# Positive Periodic Solutions of a Periodic Discrete Competitive System Subject to Feedback Controls

Authors: Ronghua Tan; Zuxiong Li; Qinglong Wang; Zhijun Liu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Species living in a fluctuating medium and human exploitation activities might result in the duration of continuous changes. Such changes can be well-approximated as feedback controls. In this contribution a periodic discrete competitive system subject to feedback controls is proposed. By using the methods of discrete inequality, fixed point theorem, and analysis techniques, a good understanding of the existence and global asymptotic stability of positive periodic solutions is obtained. Some numerical investigations are provided to verify our analytical results.


---

## Body

## 1. Introduction

Many mathematical models about bioecology play important roles for researchers to realize the interactions of ecological species. There has been increasing interest in studying the dynamical behaviors such as stability, permanence, and periodicity of competitive systems (see [1–5]). Recently, motivated by an autonomous competitive model of [6] in [7] we introduced a corresponding nonautonomous version
(1)
y
1
′
(
t
)
=
y
1
(
t
)
[
a
1
(
t
)
-
b
1
(
t
)
y
1
(
t
)
-
c
1
(
t
)
y
2
(
t
)
-
d
1
(
t
)
y
1
2
(
t
)
-
c
1
(
t
)
y
2
(
t
)
i
i
i
i
i
i
i
i
i
i
i
i
i
-
d
1
(
t
)
y
1
2
(
t
)
]
,
y
2
′
(
t
)
=
y
2
(
t
)
[
a
2
(
t
)
-
b
2
(
t
)
y
2
(
t
)
-
c
2
(
t
)
y
1
(
t
)
-
d
2
(
t
)
y
2
2
(
t
)
iiiiiiiiiiiiiii
-
d
2
(
t
)
y
2
2
(
t
)
]
,
where y
1
(
t
) and y
2
(
t
) can be interpreted as the density of two competing species at time t, respectively. a
i
(
t
) stand for the growth rates of species, b
i
(
t
) and d
i
(
t
) represent the effects of intraspecific competition, and c
i
(
t
) are the rates of interspecific competition. Furthermore, we took the influence of almost periodic environment and impulsive perturbations into account and established sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the above system (for details see [7]).Note that ecosystems in the real world are often disturbed by outside continuous forces which can lead to changes in biological coefficients such as survival rates. In the language of control, we call the disturbance functions as feedback control variables. Many good results on this direction are deliberated (see [8–12]) and some similar works on the topic have been done (see [13, 14]). This paper is concerned with a discrete model and is a continuation of the work in [7]; in this contribution we search for certain schemes (such as harvesting procedure) to ensure the system coexists under appropriate conditions. For this reason, we consider a discrete version which corresponds to differential version (i.e., system (1)), meanwhile, replacing abrupt external perturbations (impulses) in [7] by continuous external perturbations (feedback control variables), a nonautonomous discrete controlled system can be described as follows:
(2)
y
1
(
k
+
1
)
=
y
1
(
k
)
exp
⁡
[
a
1
(
k
)
-
b
1
(
k
)
y
1
(
k
)
-
c
1
(
k
)
y
2
(
k
)
-
d
1
(
k
)
y
1
2
(
k
)
iiiiiiiiiiiiii
-
d
1
(
k
)
y
1
2
(
k
)
-
e
1
(
k
)
v
1
(
k
)
]
,
y
2
(
k
+
1
)
=
y
2
(
k
)
exp
⁡
[
a
2
(
k
)
-
b
2
(
k
)
y
2
(
k
)
-
c
2
(
k
)
y
1
(
k
)
-
d
2
(
k
)
y
2
2
(
k
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
-
d
2
(
k
)
y
2
2
(
k
)
-
e
2
(
k
)
v
2
(
k
)
]
,
Δ
v
1
=
v
1
(
k
+
1
)
-
v
1
(
k
)
=
h
1
(
k
)
-
f
1
(
k
)
v
1
(
k
)
+
g
1
(
k
)
y
1
(
k
)
,
Δ
v
2
=
v
2
(
k
+
1
)
-
v
2
(
k
)
=
h
2
(
k
)
-
f
2
(
k
)
v
2
(
k
)
+
g
2
(
k
)
y
2
(
k
)
,
k
∈
Z
+
.
Here y
i
(
0
)
>
0, v
i
(
0
)
>
0. Δ
v
i
(
k
) are the first-order forward difference operators, y
i
(
k
) stand for the densities of species y
i at the kth generation, a
i
(
k
) represent the natural growth rates of species y
i at the kth generation, b
i
(
k
) and d
i
(
k
) stand for the intraspecific effects of the kth generation of species y
i on own population, and c
i
(
k
) measure the interspecific effects of the kth generation of species y
j on species y
i. The coefficients {
a
i
(
k
)
}, {
b
i
(
k
)
}, {
c
i
(
k
)
}, {
d
i
(
k
)
}, {
e
i
(
k
)
}, {
f
i
(
k
)
}, {
g
i
(
k
)
}, and {
h
i
(
k
)
} are T-periodic sequences with 0
<
f
i
(
k
)
<
1
,
i
,
j
=
1,2
;
i
≠
j. Z
+ is the set of nonnegative integers.It is well known that the discrete models governed by difference equations may be more appropriate than the continuous ones when populations have a short life expectancy or nonoverlapping generations. Also, discrete models can provide efficient computational methods of continuous models for numerical simulations. A very important ecological problem associated with the study of multispecies population interaction in a periodic environment is the positive periodic solution which plays the role played by the equilibrium of the autonomous model. In this paper, we will discuss the above discrete system (2) and focus on the existence and stability of positive periodic solution of system (2). To the best of our knowledge, there are few published papers concerning system (2).The rest structure of this paper is as follows. In the next section, we establish sufficient conditions for the existence of positive periodic solutions. In Section3, we further discuss the global asymptotic stability of positive periodic solutions. In Section 4, we carry out an example and its numerical simulations to substantiate our theoretical results. To simplify the reading of paper, we give the proofs of lemmas and theorems in appendices.


## 2. Existence

In this section, we establish sufficient conditions for the existence of positive periodic solutions of system (2). To do this, we first give two preliminary lemmas. For convenience, given a bounded sequence {
f
(
k
)
}, let f
U and f
L be defined as f
U
=
sup
⁡
k
∈
Z
+
f
(
k
), f
L
=
inf
⁡
k
∈
Z
+
f
(
k
), and(3)
M
i
=
exp
⁡
(
a
i
U
-
1
)
b
i
L
,
N
i
=
h
i
U
+
g
i
U
M
i
f
i
L
,
m
i
=
a
i
L
-
c
i
U
M
j
-
d
i
U
M
i
2
-
e
i
U
N
i
b
i
U
×
exp
⁡
(
a
i
L
-
b
i
U
M
i
-
c
i
U
M
j
-
d
i
U
M
i
2
-
e
i
U
N
i
)
,
n
i
=
(
h
i
L
+
g
i
L
m
i
)
f
i
U
,
i
≠
j
;
i
,
j
=
1,2
.Lemma 1.
Every positive solution{
y
1
(
k
)
,
y
2
(
k
)
,
v
1
(
k
)
,
v
2
(
k
)
} of system (2) satisfies limsup
⁡
k
→
+
∞
y
i
(
k
)
≤
M
i and limsup
⁡
k
→
+
∞
v
i
(
k
)
≤
N
i
,
i
=
1,2.The proof of Lemma1 is given in Appendix A.Lemma 2.
Assuming that(4)
a
1
L
-
c
1
U
M
2
-
d
1
U
M
1
2
-
e
1
U
N
1
>
0
,
a
2
L
-
c
2
U
M
1
-
d
2
U
M
2
2
-
e
2
U
N
2
>
0
hold, then every positive solution {
y
1
(
k
)
,
y
2
(
k
)
,
v
1
(
k
)
,
v
2
(
k
)
} of system (2) satisfies liminf
⁡
k
→
+
∞
y
i
(
k
)
≥
m
i and liminf
⁡
k
→
+
∞
v
i
(
k
)
≥
n
i
,
i
=
1,2.The proof of Lemma2 is given in Appendix B.Theorem 3.
If the inequalities in (4) hold, then system (2) has at least one strictly positive T-periodic solution.The proof of Theorem3 is given in Appendix C.


## 3. Global Asymptotic Stability

In this section, we further investigate the stability of positive periodic solutions of system (2). Denote
(5)
ρ
1
=
max
⁡
{
|
1
-
b
1
L
m
1
-
2
d
1
L
m
1
2
|
,
|
1
-
b
1
U
M
1
-
2
d
1
U
M
1
2
|
}
+
c
1
U
M
2
+
e
1
U
,
ρ
2
=
max
⁡
{
|
1
-
b
2
L
m
2
-
2
d
2
L
m
2
2
|
,
|
1
-
b
2
U
M
2
-
2
d
2
U
M
2
2
|
}
+
c
2
U
M
1
+
e
2
U
,
ρ
3
=
1
-
f
1
L
+
g
1
U
M
1
,
ρ
4
=
1
-
f
2
L
+
g
2
U
M
2
.Theorem 4.
In addition to (4), further suppose that ρ
i
<
1, i
=
1,2
,
3,4. Then the positive periodic solution of system (2) is globally asymptotically stable.The proof of Theorem4 is given in Appendix D.


## 4. An Example

In this section we give a numerical example and its corresponding simulations.Example 1.
Consider a discrete competitive system with feedback controls(6)
y
1
(
k
+
1
)
=
y
1
(
k
)
exp
⁡
[
0.98
+
0.003
sin
(
k
π
)
(
0.003
+
0.0001
sin
(
k
π
)
)
v
1
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.85
-
0.002
cos
⁡
(
k
π
)
)
y
1
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.002
+
0.0001
sin
⁡
(
k
π
)
)
y
2
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.004
-
0.0002
cos
⁡
⁡
(
k
π
)
)
y
1
2
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.003
+
0.0001
sin
(
k
π
)
)
v
1
(
k
)
]
,
y
2
(
k
+
1
)
=
y
2
(
k
)
exp
⁡
[
1.00
-
0.002
sin
(
k
π
)
(
0.002
+
0.0002
sin
(
k
π
)
)
v
2
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.92
+
0.001
cos
⁡
(
k
π
)
)
y
2
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.003
+
0.0001
cos
⁡
(
k
π
)
)
y
1
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.005
+
0.0003
sin
(
k
π
)
)
y
2
2
(
k
)
iiiiiiiiiiiiiiiiiiii
-
(
0.002
+
0.0002
sin
(
k
π
)
)
v
2
(
k
)
]
,
Δ
v
1
(
k
)
=
0.005
+
0.0002
sin
(
k
π
)
-
(
0.95
+
0.003
cos
⁡
(
k
π
)
)
v
1
(
k
)
+
(
0.006
+
0.0001
sin
(
k
π
)
)
y
1
(
k
)
,
Δ
v
2
(
k
)
=
0.004
+
0.0001
sin
(
k
π
)
-
(
0.98
+
0.002
sin
(
k
π
)
)
v
2
(
k
)
+
(
0.003
+
0.0002
cos
⁡
(
k
π
)
)
y
2
(
k
)
.
A computation shows that
(7)
a
1
L
=
0.977
,
a
1
U
=
0.983
,
b
1
L
=
0.848
,
b
1
U
=
0.852
,
c
1
L
=
0.0019
,
c
1
U
=
0.0021
,
d
1
L
=
0.0038
,
d
1
U
=
0.0042
,
e
1
L
=
0.0029
,
e
1
U
=
0.0031
,
f
1
L
=
0.947
,
f
1
U
=
0.953
,
g
1
L
=
0.0059
,
g
1
U
=
0.0061
,
h
1
L
=
0.0048
,
h
1
U
=
0.0052
,
a
2
L
=
0.998
,
a
2
U
=
1.002
,
b
2
L
=
0.919
,
b
2
U
=
0.921
,
c
2
L
=
0.0029
,
c
2
U
=
0.0031
,
d
2
L
=
0.0047
,
d
2
U
=
0.0053
,
e
2
L
=
0.0018
,
e
2
U
=
0.0022
,
f
2
L
=
0.978
,
f
2
U
=
0.982
,
g
2
L
=
0.0028
,
g
2
U
=
0.0032
,
h
2
L
=
0.0039
,
h
2
U
=
0.0041
.
Furthermore, it follows from (3) that
(8)
M
1
≈
1.1594
,
M
2
≈
1.0903
,
N
1
≈
0.0130
,
N
2
≈
0.0078
,
m
1
≈
1.1162
,
m
2
≈
1.0557
,
n
1
≈
0.0119
,
n
2
≈
0.0070
,
a
1
L
-
c
1
U
M
2
-
d
1
U
M
1
2
-
e
1
U
N
1
≈
0.9690
>
0
,
a
2
L
-
c
2
U
M
1
-
d
2
U
M
2
2
-
e
2
U
N
2
≈
0.9881
>
0
.
Obviously, the assumptions in (4) are satisfied, and moreover, one has
(9)
ρ
1
=
max
⁡
{
|
1
-
b
1
L
m
1
-
2
d
1
L
m
1
2
|
,
|
1
-
b
1
U
M
1
-
2
d
1
U
M
1
2
|
}
+
c
1
U
M
2
+
e
1
U
≈
0.0494
<
1
,
ρ
2
=
max
⁡
{
|
1
-
b
2
L
m
2
-
2
d
2
L
m
2
2
|
,
|
1
-
b
2
U
M
2
-
2
d
2
U
M
2
2
|
}
+
c
2
U
M
1
+
e
2
U
≈
0.0251
<
1
,
ρ
3
=
1
-
f
1
L
+
g
1
U
M
1
≈
0.0601
<
1
,
ρ
4
=
1
-
f
2
L
+
g
2
U
M
2
≈
0.0255
<
1
,
so the assumptions of Theorem 4 are also satisfied. Thus, there exists a globally asymptotically stable positive periodic solution of system (6). Figure 1 shows that system (6) exists a positive periodic solution(
y
1
*
(
k
)
,
y
2
*
(
k
)
,
v
1
*
(
k
)
,
v
2
*
(
k
)
), and the two-dimensional phase portrait of periodic system (6) is displayed in Figure 2. From Figure 3, we can see that any positive solution (
y
1
(
k
)
,
y
2
(
k
)
,
v
1
(
k
)
,
v
2
(
k
)
) tends to the periodic solution (
y
1
*
(
k
)
,
y
2
*
(
k
)
,
v
1
*
(
k
)
,
v
2
*
(
k
)
), and the two-dimensional phase portrait reflects the fact in Figure 4.Positive periodic solution of system (6). (a)–(d) Time-series of y
1
*
(
k
), y
2
*
(
k
), v
1
*
(
k
), and v
2
*
(
k
) with initial values y
1
*
(
0
)
=
1.1560, y
2
*
(
0
)
=
1.0720, v
1
*
(
0
)
=
0.0128, and v
2
*
(
0
)
=
0.0078 for k
∈
[
0,100
], respectively.
(a)
(b)
(c)
(d)Two-dimensional phase portrait. (a) Phase portrait ofy
1
*
(
k
) and y
2
*
(
k
) with initial values y
1
*
(
0
)
=
1.1560 and y
2
*
(
0
)
=
1.0720 for k
∈
[
0,100
], respectively. (b) Phase portrait of v
1
*
(
k
) and v
2
*
(
k
) with initial values v
1
*
(
0
)
=
0.0128 and v
2
*
(
0
)
=
0.0078 for k
∈
[
0,100
], respectively.
(a)
(b)Global asymptotic stability. (a)–(d) Time-series ofy
1
*
(
k
), y
2
*
(
k
), v
1
*
(
k
), and v
2
*
(
k
) with initial values y
1
*
(
0
)
=
1.1560, y
2
*
(
0
)
=
1.0720, v
1
*
(
0
)
=
0.0128, and v
2
*
(
0
)
=
0.0078 and y
1
(
k
), y
2
(
k
), v
1
(
k
), and v
2
(
k
) with initial values y
1
(
0
)
=
1.1300, y
2
(
0
)
=
1.0850, v
1
(
0
)
=
0.0123, and v
2
(
0
)
=
0.0071 for k
∈
[
0,100
], respectively.
(a)
(b)
(c)
(d)Two-dimensional phase portrait. (a) Phase portrait ofy
1
*
(
k
) and y
2
*
(
k
) with initial values y
1
*
(
0
)
=
1.1560 and y
2
*
(
0
)
=
1.0720 and y
1
(
k
) and y
2
(
k
) with initial values y
1
(
0
)
=
1.1300 and y
2
(
0
)
=
1.0850 for k
∈
[
0,100
], respectively. (b) Phase portrait of v
1
*
(
k
) and v
2
*
(
k
) with initial values v
1
*
(
0
)
=
0.0128 and v
2
*
(
0
)
=
0.0078 and v
1
(
k
) and v
2
(
k
) with initial values v
1
(
0
)
=
0.0123 and v
2
(
0
)
=
0.0071 for k
∈
[
0,100
], respectively.
(a)
(b)



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*Source: 927626-2014-06-30.xml*