# Dynamics of a Stochastic Delayed Competitive Model with Impulsive Toxicant Input in Polluted Environments

Authors: Meiling Deng; Meng Liu; Chuanzhi Bai
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

In the world today, with the rapid development of modern agriculture and industry, a large quantity of pollutants enter into ecosystems one by one, which is a threat to the persistence of the exposed populations. This paper investigates a stochastic delayed competitive system with impulsive toxicant input in a polluted environment. Under a simple condition, sufficient and necessary conditions for stability in the mean and extinction of each species are established. Some recent works are improved and extended greatly. Some numerical simulations are also included to illustrate and support the findings.


---

## Body

## 1. Introduction

In this paper, we consider the following stochastic delay competitive model in polluted environments with impulsive toxicant input:(1)
d
N
1
(
t
)
=
N
1
(
t
)
[
r
10
-
r
11
C
0
(
t
)
-
a
11
N
1
(
t
)
-
a
12
N
2
(
t
-
τ
1
)
]
d
t
+
λ
1
N
1
(
t
)
d
W
1
(
t
)
,
d
N
2
(
t
)
=
N
2
(
t
)
[
r
20
-
r
21
C
0
(
t
)
-
a
22
N
2
(
t
)
-
a
21
N
1
(
t
-
τ
2
)
]
d
t
+
λ
2
N
2
(
t
)
d
W
2
(
t
)
,
d
C
0
(
t
)
d
t
=
k
C
e
(
t
)
-
(
g
+
m
)
C
0
(
t
)
,
d
C
e
(
t
)
d
t
=
-
h
C
e
(
t
)
,
t
≠
n
γ
,
n
∈
Z
+
,
Δ
N
i
(
t
)
=
0
,
Δ
C
0
(
t
)
=
0
,
Δ
C
e
(
t
)
=
b
,
h
h
h
h
h
h
h
h
h
h
h
i
t
=
n
γ
,
n
∈
Z
+
,
i
=
1,2
,
with initial data
(2)
h
h
h
h
h
h
h
h
h
h
h
h
i
N
i
(
t
)
=
ψ
i
(
t
)
>
0
,
t
∈
[
-
τ
,
0
]
;
ψ
i
(
0
)
>
0
,
i
=
1,2
,
where τ
i
≥
0, τ
=
max
⁡
⁡
{
τ
1
,
τ
2
}, and ψ
i
(
t
) is a continuous function on [
-
τ
,
0
]. All coefficients in model (1) are positive. Δ
f
(
t
)
=
f
(
t
+
)
-
f
(
t
) and Z
+
=
{
1,2
,
…
}; N
i
(
t
) is the size of the ith population, i
=
1,2; r
i
0 is the growth rate of the ith population; r
i
1 is the response to the pollutant present in the organism of the ith population; C
0
(
t
) is the toxicant concentration in the organism; C
e
(
t
) is the toxicant concentration in the environment; k
C
e
(
t
) is the organism's net uptake of toxicant from the environment; g
C
0
(
t
)
+
m
C
0
(
t
) is the egestion and depuration rates of the toxicant in the organism; h
C
e
(
t
) is the toxicant loss from the environment itself; γ is the period of the impulsive effect about the exogenous input of toxicant; b is the amount of toxicant input at every time. W
i
(
t
) is a standard Brownian motion defined on a complete probability space (
Ω
,
F
,
P
); λ
i
2 is the intensity of the environmental noise.Recently, population models with toxicant effect have received great attention; see, for example, [1–18]. Liu and Zhang [15] considered the following competitive model in polluted environments with impulsive toxicant input:
(3)
d
N
1
(
t
)
d
t
=
N
1
(
t
)
[
r
10
-
r
11
C
0
(
t
)
-
a
11
N
1
(
t
)
-
a
12
N
2
(
t
)
]
,
d
N
2
(
t
)
d
t
=
N
2
(
t
)
[
r
20
-
r
21
C
0
(
t
)
-
a
21
N
1
(
t
)
-
a
22
N
2
(
t
)
]
,
d
C
0
(
t
)
d
t
=
k
C
e
(
t
)
-
(
g
+
m
)
C
0
(
t
)
,
d
C
e
(
t
)
d
t
=
-
h
C
e
(
t
)
,
t
≠
n
γ
,
n
∈
Z
+
.
Δ
N
i
(
t
)
=
0
,
Δ
C
0
(
t
)
=
0
,
Δ
C
e
(
t
)
=
b
,
h
h
h
h
h
h
h
h
h
h
h
i
t
=
n
γ
,
n
∈
Z
+
,
i
=
1,2
,
For model (3), the authors [15] proved the following.Lemma 1 (see [15]).
Define(4)
Γ
=
a
11
a
22
-
a
12
a
21
,
Γ
1
=
r
10
a
22
-
a
12
r
20
,
Γ
2
=
r
20
a
11
-
r
10
a
21
,
Γ
¯
1
:
=
a
22
r
11
-
a
12
r
21
,
Γ
¯
2
:
=
a
11
r
21
-
a
21
r
11
,
K
=
k
b
γ
h
(
g
+
m
)
.
Suppose that Γ
>
0, Γ
1
>
0, Γ
2
>
0.(A)
IfΓ
1
>
Γ
¯
1
K and Γ
2
<
Γ
¯
2
K, then lim
⁡
t
→
+
∞
⁡
N
2
(
t
)
=
0 and limsup
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
>
0.
(B)
IfΓ
1
<
Γ
¯
1
K and Γ
2
>
Γ
¯
2
K, then lim
⁡
t
→
+
∞
⁡
N
1
(
t
)
=
0 and limsup
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
>
0.
(C)
IfΓ
1
>
Γ
¯
1
K and Γ
2
>
Γ
¯
2
K, then limsup
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
i
(
s
)
d
s
>
0, i
=
1,2.From the work of Liu and Zhang [15], some important and interesting questions arise naturally.(Q1)
In the real world, the growth of population is inevitably affected by random environmental fluctuations. May [19] have claimed that population systems should be stochastic. Therefore, what happens if system (3) is affected by environmental fluctuations?
(Q2)
Gopalsamy [20] have pointed out that, in order to be reality, time delays should not be ignored. Hence, what happens if system (3) incorporates with time delays?
(Q3)
Can we improve the results obtained in Lemma1?The aims of this paper are to investigate the above questions. Recall that r
i
0 stands for the growth rate. In practice, we often estimate it by an average value plus an error term. Generally, by the famous central limit theorem, the error term follows a normal distribution. Hence, for short correlation time, we can replace r
i
0 with r
i
0
+
λ
i
W
˙
i
(
t
) (see, e.g., [21–29]), where W
˙
i
(
t
) is white noise and λ
i
2 is the intensity of the noise. At the same time, incorporating with time delays, we get model (1). For model (1), we will show the following.Theorem 2.
Suppose thatΓ
>
0. Define
(5)
η
i
=
r
i
0
-
r
i
1
K
-
0.5
λ
i
2
,
i
=
1,2
,
Γ
~
1
=
0.5
a
22
λ
1
2
-
0.5
a
12
λ
2
2
,
Γ
~
2
=
0.5
a
11
λ
2
2
-
0.5
a
21
λ
1
2
.
(I)
Ifη
1
<
0 and η
2
<
0, then both N
1 and N
2 go to extinction almost surely; that is,
(6)
lim
⁡
t
→
+
∞
⁡
N
i
(
t
)
=
0
a
.
s
.
,
i
=
1,2
;
(II)
ifη
1
>
0 and η
2
<
0, then N
2 goes to extinctiona.s. and N
1 is stable in the meana.s.; that is,
(7)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
=
η
1
a
11
,
a
.
s
.
;
(III)
ifη
1
<
0 and η
2
>
0, then N
1 goes to extinction a.s. and N
2 is stable in the meana.s.; that is,
(8)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
=
η
2
a
22
,
a
.
s
.
;
(IV)
ifη
1
>
0, η
2
>
0,
(A)
ifΓ
1
>
Γ
¯
1
K
+
Γ
~
1 and Γ
2
<
Γ
¯
2
K
+
Γ
~
2, then N
2 goes to extinction a.s. and N
1 is stable in the meana.s.
(9)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
=
η
1
a
11
,
a
.
s
.
;
(B)
ifΓ
1
<
Γ
¯
1
K
+
Γ
~
1 and Γ
2
>
Γ
¯
2
K
+
Γ
~
2, then N
1 goes to extinction a.s. and N
2 is stable in the meana.s.
(10)
lim
⁡
t
→
+
∞
t
-
1
∫
0
t
‍
N
2
(
s
)
d
s
=
η
2
a
22
,
a
.
s
.
;
(C)
ifΓ
1
>
Γ
¯
1
K
+
Γ
~
1 and Γ
2
>
Γ
¯
2
K
+
Γ
~
2, then both N
1 and N
2 are stable in the meana.s.
(11)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
=
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
,
a
.
s
.
,
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
=
Γ
2
-
Γ
¯
1
K
-
Γ
~
2
Γ
,
a
.
s
.Remark 3.
It is useful to point out that ifη
1
>
0, η
2
>
0, and Γ
>
0, then Γ
1
<
Γ
¯
1
+
Γ
~
1 and Γ
2
<
Γ
¯
2
+
Γ
~
2 will not hold simultaneously.Remark 4.
In comparison with most of the existing results, our key contributions in this paper are as follows.(i)
To the best of our knowledge, this paper is the first attempt to consider stochastic delay competitive model in polluted environments.
(ii)
Our conditions are much weaker. For example, the authors [15] supposed Γ
1
>
0 and Γ
2
>
0 which are dropped in this paper.
(iii)
Our results improve some recent works. For example, Lemma1 indicates that the superior limit is positive while Theorem 2 proves that the limit exists and establishes the explicit form of the limit.


## 2. Proof

For simplicity, define(12)
R
+
2
=
{
a
=
(
a
1
,
a
2
)
∈
R
2
∣
a
i
>
0
,
i
=
1,2
}
,
〈
g
(
t
)
〉
=
t
-
1
∫
0
t
‍
g
(
s
)
d
s
.Lemma 5.
For any given initial dataψ
(
θ
)
=
(
ψ
1
(
θ
)
,
ψ
2
(
θ
)
)
∈
C
(
[
-
τ
,
0
]
,
R
+
2
), there is a unique global positive solution N
(
t
)
=
(
N
1
(
t
)
,
N
2
(
t
)
) to model (1) on t
≥
-
τ a.s. and
(13)
limsup
⁡
t
→
+
∞
ln
⁡
⁡
N
i
(
t
)
ln
⁡
⁡
t
≤
1
a
.
s
.
,
i
=
1,2
.Proof.
The proof is a special case of Theorems 5.1 and 5.2 in Liu and Wang [25] and hence is omitted.Lemma 6 (see [26]).
Suppose thatx
(
t
)
∈
C
[
Ω
×
[
0
,
+
∞
)
,
R
+
].(I)
If there existδ and positive constants δ
0, T such that
(14)
ln
⁡
⁡
x
(
t
)
≤
δ
t
-
δ
0
∫
0
t
x
(
s
)
d
s
+
∑
i
=
1
n
β
i
W
i
(
t
)
for t
≥
T, where W
i
(
t
) are independent standard Brownian motions and β
i are constants, 1
≤
i
≤
n, then we have the following: if δ
≥
0, then limsup
⁡
t
→
+
∞
⁡
〈
x
(
t
)
〉
≤
δ
/
δ
0 a.s.; if δ
<
0, then lim
⁡
t
→
+
∞
⁡
x
(
t
)
=
0 a.s.
(II)
If there exist positive constantsδ
0, T, and δ such that
(15)
ln
⁡
⁡
x
(
t
)
≥
δ
t
-
δ
0
∫
0
t
x
(
s
)
d
s
+
∑
i
=
1
n
β
i
W
i
(
t
)
for t
≥
T, then liminf
⁡
t
→
+
∞
⁡
〈
x
(
t
)
〉
≥
δ
/
δ
0 a.s.To begin with, let us consider the following subsystem of (1):
(16)
d
C
0
(
t
)
d
t
=
k
C
e
(
t
)
-
(
g
+
m
)
C
0
(
t
)
,
d
C
e
(
t
)
d
t
=
-
h
C
e
(
t
)
,
t
≠
n
γ
,
n
∈
Z
+
,
Δ
C
0
(
t
)
=
0
,
Δ
C
e
(
t
)
=
b
,
t
=
n
γ
,
n
∈
Z
+
hhhhh
0
≤
C
0
(
0
)
≤
1
,
0
≤
C
e
(
0
)
≤
1
.Lemma 7 (see [12, 13]).
System (16) has a unique positive γ-periodic solution (
C
~
0
(
t
)
,
C
~
e
(
t
)
)
T and for each solution (
C
0
(
t
)
,
C
e
(
t
)
)
T of (16), C
0
(
t
)
→
C
~
0
(
t
) and C
e
(
t
)
→
C
~
e
(
t
) as t
→
∞. Moreover, C
0
(
t
)
>
C
~
0
(
t
) and C
e
(
t
)
>
C
~
e
(
t
) for all t
≥
0 if C
0
(
0
)
>
C
~
0
(
0
) and C
e
(
0
)
>
C
~
e
(
0
), i
=
1,2, where
(17)
C
~
0
(
t
)
=
C
~
0
(
0
)
e
-
(
g
+
m
)
(
t
-
n
γ
)
+
k
b
(
e
-
(
g
+
m
)
(
t
-
n
γ
)
-
e
-
h
(
t
-
n
γ
)
)
(
h
-
g
-
m
)
(
1
-
e
-
h
γ
)
,
C
~
e
(
t
)
=
b
e
-
h
(
t
-
n
γ
)
1
-
e
-
h
γ
,
C
~
0
(
0
)
=
k
b
(
e
-
(
g
+
m
)
γ
-
e
-
h
γ
)
(
h
-
g
-
m
)
(
1
-
e
-
(
g
+
m
)
γ
)
(
1
-
e
-
h
γ
)
,
C
~
e
(
0
)
=
b
1
-
e
-
h
γ
for t
∈
(
n
γ
,
(
n
+
1
)
γ
] and n
∈
Z
+. In addition,
(18)
lim
⁡
t
→
+
∞
t
-
1
∫
0
t
C
~
0
(
s
)
d
s
=
k
b
h
(
g
+
m
)
γ
=
K
.Proof of Theorem2.
It follows from Lemma7 that
(19)
lim
⁡
t
→
+
∞
t
-
1
∫
0
t
C
0
(
s
)
d
s
=
lim
⁡
t
→
+
∞
t
-
1
∫
0
t
C
~
0
(
s
)
d
s
=
K
.
That is to say, for all ε
>
0, there is T
>
0 such that
(20)
K
-
ε
≤
〈
C
0
(
t
)
〉
≤
K
+
ε
,
t
>
T
.
Applying Itô's formula to (1) leads to
(21)
d
ln
⁡
⁡
N
1
(
t
)
=
d
N
1
(
t
)
N
1
(
t
)
-
(
d
N
1
(
t
)
)
2
2
N
1
2
(
t
)
=
[
r
10
-
0.5
λ
1
2
-
r
11
C
0
(
t
)
-
a
11
N
1
(
t
)
-
a
12
N
2
(
t
-
τ
1
)
]
d
t
+
λ
1
d
W
1
(
t
)
,
d
ln
⁡
⁡
N
2
(
t
)
=
d
N
2
(
t
)
N
2
(
t
)
-
(
d
N
2
(
t
)
)
2
2
N
2
2
(
t
)
=
[
r
20
-
0.5
λ
2
2
-
r
21
C
0
(
t
)
-
a
21
N
1
(
t
-
τ
2
)
-
a
22
N
2
(
t
)
]
d
t
+
λ
2
d
W
2
(
t
)
.
In other words, we have shown that
(22)
ln
⁡
⁡
N
1
(
t
)
-
ln
⁡
⁡
N
1
(
0
)
=
(
r
10
-
0.5
λ
1
2
)
t
-
r
11
∫
0
t
C
0
(
s
)
d
s
-
a
11
∫
0
t
N
1
(
s
)
d
s
-
a
12
∫
0
t
‍
N
2
(
s
-
τ
1
)
d
s
+
λ
1
W
1
(
t
)
=
(
r
10
-
0.5
λ
1
2
)
t
-
a
12
∫
0
t
N
2
(
s
)
d
s
+
a
12
[
∫
t
-
τ
1
t
N
2
(
s
)
d
s
-
∫
-
τ
1
0
N
2
(
s
)
d
s
]
-
r
11
∫
0
t
C
0
(
s
)
d
s
-
a
11
∫
0
t
N
1
(
s
)
d
s
+
λ
1
W
1
(
t
)
,
ln
⁡
⁡
N
2
(
t
)
-
ln
⁡
⁡
N
2
(
0
)
=
(
r
20
-
0.5
λ
2
2
)
t
-
r
21
∫
0
t
C
0
(
s
)
d
s
-
a
21
∫
0
t
N
1
(
s
-
τ
2
)
d
s
-
a
22
∫
0
t
N
2
(
s
)
d
s
+
λ
2
W
2
(
t
)
=
(
r
20
-
0.5
λ
2
2
)
t
-
a
21
∫
0
t
N
1
(
s
)
d
s
+
a
21
[
∫
t
-
τ
2
t
N
1
(
s
)
d
s
-
∫
-
τ
2
0
N
1
(
s
)
d
s
]
-
r
21
∫
0
t
C
0
(
s
)
d
s
-
a
22
∫
0
t
N
2
(
s
)
d
s
+
λ
2
W
2
(
t
)
.
(I) Assume thatη
1
=
r
10
-
0.5
λ
1
2
-
r
11
K
<
0 and η
2
=
r
20
-
0.5
λ
2
2
-
r
21
K
<
0. In view of (22),
(23)
t
-
1
ln
⁡
⁡
N
1
(
t
)
N
1
(
0
)
≤
r
10
-
r
11
∫
0
t
C
0
(
s
)
d
s
-
0.5
λ
1
2
+
t
-
1
λ
1
W
1
(
t
)
.
By lim
⁡
t
→
+
∞
⁡
W
i
(
t
)
/
t
=
0 a.s., i
=
1,2 (19) and r
10
<
0.5
λ
1
2
+
r
11
K we have that
(24)
limsup
⁡
t
→
+
∞
⁡
t
-
1
ln
⁡
⁡
N
1
(
t
)
≤
r
10
-
0.5
λ
1
2
-
r
11
K
=
η
1
<
0
.
Therefore, lim
⁡
t
→
+
∞
⁡
N
1
(
t
)
=
0, a.s. Similarly, it follows from (9) that if η
2
<
0, then lim
⁡
t
→
+
∞
⁡
N
2
(
t
)
=
0, a.s.
(II) Assume thatη
1
>
0 and η
2
<
0. Since η
2
<
0, then, by (I), lim
⁡
t
→
+
∞
⁡
N
2
(
t
)
=
0, a.s. Hence, for arbitrary ε
>
0, there exists T
>
0 such that for t
≥
T
(25)
-
ε
2
≤
a
12
t
-
1
∫
0
t
N
2
(
s
-
τ
1
)
d
s
≤
ε
2
,
-
ε
2
≤
t
-
1
ln
⁡
⁡
N
1
(
0
)
≤
ε
2
.
When the above inequalities and (20) are used in (22), we can obtain that, for t
≥
T,
(26)
ln
⁡
⁡
N
1
(
t
)
≤
(
η
1
+
2
ε
)
t
-
a
11
∫
0
t
N
1
(
s
)
d
s
+
λ
1
W
1
(
t
)
,
(27)
ln
⁡
⁡
N
1
(
t
)
≥
(
η
1
-
2
ε
)
t
-
a
11
∫
0
t
N
1
(
s
)
d
s
+
λ
1
W
1
(
t
)
.
Note that η
1
>
0; we can let ε be sufficiently small such that η
1
-
2
ε
>
0. Applying (i) and (ii) in Lemma 6 to (26) and (27), respectively, one can see that
(28)
η
1
-
2
ε
a
11
≤
liminf
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
≤
limsup
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
≤
η
1
+
2
ε
a
11
,
a.s
.
It therefore follows from the arbitrariness of ε that
(29)
lim
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
=
η
1
a
11
,
a.s
.
The proof of (III) can be obtained similarly and hence is omitted.
Now, we are in the position to prove (IV). Assume thatη
1
>
0 and η
2
>
0. For i
=
1,2, consider the following stochastic equation:
(30)
d
y
i
(
t
)
=
y
i
(
t
)
[
r
i
0
-
r
i
1
C
0
(
t
)
-
a
i
i
y
i
(
t
)
]
d
t
+
λ
i
y
i
(
t
)
d
W
i
(
t
)
,
y
i
(
θ
)
=
ψ
(
θ
)
,
θ
∈
[
-
τ
,
0
]
.
By the classical stochastic comparison theorem [30], one can see that
(31)
N
1
(
t
)
≤
y
1
(
t
)
,
N
2
(
t
)
≤
y
2
(
t
)
.
Note that r
i
0
>
0.5
λ
i
2
+
r
i
1
K, i
=
1,2; an argument, identical to the argument used in the proof of (II), shows that
(32)
lim
⁡
t
→
+
∞
⁡
〈
y
i
(
t
)
〉
=
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
y
i
(
s
)
d
s
=
η
i
a
i
i
a.s.
,
i
=
1,2
.
Consequently,
(33)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
t
-
τ
1
t
y
2
(
s
)
d
s
=
lim
⁡
t
→
+
∞
⁡
(
t
-
1
∫
0
t
y
2
(
s
)
d
s
-
t
-
1
∫
0
t
-
τ
1
y
2
(
s
)
d
s
)
=
0
,
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
t
-
τ
2
t
y
1
(
s
)
d
s
=
0
,
a.s
.
This, together with (31), implies that
(34)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
t
-
τ
1
t
N
2
(
s
)
d
s
=
0
,
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
t
-
τ
2
t
N
1
(
s
)
d
s
=
0
,
a.s
.
On the other hand, computing (9) ×
a
11 − (22) ×
a
21 gives
(35)
a
11
ln
⁡
⁡
N
2
(
t
)
N
2
(
0
)
=
a
11
a
21
[
∫
t
-
τ
2
t
‍
N
1
(
s
)
d
s
-
∫
-
τ
2
0
‍
N
1
(
s
)
d
s
]
-
a
21
a
12
[
∫
t
-
τ
1
t
N
2
(
s
)
d
s
-
∫
-
τ
1
0
N
2
(
s
)
d
s
]
+
a
21
ln
⁡
⁡
N
1
(
t
)
N
1
(
0
)
+
(
Γ
2
-
Γ
~
2
)
t
-
Γ
¯
2
∫
0
t
C
0
(
s
)
d
s
-
Γ
∫
0
t
N
2
(
s
)
d
s
-
a
21
λ
1
W
1
(
t
)
+
a
11
λ
2
W
2
(
t
)
.
In view of (13) and (34), for arbitrary ε
>
0, there exits T
>
0 such that, for t
≥
T,
(36)
t
-
1
a
21
ln
⁡
⁡
N
1
(
t
)
N
1
(
0
)
<
ε
3
,
t
-
1
a
11
ln
⁡
⁡
N
2
(
0
)
<
ε
3
,
t
-
1
a
11
a
21
[
∫
t
-
τ
2
t
N
1
(
s
)
d
s
-
∫
-
τ
2
0
N
1
(
s
)
d
s
]
-
t
-
1
a
21
a
12
[
∫
t
-
τ
1
t
N
2
(
s
)
d
s
-
∫
-
τ
1
0
N
2
(
s
)
d
s
]
≤
ε
3
.
When the above inequalities and (20) are used in (35), we can see that for t
>
T
(37)
a
11
ln
⁡
⁡
N
2
(
t
)
≤
(
Γ
2
-
Γ
¯
2
K
-
Γ
~
2
+
2
ε
)
t
-
Γ
∫
0
t
N
2
(
s
)
d
s
-
a
21
λ
1
W
1
(
t
)
+
a
11
λ
2
W
2
(
t
)
.
Similarly, computing (22) ×
a
22 − (9) ×
a
12 gives
(38)
a
22
ln
⁡
⁡
N
1
(
t
)
N
1
(
0
)
=
a
22
a
12
[
∫
t
-
τ
1
t
N
2
(
s
)
d
s
-
∫
-
τ
1
0
N
2
(
s
)
d
s
]
-
a
12
a
21
[
∫
t
-
τ
2
t
N
1
(
s
)
d
s
-
∫
-
τ
2
0
N
1
(
s
)
d
s
]
+
a
12
ln
⁡
⁡
N
2
(
t
)
N
2
(
0
)
+
(
Γ
1
-
Γ
~
1
)
t
-
Γ
¯
1
∫
0
t
C
0
(
s
)
d
s
-
Γ
∫
0
t
N
1
(
s
)
d
s
+
a
22
λ
1
W
1
(
t
)
-
a
12
λ
2
W
2
(
t
)
.
Hence, for sufficiently large t,
(39)
a
22
ln
⁡
⁡
N
1
(
t
)
≤
(
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
+
2
ε
)
t
-
Γ
∫
0
t
N
1
(
s
)
d
s
+
a
22
λ
1
W
1
(
t
)
-
a
12
λ
2
W
2
(
t
)
.
(A) Assume thatΓ
1
>
Γ
¯
1
K
+
Γ
~
1 and Γ
2
<
Γ
¯
2
K
+
Γ
~
2. Since Γ
2
<
Γ
¯
2
K
+
Γ
~
2, then we can let ε be sufficiently small such that Γ
2
-
Γ
¯
2
K
-
Γ
~
2
+
2
ε
<
0. Applying (i) in Lemma 6 to (37) results in lim
⁡
t
→
+
∞
⁡
N
2
(
t
)
=
0, a.s. The proof of lim
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
=
η
1
/
a
11 a.s. is the same as that in (II) and hence is omitted. The proof of (B) is similar to that of (A) and hence is omitted.
(C) Assume thatΓ
1
>
Γ
¯
1
K
+
Γ
~
1 and Γ
2
>
Γ
¯
2
K
+
Γ
~
2. Notice that Γ
2
>
Γ
~
2; then, by (37) and Lemma 6,
(40)
limsup
⁡
t
→
+
∞
⁡
〈
N
2
(
t
)
〉
≤
Γ
2
-
Γ
¯
2
K
-
Γ
~
2
+
2
ε
Γ
,
a.s
.
It then follows from the arbitrariness of ε that
(41)
limsup
⁡
t
→
+
∞
⁡
〈
N
2
(
t
)
〉
≤
Γ
2
Γ
¯
2
K
-
Γ
~
2
Γ
,
a.s
.
Similarly, by (39), Lemma 6, and the arbitrariness of ε, we have
(42)
limsup
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
≤
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
,
a.s
.
Let ε be sufficiently small satisfying a
11
(
(
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
)
/
Γ
)
-
ε
>
0. Substituting (34), (41), and (20) into (22) yields
(43)
t
-
1
ln
⁡
⁡
N
1
(
t
)
=
t
-
1
ln
⁡
⁡
N
1
(
0
)
+
r
10
-
r
11
〈
C
0
(
t
)
〉
-
0.5
λ
1
2
-
a
11
〈
N
1
(
t
)
〉
-
a
12
〈
N
2
(
t
)
〉
+
λ
1
W
1
(
t
)
t
+
a
12
t
-
1
[
∫
t
-
τ
1
t
N
2
(
s
)
d
s
-
∫
-
τ
1
0
N
2
(
s
)
d
s
]
≥
η
1
-
2
ε
-
a
11
〈
N
1
(
t
)
〉
-
a
12
limsup
⁡
t
→
+
∞
⁡
〈
N
2
(
t
)
〉
+
λ
1
W
1
(
t
)
t
≥
η
1
-
2
ε
-
a
11
〈
N
1
(
t
)
〉
-
a
12
Γ
2
-
Γ
~
2
Γ
+
λ
1
W
1
(
t
)
t
=
a
11
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
-
2
ε
-
a
11
〈
N
1
(
t
)
〉
+
λ
1
W
1
(
t
)
t
,
for sufficiently large t. By (ii) in Lemma 6 and the arbitrariness of ε, one can observe that
(44)
liminf
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
≥
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
,
a.s
.
Similarly, when (34) and (42) and (20) are used in (9), we can see that liminf
⁡
t
→
+
∞
⁡
〈
N
2
(
t
)
〉
≥
(
Γ
2
-
Γ
¯
2
K
-
Γ
~
2
)
/
Γ, a.s. This, together with (41), (42), and (44), indicates that
(45)
lim
⁡
t
→
+
∞
⁡
〈
N
1
(
t
)
〉
=
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
,
lim
⁡
t
→
+
∞
⁡
〈
N
2
(
t
)
〉
=
Γ
2
-
Γ
¯
2
K
-
Γ
~
2
Γ
,
a.s
.


## 3. Numerical Simulations

In this section, using the classical Milstein method (see, e.g., [31]), we work out some numerical figures to support the analytical results. In Figure 1, we choose r
10
=
0.7, r
20
=
0.5, a
11
=
0.8, a
12
=
0.5, a
21
=
0.4, a
22
=
0.7, τ
1
=
10, τ
2
=
12, k
=
g
=
m
=
0.2, h
=
0.5, b
=
0.1, and γ
=
1. Then, Γ
=
0.36, Γ
1
=
0.24, Γ
2
=
0.12,Γ
1
¯
=
0.2, Γ
2
¯
=
0.4, and K
=
k
b
/
γ
h
(
g
+
m
)
=
0.1. The only difference between conditions of Figures 1(a), 1(b), 1(c), 1(d), 1(e), and 1(f) is that the values of λ
1 and λ
2 are different.(a)
In Figure1(a), we choose λ
1
=
1.1, λ
2
=
1. Then, η
1
=
r
10
-
r
11
K
-
0.5
λ
1
2
=
-
0.12, η
2
=
r
20
-
r
21
K
-
0.5
λ
2
2
=
-
0.1. By virtue of (I) in Theorem 2, both N
1 and N
2 are extinct; see Figure 1(a).
(b)
In Figure1(b), we set λ
1
=
1, λ
2
=
1. Then, η
1
=
0.1, η
2
=
-
0.1. In view of (II) in Theorem 2, N
2 is extinct and lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
‍
N
1
(
s
)
d
s
=
η
1
/
a
11
=
0.125. Figure 1(b) confirms this.
(c)
In Figure1(c), we let λ
1
=
1.2, λ
2
=
0.8. Then, η
1
=
-
0.12, η
2
=
0.08. It follows from (III) in Theorem 2 that N
1 is extinct and
(46)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
=
η
2
a
22
=
0.1143
.
See Figure 1(c).
(d)
In Figure1(d), we set λ
1
=
1, λ
2
=
0.8484. Then, η
1
=
0.1
,
η
2
=
0.04, Γ
1
=
0.24
>
Γ
¯
1
K
+
Γ
~
1
=
0.19, and Γ
2
=
0.12
<
Γ
¯
2
K
+
Γ
~
2
=
0.128. According to (A) in Theorem 2, N
2 is extinct and
(47)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
=
η
1
a
11
=
0.1
0.8
=
0.125
.
Figure 1(d) confirms this.
(e)
In Figure1(e), we choose λ
1
=
1.058, λ
2
=
0.8. Then, η
1
=
0.04, η
2
=
0.08, Γ
1
=
0.24
<
Γ
¯
1
K
+
Γ
~
1
=
0.252, and Γ
2
=
0.12
>
Γ
¯
2
K
+
Γ
~
2
=
0.072. By (B) in Theorem 2, N
1 is extinct and
(48)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
=
η
2
a
22
=
0.08
0.7
=
0.1143
.
See Figure 1(e).
(f)
In Figure1(f), we let λ
1
=
1, λ
2
=
0.8. Then, Γ
1
=
0.24
>
Γ
¯
1
K
+
Γ
~
1
=
0.21 and Γ
2
=
0.12
>
Γ
¯
2
K
+
Γ
~
2
=
0.096. It follows from (C) in Theorem 2 that
(49)
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
1
(
s
)
d
s
=
Γ
1
-
Γ
¯
1
K
-
Γ
~
1
Γ
=
0.03
0.36
=
0.0833
,
lim
⁡
t
→
+
∞
⁡
t
-
1
∫
0
t
N
2
(
s
)
d
s
=
Γ
2
-
Γ
¯
2
K
-
Γ
~
2
Γ
=
0.0667
.
See Figure 1(f).Solutions of system (1) for r
10
=
0.7, r
20
=
0.5, a
11
=
0.8, a
12
=
0.5, a
21
=
0.4, a
22
=
0.7, τ
1
=
10, τ
2
=
12, k
=
g
=
m
=
0.2, h
=
0.5, b
=
0.1, γ
=
1, C
0
(
0
)
=
C
e
(
0
)
=
0.1, and step size Δ
t
=
0.001. (a) is with λ
1
=
1.1
,
λ
2
=
1; (b) is with λ
1
=
1, λ
2
=
1; (c) is with λ
1
=
1.2, λ
2
=
0.8; (d) is with λ
1
=
1, λ
2
=
0.8484; (e) is with λ
1
=
1.058, λ
2
=
0.8; (f) is with λ
1
=
1, λ
2
=
0.8.
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Conclusions and Future Directions

This paper investigates a stochastic delay competitive model in a polluted environment with impulsive toxicant input. For each population, the critical value between stability in the mean and extinction is obtained. Some recent works are extended and improved. Our Theorem2 has some important and interesting interpretation.(1)
Time delay is harmless for stability in the mean and extinction of the stochastic model (1).
(2)
White noises can change the dynamics of the population model greatly.Some interesting problems deserve further study. One can consider some more realistic systems, for example, stochastic delayed population model with the Markov switching (see, e.g., [22, 23, 29]). It is also interesting to extend Theorem 2 to n-species case.



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*Source: 634871-2014-05-25.xml*