Tian et al. Advances in Difference Equations (2016) 2016:187
DOI 10.1186/s13662-016-0915-2
RESEARCH
Open Access
Existence and stability of a unique almost
periodic solution for a prey-predator system
with impulsive effects and multiple delays
Baodan Tian1,2* , Shouming Zhong2 and Ning Chen1
*
Correspondence:
[email protected]
1
Institute of Modeling and
Algorithm, School of Science,
Southwest University of Science
and Technology, Mianyang, 621010,
China
2
School of Mathematical Sciences,
University of Electronic Science and
Technology of China, Chengdu,
611731, China
Abstract
In this paper, a nonautonomous almost periodic prey-predator system with impulsive
effects and multiple delays is considered. By the mean-value theorem of multiple
variables, integral inequalities, differential inequalities, and other mathematical
analysis skills, sufficient conditions which guarantee the permanence of the system
are obtained. Furthermore, by constructing a series of Lyapunov functionals, we
derive that there exists a unique almost periodic solution of the system which is
uniformly asymptotically stable. Finally, a numerical example and some simulations
are presented to support our theoretical results.
Keywords: impulsive effects; delays; permanence; almost periodic solution;
asymptotical stability
1 Introduction
It is widely known that predation is a very common ecological phenomenon in the natural
world, and research on the dynamics of the prey-predator system is extremely meaningful
and important in many fields, such as protecting varieties of the biological species, maintaining ecological balance, agricultural pests control and management, etc. As research of
the prey-predator system is concerned, the earliest work dates back to the great contribution of Lotka () and Volterra (). However, based on the classic Lotka-Volterra
model, many ecologists found that the predation rate was not just simply proportional to
the product of the density of the predator and prey. It was always different for different systems. Then the conception of the functional response was proposed. Functional response
refers to the predation rate per predator with a response to changes in the prey density,
i.e., predation effects of predators on the prey.
Since then, studies depending on all kinds of functional responses sprang up quickly,
such as of Holling type [–], Michaelis-Menton type [], Beddington-DeAngelis type
[], Ivlev type [], Hassell-Varley type [], and so on. In , Crowley and Martin [] proposed a new functional response that can accommodate interference between predators.
It is similar to the well-known Beddington-DeAngelis functional response but has an additional term in the first right term equation modeling mutual interference between the
predator terms.
© 2016 Tian et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
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Tian et al. Advances in Difference Equations (2016) 2016:187
Page 2 of 23
Crowley-Martin type functional response is used for data sets that indicate feeding rate
that is affected by predator density. It is assumed that predator feeding rate decreases by
higher predator density even when prey density is high, and effects of predator interference
in feeding rate remain important all the time whenever an individual predator is handling
or searching for a prey at a given instant of time. And the formula of the per capita feeding
rate Crowley and Martin proposed in [] is as follows:
f (x, y) =
mx
.
( + αx)( + βy)
()
Here m, α, β are positive parameters that describe the effects of capture rate handling
time and the magnitude of interference among predators, respectively, on the feeding rate.
On the one hand, since models with Crowly-Martin functional response can be generated by a number of natural mechanisms and admits rich but biological dynamics. However, to the best of our knowledge, references reported on this functional response seems
to much less until recent several years (see [–]) for the form of the functional response
is relatively complex, for the form of this functional response is relatively complex, and it
is worthwhile to further study the models with it.
Recently, Liu et al. in [] considered a stochastic prey-predator system with CrowleyMartin type functional response as follows:
dx(t) = x(t)[r – b x(t) –
dy(t) = y(t)[r – b y(t) +
c y(t)
] dt + x(t)[δ dB (t) + μ dB (t)],
(+αx(t))(+βy(t))
c x(t)
] dt + y(t)[δ dB (t) + μ dB (t)],
(+αx(t))(+βy(t))
()
in which they obtained the condition of the global existence of a unique positive solution
and the stochastic permanence of the positive solution to the model. Furthermore, it is
shown that both the prey and the predator species will become extinct with probability
one if the noise is sufficiently large.
Yin et al. in [] studied the following modified Leslie-Gower predator-prey model with
Crowley-Martin functional response and spatial diffusion under a homogeneous Neumann boundary condition:
∂u
∂t
∂v
∂t
= d u + u( – u –
= d v + cv( –
v
),
(+au)(+bv)
dv
),
u+e
x ∈ , t > ,
x ∈ , t > ,
()
in which they obtained some important qualitative properties, including the existence of
the global positive solution, the dissipation and persistence of the two species, the local
and global asymptotic stability of the constant equilibria, and Hopf bifurcation around the
interior constant equilibrium.
On the other hand, it is well known that there are many natural or man-made factors in
the real world, such as earthquake, flooding, drought, crop-dusting, planting, hunting and
harvesting. These kinds of factors can bring sudden changes to an ecological system, and
the intrinsic discipline of the environment or the species in the system will often undergoes
these changes in a relatively short interval, usually we could regard it happens at some fixed
times. From the viewpoint of mathematics, such sudden changes could be described by
the impulsive effects to the system. In addition, when a prey-predator system is studied,
it is more reasonable to consider time delay during the predation, because there is often
Tian et al. Advances in Difference Equations (2016) 2016:187
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a digest time instead of transforming food into intrinsic growth rate of the predator. As
far as the impulsive system is concerned, there have appeared much excellent work in
the last decades, such as impulsive effects in ecological systems (see [–]), in epidemic
systems (see []) and in the neural network models (see [–]). Besides, there are many
important monographs in this field (see [–]).
Particularly, research on the impulsive system with delays still seems to be a hot issue
(see [–]), and these kinds of hybrid systems are usually called impulsive functional
systems. When an impulsive functional system is concerned, the difficulty both in the
impulsive differential equation and in the functional differential equation will occur at
the same time, then the dynamical behaviors, such as permanence, periodic solution, almost periodic solution, and its asymptotical stability properties, as well as bifurcations
and chaotic behaviors etc., might be richer, more complex, and more interesting.
Enlightened by the above literature, we propose an almost periodic prey-predator system with Crowly-Martin type functional response and impulsive effects in this paper, in
which digest delays are also considered in the process of predation, and the final model is
as follows:
⎧
c (t)y(t–τ )
dx
⎪
⎪ dt = x(t)[r (t) – d (t)x(t – τ ) – (+α(t)x(t–τ ))(+β(t)y(t–τ )) ],
⎪
⎪
⎨ dy = y(t)[r (t) – d (t)y(t – τ ) +
c (t)x(t–τ )
],



dt
(+α(t)x(t–τ ))(+β(t)y(t–τ ))
+
⎪
⎪
⎪ x(tk ) = ( + hk )x(tk ),
⎪
t = tk , k ∈ N,
⎩ +
y(tk ) = ( + hk )y(tk ),
t = tk , k ∈ N,
()
where x(t), y(t) are population densities of the predator and the prey at time t, respectively.
τi (i = , , , ) are all nonnegative constants, dj (t) (j = , ) denote the inner density resistance of the predator and of the prey. hjk > –, j = , , k ∈ N , when hjk > , the impulsive
effects represent planting, while hjk <  represents the impulsive denote harvesting.
Throughout the present paper, we define
f u = sup f (t),
t∈R+
f l = inf+ f (t),
t∈R
for any bounded function f (t) defined on R+ = [, +∞).
Further, we assume that
(C) ri (t), ci (t), di (t) (i = , ), α(t) and β(t) are all bounded and positive almost periodic
functions;
(C) Hi (t) = <tk <t ( + hik ), i = , , k ∈ N is almost period functions and there exist
positive constants Hiu and Hil such that Hil ≤ Hi (t) ≤ Hiu .
The rest of this article is organized as follows: In Section , we will give some definitions
and several useful lemmas for the proof of our main results. In Section , we will state and
prove our main results such as permanence of the system, existence, and the uniqueness
of an almost periodic solution which is uniformly asymptotically stable by constructing a
series of Lyapunov functionals. In the last section, we give a numerical examples to support
our theoretical results, then provide a brief discussion and summary of our main results.
2 Preliminaries
In this section, we will state the following definitions and lemmas, which will be used in
the next sections.
Tian et al. Advances in Difference Equations (2016) 2016:187
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We have K = {tk ∈ R|tk < tk+ , k ∈ N, limk→±∞ tk = ±∞}, and we thus denote the set of
all sequences that are unbounded and increasing. Let D ⊂ , = , τ = max≤i≤ {τi },
ξ ∈ R. Also, we denote PC(ξ ) is the space of all functions φ : [ξ – τ , ξ ] → having
points of discontinuity at μ , μ , . . . ∈ [ξ – τ , ξ ] of the first kind and left continuous at
these points.
For J ⊂ R, PC(J, R) is the space of all piecewise continuous functions from J to R with
points of discontinuity of the first kind tk , at which we have left continuity.
Let φ , φ ∈ PC(), denote by x(t) = x(t; , φ ), y(t) = y(t; , φ ), x, y ∈ the solution of
system () satisfying the following initial conditions:
 ≤ x(s; , φ ) = φ (s) < ∞,
x(; , φ ) = φ () > ,
 ≤ y(s; , φ ) = φ (s) < ∞,
s ∈ [–τ , ],
()
y(; , φ ) = φ () > .
Since the solution of system () with initial conditions () is a piecewise continuous function with points of discontinuity of the first kind, tk , k ∈ Z, we introduce the following
definitions for the almost periodicity.
j
Definition . (see []) The set of sequence {tk }, tjk = tk+j – tk , k, j ∈ N , tk ∈ K is said to
be uniformly almost periodic, if for arbitrary ε >  there exists a relatively dense set of
ε-almost periodic common for any sequences.
Definition . (see []) The function ϕ ∈ PC(R, R) is said to be almost periodic if the
following conditions hold:
j
() The set of sequences {tk }, tjk = tk+j – tk , k, j ∈ N , tk ∈ K is uniformly almost periodic.
() For any ε > , there exists a real number δ > , such that if the points t and t
belong to one and the same interval of continuity of ϕ(t) and satisfy the inequality
|t – t | < δ, then |ϕ(t ) – ϕ(t )| < ε.
() For any ε > , there exists a relatively dense set T such that if η ∈ T, then
|ϕ(t + η) – ϕ(t)| < ε for all t ∈ R satisfying the condition |t – tk | > ε, k ∈ N . The
elements of T are called ε-almost periods.
For the following system:
du
dt
dv
dt
= u(t)[r (t) – D (t)u(t – τ ) –
= v(t)[r (t) – D (t)v(t – τ ) +
C (t)v(t–τ )
],
(+A(t)u(t–τ ))(+B(t)v(t–τ ))
C (t)u(t–τ )
],
(+A(t)u(t–τ ))(+B(t)v(t–τ ))
()
with initial value
u(s) = φ (s),
v(s) = φ (s),
 < u(s; , φ ) = φ (s) < +∞,
φi ∈ PC(), i = , ,
 < v(s; , φ ) = φ (s) < +∞,
s ∈ (–τ , ].
()
Here the expressions of the functions A(t), B(t), Ci (t), Di (t), i = , , are given as follows:
A(t) = H (t)α(t) =
( + hk )α(t),
<tk <t
B(t) = H (t)β(t) =
<tk <t
( + hk )β(t),
Tian et al. Advances in Difference Equations (2016) 2016:187
C (t) = H (t)c (t) =
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( + hk )c (t),
<tk <t
C (t) = H (t)c (t) =
( + hk )c (t),
<tk <t
D (t) = H (t)d (t) =
( + hk )d (t),
<tk <t
D (t) = H (t)d (t) =
( + hk )d (t).
<tk <t
In the following, we will give some useful lemmas, which will be used in the next sections.
Lemma . Assume that (u(t), v(t))T is any solution of system () with the initial conditions
(), then u(t) > , v(t) >  for all t ∈ R+ .
Proof Let
M(t) = r (t) – D (t)u(t – τ ) –
C (t)v(t – τ )
,
( + A(t)u(t – τ ))( + B(t)v(t – τ ))
N(t) = r (t) – D (t)v(t – τ ) +
C (t)u(t – τ )
.
( + A(t)u(t – τ ))( + B(t)v(t – τ ))
Then from system () we have
t
t
M(s) ds = φ () exp
M(s) ds > ,
u(t) = u() exp


t
t
N(s) ds = φ () exp
N(s) ds > .
v(t) = v() exp


This completes the proof of this lemma.
Lemma . For system () and system (), we have the following results:
() If (u(t), v(t))T is a solution of system (), then
x(t), y(t)
T
=
( + hk )u(t),
<tk <t
T
( + hk )v(t)
<tk <t
is a solution of system ().
() If (x(t), y(t))T is a solution of system (), then
u(t), v(t)
T
=
( + hk )– x(t),
<tk <t
T
( + hk )( + hk )– y(t)
<tk <t
is a solution of system ().
Proof () (u(t), v(t))T is a solution of system (). That is, for any t = tk , k ∈ N ,
u(t) =
<tk <t
( + hk )– x(t),
v(t) =
<tk <t
( + hk )– y(t)
Tian et al. Advances in Difference Equations (2016) 2016:187
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is a solution of system (), which yields
( + hk )– x (t)
<tk <t
=
( + hk )– x(t) r (t) – D (t)
( + hk )– x(t – τ )
<tk <t
<tk <t
y(t – τ )
–
,
( + A(t) <tk <t ( + hk )– x(t – τ ))( + B(t) <tk <t ( + hk )– y(t – τ ))
C (t)
<tk <t ( + hk )
–
()
( + hk )– y (t)
<tk <t
=
( + hk ) y(t) r (t) – D (t)
<tk <t
+
–
( + A(t)
( + hk )– y(t – τ )
<tk <t
x(t – τ )
.
–
–
<tk <t ( + hk ) x(t – τ ))( + B(t)
<tk <t ( + hk ) y(t – τ ))
C (t)
<tk <t ( + hk )
–
()
By the previous definitions of the functions A(t), B(t), Ci (t), and Di (t), i = , , it follows
from the simplification of () and () that
dx
dt
dy
dt
= x(t)[r (t) – d (t)x(t – τ ) –
= y(t)[r (t) – d (t)y(t – τ ) +
c (t)y(t–τ )
],
(+α(t)x(t–τ ))(+β(t)y(t–τ ))
c (t)x(t–τ )
].
(+α(t)x(t–τ ))(+β(t)y(t–τ ))
()
On the other hand, when t = tk , k ∈ N , we have
x tk+ = lim+
( + hk )u(t) =
( + hj )u(tk )
t→tk
<tk <t
= ( + hk )
<tj ≤tk
( + hk )u(tk ) = ( + hk )x(tk ),
()
<tk <t
( + hk )v(t) =
( + hj )v(tk )
y tk+ = lim+
t→tk
<tk <t
= ( + hk )
<tj ≤tk
( + hk )v(tk ) = ( + hk )y(tk ).
()
<tk <t
Combined () with () and (), we can see that (x(t), y(t))T is the solution of the impulsive system ().
() Because u(t) and v(t) are continuous on each interval (tk , tk+ ], we only need to check
the continuity of u(t) and v(t) at the impulsive point t = tk , k ∈ N .
Since
u(t) =
( + hk )– x(t),
v(t) =
<tk <t
( + hk )– y(t),
<tk <t
we have
u tk+ =
( + hj )– x tk+ =
( + hj )– x(tk ) = u(tk ),
<tj ≤tk
<tj <tk
()
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u tk– =
( + hj )– x tk– =
( + hj )– x(tk ) = u(tk ).
<tj <tk
()
<tj <tk
Similarly, we can check that
v tk+ = v tk– = v(tk ).
()
Thus, u(t) and v(t) are continuous on [, +∞).
On the other hand, similar to the above proof of step (), we can verify that (u(t), v(t))T
satisfies system ().
Then (u(t), v(t))T is a solution of the non-impulsive system ().
This completes the proof of Lemma ..
Lemma . (see [])
() Assume that for w(t) > , t ≥ , we have
w (t) ≤ w(t) a – bw(t – τ )
with initial conditions w(s) = φ(s) ≥ , s ∈ [–τ , ], where a, b are positive constants,
then there exists a positive constant w∗ such that
lim sup w(t) ≤ w∗ :=
t→∞
aeaτ
.
b
() Assume that for w(t) > , t ≥ , we have
w (t) ≥ w(t) c – dw(t – τ )
with initial conditions w(s) = φ(s) ≥ , s ∈ [–τ , ], where c, d are positive constants,
then there exists a positive constant w∗ such that
lim inf w(t) ≥ w∗ :=
t→∞
ce(c–dw
d
∗ )τ
.
Let R be the plane Euclidean space with element X = (x, y)T and norm |X| = |x| + |y|,
C = C([–τ , ], R ), B ∈ R+ , and denote CB = {ϕ = (ϕ (s), ϕ (s))T ∈ C|ϕ ≤ B} with ϕ =
sups∈[–τ ,] |ϕ(s)| = sups∈[–τ ,] (|ϕ (s)| + |ϕ (s)|).
Consider the following almost periodic system with delay:
x (t) = f (t, xt ),
t ∈ R,
()
where f (t, ϕ) is continuous in (t, ϕ) ∈ R × CB and almost periodic in t uniformly for ϕ ∈ CB ,
∀ρ > , ∃M(ρ) >  such that |f (t, ϕ)| ≤ M(ρ) as t ∈ R, ϕ ∈ Cρ , while xt ∈ CB is defined as
xt (s) = x(t + s) for s ∈ [–τ , ].
The associated product system of () is in the form of
x (t) = f (t, xt ),
y (t) = f (t, yt ),
t ∈ R.
Then by the conclusions of [, ], we have Lemma ..
()
Tian et al. Advances in Difference Equations (2016) 2016:187
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Lemma . (see []) For φ, ψ ∈ CB , suppose that there exists a Lyapunov function
V (t, φ, ψ) defined on R+ × CB × CB satisfying the following three conditions:
() u(φ – ψ) ≤ V (t, φ, ψ) ≤ v(φ – ψ), where u, v ∈ P = {u : R+ → R+ |u is
continuous increasing function and u(s) → , as s → };
() |V (t, φ̄, ψ̄) – V (t, φ̂, ψ̂)| ≤ L(φ̄ – φ̂ + ψ̄ – ψ̂), where L is a positive constant;
() D+ V (t, φ, ψ)|() ≤ –γ V (t, φ, ψ), where γ is a positive constant.
Further, assume that () has a solution x(t, v, φ) such that |x(t, v, φ)| ≤ B for t ≥ v ≥ ,
B > B > . Then system () has a unique almost periodic solution which is uniformly
asymptotically stable.
Lemma . (see []) For any x, y, u, v ∈ R, we have the following important inequality:
|x – y| – |u – v| ≤ |x – u| + |y – v|.
()
3 Main results
First in this section we will give some notations as follows:
l l
u ru
rl Bl – Cu
r B – Cu
u ∗
τ ,
,
u
exp
r
τ
=
exp
–
D
u
∗
 

Bl Du
Bl
Dl
l u
Al ru + Cu
A r + Cu
rl
v∗ =
v∗ = u exp rl – Du v∗ τ .
exp
τ
 ,
l
l
l
A
D
A D
u∗ =
Theorem . Assume that (C) and (C) hold, suppose further that
(C) Cu < Bl rl ,
then any positive solution (u(t), v(t))T of system () satisfies
u∗ ≤ lim inf u(t) ≤ lim sup u(t) ≤ u∗ ,
t→+∞
t→+∞
v∗ ≤ lim inf v(t) ≤ lim sup v(t) ≤ v∗ .
t→+∞
t→+∞
Proof Let (u(t), v(t))T be any solution of system (), then from the first equation of (),
u (t) ≤ u(t) ru – Dl u(t – τ ) .
By the first conclusion of Lemma ., we have
lim sup u(t) ≤
t→+∞
ru
exp ru τ = u∗ .
l
D
()
Similarly, by the second equation of system (),
Cu u(t – τ )
Cu
u
l
u
l
v (t) ≤ v(t) r – D v(t – τ ) +
≤ v(t) r + l – D v(t – τ ) ,
 + Al u(t – τ )
A
which leads to
lim sup v(t) ≤
t→+∞
l u
Al ru + Cu
A r + Cu
= v∗ .
exp
τ

Al
Al Dl
()
Tian et al. Advances in Difference Equations (2016) 2016:187
Page 9 of 23
Therefore, there exists a T > , such that v(t) ≤ v∗ when t > T .
Thus, when t > T + τ , from the first equation of system () again we have
Cu v∗
Cu
l
u
l
u
≥ u(t) r – l – D u(t – τ ) .
u (t) ≥ u(t) r – D u(t – τ ) –
 + Bl v ∗
B
Then by the second conclusion of Lemma ., we have
rl Bl – Cu
exp
Bl Du
lim inf u(t) ≥
t→+∞
rl Bl – Cu
u ∗
τ = u∗ .
–
D
u

Bl
()
On the other hand, from the second equation of system () again, we can get
v(t) ≥ v(t) rl – Du v(t – τ ) .
Then, by the second conclusion of Lemma . again, we have
lim inf v(t) ≥
t→+∞
l
rl
u ∗
u exp r – D v τ = v∗ .
D
()
Thus, we complete the proof of this theorem by combining (), (), (), and (). Theorem . Assume that (C)-(C) hold, then any positive solution (x(t), y(t))T of system
() satisfies
Hl u∗ ≤ lim inf x(t) ≤ lim sup x(t) ≤ Hu u∗ ,
t→+∞
t→+∞
Hl v∗ ≤ lim inf y(t) ≤ lim sup y(t) ≤ Hu v∗ .
t→+∞
t→+∞
Proof Since (x(t), y(t))T is a solution of system (), then by the second conclusion of
Lemma .,
u(t), v(t)
T
=
x(t) y(t)
,
H (t) H (t)
T
()
is a solution of system ().
Then it follows from Theorem . that
u∗ ≤ lim inf
x(t)
x(t)
≤ lim inf u(t) ≤ lim sup u(t) = lim sup
≤ u∗ ,
t→+∞
H (t)
H
t→+∞
t→+∞
 (t)
()
v∗ ≤ lim inf
y(t)
y(t)
≤ lim inf v(t) ≤ lim sup v(t) = lim sup
≤ v∗ ,
t→+∞
H (t)
t→+∞
t→+∞ H (t)
()
t→+∞
t→+∞
which implies that
Hl u∗ ≤ lim inf x(t) ≤ lim sup x(t) ≤ Hu u∗ ,
()
Hl v∗ ≤ lim inf y(t) ≤ lim sup y(t) ≤ Hu v∗ .
()
t→+∞
t→+∞
t→+∞
t→+∞
This completes the proof of this theorem.
Tian et al. Advances in Difference Equations (2016) 2016:187
Page 10 of 23
Remark . Suppose that (C)-(C) hold, then system () is permanent.
Before we discuss the existence and uniformly asymptotically stability of a unique almost
periodic solution of system (), we will introduce the following notations first:
β = Dl u∗ –
α  = K τ  + K  τ  ;
K – (K + K )τ – (K + K )τ ;
α = Dl v∗ + P – (τ + τ )L – L τ – L τ ;
β = L  τ  + L  τ  ,
where
Au Cu u∗ v∗
K =
M N


K = Du u∗ ;
,
K =
Au Cu Du (u∗ ) v∗
,
M N
L =
Cu Du u∗ v∗
,
MN 

L = Du v∗ ,
M =  + Al u∗ ,
K =
Cu Du u∗ v∗
,
M N
Au (Cu v∗ ) u∗
,
M N 
u u ∗ ∗ 
B C u v
L =
,
MN 
L =
N =  + Bl v ∗ ,
P=
K =
L =
M =
Bu (Cu u∗ ) v∗
;
M N 
Bu Cu Du u∗ (v∗ )
;
MN 
Cu u∗
,
M N
M =
Cu v∗
;
MN 
Bl Cl u∗ v∗
.
( + Au u∗ )( + Bu v∗ )
Theorem . Assume that (C)-(C) hold, further assume that
(C) positive λ and λ exist, such that αβ < λλ < αβ ,
then system () admits a unique almost periodic solution which is uniformly asymptotically
stable.
Proof At first, we prove that system () has a unique uniformly asymptotically stable almost periodic solution.
In order to achieve this aim, we take a transformation
u(t) = ex (t) ,
v(t) = ey (t) ,
t ∈ R+ .
()
Then system () is transformed into following system:
⎧
⎨ x (t) = r (t) – D (t)ex (t–τ ) –
⎩ y (t) = r (t) – D (t)ey (t–τ ) +

C (t)ey (t–τ )
,
(+A(t)ex (t–τ ) )(+B(t)ey (t–τ ) )
x
(t–τ
)

C (t)e 
.
(+A(t)ex (t–τ ) )(+B(t)ey (t–τ ) )
()
Suppose that U (t) = (x (t), y (t))T and U (t) = (x (t), y (t))T are any two solutions of
system (), then the product system of () reads
⎧
y (t–τ )
 (t)e 
⎪
,
x (t) = r (t) – D (t)ex (t–τ ) – (+A(t)exC(t–τ
⎪
 ) )(+B(t)ey (t–τ ) )
⎪
⎪
⎪
x
(t–τ
)

⎪
C (t)e 
⎨ y (t) = r (t) – D (t)ey (t–τ ) +
,
(+A(t)ex (t–τ ) )(+B(t)ey (t–τ ) )
y (t–τ )
 (t)e 
⎪
,
x (t) = r (t) – D (t)ex (t–τ ) – (+A(t)exC(t–τ
⎪
⎪
 ) )(+B(t)ey (t–τ ) )
⎪
⎪
⎪
C (t)ex (t–τ )
⎩ y (t) = r (t) – D (t)ey (t–τ ) +
.



(+A(t)ex (t–τ ) )(+B(t)ey (t–τ ) )
()
Tian et al. Advances in Difference Equations (2016) 2016:187
Page 11 of 23
Denote
S∗ = φ = (xt , yt )T ∈ C [–τ , ], R | ln u∗ ≤ xt ≤ ln u∗ , ln v∗ ≤ yt ≤ ln v∗ , t ∈ R+ .
For any = (φ , φ )T = (xt , yt )T ∈ S∗ , we can choose = (ψ , ψ )T = (xt , yt )T ∈ S∗
such that
() – () = φ () – ψ () + φ () – ψ () > .

()
Consider a Lyapunov functional V (t) = V (t, , ) = V (t, (xt , yt )T , (xt , yt )T ) defined on
R × S∗ × S∗ as follows:
+
V (t) = V (t) + V (t) + V (t) + V (t),
()
where
V (t) = λ x (t) – x (t) + λ y (t) – y (t);
–τ t
x (r) – x (r) dr ds
V (t) = λ K
–τ
t+s
–τ
x (r) – x (r) dr ds
t
+ λ  K
–τ
t+s
–τ
x (r) – x (r) dr ds
t
+ λ  K
–τ –τ
t+s
–τ
x (r) – x (r) dr ds
t
+ λ  K
–τ –τ
t+s
–τ
x (r) – x (r) dr ds
t
+ λ  K
–τ –τ
–τ
t+s
x (r) – x (r) dr ds;
t
+ λ  K
–τ –τ
–τ
–τ –τ
t
y (r) – y (r) dr ds
t+s
–τ
t
+ λ  L
–τ –τ
y (r) – y (r) dr ds
t+s
–τ
t
+ λ  L
–τ –τ
–τ
–τ –τ
–τ
t
–τ
–τ t
V (t) = λ M
t–τ
y (r) – y (r) dr ds
t+s
t
+ λ  L
–τ
y (r) – y (r) dr ds
t+s
t
+ λ  L
y (r) – y (r) dr ds
t+s
+ λ  L
()
t+s
V (t) = λ L
()
y (r) – y (r) dr ds;
t+s
x (s) – x (s) ds + λ M
()
t
t–τ
y (s) – y (s) ds.
()
Tian et al. Advances in Difference Equations (2016) 2016:187
Page 12 of 23
According to the definitions of S∗ and V (t) = V (t, , ), there is some positive constant
M large enough such that
V (t) = V (t, , ) ≤ M.
()
By the structure of V (t), it is easy to see that
V (t) ≥ V (t) ≥ min{λ , λ } x (t) – x (t) + y (t) – y (t) = λ() – () > , ()
where λ = min{λ , λ } > .
Moreover, by the integrative inequality and the absolute-value inequality properties we
have
V (t) ≤

(λ + λ ) + λ τ  (K + K + K + L + L )

+ λ τ  (K + K + L + L + L ) + τ (λ M + λ M )
× sup xt (s) – xt (s) + yt (s) – yt (s)
s∈[–τ ,]
= λ̄ – ,
()
where
λ̄ :=

(λ + λ ) + λ τ  (K + K + K + L + L )

+ λ τ  (K + K + L + L + L ) + τ (λ M + λ M ) .
Let u, v ∈ C(R+ , R+ ), choose u = λs, v = λ̄s, then the first condition of Lemma . is satisfied.
¯ = (xt , yt )T , ¯ = (xt , yt )T , ˆ = (x∗t , y∗t )T , ˆ = (x∗t , y∗t )T ∈ S∗ , then from the preFor ∀
vious definitions of Vi (t), i = , , , , it is easy to calculate that
V (t, ,
¯ )
¯ – V (t, ,
ˆ )
ˆ = λ x (t) – x (t) – x∗ (t) – x∗ (t)| + λ |y (t) – y (t) – y∗ (t) – y∗ (t)
–τ t
x (r) – x (r) – x∗ (r) – x∗ (r) dr ds
+ λ  K


–τ
–τ
t+s
t
+ λ  K
–τ
–τ
t+s
x (r) – x (r) – x∗ (r) – x∗ (r) dr ds


t
+ λ  K
–τ –τ
–τ
t+s
t
+ λ  K
–τ –τ
–τ
t+s
x (r) – x (r) – x∗ (r) – x∗ (r) dr ds


x (r) – x (r) – x∗ (r) – x∗ (r) dr ds


t
+ λ  K
–τ –τ
–τ
t+s
t
+ λ  K
–τ –τ
t+s
x (r) – x (r) – x∗ (r) – x∗ (r) dr ds


x (r) – x (r) – x∗ (r) – x∗ (r) dr ds


Tian et al. Advances in Difference Equations (2016) 2016:187
–τ
t
+ λ  L
–τ –τ
t+s
–τ
y (r) – y (r) – y∗ (r) – y∗ (r) dr ds


t
+ λ  L
–τ –τ
t+s
–τ
y (r) – y (r) – y∗ (r) – y∗ (r) dr ds


t
+ λ  L
–τ –τ
–τ
t+s
t
+ λ  L
–τ –τ
–τ
t+s
–τ
–τ
t
+ λ M
t–τ
t
+ λ M
t–τ
y (r) – y (r) – y∗ (r) – y∗ (r) dr ds



t+s
t
+ λ  L
y (r) – y (r) – y∗ (r) – y∗ (r) dr ds


y (r) – y (r) – y∗ (r) – y∗ (r) dr ds
t
+ λ  L
–τ
Page 13 of 23
t+s

y (r) – y (r) – y∗ (r) – y∗ (r) dr ds


x (s) – x (s) – x∗ (s) – x∗ (s) ds


y (s) – y (s) – y∗ (s) – y∗ (s) ds.


Then it follows from inequality () in Lemma . that
V (t, ,
¯ )
¯ – V (t, ,
ˆ )
ˆ ≤ λ x (t) – x∗ (t) + x (t) – x∗ (t) + λ y (t) – y∗ (t) + y (t) – y∗ (t)
–τ t
x (r) – x∗ (r) + x (r) – x∗ (r) dr ds
+ λ  K


–τ
–τ
t+s
x (r) – x∗ (r) + x (r) – x∗ (r) dr ds


t
+ λ  K
–τ
t+s
–τ
x (r) – x∗ (r) + x (r) – x∗ (r) dr ds


t
+ λ  K
–τ –τ
–τ
t+s
x (r) – x∗ (r) + x (r) – x∗ (r) dr ds


t
+ λ  K
–τ –τ
t+s
–τ
x (r) – x∗ (r) + x (r) – x∗ (r) dr ds
t
+ λ  K
–τ –τ
–τ
–τ
–τ –τ
t
t+s
–τ
–τ –τ
t
t+s
–τ
y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


t
+ λ  L
–τ –τ
–τ
t+s
t
+ λ  L
–τ –τ
–τ
t+s
t
+ λ  L
–τ
t+s

y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


+ λ  L

t+s
+ λ  L

x (r) – x∗ (r) + x (r) – x∗ (r) dr ds
t
+ λ  K
–τ –τ

t+s
y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


Tian et al. Advances in Difference Equations (2016) 2016:187
–τ
t
+ λ  L
–τ
t
+ λ M
t–τ
t
+ λ M
t–τ
t+s
Page 14 of 23
y (r) – y∗ (r) + y (r) – y∗ (r) dr ds


x (s) – x∗ (s) + x (s) – x∗ (s) ds


y (s) – y∗ (s) + y (s) – y∗ (s) ds,


which yields
V (t, ,
¯ )
¯ – V (t, ,
ˆ )
ˆ ≤ λ̄

sup xit (s) – x∗it (s) + yit (s) – y∗it (s)
i= s∈[–τ ,]
¯ – ˆ + ¯ – ˆ .
= λ̄ ()
This means the second condition of Lemma . is satisfied.
Finally, we will prove the last condition of Lemma ..
In fact, calculating the right derivative D+ V (t) of V (t) along the solutions of system
() we have
D+ V (t) = x (t) – x (t) sgn x (t) – x (t) + y (t) – y (t) sgn y (t) – y (t) .
()
It follows from the mean-value theorem and the product system () that
x (t) – x (t)
= D (t)ex (t–τ ) – D (t)ex (t–τ ) +
–
C (t)ey (t–τ )
x
( + A(t)e  (t–τ ) )( + B(t)ey (t–τ ) )
C (t)ey (t–τ )
.
( + A(t)ex (t–τ ) )( + B(t)ey (t–τ ) )
Then
x (t) – x (t)
= –D (t)eθ (t) x (t – τ ) – x (t – τ )
+
A(t)C (t)eθ (t)+θ (t)
x (t – τ ) – x (t – τ )
( + A(t)eθ (t) ) ( + B(t)eθ (t) )
–
C (t)eθ (t)
y (t – τ ) – y (t – τ ) ,
θ
(t)
θ
(t)

( + A(t)e  )( + B(t)e  )
()
where θ (t) lie between x (t – τ ) and x (t – τ ), θ (t) lie between x (t – τ ) and x (t – τ ),
θ (t) lie between y (t – τ ) and y (t – τ ).
That is,
x (t) – x (t)
= –D (t)eθ (t) x (t) – x (t) + D (t)eθ (t)
t
t–τ
x (s) – x (s) ds
Tian et al. Advances in Difference Equations (2016) 2016:187
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A(t)C (t)eθ (t)+θ (t)
x (t) – x (t)
θ
(t)

θ
(t)


( + A(t)e ) ( + B(t)e )
t
A(t)C (t)eθ (t)+θ (t)
x (s) – x (s) ds
–
( + A(t)eθ (t) ) ( + B(t)eθ (t) ) t–τ 
+
–
C (t)eθ (t)
y (t – τ ) – y (t – τ ) .
θ
(t)
θ
(t)

( + A(t)e  )( + B(t)e  )
()
Integrating both sides of () on the interval [t – τ , t], we have
t
t–τ
x (s) – x (s) ds = –
D (s)eθ (s) x (s – τ ) – x (s – τ ) ds
t
t–τ
A(s)C (s)eθ (s)+θ (s)
x (s – τ ) – x (s – τ ) ds
θ
(s)

θ
(s)
( + A(s)e  ) ( + B(s)e  )
t
+
t–τ
C (s)eθ (s)
y (s – τ ) – y (s – τ ) ds
θ
(s)
θ
(s)



( + A(s)e )( + B(s)e )
t
–
t–τ
–τ
=–
D (t + s + τ )eθ (t+s+τ ) x (t + s) – x (t + s) ds
–τ
–τ
A(t + s + τ )C (t + s + τ )eθ (t+s+τ )+θ (t+s+τ )
θ (s) 
θ (t+s+τ ) )
–τ –τ ( + A(t + s + τ )e  ) ( + B(t + s + τ )e 
× x (t + s) – x (t + s) ds
–τ
C (t + s + τ )eθ (t+s+τ )
–
θ (t+s+τ ) )( + B(t + s + τ )eθ (t+s+τ ) )

–τ –τ ( + A(t + s + τ )e 
× y (t + s) – y (t + s) ds,
+
which implies that
t
t–τ
x (s) – x (s) ds ≤ K
+
–τ x (t + s) – x (t + s) ds
–τ
x (t + s) – x (t + s) ds
–τ
K
–τ –τ
y (t + s) – y (t + s) ds.
–τ
+ M
()
–τ –τ
Similarly, we can get
t
t–τ
x (s) – x (s)
ds ≤ K
–τ
x (t + s) – x (t + s) ds
–τ –τ
+ K
–τ x (t + s) – x (t + s) ds
–τ
–τ
+ M
–τ –τ
Thus,
x (t) – x (t) sgn x (t) – x (t)
≤ – Dl u∗ – K x (t) – x (t)
y (t + s) – y (t + s) ds.
()
Tian et al. Advances in Difference Equations (2016) 2016:187
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x (t + s) – x (t + s) ds
–τ
+ K
–τ –τ
–τ x (t + s) – x (t + s) ds + L
+ K
–τ
–τ
y (t + s) – y (t + s) ds
–τ –τ
–τ x (t + s) – x (t + s) ds + K
+ K
–τ
–τ
+ L
–τ
x (t + s) – x (t + s) ds
–τ –τ
y (t + s) – y (t + s) ds + M y (t – τ ) – y (t – τ ).
()
–τ –τ
By the same argument, from the second and the fourth equation of system () we can
derive that
y (t) – y (t)
= D (t)ey (t–τ ) – D (t)ey (t–τ ) +
C (t)ex (t–τ )
( + A(t)ex (t–τ ) )( + B(t)ey (t–τ ) )
C (t)ex (t–τ )
( + A(t)ex (t–τ ) )( + B(t)ey (t–τ ) )
= –D (t)eθ (t) y (t – τ ) – y (t – τ )
–
+
C (t)eθ (t)
x (t – τ ) – x (t – τ )
( + A(t)eθ (t) ) ( + B(t)eθ (t) )
–
B(t)C (t)eθ (t)+θ (t)
y (t – τ ) – y (t – τ ) ,
θ
(t)
θ
(t)

( + A(t)e  )( + B(t)e  )
()
where θ (t) lie between y (t – τ ) and y (t – τ ), θ (t) lie between x (t – τ ) and x (t – τ ),
θ (t) lie between y (t – τ ) and y (t – τ ).
Integrating both sides of () on the interval [t – τ , t], we have
t
t–τ
y (s) – y (s) ds
D (s)eθ (s) y (s – τ ) – y (s – τ ) ds
t
=
t–τ
C (s)eθ (s)
x (s – τ ) – x (s – τ ) ds
θ
(s)

θ
(s)
( + A(s)e  ) ( + B(s)e  )
t
+
t–τ
t
–
t–τ
–τ
=–
B(s)C (s)eθ (s)+θ (s)
y (s – τ ) – y (s – τ ) ds
θ
(s)
θ
(s)

( + A(s)e  )( + B(s)e  )
D (t + s + τ )eθ (t+s+τ ) y (t + s) – y (t + s) ds
–τ
–τ
+
–τ –τ
C (t + s + τ )eθ (t+s+τ ) [x (t + s) – x (t + s)]
ds
( + A(t + s + τ )eθ (t+s+τ ) ) ( + B(t + s + τ )eθ (t+s+τ ) )
–τ
B(t + s + τ )C (t + s + τ )eθ (t+s+τ )+θ (t+s+τ )
θ (t+s+τ ) )( + B(s)eθ (t+s+τ ) )
–τ –τ ( + A(t + s + τ )e 
× y (t + s) – y (t + s) ds,
–
()
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which leads to
t
y
(s)
–
y
(s)
ds


t–τ
≤
L
y (t + s) – y (t + s) ds + L
–τ –τ
y (t + s) – y (t + s) ds
–τ
–τ –τ
x (t + s) – x (t + s) ds.
–τ
+ M
()
–τ –τ
By the same argument, we can obtain
t
t–τ
≤
y (s) – y (s)
ds
y (t + s) – y (t + s) ds + L
–τ
L
–τ –τ
y (t + s) – y (t + s) ds
–τ –τ
x (t + s) – x (t + s) ds.
–τ
+ M
()
–τ –τ
On the other hand, it follows from () that
y (t) – y (t)
t
θ (t)
= D (t)e
y (s) – y (s) ds
t–τ
– D (t)eθ (t) y (t) – y (t) –
B(t)C (t)eθ (t)+θ (t)
y (t) – y (t)
( + A(t)eθ (t) )( + B(t)eθ (t) )
t
B(t)C (t)eθ (t)+θ (t)
y (s) – y (s) ds
+
θ
(t)
θ
(t)



( + A(t)e )( + B(t)e ) t–τ
–
C (t)eθ (t)
x (t – τ ) – x (t – τ ) .
θ
(t)

θ
(t)


( + A(t)e ) ( + B(t)e )
()
Thus, combined () with ()-(), one can deduce that
y (t) – y (t) sgn y (t) – y (t)
≤ – Dl v∗ + P y (t) – y (t) + L
–τ
+ L
–τ y (t + s) – y (t + s) ds
–τ
y (t + s) – y (t + s) ds + K
–τ –τ
–τ
+ L
–τ –τ
–τ + K
y (t + s) – y (t + s) ds + L
–τ
x (t + s) – x (t + s) ds
–τ –τ
–τ y (t + s) – y (t + s) ds
–τ
x (t + s) – x (t + s) ds + M x (t – τ ) – x (t – τ ).
–τ –τ
Therefore,
D+ V (t) ≤ –λ Dl u∗ – K x (t) – x (t) – λ Dl v∗ + P y (t) – y (t)
–τ
–τ
x (t + s) – x (t + s) ds + λ K
x (t + s) – x (t + s) ds
+ λ  K
–τ
–τ
()
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x (t + s) – x (t + s) ds + λ K
–τ
+ λ  K
–τ –τ
–τ –τ
–τ –τ
y (t + s) – y (t + s) ds + λ L
–τ
+ λ  L
y (t + s) – y (t + s) ds + λ L
+ λ  L
–τ –τ
+ λ  L
–τ
y (t + s) – y (t + s) ds
–τ –τ
–τ
x (t + s) – x (t + s) ds
–τ
–τ –τ
–τ –τ
x (t + s) – x (t + s) ds + λ K
–τ
+ λ  K
x (t + s) – x (t + s) ds
–τ
y (t + s) – y (t + s) ds + λ L
–τ –τ
y (t + s) – y (t + s) ds
–τ
–τ –τ
y (t + s) – y (t + s) ds
–τ –τ
+ λ M x (t – τ ) – x (t – τ ) + λ M y (t – τ ) – y (t – τ ).
()
On the other hand, according to the structure of Vi (t), i = , , , we can easily calculate
the right derivatives D+ V (t), D+ V (t), and D+ V (t) along the solution of system () as
follows:
D+ V (t) = (λ τ K + λ τ K + λ τ K + λ τ K + λ τ K + λ τ K ) × x (t) – x (t)
–τ
–τ
x (t + s) – x (t + s) ds
x (t + s) – x (t + s) ds – λ K
– λ  K
–τ
–τ
x (t + s) – x (t + s) ds – λ K
–τ
– λ  K
–τ –τ
x (t + s) – x (t + s) ds
–τ –τ
x (t + s) – x (t + s) ds
–τ
– λ  K
–τ
–τ –τ
x (t + s) – x (t + s) ds;
–τ
– λ  K
()
–τ –τ
D+ V (t) = (λ τ L + λ τ L + λ τ L + λ τ L + λ τ L + λ τ L ) × y (t) – y (t)
–τ
–τ
y (t + s) – y (t + s) ds
y (t + s) – y (t + s) ds – λ L
– λ  L
–τ –τ
–τ –τ
y (t + s) – y (t + s) ds – λ L
–τ
– λ  L
–τ –τ
– λ  L
– λ  L
–τ
y (t + s) – y (t + s) ds
–τ –τ
–τ y (t + s) – y (t + s) ds ds
–τ
–τ y (t + s) – y (t + s) ds;
()
–τ
D+ V (t) = λ M x (t) – x (t) – x (t – τ ) – x (t – τ )
+ λ M y (t) – y (t) – y (t – τ ) – y (t – τ ) .
()
From ()-(), by a simple reduction we can easily give the following estimations for
the right derivatives of V (t):
D+ V (t) ≤ – λ Dl u∗ – K – (λ τ K + λ τ K + λ τ K + λ τ K
+ λ τ K + λ τ K ) × x (t) – x (t) – λ Dl v∗ + P – (λ τ L
Tian et al. Advances in Difference Equations (2016) 2016:187
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+ λ τ L + λ τ L + λ τ L + λ τ L + λ τ L ) × y (t) – y (t)
= –(λ β – λ α )x (t) – x (t) – (λ α – λ β )y (t) – y (t).
If we denote δ = min{β –
λ
α ,α
λ  
λ
β },
λ 
–
()
then δ >  by the condition (C).
Thus, it follows from equalities () and () that
λ
D+ V (t) = – β – α λ x (t) – x (t)
λ
λ
– α – β λ y (t) – y (t)
λ
≤ –δ λ x (t) – x (t) + λ y (t) – y (t)
= –δV (t) = –δ
≤–
Let γ =
V (t)
V (t)
V (t)
λ|() – ()|
δV (t).
M
λ|()–()|
δ,
M
()
then γ >  and
D+ V (t) ≤ –γ V (t),
t ∈ R+ .
()
This means the last condition of Lemma . is satisfied. By Lemma ., the system admits a unique uniformly asymptotically stable almost periodic solution (x(t), y(t))T .
Thus, by the transformation (), we can conclude that system () admits a unique uniformly asymptotically stable almost periodic solution (u(t), v(t))T = (ex(t) , ey(t) )T .
Finally, we will explain that system () has a unique uniformly asymptotically stable almost periodic solution.
In fact, from Lemma ., we know that
x(t), y(t)
T
=
<tk <t
( + hk )u(t),
T
( + hk )v(t)
<tk <t
is a solution of system (). Since the condition (C) holds, similar to the proofs of
Lemma  and Theorem  in [], we can prove that (x(t), y(t))T is almost periodic.
Therefore, (x(t), y(t))T is the unique uniformly asymptotically stable almost periodic solution of system () because of the uniqueness and the uniformly asymptotical stability of
(u(t), v(t))T .
This completes the proof of this theorem.
4 Numerical simulations and discussions
In this section, we will give a numerical example to illustrate the feasibility of our analytical
results, then some discussions of the effects of impulsive perturbations and time delays to
the system are referred to in the end of the paper.
Tian et al. Advances in Difference Equations (2016) 2016:187
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Example . Consider the following nonautonomous prey-predator system with impulsive effects and delays:
⎧
⎫
dx
⎪
= x(t)[. + . sin(t) – (. + . cos(t))x(t – .) ⎪
⎪
dt
⎪
⎪
⎪
⎪
⎪
⎪
(.+. sin(t))y(t–.)
⎬
⎪
],
–
⎪
⎪
(+x(t–.))(+y(t–.))
⎪
⎪
⎨ dy = y(t)[. + . sin(t) – (. + . sin(t))y(t – .) ⎪
⎪
dt
⎪
⎪
⎭
(.+. cos(t))x(t–.)
⎪
+
],
⎪
⎪
(+x(t–.))(+y(t–.))
⎪
⎪
⎪
⎪
x(tk+ ) = ( + hk )x(tk ),
⎪
⎪
t = tk , k ∈ N,
⎪
⎩ y(t + ) = ( + h )y(t ),
k
k
k
t = tk ,
()
where r (t) = .+. sin(t), r (t) = .+. sin(t), d (t) = .+. cos(t), d (t) =
. + . sin(t), c (t) = . + . sin(t), c (t) = . + . cos(t), α(t) = β(t) = 
are all positive, bounded and almost periodic functions which satisfy the condition (C)
in the paper.
If we can set hk = –., hk = –., tk = k, t ∈ [, ], and choose Hl = ., Hl =
., Hu = , Hu = , then it is not difficult to verify
Hil ≤ Hi (t) =
( + hik ) ≤ Hul
(i = , ),
<tk <t=
which satisfies the condition (C).
Furthermore, according to the notations in Section  one can make some simple calculations as follows:
u∗ ≈ .,
u∗ ≈ .,
Al ≈ .,
Au = ,
rl = .,
v∗ ≈ .,
Bl ≈ .,
ru = .,
rl = .,
v∗ ≈ .;
Bu = ;
ru = .;
Cl ≈ .,
Cu = .,
Cl ≈ .,
Cu = .,
Dl ≈ .,
Du = .,
Dl ≈ .,
Du = ..
Then it is obvious that Cu ≈ . < Bl rl ≈ ., which satisfies the condition (C) in
Theorems . and .. Meanwhile, with the help of mathematical software such as Maple,
we can calculate that
α ≈ .,
β ≈ .,
α ≈ .,
β ≈ ..
Then we can choose positive constants λ = λ = , which satisfy
λ
α
α
≈ . <
=
≈ ..
β
λ
β
That is to say, the condition (C) in Theorem . is also satisfied. Thus, all the conditions
of Theorems . and . hold, according to the theoretical analysis, system () should be
permanent and admits a unique uniformly asymptotically stable almost periodic solution.
In order to verify this point by numerical simulations, we plot the time-series of the predator x(t) (see Figure (a)) and the prey y(t) (see Figure (b)). One can see that the population
Tian et al. Advances in Difference Equations (2016) 2016:187
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Figure 1 The dynamics of system (4) with two initial conditions x(0) = 1, y(0) = 0.2 and x(0) = 1.2,
y(0) = 0.8.
Figure 2 The dynamics of system (4) with initial condition x(0) = 1, y(0) = 0.2 but different impulsive
effects.
Figure 3 The dynamics of system (4) with initial condition x(0) = 1, y(0) = 0.5 but different cases of
time delay.
density of the predator oscillates around x(t) = . and the population density of the prey
oscillates around y(t) = ., this indicates the permanence of the system. The almost periodic phenomenon is also easily observed from the figure. Moreover, we can see that the
solution U (t) = (x (t), y (t))T with initial condition x() = ., y() = . gradually approximate the other solution of U (t) = (x (t), y (t))T with initial condition x() = , y() = .,
and it proved that the almost periodic case is asymptotically stable.
On the other hand, when the system is permanent, we will show the difference with two
different impulsive effects (see Figure ) and different cases of time delays (see Figure ).
Tian et al. Advances in Difference Equations (2016) 2016:187
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From Figure , we can see that both the mean density population of the predator and the
prey are decreasing with stronger impulsive effects, such as enhancing the harvesting rate,
the efficiency of the insecticides for the species, and so on. In addition, in order to show
the effects of the delays to the dynamics of the species, we consider two different cases of
time delays, Case  (with τ = ., τ = ., τ = ., τ = . in Figure (a)) and
Case  (with τ = ., τ = , τ = ., τ = . in Figure (b)) with the same initial condition
x() = , y() = .. We can see that the mean population density of the bigger time delays
(Case ) is bigger that the smaller ones (Case ). This suggests that a longer digest delay in
the predation may be conductive and could increase the possibility of the permanence of
the population.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11372294, 51349011), Scientific Research
Fund of Sichuan Provincial Education Department (14ZB0115, 15ZB0113), and Doctoral Research Fund of Southwest
University of Science and Technology (15zx7138).
Received: 19 January 2016 Accepted: 28 June 2016
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