# Dynamics Behaviors of a Discrete  Ratio-Dependent Predator-Prey System with

Authors: Jinghui Yang
Journal: Discrete Dynamics in Nature and Society
 (2008)

---

## Abstract

A ratio-dependent predator-prey system with Holling type III functional
response and feedback controls is proposed. By constructing a suitable Lyapunov
function and using the comparison theorem of difference equation, sufficient
conditions which ensure the permanence and global attractivity of the system are
obtained. After that, under some suitable conditions, we show that the predator
speciesy will be driven to extinction. Examples together with their numerical
simulations show that the main results are verifiable.


---

## Body

## 1. Introduction

Wang and Li
[1, 2] established verifiable criteria for the existence of globally
attractive positive periodic
solutions of the following delayed predator-prey model with Holling type III
functional response: (1.1)
N
1
′
(
t
)
=
N
1
(
t
)
(
b
1
(
t
)
−
a
1
(
t
)
N
1
(
t
−
τ
1
(
t
)
)
−
α
1
(
t
)
N
1
(
t
)
1
+
m
N
1
2
(
t
)
N
2
(
t
−
σ
(
t
)
)
)
,
N
2
′
(
t
)
=
N
2
(
t
)
(
−
b
2
(
t
)
−
a
2
(
t
)
N
2
(
t
)
+
α
2
(
t
)
N
1
2
(
t
−
τ
2
(
t
)
)
1
+
m
N
1
2
(
t
−
τ
2
(
t
)
)
)
, where N
1
(
t
)
,
N
2
(
t
) are the densities of the prey population and
predator population at time t, respectively; b
i
:
ℝ
→
ℝ
,
a
i
,
τ
i
,
σ
,
α
i
:
ℝ
→
[
0
,
+
∞
)
(
i
=
1
,
2
)  are continuous functions
of period T and ∫
0
T
b
i
(
t
)
d
t
>
0
,
α
i
(
t
)
≠
0; m is a nonnegative constant. For more works on
the predator-prey system with Holling type functional response, one could refer
to [3–11] and the references cited therein. But, recently, lots of scholars found
that when predators have to search for food (and, therefore, have to share or
compete for food), a more suitable general predator-prey theory should be based
on the so-called ratio-dependent theory, which can be roughly stated as that
the per capita predator growth rate should be the so-called ratio-dependent
functional response. This is strongly supported by numerous fields and
laboratory experiments and observations [12, 13]. In [14], Wang and Li proposed
the following ratio-dependent predator-prey system with Holling type III
functional response:(1.2)
x
′
(
t
)
=
x
(
t
)
(
a
(
t
)
−
b
(
t
)
∫
−
∞
t
k
(
t
−
s
)
x
(
s
)
d
s
)
−
c
(
t
)
x
2
(
t
)
y
(
t
)
m
2
y
2
(
t
)
+
x
2
(
t
)
,
y
′
(
t
)
=
y
(
t
)
(
e
(
t
)
x
2
(
t
−
τ
)
m
2
y
2
(
t
−
τ
)
+
x
2
(
t
−
τ
)
−
d
(
t
)
)
,where x
(
t
)
,
y
(
t
) are the densities of the prey population and
predator population at time t, respectively, a
(
t
)
,
b
(
t
)
,
c
(
t
)
,
d
(
t
),
and e
(
t
) are all positive periodic continuous
functions, and m
>
0
,
τ
≥
0 are real constants. They found that the
criteria for the permanence are exactly the same as those for the existence of
positive periodic solution of (1.2). For more works on the ratio-dependent
predator-prey system, one could refer to [15, 16] and the references cited
therein.On the other
hand, when the size of the population is rarely small or the population has
nonoverlapping generations, the discrete time models are more appropriate than
the continuous ones [17, 18]. For the discrete ratio-dependent predator-prey
model with Holling type III functional response, Fan and Li [19] considered the
following system:  (1.3)
N
1
(
k
+
1
)
=
N
1
(
k
)
exp
{
b
1
(
k
)
−
a
1
(
k
)
N
1
(
k
−
[
τ
1
]
)
−
a
1
(
k
)
N
1
(
k
)
N
2
(
k
)
N
1
2
(
k
)
+
m
2
(
k
)
N
2
2
(
K
)
}
,
N
2
(
k
+
1
)
=
N
2
(
k
)
exp
{
−
b
2
(
k
)
+
a
2
(
k
)
N
1
2
(
k
−
[
τ
2
]
)
N
1
2
(
k
−
[
τ
1
]
)
+
m
2
(
k
)
N
2
2
(
k
−
[
τ
2
]
)
}
,
sufficient conditions
which ensure the permanence of system (1.3) are obtained.However, as
pointed out by Huo and Li [20], “ecosystem in the real world is
continuously disturbed by unpredictable forces which can result in changes in
the biological parameters such as survival rates. The question of whether or
not an ecosystem can withstand those unpredictable disturbances which persist
for a finite period of time is of practical interest in ecology . In the
language of control variables, we call the disturbance functions as control
variables.” For this direction, in [15], Chen and Ji proposed the
following system:(1.4)
x
˙
1
=
x
1
[
a
1
(
t
)
−
b
(
t
)
x
−
c
1
(
t
)
x
1
+
α
(
t
)
x
2
−
e
1
(
t
)
u
1
]
,
x
˙
2
=
x
2
[
−
a
2
(
t
)
+
c
2
(
t
)
x
1
+
α
(
t
)
x
2
+
e
2
(
t
)
u
2
]
,
u
˙
1
=
h
1
(
t
)
−
d
1
(
t
)
u
1
+
f
1
(
t
)
x
1
,
u
˙
2
=
h
2
(
t
)
−
d
2
(
t
)
u
2
+
f
2
(
t
)
x
2
,where x
i
(
t
) stand for the densities of the prey and
predator, respectively, u
i
(
t
)
(
i
=
1
,
2
) is the control variable, α
(
t
)
,
b
(
t
)
,
a
i
(
t
)
,
c
i
(
t
)
,
h
i
(
t
)
,
d
i
(
t
)
,
f
i
(
t
),
and e
i
(
t
)
(
i
=
1
,
2
) are continuous and strictly positive
functions. In this paper, they considered the almost periodic solution of
system (1.4). For more work on this direction, one could refer to
[15, 21–28].To the best of
the author's knowledge, so far no scholar has considered system (1.3) with
feedback controls. This motivates us to propose and study the following
discrete ratio-dependent predator-prey system with Holling type III and
feedback controls: (1.5)
x
(
n
+
1
)
=
x
(
n
)
exp
{
a
(
n
)
−
b
(
n
)
x
(
n
)
−
c
(
n
)
x
(
n
)
y
(
n
)
x
2
(
n
)
+
m
2
(
n
)
y
2
(
n
)
−
e
1
(
n
)
u
1
(
n
)
}
,
y
(
n
+
1
)
=
y
(
n
)
exp
{
−
d
(
n
)
+
f
(
n
)
x
2
(
n
)
x
2
(
n
)
+
m
2
(
n
)
y
2
(
n
)
−
e
2
(
n
)
u
2
(
n
)
}
,
Δ
u
1
(
n
)
=
−
η
1
(
n
)
u
1
(
n
)
+
q
1
(
n
)
x
(
n
)
,
Δ
u
2
(
n
)
=
−
η
2
(
n
)
u
2
(
n
)
+
q
2
(
n
)
y
(
n
)
, where x
(
t
)
,
y
(
t
) are the densities of the prey population and
predator population at time t, respectively, for i
=
1
,
2
,
{
a
(
n
)
}
,
{
b
(
n
)
}
,
{
c
(
n
)
}
,
{
d
(
n
)
}
,
{
f
(
n
)
}
,
{
m
(
n
)
}
,
{
e
i
(
n
)
}
,
{
η
i
(
n
)
},
and {
q
i
(
n
)
} are all bounded nonnegative sequences such
that(H
0)
0
<
a
L
≤
a
(
n
)
≤
a
U
,
0
<
b
L
≤
b
(
n
)
≤
b
U
,
0
<
c
L
≤
c
(
n
)
≤
c
U
,
0
<
d
L
≤
d
(
n
)
≤
d
U
,
0
<
f
L
≤
f
(
n
)
≤
f
U
,
0
<
m
L
≤
m
(
n
)
≤
m
U
,
0
<
e
i
L
≤
e
i
(
n
)
≤
e
i
U
,
0
<
η
i
L
≤
η
i
(
n
)
≤
η
i
U
<
1
,
0
<
q
i
L
≤
q
i
(
n
)
≤
q
i
U
<
1.Here, for any bounded sequence  {
a
(
n
)
}
,
a
L
=
inf
n
∈
ℕ
{
a
(
n
)
}
,
a
U
=
sup
n
∈
ℕ
{
a
(
n
)
}.By the
biological meaning, we will focus our discussion on the positive solutions of
system (1.5). So, it is assumed that the initial conditions of system
(1.5) are
of the form(1.6)
x
(
0
)
>
0
,
y
(
0
)
>
0
,
u
i
(
0
)
>
0
,
i
=
1,2.It is not difficult to see that solutions of (1.5)
and (1.6) are well defined and satisfy
(1.7)
x
(
n
)
>
0
,
y
(
n
)
>
0
,
u
i
(
n
)
>
0
,
for
k
∈
ℕ
+
.The main purpose of this paper is to derive sufficient conditions for
the permanence, global attractivity, and extinction of system (1.5).The
organization of this paper is as follows. In Section2, we introduce some
useful lemmas. In Section 3, we will study the permanence and global
attractivity of system (1.5). In Section 4, we will study the extinction of the
predator species y. In the last section, numerical simulation is
presented to illustrate the feasibility of our main results.


## 2. Preliminaries

Now, let us
state several lemmas which will be useful in proving the main results.First, let us
consider the first-order difference equation(2.1)
y
(
k
+
1
)
=
A
y
(
k
)
+
B
,
k
=
1
,
2
…
,where A
,
B
are positive constants. The following Lemma
2.1 is a direct corollary of Theorem 6.2 of
L. Wang and M. Q. Wang [29, page
125].Lemma 2.1.
Assume that|
A
|
<
1.
For any initial y
(
0
),
there exists a unique solution y
(
k
) of (2.1), which can be expressed as follows:
(2.2)
y
(
k
)
=
A
k
(
y
(
0
)
−
y
*
)
+
y
*
,where y
*
=
B
/
(
1
−
A
).
Thus, for any solutions y
(
k
) of
system
(2.1),
(2.3)
lim
k
→
+
∞
y
(
k
)
=
y
*
.The following
comparison theorem for difference equation is Theorem 2.1 of [29, page
241].Lemma 2.2.
Letk
∈
ℕ
k
0
+
=
{
k
0
,
k
0
+
1
,
…
,
k
0
+
l
,
…
} and r
≥
0.
For any fixed k
,
g
(
k
,
r
) is a nondecreasing function with respect to r,
and for k
≥
k
0,
the following inequalities hold:
(2.4)
y
(
k
+
1
)
≤
g
(
k
,
y
(
k
)
)
,
u
(
k
+
1
)
≥
g
(
k
,
u
(
k
)
)
.If y
(
k
0
)
≤
u
(
k
0
),
then y
(
k
)
≤
u
(
k
) for all k
≥
k
0.The following
Lemmas2.3 and
2.4 can be found in [21].Lemma 2.3 (see [21]).
Assume thatx
(
n
) satisfies x
(
n
)
>
0 and
(2.5)
x
(
n
+
1
)
≤
x
(
n
)
exp
(
r
(
n
)
−
a
(
n
)
x
(
n
)
)for n
∈
ℕ,
where r
(
n
) and a
(
n
) are nonnegative sequences bounded above and
below by positive constants. Then(2.6)
limsup
n
→
+
∞
x
(
n
)
≤
1
a
L
exp
(
r
U
−
1
)
.Lemma 2.4 (see [21]).
Assume that{
x
(
n
)
} satisfies
(2.7)
x
(
n
+
1
)
≥
x
(
n
)
exp
(
r
(
n
)
−
a
(
n
)
x
(
n
)
)
,
k
≥
N
0
,
limsup
n
→
+
∞
x
(
n
)
≤
x
* and x
(
N
0
)
>
0,
where r
(
n
) and a
(
n
) are nonnegative sequences bounded above and
below by positive constants and N
0
∈
ℕ.
Then(2.8)
liminf
n
→
+
∞
x
(
n
)
≥
r
L
exp
{
r
L
−
a
U
x
*
}
a
U
.


## 3. Permanence and global attractivity

Now, we
investigate the permanence and global attractivity of system (1.5).Theorem 3.1.
Assume that(H
1)
a
L
−
c
U
2
m
L
−
e
1
U
W
1
>
0
,
(H
2)
f
L
−
d
U
−
e
2
U
W
2
>
0hold, then the species x and y are permanent, that is, for any positive
solution
(
x
(
n
)
,
y
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) of system (1.5) with the initial conditions
((1.6)),(3.1)
m
1
≤
liminf
n
→
+
∞
x
(
n
)
≤
limsup
n
→
+
∞
x
(
n
)
≤
M
1
,
m
2
≤
liminf
n
→
+
∞
y
(
n
)
≤
limsup
n
→
+
∞
y
(
n
)
≤
M
2
,
w
1
≤
liminf
n
→
+
∞
u
1
(
n
)
≤
limsup
n
→
+
∞
u
1
(
n
)
≤
W
1
,
w
2
≤
liminf
n
→
+
∞
u
2
(
n
)
≤
limsup
n
→
+
∞
u
2
(
n
)
≤
W
2
,where(3.2)
m
1
=
a
L
−
c
U
/
2
m
L
−
e
1
U
W
1
b
U
exp
{
a
L
−
b
U
M
1
−
c
U
2
m
L
−
e
1
U
W
1
}
,
m
2
=
2
m
1
(
f
L
−
d
U
−
e
2
U
W
2
)
f
L
m
U
exp
{
−
d
U
−
e
2
W
2
}
,
w
1
=
q
1
L
m
1
η
1
U
,
w
2
=
q
2
L
m
2
η
2
U
,
M
1
=
1
b
L
exp
{
a
U
−
1
}
,
M
2
=
f
U
M
1
2
d
L
m
L
exp
{
f
U
−
d
L
}
,
W
1
=
q
1
U
M
1
η
1
L
,
W
2
=
q
2
U
M
2
η
2
L
.Proof.
We divided the proof into five steps.
Step 1.
We show(3.3)
limsup
n
→
+
∞
x
(
n
)
≤
M
1
.
From the first
equation of system (1.5),(3.4)
x
(
n
+
1
)
≤
x
(
n
)
exp
{
a
(
n
)
−
b
(
n
)
x
(
n
)
}
.Then (3.3) follows immediately
from Lemma 2.3. Thus, for any positive ε,
there exists an n
0
>
0 such that(3.5)
x
(
n
)
≤
M
1
+
ε
∀
n
>
n
0
.
Step 2.
We provelimsup
n
→
+
∞
y
(
n
)
≤
M
2 by distinguishing two cases.
Case 1.
There exists ann
1
≥
n
0 such that y
(
n
1
+
1
)
≥
y
(
n
1
).
Then, by the second equation of system (1.5), we have(3.6)
−
d
(
n
1
)
+
f
(
n
1
)
x
2
(
n
1
)
x
2
(
n
1
)
+
m
2
(
n
1
)
y
2
(
n
1
)
−
e
2
(
n
1
)
u
2
(
n
1
)
≥
0,which implies(3.7)
−
d
L
+
f
U
x
(
n
1
)
2
m
L
y
(
n
1
)
≥
0.The above inequality combined
with (3.5) leads to y
(
n
1
)
≤
f
U
(
M
1
+
ε
)
/
2
m
L
d
L.
Thus from the second equation of system (1.5) again we have(3.8)
y
(
n
1
+
1
)
≤
y
(
n
1
)
exp
{
−
d
(
n
1
)
+
f
(
n
1
)
x
2
(
n
1
)
x
2
(
n
1
)
+
y
2
(
n
1
)
}
≤
y
(
n
1
)
exp
{
−
d
L
+
f
U
}
≤
f
U
(
M
1
+
ε
)
2
m
L
d
L
exp
{
−
d
L
+
f
U
}
=
def
M
2
ε
.We claim that y
(
n
)
≤
M
2
ε for all n
≥
n
1.
In fact, suppose there exists q
≥
n
1
+
2 such that y
(
q
)
>
M
2
ε.
Let q
0 be the smallest integer between n
1 and q such that y
(
q
0
)
>
M
2
ε and y
(
q
0
−
1
)
≤
M
2
ε.
Then y
(
q
0
)
>
y
(
q
0
−
1
) implies y
(
q
0
)
≤
M
2
ε,
a contradiction. This proves the claim. Setting ε
→
0 in (3.8) leads to limsup
n
→
+
∞
y
(
n
)
≤
M
2.
Case 2.
Supposey
(
n
+
1
)
≤
y
(
n
) for all n
≥
n
0.
In this case, lim
n
→
+
∞
y
(
n
) exists, denoted by y
¯.
We claim that y
¯
≤
f
U
M
1
/
2
m
L
d
L.
If not, suppose y
¯
>
f
U
M
1
/
2
m
L
d
L.
Choose σ
>
0 such that σ
<
y
¯
−
f
U
M
1
/
2
m
L
d
L.
Taking limit in the second equation of system (1.5) produces(3.9)
lim
n
→
+
∞
(
−
d
(
n
)
+
f
(
n
)
x
2
(
n
)
x
2
(
n
)
+
m
2
(
n
)
y
2
(
n
)
−
e
2
(
n
)
u
2
(
n
)
)
=
0
,which is impossible as(3.10)
−
d
(
n
)
+
f
(
n
)
x
2
(
n
)
x
2
(
n
)
+
m
2
(
n
)
y
2
(
n
)
−
e
2
(
n
)
u
2
(
n
)
≤
−
d
L
+
f
U
x
(
n
)
2
m
L
y
(
n
)
<
−
d
L
+
f
U
M
1
2
m
L
(
y
¯
−
σ
)
<
0for sufficiently large n.
This proves the claim. Since
(H
2) implies f
U
M
1
/
2
m
L
d
L
≤
M
2,
we have proved limsup
n
→
+
∞
y
(
n
)
≤
M
2.
Step 3.
We verify(3.11)
limsup
n
→
+
∞
u
1
(
n
)
≤
W
1
,
limsup
n
→
+
∞
u
2
(
n
)
≤
W
2
.For any positive ε,
there exists n
2 such that(3.12)
x
(
n
)
≤
M
1
+
ε
,
y
(
n
)
≤
M
2
+
ε
for
n
≥
n
2
.For n
≥
n
2,
(3.12) combined with the third equation of (1.5) gives(3.13)
Δ
u
1
(
n
)
≤
−
η
1
(
n
)
u
1
(
n
)
+
q
1
(
n
)
(
M
1
+
ε
)
.Thus(3.14)
u
1
(
n
+
1
)
≤
(
1
−
η
1
L
)
u
1
(
n
)
+
q
1
U
(
M
1
+
ε
)
for
n
≥
n
2
.With the help of Lemmas 2.1
and
2.2, we obtain(3.15)
limsup
n
→
+
∞
u
1
(
n
)
≤
q
1
U
(
M
1
+
ε
)
η
1
L
.Letting ε
→
0,
we immediately get(3.16)
limsup
n
→
+
∞
u
1
(
n
)
≤
W
1
.Similarly, one can show limsup
n
→
+
∞
u
2
(
n
)
≤
W
2.
Step 4.
We check(3.17)
liminf
n
→
+
∞
x
(
n
)
≥
m
1
,
limsup
n
→
+
∞
y
(
n
)
≥
m
2
.Conditions
(H
1) and (H
2) imply
that(3.18)
a
L
−
c
U
2
m
L
−
e
1
U
(
W
1
+
ε
)
>
0
,
f
L
−
d
U
−
e
2
U
(
W
1
+
ε
)
>
0hold for all small enough
positive constant ε.
For any such ε,
there exists n
3 such that(3.19)
x
(
n
)
≤
M
1
+
ε
,
u
1
(
n
)
≤
W
1
+
ε
for
n
≥
n
3
.Then, for n
≥
n
3,
it follows from (3.19) and the first equation of system (1.5)
that(3.20)
x
(
n
+
1
)
≥
x
(
n
)
exp
{
a
L
−
c
U
2
m
L
−
e
1
U
(
W
1
+
ε
)
−
b
U
x
(
n
)
}
.According to Lemma 2.4, one
has(3.21)
liminf
n
→
+
∞
x
(
n
)
≥
a
L
−
c
U
/
2
m
L
−
e
1
U
(
W
1
+
ε
)
b
U
exp
{
a
L
−
c
U
2
m
L
−
e
1
U
(
W
1
+
ε
)
−
b
U
(
M
1
+
ε
)
}
.Letting ε
→
0 leads to(3.22)
liminf
n
→
+
∞
x
(
n
)
≥
m
1
.
Now, for any small positiveε
1
<
m
1
/
2,
it follows from (3.19) and
(3.22) that there exists n
4
>
0 such that(3.23)
x
(
n
)
≥
m
1
−
ε
1
,
y
(
n
)
≤
M
2
+
ε
1
,
u
2
(
n
)
≤
W
2
+
ε
1for n
≥
n
4.
This, combined with the second equation of system (1.5), gives(3.24)
y
(
n
+
1
)
≥
y
(
n
)
exp
{
−
d
U
+
f
L
−
e
2
U
(
W
2
+
ε
1
)
−
f
L
m
U
y
(
n
)
2
(
m
1
−
ε
1
)
}
.Applying Lemma 2.4, one easily
obtains(3.25)
liminf
n
→
+
∞
y
(
n
)
≥
2
(
m
1
−
ε
1
)
(
f
L
−
d
U
−
e
2
U
(
W
2
+
ε
1
)
)
f
L
m
U
exp
{
f
L
−
d
U
−
e
2
U
(
W
2
+
ε
1
)
−
f
L
m
U
(
M
2
+
ε
1
)
2
(
m
1
−
ε
1
)
}
.Because of the arbitrariness of ε,
it is not difficult to see that liminf
n
→
+
∞
y
(
n
)
≥
m
2.
Step 5.
Finally, we only showliminf
n
→
+
∞
u
1
(
n
)
≥
w
1 as the proof of liminf
n
→
+
∞
u
2
(
n
)
≥
w
2 is similar. For any ε
2
>
0 such that ε
2
<
m
1,
there exists n
5 such that(3.26)
x
(
n
)
≥
m
1
−
ε
2
for
n
≥
n
5
.This and the third equation of
system (1.5) imply that(3.27)
Δ
u
1
(
n
)
≥
−
η
1
(
n
)
u
1
(
n
)
+
q
1
L
(
m
1
−
ε
2
)for n
≥
n
5.
Then, for n
≥
n
5,(3.28)
u
1
(
n
+
1
)
≥
(
1
−
η
1
U
)
u
1
(
n
)
+
q
1
L
(
m
1
−
ε
2
)
.It follows
from Lemmas
2.1 and
2.2 immediately that(3.29)
liminf
n
→
+
∞
u
1
(
n
)
≥
q
1
L
(
m
1
−
ε
2
)
η
1
U
.Letting ε
2
→
0 gives liminf
n
→
+
∞
u
1
(
n
)
≥
w
1.
This completes the proof of the theorem.Theorem 3.2.
Assume that (H
1) and (H
2) hold. Assume further that there exist positive
constants α
,
β
,
γ
,
ζ,
and δ such that
(H
3)
α
min
{
b
L
,
2
M
1
−
b
U
}
−
α
c
U
4
(
m
L
)
2
m
2
−
α
c
U
M
1
4
m
1
2
−
β
f
U
M
2
2
m
2
m
1
−
γ
q
1
U
>
δ
,
(H
4)
β
min
{
2
f
L
(
m
L
)
2
m
1
2
m
2
(
M
1
2
+
(
m
U
)
2
M
2
2
)
2
,
2
M
2
−
f
U
M
1
2
m
1
m
2
}
−
α
c
U
M
1
4
(
m
L
)
2
m
2
2
−
α
c
U
4
m
1
−
ζ
q
2
U
>
δ
,
(H
5)
γ
η
1
L
−
α
e
1
U
>
δ
,
(H
6)
ζ
η
2
L
−
β
e
2
U
>
δ
.Then the species x and y are globally attractive, that is, for any
positive solutions
(
x
1
(
n
)
,
y
1
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) and (
x
2
(
n
)
,
y
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) of system (1.5) with the initial conditions
((1.6)),(3.30)
lim
n
→
+
∞
|
x
1
(
n
)
−
x
2
(
n
)
|
=
0
,
lim
n
→
+
∞
|
y
1
(
n
)
−
y
2
(
n
)
|
=
0
,
lim
n
→
+
∞
|
u
1
(
n
)
−
u
1
(
n
)
|
=
0
,
lim
n
→
+
∞
|
u
2
(
n
)
−
u
2
(
n
)
|
=
0.Proof.
From conditions
(H
3)–(H
6),
there exists small enough positive constant
ε
<
min
{
m
1
/
2,
m
2
/
2
} such that(3.31)
α
min
{
b
L
,
2
(
M
1
+
ε
)
−
b
U
}
−
α
c
U
4
(
m
L
)
2
(
m
2
−
ε
)
−
α
c
U
(
M
1
+
ε
)
4
(
m
1
−
ε
)
2
−
β
f
U
(
M
2
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
−
γ
q
1
U
>
δ
,
β
min
{
2
f
L
(
m
L
)
2
(
m
1
−
ε
)
2
(
m
2
−
ε
)
(
(
M
1
+
ε
)
2
+
(
m
U
)
2
(
M
2
+
ε
)
2
)
2
,
2
(
M
2
+
ε
)
−
f
U
(
M
1
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
}
−
α
c
U
(
M
1
+
ε
)
4
(
m
L
)
2
(
m
2
−
ε
)
2
−
α
c
U
4
(
m
1
−
ε
)
−
ζ
q
2
U
>
δ
,
γ
η
1
L
−
α
e
1
U
>
δ
,
ζ
η
2
L
−
β
e
2
U
>
δ
.Since (H
1) and (H
2) hold, for any positive solutions (
x
1
(
n
)
,
y
1
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) and
(
x
2
(
n
)
,
y
2
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) of system (1.5) with the initial conditions
((1.6)), it follows from Theorem 3.1 that(3.32)
m
1
≤
liminf
n
→
+
∞
x
i
(
n
)
≤
limsup
n
→
+
∞
x
i
(
n
)
≤
M
1
,
m
2
≤
liminf
n
→
+
∞
y
i
(
n
)
≤
limsup
n
→
+
∞
y
i
(
n
)
≤
M
2
,
w
i
≤
liminf
n
→
+
∞
u
i
(
n
)
≤
limsup
n
→
+
∞
u
i
(
n
)
≤
W
i
,
w
i
≤
liminf
n
→
+
∞
u
i
(
n
)
≤
limsup
n
→
+
∞
u
i
(
n
)
≤
W
i
,
i
=
1,2.Then, there exists an n
0
>
0 such that, for all n
>
n
0,(3.33)
m
1
−
ε
≤
x
i
(
n
)
≤
M
1
+
ε
,
m
2
−
ε
≤
y
i
(
n
)
≤
M
2
+
ε
,
w
i
−
ε
≤
u
i
(
n
)
≤
W
i
+
ε
,
w
i
−
ε
≤
u
i
(
n
)
≤
W
i
+
ε
,
i
=
1,2.Let(3.34)
V
1
(
n
)
=
|
ln
x
1
(
n
)
−
ln
x
2
(
n
)
|
.Then, from the first equation of
system (1.5), we have(3.35)
V
1
(
n
+
1
)
=
|
ln
x
1
(
n
+
1
)
−
ln
x
2
(
n
+
1
)
|
≤
|
ln
x
1
(
n
)
−
ln
x
2
(
n
)
−
b
(
n
)
(
x
1
(
n
)
−
x
2
(
n
)
)
|
+
c
(
n
)
|
x
1
(
n
)
y
1
(
n
)
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
−
x
2
(
n
)
y
2
(
n
)
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
|
+
e
1
(
n
)
|
u
1
(
n
)
−
u
1
(
n
)
|
.Using the mean value theorem, we
get(3.36)
x
1
(
n
)
−
x
2
(
n
)
=
exp
(
ln
x
1
(
n
)
)
−
exp
(
ln
x
2
(
n
)
)
=
ξ
1
(
n
)
(
ln
x
1
(
n
)
−
ln
x
2
(
n
)
)
,where ξ
1
(
n
) lies between x
1
(
n
) and x
2
(
n
).
It follows from
(3.35) and
(3.36) that(3.37)
V
1
(
n
+
1
)
≤
|
ln
x
1
(
n
)
−
ln
x
2
(
n
)
|
−
(
1
ξ
1
(
n
)
−
|
1
ξ
1
(
n
)
−
b
(
n
)
|
)
|
x
1
(
n
)
−
x
2
(
n
)
|
+
|
c
(
n
)
x
1
(
n
)
y
1
(
n
)
x
2
(
n
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
|
x
1
(
n
)
−
x
2
(
n
)
)
|
+
|
c
(
n
)
m
2
(
n
)
y
1
2
(
n
)
y
2
(
n
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
|
x
1
(
n
)
−
x
2
(
n
)
)
|
+
|
c
(
n
)
x
1
2
(
n
)
x
2
(
n
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
|
y
1
(
n
)
−
y
2
(
n
)
)
|
+
|
c
(
n
)
m
2
(
n
)
y
1
(
n
)
y
2
(
n
)
x
1
(
n
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
|
y
1
(
n
)
−
y
2
(
n
)
)
|
+
e
1
(
n
)
|
u
1
(
n
)
−
u
1
(
n
)
|
.So, for n
≥
n
0,(3.38)
Δ
V
1
(
n
)
≤
−
min
{
b
L
,
2
M
1
+
ε
−
b
U
}
|
x
1
(
n
)
−
x
2
(
n
)
|
+
c
U
4
(
m
L
)
2
(
m
2
−
ε
)
|
x
1
(
n
)
−
x
2
(
n
)
|
+
c
U
(
M
1
+
ε
)
4
(
m
1
−
ε
)
2
|
x
1
(
n
)
−
x
2
(
n
)
|
+
c
U
(
M
1
+
ε
)
4
(
m
L
)
2
(
m
2
−
ε
)
2
|
y
1
(
n
)
−
y
2
(
n
)
|
+
c
U
4
(
m
1
−
ε
)
|
y
1
(
n
)
−
y
2
(
n
)
|
+
e
1
U
|
u
1
(
n
)
−
u
1
(
n
)
|
.Let(3.39)
V
2
(
n
)
=
|
ln
y
1
(
n
)
−
ln
y
2
(
n
)
|
.Then, from the second equation
of system (1.5), we have(3.40)
V
2
(
n
+
1
)
=
|
ln
y
1
(
n
+
1
)
−
ln
y
2
(
n
+
1
)
|
=
|
ln
y
1
(
n
)
−
ln
y
2
(
n
)
+
f
(
n
)
(
x
1
2
(
n
)
x
1
2
+
m
2
(
n
)
y
1
2
(
n
)
−
x
2
2
(
n
)
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
−
e
2
(
n
)
(
u
2
(
n
)
−
u
2
(
n
)
)
|
≤
|
ln
y
1
(
n
)
−
ln
y
2
(
n
)
−
f
(
n
)
m
2
(
n
)
x
1
(
n
)
(
x
1
(
n
)
y
2
(
n
)
+
y
1
(
n
)
x
2
(
n
)
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
×
(
y
1
(
n
)
−
y
2
(
n
)
)
|
+
f
(
n
)
m
2
(
n
)
y
1
(
n
)
(
x
1
(
n
)
y
2
(
n
)
+
y
1
(
n
)
x
2
(
n
)
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
×
|
x
1
(
n
)
−
x
2
(
n
)
|
+
e
2
(
n
)
|
u
1
(
n
)
−
u
1
(
n
)
|
.Using the mean value theorem, we
get(3.41)
y
1
(
n
)
−
y
2
(
n
)
=
exp
(
ln
y
1
(
n
)
)
−
exp
(
ln
y
2
(
n
)
)
=
ξ
2
(
n
)
(
ln
y
1
(
n
)
−
ln
y
2
(
n
)
)
,where ξ
2
(
n
) lies between y
1
(
n
) and y
2
(
n
).
Now, it follows
from (3.40) and
(3.41) that, for n
>
n
0,(3.42)
Δ
V
2
(
n
)
≤
−
(
1
ξ
2
(
n
)
−
|
1
ξ
2
(
n
)
−
f
(
n
)
m
2
(
n
)
x
1
(
n
)
(
x
1
(
n
)
y
2
(
n
)
+
y
1
(
n
)
x
2
(
n
)
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
)
|
y
1
(
n
)
−
y
2
(
n
)
|
+
f
(
n
)
m
2
(
n
)
y
1
(
n
)
(
x
1
(
n
)
y
2
(
n
)
+
y
1
(
n
)
x
2
(
n
)
)
(
x
1
2
(
n
)
+
m
2
(
n
)
y
1
2
(
n
)
)
(
x
2
2
(
n
)
+
m
2
(
n
)
y
2
2
(
n
)
)
|
x
1
(
n
)
−
x
2
(
n
)
|
+
e
2
(
n
)
|
u
2
(
n
)
−
u
2
(
n
)
|
≤
−
min
{
2
f
L
(
m
L
)
2
(
m
1
−
ε
)
2
(
m
2
−
ε
)
(
(
M
1
+
ε
)
2
+
(
m
U
)
2
(
M
2
+
ε
)
)
2
,
2
(
M
2
+
ε
)
−
f
U
(
M
1
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
}
×
|
y
1
(
n
)
−
y
2
(
n
)
|
+
f
U
(
M
2
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
|
x
1
(
n
)
−
x
2
(
n
)
|
+
e
2
U
|
u
2
(
n
)
−
u
2
(
n
)
|
.Let(3.43)
V
3
(
n
)
=
|
u
1
(
n
)
−
u
1
(
n
)
|
.
Then, from the
third equation of system (1.5), one can easily obtain that(3.44)
V
3
(
n
+
1
)
=
|
u
1
(
n
+
1
)
−
u
1
(
n
+
1
)
|
≤
(
1
−
η
1
(
n
)
)
|
u
1
(
n
)
−
u
1
(
n
)
|
+
q
1
(
n
)
|
x
1
(
n
)
−
x
2
(
n
)
|
.It follows from (3.44) that, for n
>
n
0,(3.45)
Δ
V
3
(
n
)
≤
−
η
1
L
|
u
1
(
n
)
−
u
1
(
n
)
|
+
q
1
U
|
x
1
(
n
)
−
x
2
(
n
)
|
.Let(3.46)
V
4
(
n
)
=
|
u
2
(
n
)
−
u
2
(
n
)
|
.
Similar to the
analysis of (3.44) and
(3.45), we can get(3.47)
Δ
V
4
(
n
)
≤
−
η
2
L
|
u
2
(
n
)
−
u
2
(
n
)
|
+
q
2
U
|
y
1
(
n
)
−
y
2
(
n
)
|
.Now, we define a Lyapunov
function as follows:(3.48)
V
(
n
)
=
α
V
1
(
n
)
+
β
V
2
(
n
)
+
γ
V
3
(
n
)
+
ζ
V
4
(
n
)
.For n
>
n
0,
it follows from (3.38), (3.42), (3.45), and (3.47) that(3.49)
Δ
V
(
n
)
≤
−
[
α
min
{
b
L
,
2
(
M
1
+
ε
)
−
b
U
}
−
α
c
U
4
(
m
L
)
2
(
m
2
−
ε
)
−
α
c
U
(
M
1
+
ε
)
4
(
m
1
−
ε
)
2
−
β
f
U
(
M
2
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
−
γ
q
1
U
]
×
|
x
1
(
n
)
−
x
2
(
n
)
|
−
[
β
min
{
2
f
L
(
m
L
)
2
(
m
1
−
ε
)
2
(
m
2
−
ε
)
(
(
M
1
+
ε
)
2
+
(
m
U
)
2
(
M
2
+
ε
)
2
)
2
,
2
(
M
2
+
ε
)
−
2
f
U
(
M
1
+
ε
)
2
(
m
1
−
ε
)
(
m
2
−
ε
)
}
−
α
c
U
(
M
1
+
ε
)
4
(
m
L
)
2
(
m
2
−
ε
)
2
−
α
c
U
4
(
m
1
−
ε
)
−
ζ
q
2
U
]
×
|
y
1
(
n
)
−
y
2
(
n
)
|
−
(
γ
η
1
L
−
α
e
1
u
)
×
|
u
1
(
n
)
−
u
1
(
n
)
|
−
(
ζ
η
2
L
−
β
e
2
U
)
×
|
u
2
(
n
)
−
u
2
(
n
)
|
≤
−
δ
(
|
x
1
(
n
)
−
x
2
(
n
)
|
+
|
y
1
(
n
)
−
y
2
(
n
)
|
+
|
u
1
(
n
)
−
u
1
(
n
)
|
+
|
u
2
(
n
)
−
u
2
(
n
)
|
)
.Summating both sides of the
above inequalities from n
0 to n,
we have(3.50)
∑
p
=
n
0
n
(
V
(
p
+
1
)
−
V
(
p
)
)
≤
−
δ
∑
p
=
n
0
n
(
|
x
1
(
p
)
−
x
2
(
p
)
|
+
|
y
1
(
p
)
−
y
2
(
p
)
|
+
|
u
1
(
p
)
−
u
1
(
p
)
|
+
|
u
2
(
p
)
−
u
2
(
p
)
|
)
,which implies(3.51)
V
(
n
0
)
≥
δ
∑
p
=
n
0
n
(
|
x
1
(
p
)
−
x
2
(
p
)
|
+
|
y
1
(
p
)
−
y
2
(
p
)
|
+
|
u
1
(
p
)
−
u
1
(
p
)
|
+
|
u
2
(
p
)
−
u
2
(
p
)
|
)
+
V
(
n
+
1
)
.It follows that(3.52)
V
(
n
0
)
δ
≥
∑
p
=
n
0
n
(
|
x
1
(
p
)
−
x
2
(
p
)
|
+
|
y
1
(
p
)
−
y
2
(
p
)
|
+
|
u
1
(
p
)
−
u
1
(
p
)
|
+
|
u
2
(
p
)
−
u
2
(
p
)
|
)
.Letting n
→
∞ gives(3.53)
V
(
n
0
)
δ
≥
∑
p
=
n
0
+
∞
(
|
x
1
(
p
)
−
x
2
(
p
)
|
+
|
y
1
(
p
)
−
y
2
(
p
)
|
+
|
u
1
(
p
)
−
u
1
(
p
)
|
+
|
u
2
(
p
)
−
u
2
(
p
)
|
)
,which implies that(3.54)
lim
n
→
∞
(
|
x
1
(
n
)
−
x
2
(
n
)
|
+
|
y
1
(
n
)
−
y
2
(
n
)
|
+
|
u
1
(
n
)
−
u
1
(
n
)
|
+
|
u
2
(
n
)
−
u
2
(
n
)
|
)
=
0
,that is,(3.55)
lim
n
→
∞
|
x
1
(
n
)
−
x
2
(
n
)
|
=
0
,
lim
n
→
∞
|
y
1
(
n
)
−
y
2
(
n
)
|
=
0
,
lim
n
→
∞
|
u
1
(
n
)
−
u
1
(
n
)
|
=
0
,
lim
n
→
∞
|
u
1
(
n
)
−
u
1
(
n
)
|
=
0.This completes the proof of
Theorem 3.2.


## 4. Extinction of the predator species

This section
is devoted to studying the extinction of the predator speciesy.Theorem 4.1.
Assume that(H
7)
−
d
L
+
f
U
<
0.Then, for any solution (
x
(
n
)
,
y
(
n
)
,
u
1
(
n
)
,
u
2
(
n
)
) of system (1.5),(4.1)
lim
n
→
+
∞
y
(
n
)
=
0.Proof.
From condition (H
7),
there exists small enough positive constant γ
>
0 such that(4.2)
−
d
L
+
f
U
<
−
γ
<
0.
For all n
∈
ℕ,
from (4.2) and the second equation of system (1.5), one can easily obtain
that(4.3)
y
(
n
+
1
)
=
y
(
n
)
exp
{
−
d
(
n
)
+
f
(
n
)
x
2
(
n
)
x
2
(
n
)
+
m
2
(
n
)
y
2
(
n
)
−
e
2
(
n
)
u
2
(
n
)
}
<
y
(
n
)
exp
{
−
d
L
+
f
U
}
<
y
(
n
)
exp
{
−
γ
}
.Therefore,(4.4)
y
(
n
+
1
)
<
y
(
0
)
exp
{
−
n
γ
}
,which yields(4.5)
lim
n
→
+
∞
y
(
n
)
=
0.The proof of Theorem 4.1 is complete.


## 5. Examples

The following
two examples show the feasibility of the main results.Example 5.1.
Consider the following
system:(5.1)
x
(
n
+
1
)
=
x
(
n
)
exp
{
1.225
+
0.025
sin
(
2
n
)
−
(
5.25
+
0.25
cos
(
n
)
)
x
(
n
)
−
(
0.0075
+
0.0025
sin
(
n
)
)
y
(
n
)
x
(
n
)
x
2
(
n
)
+
(
0.885
+
0.005
cos
(
3
n
)
)
2
y
2
(
n
)
−
(
0.002
+
0.001
cos
(
3
n
)
)
u
1
(
n
)
}
,
y
(
n
+
1
)
=
y
(
n
)
exp
{
−
(
0.1125
+
0.0025
cos
(
2
n
)
)
+
(
0.195
+
0.005
sin
(
3
n
)
)
x
2
(
n
)
x
2
(
n
)
+
(
0.885
+
0.005
cos
(
3
n
)
)
2
y
2
(
n
)
−
(
0.0015
+
0.0005
sin
(
n
)
)
u
2
(
n
)
}
,
Δ
u
1
(
n
)
=
−
(
0.925
+
0.025
sin
(
n
)
)
u
1
(
n
)
+
(
0.0375
+
0.0275
cos
(
n
)
)
x
(
n
)
,
Δ
u
2
(
n
)
=
−
(
0.925
+
0.025
cos
(
n
)
)
u
2
(
n
)
+
(
0.025
+
0.005
sin
(
n
)
)
y
(
n
)
.One could easily see that there
exist α
=
0.06
,
β
=
0.05
,
γ
=
0.01
,
ζ
=
0.001,
and δ
=
0.00001 such that(5.2)
a
L
−
c
U
2
m
L
−
e
1
U
W
1
≈
1.2
>
0
,
f
L
−
d
U
−
e
2
U
W
2
≈
0.04
>
0
,
α
min
{
b
L
,
2
M
1
−
b
U
}
−
α
c
U
4
(
m
L
)
2
m
2
−
α
c
U
M
1
4
m
1
2
−
β
f
U
M
2
2
m
1
m
2
−
γ
q
1
U
≈
0.04512
>
δ
,
β
min
{
2
f
L
(
m
L
)
2
m
1
2
m
2
M
1
2
+
(
m
U
)
2
M
2
2
,
2
M
2
−
f
U
M
1
2
m
1
m
2
}
−
α
c
U
M
1
4
(
m
L
)
2
m
2
−
α
c
U
4
m
1
−
ζ
q
2
U
≈
0.000013
>
δ
,
γ
η
1
L
−
α
e
1
U
≈
0.0072
>
δ
,
ζ
η
2
L
−
β
e
2
U
≈
0.00087
>
δ
.Clearly, conditions (H
1)–(H
6)   are satisfied. From Theorems 3.1
and 3.2, the
system is permanent and globally attractive. Numeric simulation (Figure 1)
strongly supports our results.Dynamics behaviors of
system (5.1) with initial conditions (
x
(
0
)
,
y
(
0
)
,
u
1
(
0
)
,
u
2
(
0
)
)
T
=
(
0.31
,
0.26
,
0.02
,
0.08
)
T   and (
0.18
,
0.18
,
0.01
,
0.1
)
T,
respectively.
(a)
Speciesx
,
y
(b)
Feedback controlsu
1
,
u
2Example 5.2.
Consider the following system:(5.3)
x
(
n
+
1
)
=
x
(
n
)
exp
{
0.8
+
0.1
sin
(
n
)
−
(
11.5
+
3.5
cos
(
n
)
)
x
(
n
)
−
(
0.035
+
0.025
sin
(
n
)
)
y
(
n
)
x
(
n
)
x
2
(
n
)
+
(
5.75
+
0.35
cos
(
2
n
)
)
2
y
2
(
n
)
−
(
0.025
+
0.005
cos
(
n
)
)
u
1
(
n
)
}
,
y
(
n
+
1
)
=
y
(
n
)
exp
{
−
(
0.975
+
0.025
cos
(
2
n
)
)
+
(
0.195
+
0.005
sin
(
3
n
)
)
x
2
(
n
)
x
2
(
n
)
+
(
5.75
+
0.35
cos
(
2
n
)
)
2
y
2
(
n
)
−
(
0.0015
+
0.0005
cos
(
3
n
)
)
u
2
(
n
)
}
,
Δ
u
1
(
n
)
=
−
(
0.925
+
0.025
sin
(
n
)
)
u
1
(
n
)
+
(
0.0375
+
0.0275
cos
(
n
)
)
x
(
n
)
,
Δ
u
2
(
n
)
=
−
(
0.925
−
0.025
cos
(
n
)
)
u
2
(
n
)
+
(
0.025
+
0.005
sin
(
n
)
)
y
(
n
)
.By simple computation, we can
easily have(5.4)
−
d
L
+
f
U
=
−
0.8
<
0.Thus, condition (H
7) is satisfied; from Theorem 4.1, it follows
that lim
n
→
+
∞
y
(
n
)
=
0. Numeric simulation (Figure 2) strongly
supports our result.Dynamics behaviors of system
(5.2) with initial conditions (
x
(
0
)
,
y
(
0
)
,
u
1
(
0
)
,
u
2
(
0
)
)
T
=
(
0.06
,
0.08
,
0.04
,
0.02
)
T.
(a)
Speciesx
(b)
Speciesy



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*Source: 186539-2008-12-04.xml*