z1((n+1)σ+)
z2((n+1)σ+)
z3((n+1)σ+)
=
1−d d 0
d1−d0
0 0 1
z1((n+1)σ)
z2((n+1)σ)
z3((n+1)σ)
. (45)
The stability of (
]
x1(t),
]
x2(t), 0)is determined by the eigenvalues of
L=
1−m10 0
0 1 −m20
0 0 1 −m3
φ1(ξσ)
1−d d 0
d1−d0
0 0 1
φ2(σ), (46)
which are
λ1= (1−m3)eRξσ
0a2+k1β1
]
x2(s)ds+Rσ
ξσ (a4−h3)+k2β2
]
x2(s)ds,
λ2=(1−d)(AE +D)−p(1−d)2(AE +D)2−4(1−2d)ADE
2
=(1−d)(AE +D)−p(1−d)2(AE −D)2+4d2ADE
2
<(1−d)(AE +D)−(1−d)(AE −D)
2
= (1−d)D,
λ3=(1−d)(AE +D) + p(1−d)2(AE +D)2−4(1−2d)ADE
2
=(1−d)(AE +D) + pd2(AE +D)2+ (1−2d)(AE −D)2
2
<(1−d)(AE +D) + pd2(AE +D)2+ (1−2d)(AE +D)2
2
= (1−d)(AE +D).
(47)
Mathematics 2023,11, 3254 11 of 25
Here 0
<E=e−Rξσ
02b1
]
x1(s)ds−Rσ
ξσ 2b3
]
x1(s)ds <
1. If conditions
(
38
)
and
(
39
)
hold, we
can deduce that
|λi|<
1
(i=
1, 2, 3
)
. According to the Floquet theory [
25
], the predator-
extinction periodic solution (
]
x1(t),
]
x2(t), 0)of system (6)is locally stable.
Next, we prove the global attraction. If (38)holds, that is
λ1= (1−m3)∗eRξσ
0a2+k1β1
]
x2(s)ds+Rσ
ξσ (a4−h3)+k2β2
]
x2(s)ds <1,
then we can take an ε>0 small enough such that
ζ1= (1−m3)eRξσ
0a2+k1β1(
]
x2(s)+ε)ds+Rσ
ξσ (a4−h3)+k2β2(
]
x2(s)+ε)ds <1. (48)
From the second and eighth equations of system (6), we have
dx2(t)
dt ≤ −d1x2(t), (49)
and dx2(t)
dt ≤ −(d2+h2)x2(t). (50)
Considering the following comparison equation:
dH11 (t)
dt =H11(t)(a1−b1H11 (t)),
dH21 (t)
dt =−d1H21(t),
t∈(nσ,(n+ξ)σ],
∆H11(t) = −m1H11 (t),
∆H21(t) = −m2H21 (t),)t= (n+ξ)σ,
dH11 (t)
dt =H11(t)[(a3−h1)−b3H11(t)],
dH21 (t)
dt =−(d2+h2)H21(t),
t∈((n+ξ)σ,(n+1)σ],
∆H11(t) = d(H21 (t)−H11(t)),
∆H21(t) = d(H11 (t)−H21(t)),)t= (n+1)σ,
(51)
from Lemma 3 and the comparison theorem of impulsive differential equations [
25
], we
have
x1(t)≤H11(t)
,
x2(t)≤H21(t)
, and
H11(t)→
]
x1(t)
,
H21(t)→
]
x2(t)
as
t→∞
. Then,
(x1(t)≤H11(t)≤
]
x1(t) + ε,
x2(t)≤H21(t)≤
]
x2(t) + ε,(52)
for a
t
large enough. For convenience, we assume
(
52
)
holds for all
t≥
0. From system
(
6
)
and (52), we have
dy(t)
dt ≤a2y(t) + k1β1(
]
x2(t) + ε)y(t),t∈(nσ,(n+ξ)σ],
∆y(t) = −m3y(t),t= (n+ξ)σ,
dy(t)
dt ≤(a4−h3)y(t) + k2β2(
]
x2(t) + ε)y(t),t∈((n+ξ)σ,(n+1)σ],
∆y(t) = 0, t= (n+1)σ,
(53)
and
y((n+1)σ)≤(1−m3)y(nσ+)eR(n+ξ)σ
nσa2+k1β1(
]
x2(s)+ε)ds+R(n+1)σ
(n+ξ)σ(a4−h3)+k2β2(
]
x2(s)+ε)ds ,(54)
hence, y(nσ)≤y(0+)ζ1n, so y(nσ)→0 as n→∞. Therefore, y(t)→0 as t→∞.
Mathematics 2023,11, 3254 12 of 25
Then, we prove that
x1(t)→
]
x1(t)
,
x2(t)→
]
x2(t)
, as
t→∞
. For an
ε1>
0 small
enough, there exists
t0>
0, such that 0
<y(t)<ε1
for all
t>t0
. Without loss of generality,
we assume that 0 <y(t)<ε1for all t≥0, so we have
−d1x2(t)−β1ε1x2(t)≤dx2(t)
dt ≤ −d1x2(t), (55)
and
−(d2+h2)x2(t)−β2ε1x2(t)≤dx2(t)
dt ≤ −(d2+h2)x2(t), (56)
and
H12(t)≤x1(t)≤H13 (t)
,
H22(t)≤x2(t)≤H23 (t)
and
H12(t)→
^
H12(t)
,
H13(t)→
]
x1(t)
,
H22(t)→
^
H22(t)
,
H23(t)→
]
x2(t)
as
t→∞
; here,
(H12(t)
,
H22(t))
and
(H22(t)
,
H23(t))
are the solutions of
dH12 (t)
dt =H12(t)(a1−b1H12 (t)),
dH22 (t)
dt =−d1H22(t)−β1ε1H22 (t),
t∈(nσ,(n+ξ)σ],
∆H12(t) = −m1H12 (t),
∆H22(t) = −m2H22 (t),)t= (n+ξ)σ,
dH12 (t)
dt =H12(t)[(a3−h1)−b3H12(t))],
dH22 (t)
dt =−(d2+h2)H22(t)−β2ε1H22 (t),
t∈((n+ξ)σ,(n+1)σ],
∆H12(t) = d(H22 (t)−H12(t)),
∆H22(t) = d(H12 (t)−H22(t)),)t= (n+1)σ,
(57)
and
dH13 (t)
dt =H13(t)(a1−b1H13 (t)),
dH23 (t)
dt =−d1H23(t)
t∈(nσ,(n+ξ)σ],
∆H13(t) = −m1H13 (t),
∆H23(t) = −m2H23 (t),)t= (n+ξ)σ,
dH13 (t)
dt =H13(t)[(a3−h1)−b3H13(t))],
dH23 (t)
dt =−(d2+h2)H23(t),
t∈((n+ξ)σ,(n+1)σ],
∆H13(t) = d(H23 (t)−H13(t)),
∆H23(t) = d(H13 (t)−H23(t)),)t= (n+1)σ,
(58)
respectively. Similarly to Lemma 4, the periodic solution of
(
57
)
is globally asymptotically
stable, and it can be expressed as
^
H12(t) =
a1H12∗ea1(t−nσ)
a1+b1H12∗(ea1(t−nσ)−1),t∈(nσ,(n+ξ)σ],
(a3−h1)H12∗∗ e(a3−h1)(t−(n+ξ)σ)
(a3−h1) + b3H12∗∗ (e(a3−h1)(t−(n+ξ)σ)−1),
t∈((n+ξ)σ,(n+1)σ],
^
H22(t) = (H22 ∗e−(d1+β1ε1)(t−nσ),t∈(nσ,(n+ξ)σ],
H22∗∗ e−(d2+h2+β2ε1)(t−(n+ξ)σ),t∈((n+ξ)σ,(n+1)σ],
(59)
Mathematics 2023,11, 3254 13 of 25
here
H12∗=B{(1−d)(D1+A)−[1+ (1−2d)AD1]}
C[1−(1−d)D1],
H22∗=dB{(1−d)(D1+A)−[1+ (1−2d)AD1]}
C[1−(1−d)D1][(1−d)−(1−2d)D1],
(60)
with condition (1−d)(D1+A)>[1+ (1−2d)AD1],
D1= (1−m2)e−(d1+β1ε1)ξσ−(d2+h2+β2ε1)(1−ξ)σ<1
and
H12∗∗ =(1−m1)a1ea1ξ σ H12∗
a1+b1(ea1ξσ −1)H12 ∗,
H22∗∗ = (1−m2)e−(d1+β1ε1)ξσ H22 ∗.
(61)
Therefore, we obtain the following results. For any
ε>
0, there exists a
t1>
0,
t>t1
such that (^
H12(t)−ε<x1(t)<
^
H13(t) + ε,
^
H22(t)−ε<x2(t)<
^
H23(t) + ε.(62)
Let ε1→0, so we have
(]
x1(t)−ε<x1(t)<
]
x1(t) + ε,
]
x2(t)−ε<x2(t)<
]
x2(t) + ε,(63)
for a tlarge enough, then x1(t)→
]
x1(t)and x2(t)→
]
x2(t)as t→∞.
Theorem 2. If
Az>1, (64)
and
(1−d)(A+DEz)<1, (65)
and
ln 1
1−m3
>a2ξσ + (a4−h3)(1−ξ)σ−a2+b2(ea2ξ σ −1)y∗
a2
−a4−h3+b4(e(a4−h3)(1−ξ)σ−1)y∗∗
a4−h3
(66)
hold, the prey-extinction periodic solution
(
0, 0,
g
y(t))
of system
(
6
)
is globally asymptotically stable,
where Ez=eRξσ
0−β1g
y(s)ds+Rσ
ξσ −β2g
y(s)ds, y∗and y∗∗ see (29)and (35).
Theorem 3. If
Az<1, (67)
and
(1−d)(A+D)<1(68)
hold, the trivial solution (0, 0, 0)of system (6)is globally asymptotically stable.
Because the proofs of Theorems 2 and 3 are similar to Theorem 1, we omit it here. In
the last part of this section, we study the permanence of system (6).
Mathematics 2023,11, 3254 14 of 25
Theorem 4. If (36),(37)and
ln 1
1−m3
<a2ξσ + (a4−h3)(1−ξ)σ+k1β1(1−e−d1ξ σ )
d1
x2∗
+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗
(69)
hold, the system (6)is permanent, where x2∗and x2∗∗ see (13)and (24).
Proof.
By Lemma 1,
x1(t)≤M0
,
x2(t)≤M0
,
y(t)≤M0
for all
t
s large enough. We
assume that x1(t)≤M0,x2(t)≤M0,y(t)≤M0for t≥0. Therefore,
dx1(t)
dt ≥ −d1x2(t)−β1M0x2(t), (70)
and dx2(t)
dt ≥ −(d2+h2)x2(t)−β2M0x2(t), (71)
and
x1(t)≥H14(t)
,
x2(t)≥H24(t)
, and
H14(t)→
^
H14(t)
,
H24(t)→
^
H24(t)
as
t→∞
; here,
(H14(t),H24 (t)) is the solution of the following comparison equation:
dH14 (t)
dt =H14(t)(a1−b1H14 (t)),
dH24 (t)
dt =−d1H24(t)−β1M0H24 (t),
t∈(nσ,(n+ξ)σ],
∆H14(t) = −m1H14 (t),
∆H24(t) = −m2H24 (t),)t= (n+ξ)σ,
dH14 (t)
dt =H14(t)[(a3−h1)−b3H14(t))],
dH24 (t)
dt =−(d2+h2)H24(t)−β2M0H22 (t),
t∈((n+ξ)σ,(n+1)σ],
∆H14(t) = d(H24 (t)−H14(t)),
∆H24(t) = d(H14 (t)−H24(t)),)t= (n+1)σ,
(72)
with
^
H14(t) =
a1H14∗ea1(t−nσ)
a1+b1H14∗(ea1(t−nσ)−1),t∈(nσ,(n+ξ)σ],
(a3−h1)H14∗∗ e(a3−h1)(t−(n+ξ)σ)
(a3−h1) + b3H14∗∗ (e(a2−h1)(t−(n+ξ)σ)−1),t∈((n+ξ)σ,(n+1)σ],
^
H24(t) = (H24 ∗e−(d1+β1M0)(t−nσ),t∈(nσ,(n+ξ)σ],
H24∗∗ e−(d2+h2+β2M0)(t−(n+ξ)σ),t∈((n+ξ)σ,(n+1)σ],
(73)
here
H14∗=B{(1−d)(D2+A)−[1+ (1−2d)AD2]}
C[1−(1−d)D2],
H24∗=dB{(1−d)(D2+A)−[1+ (1−2d)AD2]}
C[1−(1−d)D2][(1−d)−(1−2d)D2],
(74)
with condition (1−d)(D2+A)>[1+ (1−2d)AD2],
D2= (1−m2)e−(d1+β1M0)ξσ−(d2+h2+β2M0)(1−ξ)σ<1 (75)
Mathematics 2023,11, 3254 15 of 25
and
H14∗∗ =(1−m1)a1ea1ξ σ H14∗
a1+b1(ea1ξσ −1)H14 ∗,
H24∗∗ = (1−m2)e−(d1+β1M0)ξσ H24 ∗.
(76)
Therefore, for any ε2>0, we have
(x1(t)>
^
H14(t)−ε2,
x2(t)>
^
H24(t)−ε2,(77)
for a tlarge enough. So,
x1(t)≥a1ea1ξσ H14 ∗
a1+b1(ea1ξσ −1)H14 ∗+(a3−h1)e(a3−h1)(1−ξ)σH14∗∗
(a3−h1) + b3(e(a3−h1)(1−ξ)σ−1)H14∗∗ −ε2=Mx,
x2(t)≥e−(d1+β1M0)ξσ H24 ∗+e−(d2+h2+β2M0))(1−ξ)σH24∗∗ −ε2=My.
(78)
We only need to find
mz>
0, such that
y(t)≥mz
for a
t
large enough. We select
mz1>0, ε3>0 small enough, such that
ζ2= (1−m3)eR(n+ξ)σ
nσa2−b2mz1+k1β1(Hy(t)−ε3)ds R(n+1)σ
(n+ξ)σ(a4−h3)−b4mz1+k2β2(Hy(t)−ε3)ds >1. (79)
Next, we prove that y(t)<mz1cannot hold for all t≥0, otherwise
dx1(t)
dt =x1(t)(a1−b1x1(t)),
dx2(t)
dt ≥ −d1x2(t)−β1mz1x2(t),
t∈(nσ,(n+ξ)σ],
∆x1(t) = −m1x1(t),
∆x2(t) = −m2x2(t),)t= (n+ξ)σ,
dx1(t)
dt =x1(t)[(a3−h1)−b3x1(t))],
dx2(t)
dt ≥ −(d2+h2)x2(t)−β2mz1x2(t),
t∈((n+ξ)σ,(n+1)σ],
∆x1(t) = d(x2(t)−x1(t)),
∆x2(t) = d(x1(t)−x2(t)),)t= (n+1)σ.
(80)
By Lemma 3, we have
x1(t)≥Hx(t)
,
x2(t)≥Hy(t)
and
Hx(t)→Hx(t)
,
Hy(t)→
Hy(t)
as
t→∞
; here,
(Hx(t)
,
Hy(t))
is the solution of the following comparison equation:
dHx(t)
dt =Hx(t)(a1−b1Hx(t)),
dHy(t)
dt =−d1Hy(t)−β1mz1Hy(t),
t∈(nσ,(n+ξ)σ],
∆Hx(t) = −m1Hx(t),
∆Hy(t) = −m2Hy(t),)t= (n+ξ)σ,
dHx(t)
dt =Hx(t)[(a3−h1)−b3Hx(t))],
dHy(t)
dt =−(d2+h2)Hy(t)−β2mz1Hy(t),
t∈((n+ξ)σ,(n+1)σ],
∆Hx(t) = d(Hy(t)−Hx(t)),
∆Hy(t) = d(Hx(t)−Hy(t)),)t= (n+1)σ,
(81)
Mathematics 2023,11, 3254 16 of 25
with
Hx(t) =
a1Hx∗ea1(t−nσ)
a1+b1Hx∗(ea1(t−nσ)−1),t∈(nσ,(n+ξ)σ],
(a3−h1)Hx∗∗e(a3−h1)(t−(n+ξ)σ)
(a3−h1) + b3Hx∗∗(e(a3−h1)(t−(n+ξ)σ)−1),
t∈((n+ξ)σ,(n+1)σ],
Hy(t) =
Hy∗e−(d1+β1mz1)(t−nσ),t∈(nσ,(n+ξ)σ],
Hy∗∗e−(d2+h2+β2mz1)(t−(n+ξ)σ),t∈((n+ξ)σ,(n+1)σ],
(82)
here
Hx∗=B[(1−A+dA)(D3−1)−dD3(1−A)]
C[1−(1−d)D3],
Hy∗=dB[(1−A+dA)(D3−1)−dD3(1−A)]
C[1−(1−d)D3][(1−d) + (2d−1)D3],
(83)
with (1−A+dA)(D3−1)>dD3(1−A),
D3= (1−m2)e−(d1+β1mz1)ξσ−(d2+h2+β2mz1)(1−ξ)σ<1 (84)
and
Hx∗∗ =(1−m1)a1ea1ξ σ Hx∗
a1+b1(ea1ξσ −1)Hx∗,
Hy∗∗ = (1−m2)e−(d1+β1mz1)ξ σ Hy∗.
(85)
There exists a T1>0 such that for t≥T1,
(x1(t)≥Hx(t)≥Hx(t)−ε3,
x2(t)≥Hy(t)≥Hy(t)−ε3,(86)
and
dy(t)
dt ≥a2y(t)−b2mz1y(t) + k1β1(Hy(t)−ε3)y(t),t∈(nσ,(n+ξ)σ],
∆y(t) = −m3y(t),t= (n+ξ)σ,
dy(t)
dt ≥(a4−h3)y(t)−b4mz1y(t) + k2β2(Hy(t)−ε3)y(t),t∈((n+ξ)σ,(n+1)σ],
∆y(t) = 0, t= (n+1)σ.
(87)
Let
N1∈N
and
N1τ>T1
, integrating system
(
87
)
on
(nσ
,
(n+
1
)σ]
,
n≥N1
, and we
have
y((n+1)σ)≥(1−m3)y(nτ)eR(n+ξ)σ
nτa2−b2mz1+k1β1(Hy(t)−ε3)ds+R(n+1)σ
(n+ξ)σ(a4−h3)−b4mz1+k2β2(Hy(t)−ε3)ds
=y(nσ)ζ2,
(88)
then
z((N1+k)σ)≥z(N1σ)ζ2k→∞
as
k→∞
, which is in contradiction to the bounded-
ness of
y(t)
. Hence, there exists a
t1>
0 such that
y(t1)≥mz1
. If
y(t)≥mz1
, which holds
for all t>t1, then we are done. Otherwise, y(t)<mz1for some t>t1.
Let t∗=inf
t≥t1
{y(t)<mz1}; there are two possible cases for t∗.
Case1 t∗= (n1+ξ)σ
,
n1∈Z+
, we have
y(t)≥mz1
for
t∈[t1
,
t∗]
. Since
y(t)
is continuous,
we can obtain y(t∗) = mz1. Select n2,n3∈Z+, such that
(1−m3)n2en2ρσζ2n3>(1−m3)n2e(n2+1)ρσ ζ2n3>1, (89)
here
ρ=min{a2−b2mz1
,
a4−b4mz1−h3}<
0. By setting
T0= (n2+n3)σ
, it can be
claimed that there exists
t2∈(t∗
,
t∗+T0]
such that
y(t2)≥mz1
. Otherwise,
y(t)<mz1
,
t∈(t∗,t∗+T0].
Consider
(
4.46
)
with initial value
Hx(t∗+) = x1(ξ+)
,
Hy(t∗+) = x2(ξ+)
;
Mathematics 2023,11, 3254 17 of 25
we have
x2(t)≥Hy(t)≥Hy(t)−ε3
for
t∗+n2σ≤t≤t∗+T0
. And this implies that
(
87
)
will hold for t∈[t∗+n2σ,t∗+T0], then
y(t∗+T0)≥y(t∗+n2σ)ζ2n3.(90)
From system (6), we have
dy(t)
dt ≥ρy(t),t6= (n+ξ)σ,
∆y(t) = −m3y(t),t= (n+ξ)σ.
(91)
Integrating (91)on [t∗,t∗+n2σ], we have
y(t∗+n2σ)≥(1−m3)n2mz1en2ρσ.(92)
Then, by (90)and (92), we have
y(t∗+T0)≥(1−m3)n2mz1en2ρσζ2n3>mz1, (93)
which contradicts the priori condition of y(t)<mz1.
Let
t=inf
t>t∗{y(t)≥mz1}
, then
y(t) = mz1
. Since
(
87
)
holds for
t∈(t∗
,
t]
and to
integrate in (t∗,t], we obtain
y(t)≥y(t∗+)eσ(t−t∗)≥(1−m3)n2+n3mz1e(n2+n3)ρσ ,e
m. (94)
Since
y(t)≥e
m
for
t∈(t∗
,
t]
, and the same argument can be continued for
t>t
,
y(t)≥e
mfor all t>t1.
Case2 t∗6= (n1+ξ)σ
,
n∈Z+
, then
y(t)≥mz1
for
t∈[t1
,
t∗)
and
y(t∗) = mz1
. Sup-
pose
t∗∈((n10+ξ)σ
,
(n10+ξ+
1
)σ)
,
n10∈Z+
, then there are two possible cases for
t∈(t∗,(n10+ξ+1)σ).
Case2a y(t)≤mz1
for all
t∈(t∗
,
(n10+ξ+
1
)σ)
. Similar to Case 1, we can prove that there
must be a t20∈[(n10+ξ+1)σ,(n10+ξ+1)σ+T0], such that y(t20)>mz1.
Let
e
t=inf
t>t∗{y(t)>mz1}
, then
y(t)≤mz1
for
t∈(t∗
,
e
t)
and
y(e
t) = mz1
. Note that
(
66
)
holds for
t∈(t∗,e
t), so we have
y(t)≥eρ(t−t∗)≥(1−m3)n2+n3mz1e(n2+n3+1)ρσ ,e
m0<e
m. (95)
And the same argument can be continued for t>e
t, since y(e
t)≥mz1.
Case2b
There is a
t∗∈(t∗
,
(n10+ξ+
1
)σ)
, such that
y(t)>mz1
. Let
ˆ
t=inf
t>t∗{y(t)>mz1}
,
then
y(t)≤mz1
for
t∈[t∗
,
ˆ
t)
and
y(ˆ
t) = mz1
.
(
91
)
holds for
t∈[t∗
,
ˆ
t)
, and integrating it
on [t∗,ˆ
t), we have
y(t)≥y(t∗)eρ(t−t∗)≥mz1eρ(t−t∗)≥mz1eρσ >e
m. (96)
Because
y(ˆ
t)≥mz1
, the same arguments can be continued for
t>ˆ
t
. Hence,
y(t)≥e
m
for all t≥t1.
5. Numerical Simulations and Discussion
This section is devoted to confirming the theoretical results obtained in the above
sections through numerical simulations. Since the theoretical results depend on harvesting,
the simulations are implemented by considering different values of transient impulsive
harvesting rate mi(i=1, 2, 3)and nontransient impulsive harvesting rate hi(i=1, 2, 3).
Mathematics 2023,11, 3254 18 of 25
Example 1.
For biological considerations, all the parameters are assumed to be positive. And
referring to references [
26
,
27
], the model parameters are set to
a1=
0.7,
b1=
0.65,
d1=0.3,
β1=
0.3,
a2=
0.4,
b2=
0.35,
k1=
0.4,
m1=
0.2,
m2=
0.2,
m3=
0.4,
a3=
0.8,
h1=0.1,
b3=
0.5,
d2=
0.3,
h2=
0.1,
β2=
0.6,
a4=
0.6,
h3=
0.1,
b4=
0.4,
k2=
0.5,
d=0.55,
l=
0.56,
σ=
2. Then,
(
1
−d)(A+D) =
1.6408
>
0.8696
=
1
+ (
1
−
2
d)AD
,
(
1
−
2
d)AD =−
0.1304
<
1,
ln 1
1−m3=
0.5108
<
0.9966
=a2ξσ + (a4−h3)(
1
−ξ)σ+
k1β1(1−e−d1ξσ )
d1x2∗+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗
, the conditions of Theorem 4, are satisfied with ini-
tial value
x1(0) = 1, x2(0) = 1, y(0) = 0.5,
and system
(
6
)
is permanent (see Figure 1). That is,
the prey and predator populations will coexist.
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 1.
Dynamical behavior of the permanence of system
(
6
)
: (
a
–
c
) time series of populations x, y,
and z; (d) phase portrait of system (6).
5.1. The Effect of the Transient Impulsive Harvesting on Populations
Example 2.
Let
m3=
0.7 and keep fixed the values of other parameters, as in Figure 1. Then,
(
1
−d)(A+D) =
1.6408
>
0.8696
=
1
+ (
1
−
2
d)AD
,
(
1
−
2
d)AD =−
0.1304
<
1,
(
1
−d)(AE +D) =
0.5682
<
1,
ln 1
1−m3=
1.2040
>
0.9966
=a2ξσ + (a4−h3)(
1
−ξ)σ+
k1β1(1−e−d1ξσ )
d1x2∗+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗
, and conditions (36)–(39) hold. From Theorem 2,
the predator-extinction periodic solution
(
]
x1(t)
,
]
x2(t)
, 0
)
of system
(
6
)
is globally asymptotically
stable (see Figure 2).
Mathematics 2023,11, 3254 19 of 25
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 2.
Dynamical behavior of system
(
6
)
on predator-extinction periodic solution with
m3=0.7: (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
Example 3.
Let
m1=
0.6,
m2=
0.5, and keep fixed the values of other parameters, as in
Figure 1
. Then,
Az=
1.4582
>
1,
(
1
−d)(A+DEz) =
0.7991
<
1,
ln 1
1−m3=
0.5108
>
−
1.8039
=a2ξσ + (a4−h3)(
1
−ξ)σ−a2+b2(ea2ξσ −1)y∗
a2−a4−h3+b4(e(a4−h3)(1−ξ)σ−1)y∗∗
a4−h3
, and con-
ditions
(64)–(66)
hold. From Theorem 2, the prey-extinction periodic solution
(
0, 0,
g
y(t))
of sys-
tem (6)is globally asymptotically stable (see Figure 3).
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
Figure 3. Cont.
Mathematics 2023,11, 3254 20 of 25
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 3.
Dynamical behavior of system
(
6
)
on prey-extinction periodic solution with
m1=
0.6,
m2=0.5: (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
Example 4.
Let
m1=
0.6,
m2=
0.5,
m3=
0.7, and keep fixed the values of other parameters, in
as Figure 1. Then,
Az=
0.7291
<
1,
(
1
−d)(A+D) =
0.8430
<
1, and conditions
(
67
)
and
(
68
)
hold. From Theorem 3, the trivial solution
(
0, 0, 0
)
of system
(
6
)
is globally asymptotically
stable (see Figure 4).
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 4.
Dynamical behavior of system
(
6
)
on trivial solution with
m1=
0.6,
m2=
0.5,
m3=0.7 : (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
Comparing Figures 1and 2, we can know that when
m3=
0.4, the prey and predator
populations coexist, while when
m3=
0.7, the predator population goes extinct. Comparing
Figures 1and 3, we can know that when
m1=
0.2,
m2=
0.2, the prey and predator
populations coexist, while when
m1=
0.6,
m2=
0.5, the prey populations go extinct. From
Figure 4, we can see that all the populations go extinct as m1=0.6, m2=0.5, m3=0.7.
Mathematics 2023,11, 3254 21 of 25
5.2. The Effect of Nontransient Impulsive Harvesting on Populations
Example 5.
Let
h3=
0.9, and keep fixed the values of other parameters, as in Figure 1. Then,
(
1
−d)(A+D) =
1.6408
>
0.8696
=
1
+ (
1
−
2
d)AD
,
(
1
−
2
d)AD =−
0.1304
<
1,
(
1
−d)(AE +D) =
0.5682
<
1,
ln 1
1−m3=
0.5108
>
0.2926
=a2ξσ + (a4−h3)(
1
−ξ)σ+
k1β1(1−e−d1ξσ )
d1x2∗+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗
, and conditions
(
36
)
–
(
39
)
hold. From Theorem 2,
the predator-extinction periodic solution
(
]
x1(t)
,
]
x2(t)
, 0
)
of system
(
6
)
is globally asymptotically
stable (see Figure 5).
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 5.
Dynamical behavior of system
(
6
)
on predator-extinction periodic solution with
h3=0.9: (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
Example 6.
Let
h1=
0.9,
h2=
0.9, and keep fixed the values of other parameters, as in
Figure 1.
Then
Az=
1.4582
>
1,
(
1
−d)(A+DEz) =
0.7768
<
1,
ln 1
1−m3=
0.5108
>
−
1.8039
=a2ξσ + (a4−h3)(
1
−ξ)σ−a2+b2(ea2ξσ −1)y∗
a2−a4−h3+b4(e(a4−h3)(1−ξ)σ−1)y∗∗
a4−h3
, and con-
ditions
(64)–(66)
hold. From Theorem 2, the prey-extinction periodic solution
(
0, 0,
g
y(t))
of
system (6)is globally asymptotically stable (see Figure 6).
Example 7.
Let
h1=
0.9,
h2=
0.9,
h3=
0.9, and keep fixed the values of other parameters, as in
Figure 1
. Then,
Az=
0.7212
<
1,
(
1
−d)(A+D) =
0.8115
<
1, and conditions
(
67
)
and
(
68
)
hold. From Theorem 3, the trivial solution
(
0, 0, 0
)
of system
(
2.1
)
is globally asymptotically stable
(see Figure 7).
Mathematics 2023,11, 3254 22 of 25
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80 90
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 6.
Dynamical behavior of system
(
6
)
on prey-extinction periodic solution with
h1=
0.9,
h2=0.9: (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
Comparing Figures 1and 4, we can know that when
h3=
0.1, the prey and predator
populations coexist, while when
h3=
0.9, the predator population go extinct. Comparing
Figures 1and 5, we can know that when
h1=
0.1,
h2=
0.1, the prey and predator
populations coexist, while when
h1=
0.9,
h2=
0.9, the prey populations go extinct. From
Figure 7, we can see that all the populations go extinct as h1=0.9, h2=0.9, h3=0.9.
Figures 1–7show the global asymptotic stability of the boundary periodic solutions
and the permanent extinction of system
(
6
)
under the control of the transient/nontransient
impulse harvesting rate, respectively. It is clear that with increasing transient/ nontransient
impulsive harvesting rate, predator or prey populations cannot survive due to higher
harvesting rate. The values of
m3
,
h3
, will not only directly affect the survival of the
predator but also have an indirect effect on the prey. When
m3
or
h3
keeps increasing and
exceeding the threshold, the predator population goes extinct and the population density of
the prey populations increase accordingly. Similarly, The decrease in the density of predator
population is observed as the prey populations go extinct, which
is biologically reasonable.
Mathematics 2023,11, 3254 23 of 25
(a) (b)
0 10 20 30 40 50 60 70 80
t
0
0.2
0.4
0.6
0.8
1
x1(t)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x2(t)
(c) (d)
0 10 20 30 40 50 60 70 80
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(t)
Figure 7.
Dynamical behavior of system
(
6
)
on trivial solution with
h1=
0.9,
h2=
0.9,
h3=0.9: (a–c) time series of populations x, y, and z; (d) phase portrait of system (6).
6. Conclusions
In this paper, we propose a new predator–prey model to study the effects of tran-
sient/nontransient harvesting and pulse diffusion between prey on the prey and predator’s
survival. Here, the predators live in their territory, which is patch 2, but the prey can impul-
sively diffuse between two patches. We focus on analyzing the dynamics of the investigated
system generated by transient and nontransient impulsive harvesting to understand how
predator and prey populations change when the system has an effect of harvesting. The
main results of the present study are provided below:
1. All solutions of system (6)are uniformly ultimately bounded.
2.
If (36)–(39) hold, the solution
(
]
x1(t)
,
]
x2(t)
, 0
)
of system
(
6
)
is globally asymptotically
stable.
3.
If (64)–(66) hold, the solution
(
0, 0,
g
y(t))
of system
(
6
)
is globally asymptotically stable.
4. If (67)–(68) hold, the trivial solution of system (6)is globally asymptotically stable.
5. The permanent conditions of system (6)have also been established, that is
(1−d)(A+D)>[1+ (1−2d)AD],(1−2d)AD <1,
and
ln 1
1−m3
<a2ξσ + (a4−h3)(1−ξ)σ+k1β1(1−e−d1ξ σ )
d1
x2∗
+k2β2(1−e−(d2+h2)(1−ξ)σ)
(d2+h2)x2∗∗.
In addition, from numerical simulations and theorems, we can deduce that there exist
a predator transient impulsive harvesting threshold
m3∗
and a nontransient impulsive
harvesting threshold
h3∗
. When
m3>m3∗
or
h3>h3∗
, the predator population
z
goes
Mathematics 2023,11, 3254 24 of 25
extinct. When
m3<m3∗
or
h3<h3∗
, system
(
6
)
is permanent. In addition, there must exist
thresholds
m1∗
,
m2∗
and
h1∗
,
h2∗
. When
m1>m1∗
and
m2>m2∗
, or
h1>h1∗
and
h2>h2∗
,
the prey populations
x
and
y
go extinct. When
m1<m1∗
and
m2<m2∗
, or
h1<h1∗
and
h2<h2∗
, system
(
6
)
is permanent. Therefore, we must choose a suitable harvesting rate
smaller than the value of the harvesting threshold when hunting both prey and predator for
economic interest. Reducing the amount of transient or nontransient impulsive harvesting
is significant for preventing population extinction so as to maintain ecological balance.
In future work, we can continue to study the optimal harvest strategy of
system (6)
to ex-
plore the maximum sustainable yield and the corresponding harvest effort of
system (6)[28,29].
We can also consider impulsive delayed harvesting or stage structure of prey/predator
populations, which will lead to richer dynamics [
30
]. In addition, trying to solve system
(
6
)
using an intelligent computational solver, or different numerical methods such as the
Galerkin method or Legendre wavelet algorithm will also be interesting work [31–33].
Author Contributions:
Q.Q., conceptualization, formal analysis, writing—original draft; X.D., vali-
dation; J.J., writing—review and editing. All authors have read and agreed to the published version
of the manuscript.
Funding:
This paper was supported by National Natural Science Foundation of China (12261018,
11761019, 11361014), the Science Technology Foundation of Guizhou Education Department (20175736-
001), and the Project of High Level Creative Talents in Guizhou Province (No.20164035).
Data Availability Statement: Not applicable.
Acknowledgments:
The authors thank the editor and anonymous referees for useful comments that
led to a great improvement of the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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