# On a Periodic Time-Dependent Model of Population Dynamics with Stage Structure and Impulsive Effects

Authors: Kaiyuan Liu; Lansun Chen
Journal: Discrete Dynamics in Nature and Society
 (2008)

---

## Abstract

We consider a periodic time-dependent predator-prey system with stage structure and impulsive harvesting, in which the prey has a life history that takes them through two stages, immature and mature. A set of sufficient and necessary conditions which guarantee the permanence of the system are obtained. Finally, we give a brief discussion of our results.


---

## Body

## 1. Introduction

In the natural
world, there are many species whose individual members have a life history that
takes them through two stages, immature and mature. In particular, we have
mammalian populations and some amphibious animals in mind, which exhibit these
two stages. From the view point of mathematics, the description of the stage
structure of the population in the life history is also an interesting problem
in population dynamics. The permanence and extinction of species are
significant concepts for those stage-structured population dynamical systems.
Recently, stage structure models have been  studied by many authors [1–3]. This is not only because
they are much more simple than the models governed by partial differential
equations but also because they can exhibit
phenomena similar to those of partial differential models [4], and many important
physiological parameters can be incorporated. Much research has been devoted to
the models concerning single-species population growth with the stage structure
of immature and mature [5, 6]. Two species models with stage structure were
investigated by Wang and Chen (1997), Magnusson (1999), Xiao and Chen (2003),
as well as Cui and Song (2004). Also, Zhang, Chen,
and Neumann (2000) proposed the following autonomous stage structure
predator-prey system:(1.1)
x
1
′
=
α
x
2
−
r
1
x
1
−
β
x
1
−
η
x
1
2
−
β
1
x
1
x
3
,
x
2
′
=
β
x
1
−
r
2
x
2
,
x
3
′
=
x
3
(
−
r
+
k
β
1
x
1
−
η
1
x
3
)
,where α, β, β
1, η, η
1, r, r
1, r
2, and k are all positive constants, k is a digesting constant.On the other
hand, since biological and environmental parameters are naturally subject to
fluctuation in time, the effects of a periodically varying environment are
considered as important selective forces on systems in a fluctuating
environment. So more realistic and interesting models should
take into account the seasonality of the changing environment
[7, 8]. This motivated Cui and Y.
Takeuchi (2006) to consider the following periodically nonautonomous predator-prey
model with stage structure for prey:(1.2)
x
˙
1
=
a
(
t
)
x
2
−
b
(
t
)
x
1
−
d
(
t
)
x
1
2
−
p
(
t
)
ϕ
(
t
,
x
1
)
x
1
y
,
x
˙
2
=
c
(
t
)
x
1
−
f
(
t
)
x
2
2
,
y
˙
=
y
[
−
g
(
t
)
+
h
(
t
)
ϕ
(
t
,
x
1
)
x
1
−
q
(
t
)
y
]
,in which a
(
t
), b
(
t
), c
(
t
), d
(
t
), f
(
t
), g
(
t
), h
(
t
), p
(
t
), q
(
t
),
and ϕ
(
t
,
x
1
) are all continuous ω-periodic functions; a
(
t
), b
(
t
), c
(
t
), d
(
t
), f
(
t
), h
(
t
), p
(
t
), q
(
t
) are all positive; g
(
t
), ϕ
(
t
,
x
1
) are nonnegative; x
1 and x
2 denote the density of immature and mature
population (prey), respectively; and y is the density of the predator that only prey
on x
1 (immature prey). They provided a set of
sufficient and necessary conditions to guarantee the permanence of the above
system.Systems with
impulsive effects describing evolution processes are characterized by the fact
that at certain moments of time, they experience a change of state abruptly.
Processes of such type are studied in almost every domain of applied science.
Impulsive equations [9, 10] have been recently used in population dynamics in
relation to impulsive vaccination [11, 12], population ecology [13, 14], the chemotherapeutic
treatment of disease [15], birth pulses [16], as well as the theory of
the chemostat [17].Let us assume
that the predator population is affected by harvesting (e.g., fishing or
hunting). Further, as we all know that the harvesting does not occur
continuously, that is, the harvesting occurs in regular pulses, then let us
assume that at some fixed moments, the predator
population in system (1.2) is subject to a perturbation which incorporates the
proportional decrease. After a perturbation at step τ
k
>
0
⁢
(
k
∈
N
),
the size of the population y
(
τ
k
+
) becomes(1.3)
y
(
τ
k
+
)
=
(
1
−
u
k
)
y
(
τ
k
)
,where y
(
τ
k
) is the size of the predator population at step τ
k before the impulsive perturbation, and 0
<
u
k
<
1 represents the rate at which the predator is
harvested.In this
article, we extend the model (1.2) to the case when the predator population is
omnivorous and affected by impulsive effects, which is governed by the
following system:(1.4)
x
˙
1
=
a
(
t
)
x
2
−
b
(
t
)
x
1
−
d
(
t
)
x
1
2
−
p
1
(
t
)
ϕ
1
(
t
,
x
1
)
x
1
2
y
,

t
⁢
≠
τ
k
,
x
˙
2
=
c
(
t
)
x
1
−
f
(
t
)
x
2
2
−
p
2
(
t
)
ϕ
2
(
t
,
x
2
)
x
2
2
y
,

t
⁢
≠
τ
k
,
y
˙
=
y
[
g
(
t
)
+
∑
i
=
1
2
h
i
(
t
)
ϕ
i
(
t
,
x
i
)
x
i
2
−
q
(
t
)
y
(
t
)
]
,

t
⁢
≠
τ
k
,
Δ
y
(
t
)
=
−
u
k
y
(
t
)
,

t
=
τ
k
,
⁢
k
∈
N
,with the initial value
conditions(1.5)
x
1
(
0
)
=
x
10
≥
0,


x
2
(
0
)
=
x
20
≥
0,


y
(
0
)
=
y
0
≥
0,in which x
1 and x
2 are the densities of immature and mature prey,
respectively; and y is the density of the predator that can prey
on x
1
and
x
2.
When its favorite food is severely scarce, population y can eat other
resources: Δ
y
(
t
)
=
y
(
t
+
)
−
y
(
t
).
Also, there exists a positive integer q such that(1.6)
u
k
+
q
=
u
k
,


τ
k
+
q
=
τ
k
+
ω
,
0
=
τ
0
<
τ
1
<
τ
2
<
⋯
<
τ
k
<
τ
k
+
1
<
⋯
,

k
∈
N
. Here, x
i
2
ϕ
i
(
t
,
x
i
),
the number of the prey x
i consumed per predator in unit time, is called
the predator functional response. We assume that there exists a positive
constant L such that(1.7)
0
<
ϕ
i
(
t
,
x
i
)
<
L
;

∂
∂
x
i
(
x
i
2
ϕ
i
(
t
,
x
i
)
)
≥
0

for
x
i
>
0
,
i
=
1,2.The last condition in (1.7)
implies that as the prey population increases, the
consumption rate of prey consumed per predator
increases. The birth rate of the immature prey
population is proportional to the existing mature prey population with a
proportionality function a
(
t
).
For the immature prey population, the death rate is proportional to the
existing immature prey population with a proportionality function b
(
t
).
The variable parameter d
(
t
) represents that the immature prey population
is density restriction. The transition rate from immature individuals to the
mature individuals is assumed to be proportional to the existing immature
population with proportionality coefficient c
(
t
).
The death rate of the mature population is of a logistic nature with
proportionality coefficient f
(
t
).
Also, p
i
(
t
) and h
i
(
t
)
(
i
=
1
,
2
) are the coefficients that relate to conversion
rates of the immature and, respectively, mature prey biomass into predator
biomass. The coefficients in (1.4) are all continuous ω-periodic for t
≥
0.
In fact, a
(
t
), b
(
t
), c
(
t
), d
(
t
), f
(
t
), p
i
(
t
), h
i
(
t
),
and q
(
t
) are all strictly positive, and ϕ
i
(
t
,
x
i
) is nonnegative (
i
=
1
,
2
).The organization
of this paper is as follows. In Section2, we provide some preliminary results
which will be useful. In Section 3, we investigate the permanence and
extinction of system (1.4) by using analysis technique. In the last section, we
give a biological example and a brief discussion of our result.


## 2. Preliminary Results

Before stating
and proving our main results, we give the following definitions, notations, and
lemmas which will be useful in the following section.LetR
+
=
[
0
,
∞
), R
+
n
=
{
(
x
1
,
…
,
x
n
)
:
x
i
>
0
,
i
=
1
,
…
,
n
} and α
(
t
) be a continuous ω-periodic function defined on [
0
,
+
∞
),
then we set(2.1)
A
ω
(
α
)
=
ω
−
1
∫
0
ω
α
(
t
)
d
t
,


α
U
=
max
⁡
t
∈
[
0
,
ω
]
α
(
t
)
,


α
L
=
min
⁡
t
∈
[
0
,
ω
]
α
(
t
)
.
Lemma 2.1.
System (1.4)
is dissipative.
Proof.
Define a
functionV
(
t
,
x
1
,
x
2
,
y
) such that(2.2)
V
(
t
,
x
1
,
x
2
,
y
)
=
h
(
x
1
(
t
)
+
x
2
(
t
)
)
+
p
y
(
t
)
,in which h
=
max
⁡
t
∈
[
0
,
ω
]
{
h
1
(
t
)
,
h
2
(
t
)
}, p
=
min
⁡
t
∈
[
0
,
ω
]
{
p
1
(
t
)
,
p
2
(
t
)
}.
After a simple computations, we have(2.3)
D
+
V
|
(
4
)
+
w
¯
V
<
(
h
a
U
+
w
¯
)
x
2
−
h
f
L
x
2
2
+
(
h
c
U
+
w
¯
−
h
b
L
)
x
1
−
h
d
L
x
1
2
+
(
w
¯
+
p
g
U
)
y
−
p
q
L
y
2
,in which w
¯ is a positive constant. Obviously, the
right-hand side of the above inequality is bounded
above for all (
x
1
(
t
)
,
x
2
(
t
)
,
y
(
t
)
)
∈
R
+
3.
Hence,(2.4)
D
+
V
|
(
4
)
+
w
¯
V
<
λ
,where λ is a positive constant. When t
=
τ
k,
we get(2.5)
V
(
τ
k
+
)
≤
V
(
τ
k
)
.According to Lemma 2.2 in [9, page 23 ], we derive that(2.6)
V
(
t
)
=
V
(
0
)
e
−
w
¯
t
+
∫
0
t
λ
e
−
w
¯
(
t
−
s
)
d
s
<
2
λ
w
¯
,

as
t
⟶
∞
,which implies that system (1.4) is
dissipative. This completes the proof.Next, we consider the following two
subsystems of system (1.4):(2.7)
x
˙
1
=
a
(
t
)
x
2
−
b
(
t
)
x
1
−
d
(
t
)
x
1
2
,
x
˙
2
=
c
(
t
)
x
1
−
f
(
t
)
x
2
2
,
(2.8)
y
˙
=
y
[
g
(
t
)
−
q
(
t
)
y
(
t
)
]
,

t
⁢
≠
τ
k
,
Δ
y
(
t
)
=
−
u
k
y
(
t
)
,

t
=
τ
k
,
⁢
k
∈
N
.
Lemma 2.2 ([18]).
The system (2.7) has a
positive ω-periodic solution (
x
1
*
(
t
)
,
x
2
*
(
t
)
) which is globally asymptotically stable with
respect to R
+
2.
Lemma 2.3.
If the following
conditions(2.9)
A
ω
(
g
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
>
exp
⁡
(
−
ω
A
ω
(
g
)
)hold, then system (2.8) has a
unique ω-periodic solution:(2.10)
y
*
(
t
)
=
(
∏
0
<
τ
k
<
t
(
1
/
(
1

−

u
k
)
)
exp
⁡
(
−
∫
0
t
g
(
ξ
)
d
ξ
)
∫
0
ω
∏
s
≤
τ
k
<
ω
(
1
/
(
1

−

u
k
)
)
exp
⁡
(
−
∫
s
ω
g
(
ξ
)
d
ξ
)
q
(
s
)
d
s
1
−
∏
k
=
1
q
(
1
/
(
1
−
u
k
)
)
exp
⁡
(
−
∫
0
ω
g
(
ξ
)
d
ξ
)


+
∫
0
t
∏
s
≤
τ
k
<
t
1
1
−
u
k
exp
⁡
(
−
∫
s
t
g
(
ξ
)
d
ξ
)
q
(
s
)
d
s
)
−
1and for every solution y
(
t
) of
system (2.8),(2.11)
|
y
(
t
)
−
y
*
(
t
)
|
⟶
0

as
t
⟶
∞
.
Proof.
Lety
(
t
)
=
1
/
z
(
t
) and obtain the linear nonhomogeneous impulsive
equation(2.12)
z
˙
(
t
)
=
−
g
(
t
)
z
(
t
)
+
q
(
t
)
,

t
⁢
≠
τ
k
,
z
(
t
+
)
=
1
1
−
u
k
z
(
t
)
,

t
=
τ
k
.Let W
(
t
,
s
)
=
∏
s
≤
τ
k
<
t
(
1
/
(
1
−
u
k
)
)
exp
⁡
(
−
∫
s
t
g
(
ξ
)
d
ξ
) be the Cauchy matrix for the relevant
homogeneous equation, then the solution of (2.12) has the form(2.13)
z
(
t
)
=
W
(
t
,0
)
z
(
0
)
+
∫
0
t
W
(
t
,
s
)
q
(
s
)
d
s
.The solution z
(
t
) will be ω-periodic if z
(
ω
)
=
z
(
0
),
or if(2.14)
(
1
−
W
(
ω
,0
)
)
z
(
0
)
=
∫
0
ω
W
(
ω
,
s
)
q
(
s
)
d
s
.In view of conditions (2.9), (2.14) has a unique solution(2.15)
z
(
0
)
=
∫
0
ω
W
(
ω
,
s
)
q
(
s
)
d
s
1
−
W
(
ω
,0
)
.Then, (2.13) has a unique ω-periodic solution(2.16)
z
*
(
t
)
=
W
(
t
,0
)
∫
0
ω
W
(
ω
,
s
)
q
(
s
)
d
s
1
−
W
(
ω
,0
)
+
∫
0
t
W
(
t
,
s
)
q
(
s
)
d
s
.Hence, (2.8) has a unique ω-periodic solution(2.17)
y
*
(
t
)
=
(
W
(
t
,0
)
∫
0
ω
W
(
ω
,
s
)
q
(
s
)
d
s
1
−
W
(
ω
,0
)
+
∫
0
t
W
(
t
,
s
)
q
(
s
)
d
s
)
−
1
=
(
∏
0
≤
τ
k
<
t
(
1
/
(
1

−

u
k
)
)
exp
⁡
(
−
∫
0
t
g
(
ξ
)
d
ξ
)
∫
0
ω
∏
s
≤
τ
k
<
ω
(
1
/
(
1

−

u
k
)
)
exp
⁡
(
−
∫
s
ω
g
(
ξ
)
d
ξ
)
q
(
s
)
d
s
1
−
∏
k
=
1
q
(
1
/
(
1
−
u
k
)
)
exp
⁡
(
−
∫
0
ω
g
(
ξ
)
d
ξ
)


+
∫
0
t
∏
s
≤
τ
k
<
t
1
1
−
u
k
exp
⁡
(
−
∫
s
t
g
(
ξ
)
d
ξ
)
q
(
s
)
d
s
)
−
1
.From (2.12), we derive
that(2.18)
(
z
(
t
)
−
z
*
(
t
)
)
′
=
−
g
(
t
)
(
z
(
t
)
−
z
*
(
t
)
)
,

τ
k
<
t
≤
τ
k
+
1
.Let(2.19)
κ
≐
∏
k
=
1
q
1
1
−
u
k
exp
⁡
(
−
∫
0
ω
g
(
t
)
d
t
)
.Since conditions (2.9)
hold, we note that 0
<
κ
<
1. Then, from Lemma 2.1, the solution of (2.8) is
governed by(2.20)
|
y
(
t
)
−
y
*
(
t
)
|
≤
2
λ
w
¯
y
*
U
|
z
(
0
+
)
−
z
*
(
0
+
)
|
∏
0
<
τ
k
<
t
1
1
−
u
k
exp
⁡
(
−
∫
0
t
g
(
s
)
d
s
)
,
≤
2
λ
w
¯
y
*
U
|
z
(
0
+
)
−
z
*
(
0
+
)
|
κ
[
t
/
ω
]
+
1
⟶
0

as
t
⟶
∞
,in which λ and w
¯ are defined in Lemma 2.1. This completes the
proof.


## 3. Permanence and Extinction of System

Theorem 3.1.
System
(1.4) is permanent if and only if(3.1)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
>
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
)
,in which (
x
1
*
(
t
)
,
x
2
*
(
t
)
) is the positive ω-periodic solution of system (2.7).We need the following
lemmas to prove Theorem 3.1.Lemma 3.2.
There exist positive constantsM
x and M
y such that(3.2)
limsup
⁡
t
→
∞
⁢
x
i
(
t
)
≤
M
x

(
i
=
1
,
2
)
,


limsup
⁡
t
→
∞
⁢
y
(
t
)
≤
M
y
.
Proof.
In fact, from Lemmas2.1 and 2.3, the proof of the lemma is obvious. This completes the proof.
Lemma 3.3.
There exists a positive constantρ
x
⁢
(
ρ
x
<
M
x
)  such that(3.3)
liminf
⁡
t
→
∞
⁢
x
i
(
t
)
≥
ρ
x

(
i
=
1
,
2
)
.
Proof.
From
Lemma3.2, one notes that there exists a positive constant T
0 such that(3.4)
0
<
y
(
t
)
≤
M
y

for
t
≥
T
0
,we then have(3.5)
x
˙
1
≥
a
(
t
)
x
2
−
b
(
t
)
x
1
−
(
d
(
t
)
+
p
1
(
t
)
L
M
y
)
x
1
2
,
x
˙
2
≥
c
(
t
)
x
1
−
(
f
(
t
)
+
p
2
(
t
)
L
M
y
)
x
2
2for t
≥
T
0
. According to Lemma 2.2, the following
auxiliary equations:(3.6)
u
˙
1
=
a
(
t
)
u
2
−
b
(
t
)
u
1
−
(
d
(
t
)
+
p
1
(
t
)
L
M
y
)
u
1
2
,
u
˙
2
=
c
(
t
)
u
1
−
(
f
(
t
)
+
p
2
(
t
)
L
M
y
)
u
2
2has a globally asymptotically
stable ω-periodic solution (
u
1
*
(
t
)
,
u
2
*
(
t
)
).
Let (
u
1
(
t
)
,
u
2
(
t
)
) be the solution of (3.6) with (
u
1
(
0
)
,
u
2
(
0
)
)
=
(
x
1
(
0
)
,
x
2
(
0
)
).
By the vector comparison theorem [19], we obtain(3.7)
x
i
(
t
)
≥
u
i
(
t
)
,

i
=
1
,
2
⁢
∀
t
≥
0.According
to the global asymptotic stability of (
u
1
*
(
t
)
,
u
2
*
(
t
)
),
for any positive constant ε
(
≤
min
⁡
t
∈
[
0
,
ω
]
{
u
i
*
(
t
)
/
3,
⁢
i
=
1
,
2
}
),
there exists a T
1
(
>
T
0
) such that for all t
≥
T
1,(3.8)
|
u
i
(
t
)
−
u
i
*
(
t
)
|
<
ε
,

i
=
1,2.Hence, for all t
≥
T
1,
we derive that(3.9)
u
i
(
t
)
≥
u
i
*
(
t
)
−
ε
,

i
=
1,2.Let(3.10)
ρ
x
=
max
⁡
t
∈
[
0
,
ω
]
{
u
i
*
(
t
)
2
,
⁢
i
=
1
,
2
}
,then(3.11)
x
i
(
t
)
≥
ρ
x
,

i
=
1,2.Consequently,(3.12)
liminf
⁡
t
→
∞
⁢
x
i
(
t
)
≥
ρ
x
,

i
=
1,2.This completes the proof.
Lemma 3.4.
Suppose that (3.1) holds, then there
exists a positive constant ϱ
y
⁢
(
ϱ
y
<
M
y
) such that(3.13)
limsup
⁡
t
→
∞
⁢
y
(
t
)
≥
ϱ
y
.
Proof.
In view of (3.1), we can choose a
positive constant ε
0 such that(3.14)
A
ω
(
ψ
ε
0
(
t
)
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
>
exp
⁡
(
−
ω
A
ω
(
ψ
ε
0
(
t
)
)
)
,in which(3.15)
ψ
ε
0
(
t
)
=
g
(
t
)
+
∑
i
=
1
2
h
i
(
t
)
ϕ
i
(
t
,
(
x
i
*
(
t
)
−
ε
0
)
)
(
x
i
*
(
t
)
−
ε
0
)
2
−
q
(
t
)
ε
0
.Consider the following system
with a positive parameter μ:
(3.16)
x
˙
1
=
a
(
t
)
x
2
−
b
(
t
)
x
1
−
[
d
(
t
)
+
L
μ
p
1
(
t
)
]
x
1
2
,
x
˙
2
=
c
(
t
)
x
1
−
[
f
(
t
)
+
L
μ
p
2
(
t
)
]
x
2
2
.By Lemma 2.2, system (3.16) has a
positive ω-periodic solution (
x
1
μ
*
(
t
)
,
x
2
μ
*
(
t
)
),
which is globally asymptotically stable. Let (
x
1
μ
(
t
)
,
x
2
μ
(
t
)
) be the solution of (3.16) with initial condition x
i
μ
(
0
)
=
x
i
*
(
0
), i
=
1
,
2,
where (
x
1
*
(
t
)
,
x
2
*
(
t
)
) is the positive periodic solution of (2.7).
Hence, for the above ε
0,
there exists T
2
>
T
1 such that(3.17)
|
x
i
μ
(
t
)
−
x
i
μ
*
(
t
)
|
<
ε
0
4for t
≥
T
2, i
=
1,2. According to the continuity of the solution in
the parameter μ,
we have x
i
μ
(
t
)
→
x
i
*
(
t
)
(
i
=
1
,
2
) uniformly in [
T
2
,
T
2
+
ω
] as μ
→
0.
Hence, for ε
0
>
0,
there exists μ
0
=
μ
0
(
ε
0
)
(
0
<
μ
0
<
ε
0
) such that(3.18)
|
x
i
μ
(
t
)
−
x
i
*
(
t
)
|
<
ε
0
4
,

0
≤
μ
≤
μ
0
,
t
∈
[
T
2
,
T
2
+
ω
], i
=
1,2. Thus, from (3.17) and (3.18), we
get(3.19)
|
x
i
μ
*
(
t
)
−
x
i
*
(
t
)
|
<
ε
0
2
,

0
≤
μ
≤
μ
0
,
t
∈
[
T
2
,
T
2
+
ω
], i
=
1,2. Since x
i
μ
*
(
t
) and x
i
*
(
t
) are all ω-periodic, we have(3.20)
|
x
i
μ
*
(
t
)
−
x
i
*
(
t
)
|
<
ε
0
2
,

0
≤
μ
≤
μ
0
,
⁢
t
≥
0,
⁢
i
=
1,2.
Choose a constantμ
1
⁢
(
0
<
μ
1
<
μ
0
,
⁢
μ
1
<
ε
0
),
from (3.20), we derive(3.21)
x
i
μ
1
*
(
t
)
≥
x
i
*
(
t
)
−
ε
0
2
,

t
≥
0,
⁢
i
=
1,2.Suppose that (3.13) is not true,
then for the above ε
0,
there exists a ν
∈
R
+
3 such that(3.22)
limsup
⁡
t
→
∞
⁢
y
(
t
)
<
μ
1
,where (
x
1
(
t
)
,
x
2
(
t
)
,
y
(
t
)
) is the solution of (1.4) with the initial
condition (
x
10
,
x
20
,
y
0
)
=
ν.
So there exists a constant T
3
(
>
T
2
) such that(3.23)
y
(
t
)
<
μ
1
,

t
≥
T
3
,then we derive
that(3.24)
x
˙
1
≥
a
(
t
)
x
2
−
b
(
t
)
x
1
−
(
d
(
t
)
+
L
μ
1
p
1
(
t
)
)
x
1
2
,
x
˙
2
≥
c
(
t
)
x
1
−
(
f
(
t
)
+
L
μ
1
p
2
(
t
)
)
x
2
2for t
≥
T
3.
Let (
x
1
μ
1
,
x
2
μ
1
) be the solution of (3.16) with μ
=
μ
1 and (
x
1
μ
1
(
T
3
)
,
x
2
μ
1
(
T
3
)
)
=
(
x
1
(
T
3
)
,
x
2
(
T
3
)
),
then by the vector comparison theorem, we obtain(3.25)
x
i
(
t
)
≥
x
i
μ
1
(
t
)
,

i
=
1
,
2
,
t
≥
T
3
. By the global asymptotic stability of (
x
1
μ
1
*
(
t
)
,
x
2
μ
1
*
(
t
)
),
for the given ε
0
>
0,
there exists T
4
>
T
3 such that(3.26)
x
i
μ
1
(
t
)
>
x
i
μ
1
*
(
t
)
−
ε
0
2
,

t
≥
T
4
,
⁢
i
=
1
,
2
, and hence, by (3.21), we get(3.27)
x
i
(
t
)
>
x
i
*
(
t
)
−
ε
0
,

t
≥
T
4
,
⁢
i
=
1,2.Since 0
<
y
(
t
)
<
μ
1
<
ε
0 together with (1.4) and (1.7), we
have(3.28)
y
˙
≥
y
[
g
(
t
)
+
∑
i
=
1
2
h
i
(
t
)
ϕ
i
(
t
,
(
x
i
*
(
t
)
−
ε
0
)
)
(
x
i
*
(
t
)
−
ε
0
)
2
−
q
(
t
)
ε
0
]
,

t
≠
τ
k
,
y
(
t
+
)
=
(
1
−
u
k
)
y
(
t
)
,

t
=
τ
k
,
⁢
k
∈
N
.Hence, it follows from Lemma 2.2 in [9, page 23] that(3.29)
y
(
t
)
≥
y
0
∏
0
<
τ
k
<
t
(
1
−
u
k
)
exp
⁡
(
∫
0
t
ψ
ε
0
(
s
)
d
s
)
,that is,(3.30)
y
(
t
)
≥
y
0
[
∏
0
<
τ
k
<
ω
(
1
−
u
k
)
exp
⁡
(
ω
A
ω
(
ψ
ε
0
)
)
]
[
t
/
ω
]
.By (3.14), we know that y
(
t
)
→
∞ as t
→
∞,
which leads to a contradiction. This completes the proof.
Lemma 3.5.
Assume that (3.1) holds, then there
exists a positive constant δ
y
⁢
(
δ
y
<
M
y
) such that any solution (
x
1
,
x
2
,
y
) of system (1.4) with initial conditions
satisfies(3.31)
liminf
⁡
t
→
∞
⁢
y
(
t
)
≥
δ
y
.
Proof.
Suppose that (3.31) is not true, there
must exists a time sequence {
t
t
k
(
k
^
)
}
k
=
1
∞
⊂
R
+, k
^
∈
Z such that(3.32)
liminf
⁡
t
k
(
k
^
)
→
∞
⁢
y
(
t
k
k
^
)
<
ϱ
y
(
k
^
+
1
)
2
,and by Lemma 3.4, we have lim sup
⁡
t
k
(
k
^
)
→
∞
y
(
t
k
(
k
^
)
)
≥
ϱ
y, k
^
=
1
,
2
,
….
Hence, for each k
^,
we choose two time sequences {
s
q
(
k
^
)
} and {
t
q
(
k
^
)
},
satisfying 0
<
s
1
(
k
^
)
<
t
1
(
k
^
)
<
s
2
(
k
^
)
<
t
2
(
k
^
)
<
⋯
<
s
q
(
k
^
)
<
t
q
(
k
^
)
<
⋯ and s
q
(
k
^
)
→
∞ as q
→
∞,
as well as(3.33)
y
(
s
q
(
k
^
)
)
=
ϱ
y
k
^
+
1
,


y
(
t
q
(
k
^
)
)
=
∏
0
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
(
k
^
+
1
)
2
,
(3.34)
∏
0
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
(
k
^
+
1
)
2
<
y
(
t
)
<
ϱ
y
k
^
+
1
,

t
∈
(
s
q
(
k
^
)
,
t
q
(
k
^
)
)
.By Lemma 3.2, for a given
positive integer k
^,
there exists T
(
k
^
)
>
0 such that x
i
(
t
)
≤
M
x
⁢
(
i
=
1
,
2
)  and y
(
t
)
≤
M
y for all t
≥
T
(
k
^
).
In view of s
q
(
k
^
)
→
∞ as q
→
∞,
there exists a positive integer K
1
(
k
^
) such that s
q
(
k
^
)
>
T
(
k
^
) as q
≥
K
1
(
k
^
).
Hence, for any t
≥
T
(
k
^
),
we have(3.35)
y
˙
≥
y
(
−
|
g
(
t
)
|
−
M
y
q
(
t
)
)
,

t
≠
τ
k
,
y
(
t
+
)
=
(
1
−
u
k
)
y
(
t
)
,

t
=
τ
k
,
⁢
k
∈
N
.Integrating the above inequality
from s
q
(
k
^
) to t
q
(
k
^
),
for any q
≥
K
1
(
k
^
),
then we have(3.36)
y
(
t
q
(
k
^
)
)
≥
y
(
s
q
(
k
^
)
)
∏
s
q
(
k
^
)
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
exp
⁡
(
∫
s
q
(
k
^
)
t
q
(
k
^
)
[
−
|
g
(
t
)
|
−
M
y
q
(
t
)
]
d
t
)
.Obviously, it follows from (3.33)
that(3.37)
∫
s
q
(
k
^
)
t
q
(
k
^
)
[
|
g
(
t
)
|
+
M
y
q
(
t
)
]
d
t
>
ln
⁡
(
k
^
+
1
)

for
q
≥
K
1
(
k
^
)
.Hence, in view of the
periodicity of g
(
t
) and q
(
t
),
we get(3.38)
t
q
(
k
^
)
−
s
q
(
k
^
)
⟶
∞

as
k
^
⟶
∞
,
⁢
q
≥
K
1
(
k
^
)
.By (3.14), (3.33), and (3.38), there
are positive constants T and N
0 such that(3.39)
y
(
s
q
(
k
^
)
)
=
∏
0
<
τ
k
≤
s
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
k
^
+
1
<
ε
0
,
(3.40)
t
q
(
k
^
)
−
s
q
(
k
^
)
>
2
T
,
(3.41)
∏
0
<
τ
k
<
κ
(
1
−
u
k
)
exp
⁡
(
∫
0
κ
ψ
ε
0
(
t
)
d
t
)
>
1for k
^
≥
N
0
, for q
≥
K
1
(
k
^
),
and for all κ
>
T.
However, (3.39) implies that(3.42)
y
(
t
)
<
ε
0
,

t
∈
[
s
q
(
k
^
)
,
t
q
(
k
^
)
]for k
^
≥
N
0
,
q
≥
K
1
(
k
^
)
. Then, we get(3.43)
x
˙
1
≥
a
(
t
)
x
2
−
b
(
t
)
x
1
−
(
d
(
t
)
+
L
ε
0
p
1
(
t
)
)
x
1
2
,
x
˙
2
≥
c
(
t
)
x
1
−
(
f
(
t
)
+
L
ε
0
p
2
(
t
)
)
x
2
2
.Let (
x
1
ε
0
,
x
2
ε
0
) be the solution of (3.16) with μ
=
ε
0 and (
x
1
ε
0
(
T
3
)
,
x
2
ε
0
(
T
3
)
)
=
(
x
1
(
T
3
)
,
x
2
(
T
3
)
),
then by the vector comparison theorem, we obtain(3.44)
x
i
(
t
)
≥
x
i
ε
0
(
t
)
,

i
=
1
,
2
,
t
∈
[
s
q
(
k
^
)
,
t
q
(
k
^
)
]
.From lim
⁡
q
→
∞
s
q
(
k
^
)
=
∞ and Lemmas 3.2 and 3.3, we obtain that
for any k
^,
there is a K
2
(
k
^
)
>
K
1
(
k
^
) such that for any q
≥
K
2
(
k
^
),(3.45)
ρ
x
≤
x
i
(
s
q
(
k
^
)
)
≤
M
x
,

i
=
1,2.
Forμ
=
ε
0, (3.16) has a globally asymptotically stable positive ω-periodic solution (
x
1
ε
0
*
(
t
)
,
x
2
ε
0
*
(
t
)
).
By the global asymptotic stability of (
x
1
ε
0
*
(
t
)
,
x
2
ε
0
*
(
t
)
),
for the given ε
0
>
0, there exists T
5
>
T
4,
and T
5 is dependent of any k
^ and q such that(3.46)
x
i
ε
0
(
t
)
>
x
i
ε
0
*
(
t
)
−
ε
0
2
,

t
≥
T
5
+
s
q
(
k
^
)
,
⁢
i
=
1
,
2
,
q
≥
K
2
(
k
^
)
,and hence, by (3.21), we
get(3.47)
x
i
ε
0
(
t
)
>
x
i
*
(
t
)
−
ε
0
,

t
≥
T
5
+
s
q
(
k
^
)
,
⁢
i
=
1
,
2
,
q
≥
K
2
(
k
^
)
.By (3.44), there is
an N
1
≥
N
0 such that t
q
(
k
^
)
−
s
q
(
k
^
)
≥
2
T for all k
^
≥
N
1 and q
≥
K
2
(
k
^
),
where T
≥
T
5.
Hence, from (3.44) and (3.47), we obtain(3.48)
x
i
(
t
)
≥
x
i
*
(
t
)
−
ε
0
,

i
=
1
,
2for all t
∈
[
T
+
s
q
(
k
^
)
,
t
q
(
k
^
)
], k
^
≥
N
1,
and q
≥
K
2
(
k
^
).
Since, for any t
∈
[
T
+
s
q
(
k
^
)
,
t
q
(
k
^
)
], k
≥
N
1 and q
≥
K
2
(
k
^
),
by (1.4) and (3.42), we have (3.49)
y
˙
≥
y
[
g
(
t
)
+
∑
i
=
1
2
h
i
(
t
)
ϕ
i
(
t
,
(
x
i
*
(
t
)
−
ε
0
)
)
(
x
i
*
(
t
)
−
ε
0
)
2
−
q
(
t
)
ε
0
]
,

t
⁢
≠
τ
k
,
⁢
t
∈
[
T
+
s
q
(
k
^
)
,
t
q
(
k
^
)
]
,
y
(
t
+
)
=
(
1
−
u
k
)
y
(
t
)
,

t
=
τ
k
,
k
∈
N
.Integrating from T
+
s
q
(
k
^
) to t
q
(
k
^
) for any k
≥
N
1 and q
≥
K
2
(
k
^
),
we obtain(3.50)
y
(
t
q
(
k
^
)
)
≥
y
(
s
q
(
k
^
)
+
T
)
∏
s
q
(
k
^
)
+
T
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
exp
⁡
(
∫
s
q
(
k
^
)
+
T
t
q
(
k
^
)
ψ
ε
0
(
t
)
d
t
)
.Hence, by (3.34), (3.40), and (3.41),
we finally derive that(3.51)
∏
0
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
(
k
^
+
1
)
2
≥
∏
0
<
τ
k
≤
t
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
(
k
^
+
1
)
2
∏
s
q
(
k
^
)
+
T
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
exp
⁡
(
∫
s
q
(
k
^
)
+
T
t
q
(
k
^
)
ψ
ε
0
(
t
)
d
t
)
>
∏
0
<
τ
k
<
t
q
(
k
^
)
(
1
−
u
k
)
ϱ
y
(
k
^
+
1
)
2
.This leads to a contradiction.
This completes the proof.According to Lemmas 3.2–3.5, we can
directly prove the sufficient part of the theorem. Next, we are ready to show
the necessity of this theorem.Consider the case of(3.52)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
≤
0,it is then
easy to derive that the predator population y
(
t
) must be extinct because of sterilization and
impulsive harvesting. Suppose that(3.53)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
≤
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
)
.We will show
that(3.54)
lim
⁡
t
→
∞
⁢
y
(
t
)
=
0.In fact, from (1.7) and (3.53), we
know that for any given 0
<
ϵ
<
1,
there exists ϵ
1
>
0 such that(3.55)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
(
x
i
*
+
ϵ
1
)
)
(
x
i
*
+
ϵ
1
)
2
−
q
ϵ
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
<
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
(
x
i
*
+
ϵ
1
)
)
(
x
i
*
+
ϵ
1
)
2
−
q
ϵ
)
)
.Note that q
(
t
)
>
0 for t
≥
0.
Since(3.56)
x
˙
1
≤
a
(
t
)
x
2
−
b
(
t
)
x
1
−
d
(
t
)
x
1
2
,
x
˙
2
≤
c
(
t
)
x
1
−
f
(
t
)
x
2
2
,for the given ϵ
1
>
0,
it is easy to show that there exists T
(
1
)
>
0 such that(3.57)
x
i
(
t
)
≤
x
i
*
(
t
)
+
ϵ
1

for
t
>
T
(
1
)
,
⁢
i
=
1
,
2
,we then have(3.58)
∏
k
=
1
q
(
1
−
u
k
)
<
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
)
(
x
i
)
2
−
q
ϵ
)
)
.We now show that there must
exist T
(
2
)
(
>
T
(
1
)
) such that y
(
T
(
2
)
)
<
ϵ.
Otherwise, by the last two equations in system (1.4), we have(3.59)
ϵ
≤
y
(
t
)
≤
y
(
T
(
1
)
)
∏
T
(
1
)
<
τ
k
<
t
(
1
−
u
k
)
exp
⁡
(
∫
T
(
1
)
t
(
g
(
s
)
+
∑
i
=
1
2
h
i
(
s
)
ϕ
i
(
s
,
x
i
(
s
)
)
x
i
2
(
s
)
−
q
(
s
)
ϵ
)
d
s
)
⟶
0

as
t
⟶
∞
.This implies that ϵ
≤
0,
which is a contradiction.LetM
(
ϵ
)
=
max
⁡
t
∈
[
0
,
ω
]
{
g
(
t
)
+
∑
i
=
1
2
h
i
(
t
)
ϕ
i
(
s
,
x
i
(
t
)
)
x
i
2
(
t
)
+
q
(
t
)
ϵ
}.
Note that M
(
ϵ
) is bounded. We then show that(3.60)
y
(
t
)
≤
ϵ
exp
⁡
(
M
(
ϵ
)
ω
)

for
t
≥
T
(
2
)
.Otherwise, there exists T
(
3
)
>
T
(
2
) such that T
(
3
)
∈
(
T
(
2
)
+
P
1
ω
,
T
(
2
)
+
(
P
1
+
1
)
ω
]
, in which P
1
∈
Z
+
=
{
0
,
1
,
2
,
…
},
and y
(
T
(
3
)
)
>
ϵ
exp
⁡
(
M
(
ϵ
)
ω
)
. By (3.58), we have(3.61)
ϵ
exp
⁡
(
M
(
ε
)
ω
)


<
y
(
T
(
3
)
)


=
y
(
T
(
2
)
)
∏
T
(
2
)
<
τ
k
<
T
(
3
)
(
1
−
u
k
)
exp
(
∫
T
(
2
)
T
(
3
)
(
g
(
s
)
+
∑
i
=
1
2
h
i
(
s
)
ϕ
i
(
s
,
x
i
(
s
)
)
x
i
2
(
s
)
−
q
(
s
)
ϵ
)
d
s
)


≤
y
(
T
(
2
)
)
[
∏
k
=
1
q
(
1
−
u
k
)
exp
(
∫
0
ω
(
g
(
s
)
+
⁡
∑
i
=
1
2
h
i
(
s
)
ϕ
i
(
s
,
x
i
(
s
)
)
x
i
2
(
s
)
−
q
(
s
)
ϵ
)
d
s
)
]
P
1



×
∏
T
(
2
)
+
P
1
ω
<
τ
k
<
T
(
3
)
(
1
−
u
k
)
exp
⁡
(
∫
0
ω
(
g
(
s
)
+
∑
i
=
1
2
h
i
(
s
)
ϕ
i
(
s
,
x
i
(
s
)
)
x
i
2
(
s
)
−
q
(
s
)
ϵ
)
d
s
)


<
ϵ
exp
⁡
(
M
(
ε
)
ω
)
,which leads to a contradiction.
This  implies that (3.60) holds. In view of the arbitrariness of ϵ,
we get that y
(
t
)
→
0 as t
→
∞.
This completes the proof.Remark 3.6.
From the proof of Theorem3.1 above, we note that the predator
population y in system (1.4) will be extinct provided
that(3.62)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
≤
0,or excessive harvesting, that
is,(3.63)
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
>
0
,
∏
k
=
1
q
(
1
−
u
k
)
≤
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
)
.


## 4. Discussion

Our result
could provide a useful insight into the conservation of beneficial animals,
especially rare animals. As an example, we depict the case of Oreolalax
omeimontis tadpole, Oreolalax omeimontis, and red-eared slider (T. scripta
elegans). Oreolalax omeimontis is a rare species of frog found near Mt. Omei in
Sichuan (China). The red-eared slider is a native of the Mississippi Valley
area of the United States [8, 20]. Since the 1970s, large numbers of red-eared sliders
have been produced on turtle farms in the USA for the international pet trade.
Red-eared Turtles are traded as pet animals and have been introduced to many
countries. They are omnivorous and will eat insects, tadpoles, crayfish,
shrimp, worms, snails, amphibians and small fish, as well as aquatic plants,
and they hardly may be controlled by a natural
enemy. In our model, the variables x
1
(
t
) and x
2
(
t
) represent the density of Oreolalax omeimontis
tadpole and Oreolalax omeimontis at time t, respectively. The variable y
(
t
) describes the density of the red-eared slider
at time t. It is well known that Oreolalax omeimontis is beneficial to humans
because they eat so many insect pests. Ironically, although red-eared sliders
have been widely introduced throughout the world, its detrimental effects have
been reported by many researchers [21, 22]. The red-eared slider is one of main enemies of
Oreolalax omeimontis. To protect these beneficial and rare toads, we must
control the amount of red-eared sliders in the habitat of Oreolalax omeimontis.
From a control point of view, our aim is to keep red-eared sliders at an
acceptably low level with a minimum use of artificial control measures, not to
eradicate all red-eared sliders. Hence, in the above example, the problem of
nonextinction of populations becomes a description of reasonable harvesting
rates u
k
⁢
(
k
=
1
,
2
⋯
q
).According to
our main result, system (1.4) is permanent if and only if the growth of the
predator y by foraging prey populations x
1 and x
2 plus its intrinsic rate of increase
is positive on average during the period ω,
and harvesting rates u
k
⁢
(
k
=
1
,
2
,
⋯
q
) of red-eared sliders during the period ω are small enough to satisfy
that(4.1)
∏
k
=
1
q
(
1
−
u
k
)
>
exp
⁡
(
−
ω
A
ω
(
g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
)
)
.These seem to be reasonable from
a biological point of view; but it should be noted that condition (3.1) allows
the predator to be g
+
∑
i
=
1
2
h
i
ϕ
i
(
t
,
x
i
*
)
x
i
*
2
<
0 for some time intervals among 0
≤
t
≤
ω
. That is, under reasonable harvesting rates,
the predator can survive together with prey populations even if the growth rate
of the former is negative for some seasons during a period. We hope that our
result can be used to help protect beneficial animals found in their habitats.



---

*Source: 389727-2008-04-14.xml*