# Extinction in a Nonautonomous Discrete Lotka-Volterra Competitive System with Time Delay

Authors: Meng Hu; Jingyuan Wei
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

This paper is concerned with a nonautonomous discrete Lotka-Volterra competitive system with time delay. By using some analytical techniques, we prove that, under certain conditions, one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.


---

## Body

## 1. Introduction

In this paper, we study the following Lotka-Volterra competitive system of difference equations(1)
x
1
n
+
1
=
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
,
x
2
n
+
1
=
x
2
n
exp
⁡
r
2
n
1
-
μ
1
n
x
1
n
-
x
2
n
-
τ
k
2
n
,where x
1
(
n
)
,
x
2
(
n
) are population density of species x
1 and x
2 at time n, respectively, and {
r
i
(
n
)
}, {
k
i
(
n
)
}, and {
μ
i
(
n
)
} for i
=
1,2 are bounded positive sequences such that(2)
0
<
k
i
∗
≤
k
i
n
≤
k
i
∗
,
0
<
r
i
∗
≤
r
i
n
≤
r
i
∗
,
0
<
μ
i
∗
≤
μ
i
n
≤
μ
i
∗
,
n
∈
N
.Here, for any bounded sequence {
a
(
n
)
}
,
a
∗
=
s
u
p
n
∈
N
a
(
n
) and a
∗
=
i
n
f
n
∈
N
a
(
n
).We consider system (1) with the following initial conditions:(3)
x
1
0
>
0
,
x
2
ϑ
=
φ
ϑ
>
0
,
ϑ
∈
N
-
τ
,
0
=
-
τ
,
-
τ
+
1
,
…
,
0
.It is not difficult to see that solutions of (1) are well defined for all n
≥
0 and satisfy x
i
(
n
)
>
0
,
i
=
1,2.During the last decades, the study of extinction and permanence of the species has become one of the most important topics in population dynamics, and most of the studies are based on the traditional Lotka-Volterra competitive systems; see, for example, [1–15].Consider the following nonautonomous Lotka-Volterra system of differential equations:(4)
x
i
′
t
=
x
i
t
b
i
t
-
∑
j
=
1
n
a
i
j
t
x
j
t
,
i
=
1,2
,
⋯
,
n
,
n
≥
2
,where x
i
(
t
) is population density of the ith species at time t and a
i
j
(
t
) and b
i
(
t
), i
,
j
=
1,2
,
⋯
,
n, are continuous bounded functions defined on R.Assume that(5)
a
i
j
l
>
0
,
a
i
j
u
<
+
∞
,
i
,
j
=
1,2
,
⋯
,
n
,
(6)
b
i
l
>
0
,
b
i
u
<
+
∞
,
i
=
1,2
,
⋯
,
n
;that is, the coefficients of system (4) are bounded above and below by strictly positive reals. Here, for any bounded function f
(
t
), f
u
=
s
u
p
t
∈
R
f
(
t
) and f
l
=
i
n
f
t
∈
R
f
(
t
).Montes de Oca and Zeeman [11] studied system (4), under which the functions a
i
j
(
t
) and b
i
(
t
) were assumed to satisfy conditions (5) and (6). It was shown that if, for each k
>
1, there exists i
k
<
k such that for any j
≤
k the inequality(7)
b
k
u
a
k
j
l
<
b
i
k
l
a
i
k
j
uholds, then every solution (
x
1
(
t
)
,
x
2
(
t
)
,
⋯
,
x
n
(
t
)
) of system (4) with x
i
(
t
0
)
>
0, i
=
1,2
,
⋯
,
n, for some t
0
∈
R has the property (8)
l
i
m
t
⟶
+
∞
x
1
t
-
u
1
t
=
0
,
l
i
m
t
⟶
+
∞
x
j
t
=
0
,
j
=
2,3
,
⋯
,
n
,where u
1
(
t
) is the unique solution of the logistic differential equation(9)
u
′
t
=
u
t
b
1
t
-
a
11
t
u
twhich is bounded above and below by strictly positive reals for all t
∈
R.Zeeman [12], Ahmad [13], Teng [14], and Zhao et al. [15] have also studied the extinction of species in system (4), especially in [15], and Zhao et al. obtained the same results as [11–13] did under the weaker assumption that, for each k
>
1, there exists i
k
<
k such that for any j
≤
k the inequality(10)
s
u
p
t
∈
t
0
,
+
∞
b
k
t
b
i
k
t
<
i
n
f
t
∈
t
0
,
+
∞
a
k
j
t
a
i
k
j
tholds for some t
0
∈
R.In addition, the nonautonomous discrete population models also received much attention from many scholars in the last decades, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have nonoverlapping generations. One of the famous models is the discrete Lotka-Volterra competitive system. Owing to its theoretical and practical significance, various discrete Lotka-Volterra competitive systems have been studied; see, for example, [16–18]. In these literatures, systems which have been studied are all delayed systems. Research has shown that time delays have a great destabilizing influence on species populations [19]. However, there are seldom results on the extinction and stability of species in a discrete population dynamic system, especially for a population dynamic system with time delay.Motivated by the above works, the main purpose of this paper is to study the extinction and stability of system (1) and derive some sufficient conditions which guarantee one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.The organization of this paper is as follows. In Section2, preliminary results are presented. In Sections 3 and 4, the main results are stated and proved. In Section 5, two examples together with their numerical simulations are given to illustrate the feasibility of the obtained results. In the last section, a brief discussion is stated.


## 2. Preliminaries

In this section, we shall develop some preliminary results, which will be used to prove the main results.Lemma 1 (see [20]).
Assume that{
x
(
n
)
} satisfies (11)
x
n
+
1
≥
x
n
exp
⁡
r
n
1
-
a
x
n
,
n
≥
n
0
,
l
i
m
s
u
p
n
⟶
+
∞
x
(
n
)
≤
x
∗ and x
(
n
0
)
>
0, where a is a positive constant such that a
x
∗
>
1 and n
0
∈
N. Then (12)
lim
inf
n
⟶
+
∞
⁡
x
n
≥
1
a
exp
⁡
r
∗
1
-
a
x
∗
.Lemma 2 (see [20]).
Assume that{
x
(
n
)
} satisfies x
(
n
)
>
0 and (13)
x
n
+
1
≤
x
n
exp
⁡
r
n
1
-
a
x
n for n
∈
[
n
1
,
+
∞
), where a is a positive constant. Then (14)
lim
sup
n
⟶
+
∞
⁡
x
n
≤
1
a
r
∗
exp
⁡
r
∗
-
1
.Lemma 3.
For every solution(
x
1
(
n
)
,
x
2
(
n
)
)
T of (1) we have (15)
l
i
m
s
u
p
n
⟶
+
∞
x
i
n
≤
x
i
∗
,
i
=
1,2
, where x
1
∗
=
k
1
∗
/
r
1
∗
exp
⁡
(
r
1
∗
-
1
)
,
x
2
∗
=
k
2
∗
/
r
2
∗
exp
⁡
(
r
2
∗
(
τ
+
1
)
-
1
).Proof.
Letx
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be any positive solution of system (1). From the first equation of (1), (16)
x
1
n
+
1
≤
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
n
≤
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
∗
. By Lemma 2, we have (17)
l
i
m
s
u
p
n
⟶
+
∞
x
1
n
≤
k
1
∗
r
1
∗
exp
⁡
r
1
∗
-
1
≜
x
1
∗
.
From the second equation of (1), we have (18)
x
2
n
+
1
≤
x
2
n
exp
⁡
r
2
n
, and, then,(19)
x
2
n
-
τ
≥
x
2
n
exp
⁡
-
τ
r
2
∗
. Substituting (19) into the second equation of (1), then(20)
x
2
n
+
1
≤
x
2
n
exp
r
2
n
1
-
x
2
n
-
τ
k
2
n
≤
⁡
x
2
n
exp
⁡
r
2
n
1
-
exp
⁡
-
τ
r
2
∗
k
2
∗
x
2
n
. By Lemma 2, we have (21)
l
i
m
s
u
p
n
⟶
+
∞
x
2
n
≤
k
2
∗
r
2
∗
exp
⁡
r
2
∗
τ
+
1
-
1
≜
x
2
∗
. This completes the proof.


## 3. Extinction ofx
2 and Stability of x
1

In this section, we firstly present the extinction of the speciesx
2.Theorem 4.
Assume that the inequality(22)
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
r
1
n
<
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n holds, and then the species x
2 will be driven to extinction; that is, for any positive solution (
x
1
(
n
)
,
x
2
(
n
)
)
T of system (1), x
2
(
n
)
⟶
0 exponentially as n
⟶
+
∞.Proof.
Letx
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be a solution of system (1) with initial conditions (3). First we show that x
2
(
n
)
⟶
0 exponentially as n
⟶
+
∞.
From (1), we have(23)
l
n
x
1
n
+
1
-
l
n
x
1
n
=
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
,
l
n
x
2
n
+
1
-
l
n
x
2
n
=
r
2
n
1
-
μ
1
n
x
1
n
-
x
2
n
-
τ
k
2
n
. By inequality (22), we can choose α
,
β
,
ε
>
0 such that(24)
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
r
1
n
<
α
β
-
ε
<
α
β
<
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n
, and then there exists an N
1
>
0 such that, for all n
>
N
1,(25)
r
2
n
β
-
r
1
n
α
<
-
ε
β
r
1
n
<
-
ε
β
r
1
∗
<
0
;
(26)
α
r
1
n
-
β
r
2
n
μ
1
n
k
1
n
<
0
;
(27)
α
r
1
n
μ
2
n
k
2
n
-
β
r
2
n
<
0
. It follows from (23) and (25)-(27) that(28)
β
l
n
x
2
n
+
1
-
l
n
x
2
n
-
α
l
n
x
1
n
+
1
-
l
n
x
1
n
=
r
2
n
β
-
r
1
n
α
-
β
r
2
n
μ
1
n
-
α
r
1
n
k
1
n
x
1
n
-
β
r
2
n
k
2
n
-
α
r
1
n
μ
2
n
x
2
n
-
τ
≤
r
2
n
β
-
r
1
n
α
<
-
ε
β
r
1
∗
<
0
. Summating both sides of inequality (28) from 0 to n
-
1, then(29)
β
l
n
x
2
n
-
l
n
x
2
0
-
α
l
n
x
1
n
-
l
n
x
1
0
<
-
ε
β
r
1
∗
n
. So, we can get(30)
x
2
n
<
x
1
n
x
1
0
α
x
2
0
β
1
/
β
exp
⁡
-
ε
r
1
∗
n
<
x
1
∗
x
1
0
α
x
2
0
β
1
/
β
exp
⁡
-
ε
r
1
∗
n
. Therefore, we have x
2
(
n
)
⟶
0 exponentially as n
⟶
+
∞. This completes the proof.Lemma 5.
Under the assumption of Theorem4. Let x
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be any positive solution of system (1), and then there exists a positive constant x
1
∗ such that (31)
l
i
m
i
n
f
n
⟶
+
∞
x
1
n
≥
x
1
∗
, where x
1
∗
=
k
1
∗
exp
⁡
[
r
1
∗
(
1
-
x
1
∗
/
k
1
∗
)
] is a constant independent of any positive solution of system (1); i.e., the first species x
1 of system (1) is permanent.Proof.
By Lemma3 and Theorem 4, (32)
l
i
m
s
u
p
n
⟶
+
∞
x
1
n
≤
x
1
∗
,
l
i
m
n
⟶
+
∞
x
2
n
=
0
, and, for arbitrarily small positive constant ε
(
0
<
ε
<
1
/
μ
2
∗
), there exists an N
2
>
0 such that (33)
x
1
n
<
x
1
∗
+
ε
,
x
2
n
<
ε for all n
>
N
2.
From the first equation of (1), for n
>
N
2
+
τ, (34)
x
1
n
+
1
=
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
>
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
∗
-
μ
2
∗
ε
=
x
1
n
exp
⁡
r
1
n
1
-
μ
2
∗
ε
1
-
x
1
n
1
-
μ
2
∗
ε
k
1
∗
. Let ε
⟶
0, and then (35)
x
1
n
+
1
≥
x
1
n
exp
⁡
r
1
n
1
-
x
1
n
k
1
∗
. It is easy to check that the inequality x
1
∗
/
k
1
∗
>
1 holds. By Lemma 1, we have (36)
l
i
m
i
n
f
n
⟶
+
∞
x
1
n
≥
k
1
∗
exp
⁡
r
1
∗
1
-
x
1
∗
k
1
∗
≜
x
1
∗
. This completes the proof.Consider the following discrete logistic equation:(37)
x
n
+
1
=
x
n
exp
⁡
r
1
n
1
-
x
n
k
1
n
.Lemma 6 (see [21]).
Assume that{
r
1
(
n
)
} and {
k
1
(
n
)
} satisfy (2), and then any positive solution {
x
(
n
)
} of (37) satisfies (38)
x
1
∗
<
l
i
m
i
n
f
n
⟶
+
∞
x
n
≤
l
i
m
s
u
p
n
⟶
+
∞
x
n
≤
x
1
∗
.Theorem 7.
Under the assumptions of Theorem4 and Lemmas 5 and 6, furthermore, suppose that(39)
k
1
∗
k
1
∗
exp
⁡
r
1
∗
-
1
≤
2
. Let x
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be any positive solution of system (1), and then the species x
2 will be driven to extinction; that is, x
2
(
n
)
⟶
0 as n
⟶
+
∞, and x
1
(
n
)
⟶
x
(
n
) as n
⟶
+
∞, where x
(
n
) is any positive solution of (37).Proof.
Letx
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be a solution of system (1) with initial conditions (3). From Lemmas 3 and 5, x
1
(
n
) is bounded above and below by positive constants on [
0
,
+
∞
). To finish the proof of Theorem 7, it is enough to show that x
1
(
n
)
⟶
x
(
n
) as n
⟶
+
∞, where x
(
n
) is any positive solution of (37).
Let(40)
y
n
=
l
n
x
1
n
-
l
n
x
n
, and then (41)
x
1
n
=
x
n
exp
⁡
y
n
. From the first equation of (1) and (37), we have (42)
l
n
x
1
n
+
1
-
l
n
x
1
n
=
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
,
l
n
x
n
+
1
-
l
n
x
n
=
r
1
n
1
-
x
n
k
1
n
, and then(43)
y
n
+
1
=
l
n
x
1
n
+
1
-
l
n
x
n
+
1
=
l
n
x
1
n
+
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
-
l
n
x
n
-
r
1
n
1
-
x
n
k
1
n
=
y
n
-
r
1
n
k
1
n
x
1
n
-
x
n
-
r
1
n
μ
2
n
x
2
n
-
τ
=
y
n
-
r
1
n
k
1
n
x
n
exp
⁡
y
n
-
1
-
r
1
n
μ
2
n
x
2
n
-
τ
. By the mean value theorem, e
x
p
{
y
(
n
)
}
-
1
=
e
x
p
{
θ
y
(
n
)
}
y
(
n
), where θ
∈
(
0,1
). It follows from (43) that(44)
y
n
+
1
=
y
n
1
-
r
1
n
k
1
n
x
n
exp
⁡
θ
y
n
-
r
1
n
μ
2
n
x
2
n
-
τ
. Notice that θ
∈
(
0,1
) and (40) implies that x
(
n
)
e
x
p
{
θ
y
(
n
)
} lies between x
(
n
) and x
1
(
n
). From Lemmas 3, 5, and 6 and Theorem 4, for arbitrarily small ε
>
0, there exists an N
3
>
0 such that (45)
x
1
∗
-
ε
<
x
1
n
<
x
1
∗
+
ε
,
x
2
n
<
ε
,
x
1
∗
-
ε
<
x
n
<
x
1
∗
+
ε for all n
>
N
3. Therefore,(46)
y
n
+
1
=
λ
y
n
+
r
1
∗
μ
2
∗
ε
, where (47)
λ
=
max
⁡
1
-
r
1
∗
k
1
∗
x
1
∗
+
ε
,
1
-
r
1
∗
k
1
∗
x
1
∗
-
ε
. Repeated iteration of (46) is(48)
y
n
+
1
<
λ
n
-
N
3
y
N
3
+
1
-
λ
n
-
N
3
1
-
λ
r
1
∗
μ
2
∗
ε
. If (
k
1
∗
/
k
1
∗
)
e
x
p
(
r
1
∗
-
1
)
<
2, then -
1
<
1
-
r
1
∗
/
k
1
∗
x
1
∗. For the above small enough ε, there is -
1
<
1
-
r
1
∗
/
k
1
∗
(
x
1
∗
+
ε
). On the other hand, 1
-
⁡
r
1
∗
/
k
1
∗
(
x
1
∗
+
ε
)
<
1
-
r
1
∗
/
k
1
∗
(
x
1
∗
-
ε
)
<
1; that is, 0
<
λ
<
1. From (48), we can obtain (49)
l
i
m
n
⟶
+
∞
y
n
=
0
; that is, (50)
l
i
m
n
⟶
+
∞
x
1
n
-
x
n
=
0
. This completes the proof.


## 4. Extinction ofx
1 and Stability of x
2

In this section, we firstly present the extinction of the speciesx
1.Theorem 8.
Assume that the inequality(51)
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
r
1
n
>
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n holds, and then the species x
1 will be driven to extinction; that is, for any positive solution (
x
1
(
n
)
,
x
2
(
n
)
)
T of system (1), x
1
(
n
)
⟶
0 exponentially as n
⟶
+
∞.Proof.
Letx
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be a solution of system (1) with initial conditions (3). First we show that x
1
(
n
)
⟶
0 exponentially as n
⟶
+
∞.
From (1), we have(52)
l
n
x
1
n
+
1
-
l
n
x
1
n
=
r
1
n
1
-
x
1
n
k
1
n
-
μ
2
n
x
2
n
-
τ
,
l
n
x
2
n
+
1
-
l
n
x
2
n
=
r
2
n
1
-
μ
1
n
x
1
n
-
x
2
n
-
τ
k
2
n
. By inequality (51), we can choose ξ
,
η
,
ε
>
0 such that(53)
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
r
1
n
>
ξ
η
+
ε
>
ξ
η
>
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n
, and then there exists an N
4
>
0 such that, for all n
>
N
4,(54)
r
2
n
η
-
r
1
n
ξ
>
ε
η
r
1
n
>
ε
η
r
1
∗
>
0
;
(55)
ξ
r
1
n
-
η
r
2
n
μ
1
n
k
1
n
>
0
;
(56)
ξ
r
1
n
μ
2
n
k
2
n
-
η
r
2
n
>
0
. It follows from (52) and (54)-(56) that(57)
ξ
l
n
x
1
n
+
1
-
l
n
x
1
n
-
η
l
n
x
2
n
+
1
-
l
n
x
2
n
=
r
1
n
ξ
-
r
2
n
η
-
ξ
r
1
n
k
1
n
-
η
r
2
n
μ
1
n
x
1
n
-
ξ
r
1
n
μ
2
n
-
η
r
2
n
k
2
n
x
2
n
-
τ
≤
r
1
n
ξ
-
r
2
n
η
<
-
ε
η
r
1
∗
<
0
. Summating both sides of inequality (57) from 0 to n
-
1, then(58)
ξ
l
n
x
1
n
-
l
n
x
1
0
-
η
l
n
x
2
n
-
l
n
x
2
0
<
-
ε
η
r
1
∗
n
. So, we can get(59)
x
1
n
<
x
2
n
x
2
0
η
x
1
0
ξ
exp
⁡
-
ε
η
r
1
∗
n
1
/
ξ
<
x
1
0
x
2
∗
x
2
0
η
exp
⁡
-
ε
η
r
1
∗
n
1
/
ξ
. Therefore, we have x
1
(
n
)
⟶
0 exponentially as n
⟶
+
∞. This completes the proof.Lemma 9.
Under the assumption of Theorem8, let x
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be any positive solution of system (1), and then there exists a positive constant x
2
∗ such that (60)
l
i
m
i
n
f
n
⟶
+
∞
x
2
n
≥
x
2
∗
, where (61)
x
2
∗
=
k
2
∗
exp
⁡
τ
r
2
∗
1
-
x
2
∗
k
2
∗
exp
⁡
r
1
∗
1
-
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
/
k
2
∗
k
2
∗
x
2
∗ is a constant independent of any positive solution of system (1); i.e., the second species x
2 of system (1) is permanent.Proof.
By Lemma3 and Theorem 8, (62)
l
i
m
n
⟶
+
∞
x
1
n
=
0
,
l
i
m
s
u
p
n
⟶
+
∞
x
2
n
≤
x
2
∗
, for arbitrarily small positive constant ε
(
0
<
ε
<
1
/
μ
2
∗
), there exists an N
5
>
0 such that (63)
x
1
n
<
ε
,
x
2
n
<
x
2
∗
+
ε for all n
>
N
5.
From the second equation of (1), for n
>
N
5
+
τ, (64)
x
2
n
+
1
=
x
2
n
exp
⁡
r
2
n
1
-
μ
1
n
x
n
-
x
2
n
-
τ
k
2
n
>
x
2
n
exp
⁡
r
2
n
1
-
μ
1
∗
ε
-
x
2
∗
+
ε
k
2
∗
. Let ε
⟶
0, and then(65)
x
2
n
+
1
≥
x
2
n
exp
⁡
r
2
n
1
-
x
2
∗
k
2
∗
. Noting the fact that e
x
p
(
x
-
1
)
/
x
≥
1 for x
>
0, we obtain (66)
1
-
x
2
∗
k
2
∗
=
1
-
k
2
∗
k
2
∗
exp
⁡
r
2
∗
τ
+
1
-
1
r
2
∗
≤
0
. Therefore, from (65), we have(67)
x
2
n
+
1
≥
x
2
n
exp
⁡
r
2
∗
1
-
x
2
∗
k
2
∗
. Using (67), one could easily obtain that(68)
x
2
n
-
τ
≤
x
2
n
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
k
2
∗
. Substituting (68) into the second equation (1), for n
>
N
5
+
τ, deduces (69)
x
2
n
+
1
>
x
2
n
exp
⁡
r
2
n
1
-
μ
1
∗
ε
-
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
/
k
2
∗
k
2
∗
x
2
n
. Let ε
⟶
0, and then (70)
x
2
n
+
1
≥
x
2
n
exp
⁡
r
2
n
1
-
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
/
k
2
∗
k
2
∗
x
2
n
. It is easy to check that the inequality (71)
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
/
k
2
∗
k
2
∗
x
2
∗
>
1 holds. By Lemma 1, we have (72)
l
i
m
i
n
f
n
⟶
+
∞
x
2
n
≥
k
2
∗
exp
⁡
τ
r
2
∗
1
-
x
2
∗
k
2
∗
exp
⁡
r
2
∗
1
-
exp
⁡
-
τ
r
2
∗
1
-
x
2
∗
/
k
2
∗
k
2
∗
x
2
∗
≜
x
2
∗
. This completes the proof.Consider the following discrete logistic equation:(73)
x
n
+
1
=
x
n
exp
⁡
r
2
n
1
-
x
n
-
τ
k
2
n
.Lemma 10 (see [22]).
Assume that{
r
2
(
n
)
} and {
k
2
(
n
)
} satisfy (2), and then any positive solution x
(
n
) of (73) satisfies(74)
x
2
∗
<
l
i
m
i
n
f
n
⟶
+
∞
x
n
≤
l
i
m
s
u
p
n
⟶
+
∞
x
n
≤
x
2
∗
.Theorem 11.
Under the assumptions of Theorem8 and Lemmas 9 and 10, furthermore, suppose that(75)
k
2
∗
k
2
∗
exp
⁡
r
2
∗
1
+
τ
-
1
≤
2
1
+
2
τ
. Let x
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be any positive solution of system (1), and then the species x
1 will be driven to extinction; that is, x
1
(
n
)
⟶
0 as n
⟶
+
∞ and x
2
(
n
)
⟶
x
(
n
) as n
⟶
+
∞, where x
(
n
) is any positive solution of (73).Proof.
Letx
~
(
n
)
=
(
x
1
(
n
)
,
x
2
(
n
)
)
T be a solution of system (1) with initial conditions (3). To finish the proof of Theorem 11, it is enough to show that x
2
(
n
)
⟶
x
(
n
) as n
⟶
+
∞, where x
(
n
) is any positive solution of (73).
From Lemmas3 and 9, Theorem 8, and (74), for arbitrarily small positive constant ε
>
0, there exists an N
6
>
0 such that, for all n
>
N
6,(76)
x
2
∗
-
ε
<
x
2
n
,
x
n
<
x
2
∗
+
ε
,
x
1
n
<
ε
.
Let(77)
u
n
=
l
n
x
2
n
-
l
n
x
n
-
∑
s
=
n
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
. From the second equation of (1) and (73), we have (78)
l
n
x
2
n
+
1
-
l
n
x
2
n
=
r
2
n
1
-
μ
1
n
x
1
n
-
x
2
n
-
τ
k
2
n
,
l
n
x
n
+
1
-
l
n
x
n
=
r
2
n
1
-
x
n
-
τ
k
2
n
, and then (79)
u
n
+
1
=
l
n
x
2
n
-
l
n
x
n
-
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
-
∑
s
=
n
+
1
-
τ
n
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
-
r
2
n
μ
1
n
x
1
n
. Therefore, we have (80)
Δ
u
n
=
u
n
+
1
-
u
n
=
-
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
-
r
2
n
μ
1
n
x
1
n
, and (81)
u
n
+
1
+
u
n
=
2
l
n
x
2
n
-
l
n
x
n
-
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
-
∑
s
=
n
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
-
∑
s
=
n
+
1
-
τ
n
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
-
r
2
n
μ
1
n
x
1
n
=
2
l
n
x
2
n
-
l
n
x
n
-
2
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
-
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
-
2
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
-
r
2
n
μ
1
n
x
1
n
.
Define(82)
V
1
n
=
u
2
n
, and then(83)
Δ
V
1
n
=
u
2
n
+
1
-
u
2
n
=
Δ
u
n
u
n
+
1
+
u
n
=
-
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
-
r
2
n
μ
n
x
1
n
u
n
+
1
+
u
n
=
-
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
u
n
+
1
+
u
n
-
r
2
n
μ
1
n
x
1
n
u
n
+
1
+
u
n
=
-
2
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
l
n
x
2
n
-
l
n
x
n
+
2
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
x
n
x
2
n
-
τ
-
x
n
-
τ
+
r
2
2
n
+
τ
k
2
2
n
+
τ
x
2
n
-
x
n
2
+
2
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
x
2
n
-
x
n
+
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
-
u
n
+
1
+
u
n
r
2
n
μ
1
n
x
1
n
. Using the mean value theorem, then (84)
l
n
x
2
n
-
l
n
x
n
=
1
ζ
n
x
2
n
-
x
n
, where ζ
(
n
) lies between x
2
(
n
) and x
(
n
).
Noting the fact that2
a
b
≤
a
2
+
b
2, it follows from (83) that(85)
Δ
V
1
n
≤
-
2
r
2
n
+
τ
k
2
n
+
τ
1
ζ
n
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
2
+
r
2
2
n
+
τ
k
2
2
n
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
-
u
n
+
1
+
u
n
r
2
n
μ
1
n
x
1
n
. From (76) and (85), let ε be a sufficient small positive constant, for n
>
N
6, and we have(86)
Δ
V
1
n
≤
-
2
r
2
n
+
τ
k
2
n
+
τ
1
ζ
n
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
2
+
r
2
2
n
+
τ
k
2
2
n
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
. Let (87)
V
2
n
=
∑
s
=
n
-
τ
n
-
1
r
2
s
+
2
τ
k
2
s
+
2
τ
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
2
,
V
3
n
=
∑
l
=
n
n
-
2
+
τ
r
2
l
+
τ
k
2
l
+
τ
∑
s
=
l
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
s
-
x
s
2
, and then(88)
Δ
V
2
n
=
r
2
n
+
2
τ
k
2
n
+
2
τ
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
2
-
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
τ
-
x
n
-
τ
2
,
Δ
V
3
n
=
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
n
-
1
+
τ
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
-
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
. Define (89)
V
n
=
V
1
n
+
V
2
n
+
V
3
n
. It follows from (86) and (88) that(90)
Δ
V
n
≤
-
2
r
2
n
+
τ
k
2
n
+
τ
1
ζ
n
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
r
2
n
k
2
n
x
2
n
-
x
n
2
+
r
2
2
n
+
τ
k
2
2
n
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
2
τ
k
2
n
+
2
τ
r
2
n
+
τ
k
2
n
+
τ
x
2
n
-
x
n
2
+
r
2
n
+
τ
k
2
n
+
τ
∑
s
=
n
+
1
n
-
1
+
τ
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
=
-
r
2
n
+
τ
k
2
n
+
τ
2
ζ
n
-
r
2
n
k
2
n
-
r
2
n
+
τ
k
2
n
+
τ
-
∑
s
=
n
+
1
-
τ
n
-
1
r
2
s
+
τ
k
2
s
+
τ
-
r
2
n
+
τ
k
2
n
+
τ
-
∑
s
=
n
+
1
n
-
1
+
τ
r
2
s
+
τ
k
2
s
+
τ
x
2
n
-
x
n
2
. From (75), we have (91)
2
x
2
∗
-
1
+
2
τ
r
2
∗
k
2
∗
>
0
, and then, for the above small enough ε, there exists a positive constant q
>
0 such that(92)
2
x
2
∗
+
ε
-
1
+
2
τ
r
2
∗
k
2
∗
>
q
>
0
. From (76), (90), and (92), for n
>
N
6, we have(93)
Δ
V
n
≤
-
r
2
n
+
τ
k
2
n
+
τ
2
x
1
∗
+
ε
-
1
+
2
τ
r
2
∗
k
2
∗
x
2
n
-
x
n
2
<
-
q
r
2
∗
k
2
∗
x
2
n
-
x
n
2
. Summating both sides of (93) from N
6
+
τ to n, we have (94)
∑
j
=
N
6
+
τ
n
V
j
+
1
-
V
j
≤
-
q
r
2
∗
k
2
∗
∑
j
=
N
6
+
τ
n
x
2
j
-
x
j
2
, then (95)
V
n
+
1
+
q
r
2
∗
k
2
∗
∑
j
=
N
6
+
τ
n
x
2
j
-
x
j
2
≤
V
N
6
+
τ
, and so, (96)
∑
j
=
N
6
+
τ
n
x
2
j
-
x
j
2
≤
V
N
6
+
τ
q
k
2
∗
r
2
∗
. It follows from (76) that V
i
(
N
6
+
τ
)
,
i
=
1,2
,
3 are all bounded. Hence, (97)
∑
j
=
N
6
+
τ
n
x
2
j
-
x
j
2
≤
V
N
6
+
τ
q
k
2
∗
r
2
∗
<
+
∞
, and then (98)
l
i
m
s
u
p
n
⟶
+
∞
∑
j
=
N
6
+
τ
n
x
2
j
-
x
j
2
≤
V
N
6
+
τ
q
k
2
∗
r
2
∗
<
+
∞
. This implies that (99)
l
i
m
n
⟶
+
∞
x
2
n
-
x
n
2
=
0
. This completes the proof.


## 5. Numerical Examples and Simulations

In this section, we give two examples to illustrate the feasibility of our results.Example 1.
Consider the following system:(100)
x
1
n
+
1
=
x
1
n
exp
⁡
0.5
-
0.2
sin
⁡
n
1
-
x
1
n
2
+
0.2
cos
⁡
n
-
x
2
n
-
2
,
x
2
n
+
1
=
x
2
n
exp
⁡
0.2
-
0.1
cos
⁡
n
1
-
x
1
n
-
x
2
n
-
2
0.5
+
0.1
sin
⁡
n
; that is (101)
r
1
n
=
0.5
-
0.2
sin
⁡
n
,
k
1
n
=
2
+
0.2
cos
⁡
n
,
μ
2
n
=
1
,
r
2
n
=
0.2
-
0.1
cos
⁡
n
,
μ
1
n
=
1
,
k
2
n
=
0.5
+
0.1
sin
⁡
n
.
By a direct calculation, we can get(102)
x
1
∗
=
2.3283
,
x
1
∗
=
1.4657
,
x
2
∗
=
1.8097
;
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
r
1
n
=
0.3709
<
0.4078
=
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n
;
k
1
∗
k
1
∗
exp
⁡
r
1
∗
-
1
=
0.9054
<
2
; that is, the conditions of Theorem 7 hold, and so species x
2 will be driven to extinction while species x
1 is asymptotic to any positive solution of(103)
x
1
n
+
1
=
x
1
n
exp
⁡
0.15
+
0.05
sin
⁡
n
1
-
x
1
n
1.2
+
0.2
cos
⁡
n
.The solutions of systems (100) and (103) corresponding to initial values are displayed in Figure 1.Figure 1
Dynamic behaviors of speciesx
1 and x
2 in (100) with initial values x
1
(
0
)
=
1,3
,
5 and x
2
(
ϑ
)
=
1,3
,
5 for ϑ
=
-
2
,
-
1,0, respectively; x is a solution of (103).Example 2.
Consider the following system:(104)
x
1
n
+
1
=
x
1
n
exp
⁡
0.15
+
0.05
sin
⁡
n
1
-
x
1
n
1.2
+
0.2
cos
⁡
n
-
x
2
n
-
1
,
x
2
n
+
1
=
x
2
n
exp
⁡
0.17
+
0.07
cos
⁡
n
1
-
0.3
x
1
n
-
x
2
n
-
1
2.1
+
0.1
sin
⁡
n
; that is, (105)
r
1
n
=
0.15
+
0.05
sin
⁡
n
,
k
1
n
=
1.2
+
0.2
cos
⁡
n
,
μ
2
n
=
1
,
r
2
n
=
0.17
+
0.07
cos
⁡
n
,
μ
1
n
=
0.3
,
k
2
n
=
2.1
+
0.1
sin
⁡
n
. By a direct calculation, we can get (106)
x
1
∗
=
3.1453
,
x
2
∗
=
5.4498
,
x
2
∗
=
1.7240
;
l
i
m
i
n
f
n
⟶
+
∞
r
2
n
r
1
n
=
1.0712
>
0.9552
=
l
i
m
s
u
p
n
⟶
+
∞
r
2
n
μ
1
n
k
1
n
r
1
n
,
r
2
n
r
1
n
μ
2
n
k
2
n
;
k
2
∗
k
2
∗
exp
⁡
r
2
∗
1
+
τ
-
1
=
0.6540
<
0.6667
=
2
1
+
2
τ
; that is, the conditions of Theorem 11 hold, and so species x
1 will be driven to extinction while species x
2 is asymptotic to any positive solution of(107)
x
2
n
+
1
=
x
2
n
exp
⁡
0.17
+
0.07
cos
⁡
n
1
-
x
2
n
-
1
2.1
+
0.1
sin
⁡
n
.The solutions of systems (104) and (107) corresponding to initial values are displayed in Figure 2.Figure 2
Dynamic behaviors of speciesx
1 and x
2 in (104) with initial values x
1
(
0
)
=
1,3
,
5 and x
2
(
ϑ
)
=
1,3
,
5 for ϑ
=
-
1,0, respectively; x is a solution of (107).


## 6. Conclusion

A nonautonomous discrete Lotka-Volterra competitive system with time delay has been studied in this paper. It is shown that if the coefficients are bounded above and below by positive constants and satisfy certain inequalities, then one of the species will be driven to extinction while the other one will stabilize at a certain solution of a nonlinear single species model.This paper provided an effective method for the further study on permanence and extinction of population dynamic systems with time delay. In fact, our techniques in this paper are applicable to a pure delayed discreten-species Lotka-Volterra competitive system. Furthermore, one may consider a discrete Lotka-Volterra competitive system with infinite delay, which we leave for future work.As we know, system (1) is a basic model, and, based on system (1), we can establish different types of Lotka-Volterra competitive systems according to the ecological significance, such as plankton allelopathy systems and functional response systems, by using the same methods and analytical techniques, and similar results can be obtained. From the obtained results, we not only can reveal the inherent law of the system and predict the development of the population but also can control or adjust the ecological development of the population in a better way. Besides, the results obtained in this paper also can be applied to economic systems, such as systems of industrial clusters and financial ecology; one may see [23].



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