Xu and Wu Advances in Difference Equations 2014, 2014:264
http://www.advancesindifferenceequations.com/content/2014/1/264
RESEARCH
Open Access
Dynamics of a nonautonomous
Lotka-Volterra predator-prey dispersal system
with impulsive effects
Lijun Xu1 and Wenquan Wu2*
*
Correspondence:
[email protected]
2
Department of Mathematics, Aba
Teachers College, Wenchuan,
Sichuan 623002, China
Full list of author information is
available at the end of the article
Abstract
By applying the comparison theorem, Lyapunov functional, and almost periodic
functional hull theory of the impulsive differential equations, this paper gives some
new sufficient conditions for the uniform persistence, global asymptotical stability,
and almost periodic solution to a nonautonomous Lotka-Volterra predator-prey
dispersal system with impulsive effects. The main results of this paper extend some
corresponding results obtained in recent years. The method used in this paper
provides a possible method to study the uniform persistence, global asymptotical
stability, and almost periodic solution of the models with impulsive perturbations in
biological populations.
MSC: 34K14; 34K20; 34K45; 92D25
Keywords: uniform persistence; diffusion; comparison theorem; predator-prey;
impulse
1 Introduction
Because of the ecological effects of human activities and industry, more and more habitats
are broken into patches and some of them are polluted. Negative feedback crowding or the
effect of the past life history of the species on its present birth rate are common examples
illustrating the biological meaning of time delays and justifying their use in these systems.
Recently, diffusions have been introduced into Lotka-Volterra type systems. The effect of
an environment change in the growth and diffusion of a species in a heterogeneous habitat
is a subject of considerable interest in the ecological literature [–�].
As was pointed out by Berryman [п™Ђ], the dynamic relationship between predators and
their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. In recent
years, the predator-prey system has been extensively studied by many scholars, many excellent results were obtained concerned with the persistent property and positive periodic
solution of the system; see [–�] and the references cited therein.
Considering the effect of almost periodically varying environment is an important selective forces on systems in a fluctuating environment, Meng and Chen [�] studied
the case of combined effects: dispersion, time delays, almost periodicity of the environment. Namely, they investigated the following general nonautonomous Lotka-Volterra
В©2014 Xu and Wu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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type predator-prey dispersal system:
вЋ§
x˙  (t) = x (t)[r (t) – a (t)x (t) – b (t)x (t – τ (t))
вЋЄ
вЋЄ
вЋЄ
�
c(t)y(t)
вЋЄ
вЋЄ
– –σ k (t, s)x (t + s) ds – +α(t)x
] + ni=� Di (t)[xi (t) – x (t)],
вЋЄ
вЋЄ
п›њ (t)
вЋЄ
⎨ x˙ (t) = x (t)[r (t) – a (t)x (t) – b (t)x (t – τ (t))
i
i
i
i
i
i
i
i
�
n
вЋЄ
– –σi ki (t, s)xi (t + s) ds] + j= Dji (t)[xj (t) – xi (t)], i = �, �, . . . , n,
вЋЄ
вЋЄ
вЋЄ
f (t)xп›њ (t)
вЋЄ
вЋЄ
Л™
y
(t)
=
y(t)[–r
n+ (t) + +α(t)x (t) – an+ (t)y(t) – bn+ (t)y(t – τn+ (t))
вЋЄ
вЋЄ
вЋ©
�
– –σn+ kn+ (t, s)y(t + s) ds].
(п›њ.п›њ)
By using the comparison theorem and functional hull theory of almost periodic system,
the authors [�] obtained some sufficient conditions for the uniform persistence, global
asymptotical stability, and almost periodic solution to system (п›њ.п›њ).
However, the ecological system is often deeply perturbed by human exploitation activities such as planting and harvesting and so on, which makes it unsuitable to be considered continually. To obtain a more accurate description of such systems, we need to consider impulsive differential equations. In recent years, the impulsive differential equations
have been intensively investigated (see [�–�] for more details). To the best of the authors’ knowledge, in the literature, there are few papers concerning the permanence, global
asymptotical stability, and almost periodic solution to the Lotka-Volterra type predatorprey dispersal system with impulsive effects. Therefore, we consider the following LotkaVolterra type predator-prey dispersal system with impulsive effects:
вЋ§
x˙  (t) = x (t)[r (t) – a (t)x (t) – b (t)x (t – τ (t))
вЋЄ
вЋЄ
вЋЄ
�
c(t)y(t)
вЋЄ
вЋЄ
– –σ k (t, s)x (t + s) ds – +α(t)x
] + ni=� Di (t)[xi (t) – x (t)],
вЋЄ
вЋЄ
п›њ (t)
вЋЄ
вЋЄ
вЋЄ
x˙ i (t) = xi (t)[ri (t) – ai (t)xi (t) – bi (t)xi (t – τi (t))
вЋЄ
вЋЄ
вЋЄ
�
вЋЁ
– –σi ki (t, s)xi (t + s) ds] + nj= Dji (t)[xj (t) – xi (t)], i = �, �, . . . , n,
f (t)xп›њ (t)
вЋЄ
– an+ (t)y(t) – bn+ (t)y(t – τn+ (t))
y˙ (t) = y(t)[–rn+ (t) + +α(t)x
вЋЄ
вЋЄ
п›њ (t)
вЋЄ
�
вЋЄ
вЋЄ
k
(t,
s)y(t
+
s)
ds],
t = tk ,
–
вЋЄ
–σn+ n+
вЋЄ
вЋЄ
вЋЄ
вЋЄ xj (tk ) = hjk xj (tk ), j = п›њ, п�є, . . . , n,
вЋЄ
вЋЄ
вЋ©
y(tk ) = hn+п›њ,k y(tk ), k в€€ Z,
(п›њ.п�є)
where xп›њ and y are population density of prey species x and predator species y in patch
, and xi is density of prey species x in patch i; predator species y is confined to patch ,
while the prey species x can disperse among n patches; Dij (t) is the dispersion rate of the
species from patch j to patch i, the terms bi (t)xi (t – τi (t)) (i = , �, . . . , n), bn+ (t)y(t – τn+ (t)),
�
�
–σi ki (t, s)xi (t + s) ds (i = , �, . . . , n) and –σn+ kn+ (t, s)y(t + s) ds represent the negative feedback crowding and the effect of all the past life history of the species on its present birth
rate, respectively; xi (tk ) = xi (tk+ ) – xi (tk– ), xi (tk+ ) and xi (tk– ) represent the right and the left
limit of xi (tk ), xi (tk– ) = xi (tk ), k ∈ Z, i = , �, . . . , n. Related to a continuous function f , we
use the following notations: f l = infsв€€R f (s), f u = supsв€€R f (s).
In system (п›њ.п�є), we always assume that for all i = п›њ, п�є, . . . , n + п›њ, j = п›њ, п�є, . . . , n:
(H ) ri (t), ai (t), bi (t), c(t), f (t), α(t) and Dij (t) (Dii (t) = �) are nonnegative and continuous
almost periodic functions for all t ∈ R, and ali + bli > �.
(H� ) ki (t, s) are defined on R × (–∞, �] and nonnegative and continuous almost periodic
functions with respect to t ∈ R and integrable with respect to s on (–∞, �] such that
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�
–σi
�
ki (t, s) ds is continuous and bounded with respect to t ∈ R, � < –σi (–s)kiu (s) ds <
+в€ћ.
(H� ) τi (t) is continuous and differentiable bounded almost periodic functions on R, and
inft∈R { – τ˙i (t)} > �.
(H� ) The sequences {hik } are almost periodic and hik > –.
j
j
(H� ) The set of sequences {tk }, tk = tk+j – tk , k ∈ Z, j ∈ Z is uniformly almost periodic and
θ := infk∈Z tk > �.
The main purpose of this paper is to establish some new sufficient conditions which
guarantee the uniform persistence, global asymptotical stability, and almost periodic solution of system (п›њ.п�є) by using the comparison theorem, the Lyapunov functional, and
almost periodic functional hull theory of the impulsive differential equations [�, ] (see
Theorem �., Theorem �., and Theorem �. in Sections �-�).
The organization of this paper is as follows. In Section �, we give some basic definitions and necessary lemmas which will be used in later sections. In Section �, by using
the comparison theorem of the impulsive differential equations, we give the permanence
of system (п›њ.п�є). In Section п�ј, we study the global asymptotical stability of system (п›њ.п�є) by
constructing a suitable Lyapunov functional. In Section �, some new sufficient conditions
are obtained for the existence, uniqueness, and global asymptotical stability of the positive
almost periodic solution of system (п›њ.п�є).
2 Preliminaries
Now, let us state the following definitions and lemmas, which will be useful in proving our
main result.
Let Rn be the n-dimensional Euclidean space with norm x = ni=п›њ |xi |. By I, I =
{{tk } ∈ R : tk < tk+ , k ∈ Z, limk→±∞ tk = ±∞}, we denote the set of all sequences that
are unbounded and strictly increasing with distance ПЃ({tk(п›њ) }, {tk(п�є) }). Let вЉ‚ R, = в€…,
τ := supt∈R {τi (t) : i = , �, . . . , n}, ξ� ∈ R, introduce the following notations:
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 being left continuous at these points.
For J вЉ‚ R, PC(J, R) is the space of all piecewise continuous functions from J to R with
points of discontinuity of the п¬Ѓrst kind tk , at which it is left continuous.
Let φi , ϕ ∈ PC(�). Denote by xi (t) = xi (t; �, φi ), y(t) = y(t; �, ϕ), xi , y ∈ , i = , �, . . . , n the
solution of system (п›њ.п�є) satisfying the initial conditions
� ≤ xi (s; �, φi ) = φi (s) < +∞,
� ≤ y(s; �, ϕ) = ϕ(s) < +∞,
s ∈ [–τ , �],
s ∈ [–τ , �],
xi (� + �; �, φi ) = φi (�) > �;
y(� + �; �, ϕ) = ϕ(�) > �.
Remark п�є.п›њ The problems of existence, uniqueness, and continuity of the solutions of
impulsive differential equations have been investigated by many authors. Efficient sufficient conditions which guarantee the existence of the solutions of such systems are given
in [п›њп�ї, п›њп™Ђ].
Since the solution of system (.�) is a piecewise continuous function with points of discontinuity of the first kind tk , k ∈ Z we adopt the following definitions for almost periodicity.
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Let T, P в€€ I, s(T в€Є P) : I в†’ I be a map such that the set s(T в€Є P) forms a strictly increasing
sequence and if D ⊂ R and > �, θ (D) = {t + : t ∈ D}, F (D) = {θ (D) : > �}.
By П† = (П•(t), T) we denote the element from the space PC Г— I, and for every sequence of
real numbers {αn } we let θαn φ denote the sets {ϕ(t – αn ), T – αn } ⊂ PC × I, where T – αn =
{tk – αn : k ∈ Z, n = , �, . . .}.
j
j
Definition �. ([]) The set of sequences {tk }, tk = tk+j – tk , k ∈ Z, j ∈ Z, {tk } ∈ I is said
to be uniformly almost periodic if for arbitrary > � there exists a relatively dense set of
-almost periods common for any sequences.
Definition �.� ([]) The function ϕ ∈ PC(R, R) is said to be almost periodic, if the following hold:
j
j
() The set of sequences {tk }, tk = tk+j – tk , k ∈ Z, j ∈ Z, {tk } ∈ I is uniformly almost
periodic.
(�) For any > � there exists a real number δ > � such that if the points t and t belong
to one and the same interval of continuity of П•(t) and satisfy the inequality
|t – t | < δ, then |ϕ(t ) – ϕ(t )| < .
(�) For any > � there exists a relatively dense set T such that if η ∈ T, then
|ϕ(t + η) – ϕ(t)| < for all t ∈ R satisfying the condition |t – tk | > , k ∈ Z.
The elements of T are called -almost periods.
j
j
Lemma �. ([]) The set of sequences {tk }, tk = tk+j – tk , k ∈ Z, j ∈ Z, {tk } ∈ I is uniformly
almost periodic if and only if from each infinite sequence of shifts {tk –αn }, k ∈ Z, n = , �, . . . ,
О±n в€€ R, we can choose a subsequence which is convergent in I.
Definition �.� ([]) The sequence φn , φn = (ϕn (t), Tn ) ∈ PC × I is uniformly convergent
to φ, φ = (ϕ(t), T) ∈ PC × I if and only if for any > � there exists n� > � such that
ПЃ(T, Tn ) < ,
ϕn (t) – ϕ(t) <
hold uniformly for n ≥ n� and t ∈ R \ F (s(Tn ∪ T)).
Definition �.� ([]) The function φ ∈ PC is said to be an almost periodic piecewise continuous function with points of discontinuity of the first kind from the set T if for every
sequence of real numbers {О±m } there exists a subsequence {О±n } such that ОёО±n П† is compact
in PC Г— I.
Lemma п�є.п�є ([п›њп™Ђ]) Let {tk } в€€ I. Then there exists a positive integer A such that on each
interval of length п›њ, we have no more than A elements of the sequence {tk }, i.e.,
i(s, t) ≤ A(t – s) + A,
where i(s, t) is the number of the points tk in the interval (s, t).
Lemma �.� Let {tk } ∈ I. Then
i(s, t) ≥
t–s
– ,
Оё
where i(s, t) is the number of the points tk in the interval (s, t).
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Proof The proof of this lemma is easy and we omit it. This completes the proof.
3 Uniform persistence
In this section, we establish a uniform persistence result for system (п›њ.п�є).
Lemma �. ([�]) Assume that x ∈ PC(R) with points of discontinuity at t = tk and is left
continuous at t = tk for k в€€ Z+ , and
x˙ (t) ≤ f (t, x(t)), t = tk ,
x(tk+ ) ≤ Ik (x(tk )), k ∈ Z+ ,
(�.)
where f в€€ C(R Г— R, R), Ik в€€ C(R, R) and Ik (x) is nondecreasing in x for k в€€ Z+ . Let uв€— (t) be
the maximal solution of the scalar impulsive differential equation
вЋ§
вЋЄ
Л™ = f (t, u(t)), t = tk ,
вЋЁ u(t)
u(tk+ ) = Ik (u(tk )) ≥ �, k ∈ Z+ ,
вЋЄ
вЋ© +
u(t� ) = u�
(�.�)
existing on [t� , ∞). Then x(t�+ ) ≤ u� implies x(t) ≤ u∗ (t) for t ≥ t� .
Remark �. If the inequalities (�.) in Lemma �. is reversed and u∗ (t) is the minimal
solution of system (�.�) existing on [t� , ∞), then x(t�+ ) ≥ u� implies x(t) ≥ u∗ (t) for t ≥ t� .
Lemma �.� Assume that aθ > ξ l , b > �, hk > –, and x(t) > � is a solution of the following
impulsive logistic equation:
x˙ (t) = x(t)[a – bx(t)], t = tk ,
x(tk ) = hk x(tk ), k в€€ Z,
(�.�)
then
l
eξ (aθ – ξ l )
,
lim sup x(t) ≤
bОё
t→+∞
where Оѕ l := ln infkв€€Z
п›њ
п›њ+hk
.
Proof Let u = x , then system (�.�) changes to
du(t)
= –au(t) + b, t
dt
u(tk )
+
u(tk ) = п›њ+h
, k в€€ Z.
k
= tk ,
Similar to the proof in [], we can obtain from Lemma �.�
t
u(t) = W (t, �)u(�) + b
W (t, s) ds
�
=
tk ∈[�,t]
=
п›њ
e–at u(�) + b
п›њ + hk
п›њ
п›њ + hk
t
θ –
t
� t ∈[s,t]
k
t
e–at u(�) + b
�
п›њ
e–a(t–s) ds
п›њ + hk
п›њ
п›њ + hk
t–s –
Оё
e–a(t–s) ds
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t
Оѕl
l
≥ e–ξ e–(a– θ )t u(�) + b
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Оѕl
l
e–ξ e–(a– θ )(t–s) ds
�
=
Оѕl
l
l
Оѕ
l
e–ξ e–(a– θ )t u(�) +
be–ξ [ – e–(a– θ )t ]
a–
Оѕl
Оё
,
(�.�)
where
e–a(t–s) ,
W (t, s) =
k+  –a(t–s)
,
j=m п›њ+hj e
tk– < s < t < tk ;
tm– < s ≤ tm < tk < t ≤ tk+ .
Then
l
lim sup x(t) = lim sup u(t)
t→+∞
–
t→+∞
eξ (aθ – ξ l )
.
bОё
≤
This completes the proof.
Lemma �.� Assume that a > ξ u A, b > �, hk > – and x(t) > � is a solution of the following
impulsive logistic equation:
x˙ (t) = x(t)[a – bx(t)], t = tk ,
x(tk ) = hk x(tk ), k в€€ Z,
(�.�)
then
lim inf x(t) ≥
t→+∞
a – ξ uA
,
beОѕ u A
where A is defined as that in Lemma �.�, ξ u := ln supk∈Z
п›њ
.
п›њ+hk
Proof Let u = x , then system (�.�) changes to
du(t)
= –au(t) + b, t
dt
u(tk )
+
u(tk ) = п›њ+h
, k в€€ Z.
k
= tk ,
Similar to the proof as that in (�.�), we can obtain from Lemma �.�
t
W (t, s) ds
u(t) = W (t, �)u(�) + b
�
≤
tk ∈[�,t]
≤
п›њ
e–at u(�) + b
п›њ + hk
п›њ
п›њ + hk
t
� t ∈[s,t]
k
At+A
u
≤ eξ A e–(a–ξ
t
e–at u(�) + b
�
u A)t
t
u
п›њ
e–a(t–s) ds
п›њ + hk
п›њ
п›њ + hk
eξ A e–(a–ξ
u(�) + b
A(t–s)+A
e–a(t–s) ds
u A)(t–s)
�
u
u
= eξ A e–(a–ξ
u A)t
u(�) +
beξ A [ – e–(a–ξ
a – ξ uA
u A)t
]
,
ds
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which implies that
lim inf x(t) = lim inf u(t)
t→+∞
–
t→+∞
≥
a – ξ uA
.
beОѕ u A
This completes the proof.
Lemma �.� Assume that aθ > ξ l and for x(t) > �, we have
x˙ (t) ≤ x(t)[a – b� x(t) – b x(t – τ (t))],
x(tk ) ≤ hk x(tk ), k ∈ Z,
t = tk ,
(�.�)
where
a > �,
b� , b ≥ �,
b� + b > �.
Then there exists a positive constant M such that
l
lim sup x(t) ≤
t→+∞
eξ (aθ – ξ l )
:= M,
BОё
where B = b� + inft∈R b
tk ∈[t–τ (t),t) ( + hk )
– –aτ (t)
e
.
Proof From system (�.�), we have
x˙ (t) ≤ ax(t), t = tk ,
x(tk ) ≤ hk x(tk ), k ∈ Z,
is equivalent to
d
[x(t)e–at ] ≤ �,
dt
t = tk ,
x(tk ) ≤ hk x(tk ), k ∈ Z+ .
(�.�)
For some t ∈ [�, +∞) and t = tk , k ∈ Z+ , consider interval [t – τ (t), t). Assume that t < t� <
· · · < tj are the impulse points in [t – τ (t), t). Integrating the first inequality of system (�.�)
from t – τ (t) to t leads to
x(t )e–at ≤ x t – τ (t) e–a(t–τ (t)) .
Integrating the first inequality of system (�.�) from t to t� leads to
x(t� )e–at� ≤ x t+ e–at ≤ ( + h )x(t )e–at ≤ ( + h )x t – τ (t) e–a(t–τ (t)) .
Repeating the above process, integrating the first inequality of system (�.�) from tj to t
leads to
x(t)e–at ≤ x tj+ e–atj ≤ ( + hj )x(tj )e–atj ≤
( + hk )x t – τ (t) e–a(t–τ (t)) .
tk ∈[t–τ (t),t)
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Then
x t – τ (t) ≥
( + hk )– e–aτ (t) x(t).
(�.)
tk ∈[t–τ (t),t)
Substituting (�.) into system (�.�) leads to
x˙ (t) ≤ x(t)[a – Bx(t)], t = tk ,
x(tk ) ≤ hk x(tk ), k ∈ Z.
Consider the auxiliary system
вЋ§
вЋЄ
⎨ z˙ (t) = z(t)[a – Bz(t)], t = tk ,
z(tk+ ) = (п›њ + hk )z(tk ), k в€€ Z,
вЋЄ
вЋ© +
z(� ) = x(�+ ).
(�.)
By Lemma �., x(t) ≤ z(t), where z(t) is the solution of system (�.). By Lemma �.�, we
have from (�.)
l
lim sup x(t) ≤ lim sup z(t) ≤
t→+∞
t→+∞
eξ (aθ – ξ l )
.
BОё
This completes the proof.
Lemma �.� Assume that a > ξ u A, for x(t) > � and lim supt→+∞ x(t) ≤ M, we have
x˙ (t) ≥ x(t)[a – b� x(t) – b x(t – τ (t))],
x(tk ) = hk x(tk ),
t = tk ,
(�.�)
where
a > K + Оѕ u A,
b� , b ≥ �,
b := b� + b > �,
k в€€ Z.
Then there exists a positive constant N such that
lim inf x(t) ≥
t→+∞
a – ξ uA
:= N,
DeОѕ u A
where
D := b� + sup b
tв€€R
( + hk )– e–[a–bM]τ (t) .
tk ∈[t–τ (t),t)
Proof According to the assumption, for ∀  > �, there exists T > � such that
x(t) ≤ M +
п›њ
for t ≥ T .
From system (�.�), we have
x˙ (t) ≥ [a – b(M +  )]x(t) := L  x(t),
x(tk ) = hk x(tk ) + dk , k в€€ Z,
t = t k , t ≥ T ,
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Page 9 of 23
is equivalent to
d
[x(t)e–L  t ] ≥ �,
dt
t = t k , t ≥ T ,
x(tk ) = hk x(tk ) + dk , k в€€ Z.
(�.)
Similar to the arguments in (�.), we obtain
x t – τ (t) ≤
( + hk )– e–L  τ (t) x(t).
(�.�)
tk ∈[t–τ (t),t)
Let
( + hk )– e–[a–b(M+
D  := b� + sup b
tв€€R
п›њ )]П„ (t)
.
tk ∈[t–τ (t),t)
Substituting (�.�) into system (�.�) leads to
x˙ (t) ≥ x(t)[a – D  x(t)], t = tk , t ≥ T ,
x(tk ) = hk x(tk ), k в€€ Z.
Consider the auxiliary system
вЋ§
вЋЄ
⎨ z˙ (t) = z(t)[a – D  z(t)], t = tk , t ≥ T ,
z(tk+ ) = (п›њ + hk )z(tk ), k в€€ Z,
вЋЄ
вЋ©
z(Tп›њ+ ) = x(Tп›њ+ ).
(�.�)
By Remark �., x(t) ≥ z(t), where z(t) is the solution of system (�.�). By Lemma �.�, we
have from (�.�)
lim inf x(t) ≥ lim inf z(t) ≥
t→+∞
t→+∞
a – ξ uA
.
DeОѕ u A
This completes the proof.
Let
ru := max riu ,
≤i≤n
Оѕ l := ln inf
kв€€Z
al := min ali ,
huk := max hik ,
≤i≤n
п›њ
,
п›њ + huk
≤i≤n
l
Оѕn+п›њ
:= ln inf
kв€€Z
k в€€ Z,
п›њ
.
п›њ + hn+п›њ,k
Proposition �. Every solution x(t) = (x (t), x� (t), . . . , xn (t), y(t))T of system (.�) satisfies
l
lim sup xi (t) ≤ Mi :=
t→∞
eξ (ru θ – ξ l )
,
al Оё
l
lim sup y(t) ≤ Mn+ :=
l
)
eξn+ (ryu θ – ξn+
t→∞
if the following condition holds:
Bn+п›њ Оё
,
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l
(H� ) ru θ > ξ l , ryu θ > ξn+
, j = п›њ, п�є, . . . , n,
where Bn+п›њ := aln+п›њ + inftв€€R bln+п›њ
tk ∈[t–τn+ (t),t) ( + hn+,k )
u П„
– –rn+
n+п›њ (t)
e
, ryu :=
f u Mп›њ
.
п›њ+О± l Mп›њ
Proof Define V (t) = max{x (t), x� (t), . . . , xn (t)} for t ≥ �. For any t � ≥ � and t � = tk , k ∈ Z,
there must exist i ∈ {, �, . . . , n} and δ > t � small enough such that V (t � ) = xi (t � ) and xj (s) ≤
xi (s) for ∀s ∈ [t � , δ), j = i, i, j ∈ {, �, . . . , n}. Calculating the upper right derivative of V (t� )
from the positive solution for system (п›њ.п�є), we have
D+ V t � = x˙ i t � ≤ xi t � riu – ali xi t �
≤ V t � r u – al V t � .
By the arbitrariness of t � , we have
D+ V (t) ≤ V (t) r u – al V (t) ,
в€Ђt = tk , k в€€ Z.
(�.�)
Observe that xi (tk+ ) = ( + hik )xi (tk ) and  + hik > �, k ∈ Z. For arbitrary impulse point tk ,
there exists i� ∈ {, �, . . . , n} such that V (tk ) = max{x (tk ), x� (tk ), . . . , xn (tk )} = xi� (tk ), that is,
V tk+ = xi� tk+ = ( + hi� k )xi� (tk ) ≤  + huk V (tk ),
k в€€ Z.
By Lemma �.�, we obtain from (�.�)-(�.�)
lim sup xi (t) ≤ lim sup V (t) ≤ Mi ,
t→∞
i = п›њ, п�є, . . . , n.
t→∞
For any positive constant
xi (t) ≤ Mi +
п�є
> �, there exists T� > � such that
for t ≥ T� , i = , �, . . . , n.
п�є
In view of system (п›њ.п�є), it follows that
u
f (Mп›њ + п�є )
l
l
y˙ (t) ≤ y(t)[ +α
l (M + ) – an+ y(t) – bn+ y(t – τn+ (t))],
п›њ
п�є
y(tk ) = hn+п›њ,k y(tk ),
t = tk ,
k в€€ Z,
which implies from Lemma �.� that
lim sup y(t) ≤ Mn+ .
t→∞
This completes the proof.
Define
Оѕiu := ln sup
kв€€Z
п›њ
,
п›њ + hik
i = п›њ, п�є, . . . , n, n + п›њ.
Proposition �.� Assume that the following condition (H� ) holds:
n
p := rl –
�
Dui –
i=п�є
–σ
ku (s) dsM – cu Mn+ ≥ ξu A,
(�.�)
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n
pi := ril –
�
Duji –
j=п›њ
u
pn+ := –rn+
+
–σi
Page 11 of 23
kiu (s) dsMi ≥ ξiu A,
�
f l Nп›њ
–
п›њ + О± u Nп›њ
–σn+
i = п�є, . . . , n,
u
u
kn+п›њ
(s) dsMn+ ≥ ξn+
A,
then every solution x(t) = (x (t), x� (t), . . . , xn (t), y(t))T of system (.�) satisfies
lim inf xi (t) ≥ Ni :=
pi – ξiu A
u
Qi eОѕi A
t→+∞
lim inf y(t) ≥ Nn+ :=
t→+∞
,
u
A
pn+ – ξn+
u
Qn+п›њ eОѕn+п›њ A
,
where
u
tв€€R
Proof For в€Ђ
�
u
( + hik )– e–[pi –(ai +bi )Mi ]τi (t) ,
Qi := aui + sup bui
i = п›њ, п�є, . . . , n + п›њ.
tk ∈[t–τi (t),t)
> �, there exists T� > � such that
xi (t) ≤ Mi +
�,
y(t) ≤ Mn+ +
�
for t ≥ T� , i = , �, . . . , n.
From system (.�), for t ≥ T� , we have
вЋ§
вЋЄ
x˙  (t) ≥ x (t)[rl – ni=� Dui – au x (t) – bu x (t – τ (t))
вЋЄ
вЋЄ
вЋЄ
�
вЋЄ
– –σ ku (s) ds(M + � ) – cu (Mn+ + � )],
вЋЄ
вЋЁ
x˙ i (t) ≥ xi (t)[ril – nj= Duji – aui xi (t) – bui xi (t – τi (t))
вЋЄ
вЋЄ
�
вЋЄ
– –σi kiu (s) ds(Mi + � )], i = �, �, . . . , n, t = tk ,
вЋЄ
вЋЄ
вЋЄ
вЋ©
xi (tk ) = hik xi (tk ), k в€€ Z.
By Lemma �.� and the arbitrariness of
lim inf xi (t) ≥ Ni ,
п�ј
we have
i = п›њ, п�є, . . . , n.
t→+∞
Then for в€Ђ
�,
> �, there exists T� > � such that
x (t) ≥ N –
п�ј,
y(t) ≤ Mn+ +
п�ј
for t ≥ T� .
From system (.�), for t ≥ T� , we have
вЋ§
f (t)(N – � )
u
u
u
вЋЄ
⎨ y˙ (t) ≥ y(t)[–rn+ + +α(t)(N – � ) – an+ y(t) – bn+ y(t – τn+ (t))
�
u
– –σn+ kn+
(s) ds(Mn+п›њ + п�ј )], t = tk ,
вЋЄ
вЋ©
y(tk ) = hn+п›њ,k y(tk ), k в€€ Z.
By Lemma �.� and the arbitrariness of
lim inf y(t) ≥ Nn+ .
t→+∞
This completes the proof.
п�ј,
we have
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Remark �.� When hik (i = , �, . . . , n + , k ∈ Z) ≡ � in system (.�), then Propositions �.
and �.� improve the corresponding results in [�]. So Propositions �. and �.� extend and
improve the corresponding results in [п›њп�ѕ].
Remark �.� In view of Propositions �. and �.�, the distance θ between impulse points,
the values of impulse coefficients hik (i = , �, . . . , n + , k ∈ Z) and the number A of the
impulse points in each interval of length  have negative effect on the uniform persistence
of system (п›њ.п�є).
By Propositions �. and �.�, we have:
Theorem �. Assume that (H )-(H� ) hold, then system (.�) is uniformly persistent.
Remark �.� Theorem �. gives the sufficient conditions for the uniform persistence of
system (.�). Therefore, Theorem �. provides a possible method to study the permanence of the models with almost periodic impulsive perturbations in biological populations.
4 Global asymptotical stability
The main result of this section concerns the global asymptotical stability of positive solution of system (п›њ.п�є).
Theorem �. Assume that (H )-(H� ) hold. Suppose further that
(Hп™Ђ ) there exist positive constants О»i such that
inf λ a (t) –
tв€€R
–
λ b (δ– (t))
– λ
 – τ˙ (δ– (t))
О±(t)c(t)Mn+п›њ
–
[п›њ + О±(t)Nп›њ ]п�є
inf λi ai (t) –
tв€€R
n
–
j=п›њ
n
j=п›њ
О»j Dij (t)
О»n+п›њ f (t)
–
Nj
п›њ + О±(t)Nп›њ
tв€€R
kn+ (t – s, s) ds –
–σn+
> �,
�
ki (t – s, s) ds
–σi
> �,
–
О»n+п›њ bn+п›њ (Оґn+п›њ
(t))
–
 – τ˙n+ (δn+ (t))
�
– λn+
k (t – s, s) ds
–σ
О»п›њ Djп›њ (t)
О»n+п›њ f (t)
–
Nп›њ
п›њ + О±(t)Nп›њ
λi bi (δi– (t))
– λi
 – τ˙i (δi– (t))
inf λn+ an+ (t) –
�
c(t)
> �,
п›њ + О±(t)Nп›њ
where δj– is an inverse function of τj , i = �, . . . , n, j = , �, . . . , n + .
Then system (п›њ.п�є) is globally asymptotically stable.
Proof Suppose that X(t) = (xп›њ (t), . . . , xn (t), y(t))T and X в€— (t) = (xв€—п›њ (t), . . . , xв€—n (t), yв€— (t))T are any
two solutions of system (п›њ.п�є).
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By Theorem �. and (H ), for
� < Ni –
п�Ѕ
≤ xi (t) ≤ Mi +
tв€€R
О±(t)c(t)(Mn+п›њ + п�Ѕ )
–
[ + α(t)(N – � )]�
inf λi ai (t) –
tв€€R
n
–
j=п›њ
п�Ѕ,
� < Nn+ –
n
j=п›њ
п�Ѕ )]
>
for t ≥ T� ,
п�Ѕ
k (t – s, s) ds
–σ
п�Ѕ )]
>
,
�
ki (t – s, s) ds
–σi
п�Ѕ )]
>
–
О»n+п›њ bn+п›њ (Оґn+п›њ
(t))
– λn+
–
 – τ˙n+ (δn+ (t))
c(t)
[ + α(t)(N –
≤ y(t) ≤ Mn+ +
> � such that
�
О»j Dij (t)
О»n+п›њ f (t)
–
Nj – � [ + α(t)(N –
tв€€R
п�Ѕ
О»п›њ Djп›њ (t)
О»n+п›њ f (t)
–
N – � [ + α(t)(N –
λi bi (δi– (t))
– λi
 – τ˙i (δi– (t))
inf λn+ an+ (t) –
–
> � small enough, there exist T� > � and
λ b (δ– (t))
– λ
 – τ˙ (δ– (t))
inf λ a (t) –
–
п�Ѕ
Page 13 of 23
,
�
kn+ (t – s, s) ds
–σn+
,
where i = �, �, . . . , n.
Construct a Lyapunov functional as follows:
V (t) = V (t) + V� (t) + V� (t),
∀t ≥ T� ,
where
n
λi ln xi (t) – ln x∗i (t) + λn+ ln y(t) – ln y∗ (t) ,
Vп›њ (t) =
i=п›њ
n
t
V� (t) =
t–τi (t)
i=п›њ
t
+
t–τn+ (t)
n
V� (t) =
λi bi (δi– (s))
xi (s) – x∗i (s) ds
 – τ˙i (δi– (s))
–
О»n+п›њ bn+п›њ (Оґn+п›њ
(s))
y(s) – y∗ (s) ds,
–
 – τ˙n+ (δn+ (s))
�
t
–σi
t+s
О»i
i=п›њ
�
ki (l – s, s) xi (l) – x∗i (l) dl ds
t
+ О»n+п›њ
–σn+
kn+ (l – s, s) y(l) – y∗ (l) dl ds.
t+s
For t = tk , k в€€ Z, calculating the upper right derivative of Vп›њ (t) along the solution of
system (п›њ.п�є), it follows that
n
D+ Vп›њ (t) =
О»i
i=п›њ
+ О»n+п›њ
xЛ™ i (t) xЛ™ в€—i (t)
–
sgn xi (t) – x∗i (t)
xi (t) xв€—i (t)
yЛ™ (t) yЛ™ в€— (t)
sgn y(t) – y∗ (t)
–
y(t) yв€— (t)
n
λi –ai (t) xi (t) – x∗i (t) + bi (t) xi t – τi (t) – x∗i t – τi (t)
≤
i=п›њ
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�
Page 14 of 23
ki (t, s) xi (t + s) – x∗i (t + s) ds
+
–σi
n
+ λ sgn x (t) – x∗ (t)
Djп›њ (t)
[xj (t)x∗ (t) – x (t)x∗j (t)]
xп›њ (t)xв€—п›њ (t)
j=п�є
n
n
λi sgn xi (t) – x∗i (t)
+
i=п�є
Dji (t)
[xj (t)x∗i (t) – xi (t)x∗j (t)]
xi (t)xв€—i (t)
j=п›њ
+ λ sgn x (t) – x∗ (t) –
c(t)yв€— (t)
c(t)y(t)
+
п›њ + О±(t)xп›њ (t) п›њ + О±(t)xв€—п›њ (t)
f (t)xв€—п›њ (t)
f (t)xп›њ (t)
–
п›њ + О±(t)xп›њ (t) п›њ + О±(t)xв€—п›њ (t)
+ О»n+п›њ
– λn+ an+ (t) y(t) – y∗ (t) + λn+ bn+ (t) y t – τn+ (t) – y∗ t – τn+ (t)
�
kn+ (t, s) y(t + s) – y∗ (t + s) ds
+ О»n+п›њ
–σn+
n
n
λi ai (t) xi (t) – x∗i (t) +
≤–
i=п›њ
i=п›њ
n
+
λi bi (t) xi t – τi (t) – x∗i t – τi (t)
�
О»i
–σi
i=п›њ
n
ki (t, s) xi (t + s) – x∗i (t + s) ds
О»п›њ Djп›њ (t)
x (t) – x∗ (t) +
N – �
+
j=п›њ
n
n
i=п�є j=п›њ
О»j Dij (t)
xi (t) – x∗i (t)
Nj – �
+
О±(t)c(t)(Mn+п›њ + п�Ѕ )
c(t)
x (t) – x∗ (t) +
[ + α(t)(N – � )]�
[ + α(t)(N –
+
О»n+п›њ f (t)
[ + α(t)(N –
п�Ѕ )]
п�Ѕ )]
y(t) – y∗ (t)
x (t) – x∗ (t)
– λn+ an+ (t) y(t) – y∗ (t) + λn+ bn+ (t) y t – τn+ (t) – y∗ t – τn+ (t)
�
+ О»n+п›њ
kn+ (t, s) y(t + s) – y∗ (t + s) ds.
(п�ј.п›њ)
–σn+
Here we use the following inequality which has been proved in [п›њп�ѕ]:
n
sgn xi (t) – x∗i (t)
Dji (t)
j=п›њ
[xj (t)x∗i (t) – xi (t)x∗j (t)]
xi (t)xв€—i (t)
n
≤
j=п›њ
Dji (t)
xj (t) – x∗j (t) .
Ni – �
Moreover, we obtain
n
D+ V� (t) =
i=п›њ
–
О»n+п›њ bn+п›њ (Оґn+п›њ
λi bi (δi– (t))
(t))
в€—
+
(t)
–
x
(t)
x
y(t) – y∗ (t)
i
i
–
–
 – τ˙i (δi (t))
 – τ˙n+ (δn+ (t))
n
λi bi (t) xi t – τi (t) – x∗i t – τi (t)
–
i=п›њ
– λn+ bn+ (t) y t – τn+ (t) – y∗ t – τn+ (t) ,
n
D+ V� (t) =
�
О»i
i=п›њ
–σi
ki (t – s, s) xi (t) – x∗i (t) ds
(п�ј.п�є)
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�
Page 15 of 23
kn+ (t – s, s) y(t) – y∗ (t) ds
+ О»n+п›њ
–σn+
n
�
–
О»i
–σi
i=п›њ
�
– λn+
ki (t, s) xi (t + s) – x∗i (t + s) ds
kn+ (t, s) y(t + s) – y∗ (t + s) ds.
(�.�)
–σn+
From (�.)-(�.�), one has
D+ V (t) ≤ – λ a (t) –
–
λ b (δ– (t))
– λ
 – τ˙ (δ– (t))
О±(t)c(t)(Mn+п›њ + п�Ѕ )
–
[ + α(t)(N – � )]�
n
–
λi ai (t) –
i=п�є
n
j=п›њ
j=п›њ
– λn+ an+ (t) –
c(t)
[ + α(t)(N –
k (t – s, s) ds
–σ
О»п›њ Djп›њ (t)
О»n+п›њ f (t)
–
N – � [ + α(t)(N –
λi bi (δi– (t))
– λi
 – τ˙i (δi– (t))
О»j Dij (t)
О»n+п›њ f (t)
–
Nj – � [ + α(t)(N –
–
–
n
�
п�Ѕ )]
x (t) – x∗ (t)
�
ki (t – s, s) ds
–σi
xi (t) – x∗i (t)
–
О»n+п›њ bn+п›њ (Оґn+п›њ
(t))
– λn+
–
 – τ˙n+ (δn+ (t))
п�Ѕ )]
п�Ѕ )]
�
kn+ (t – s, s) ds
–σn+
y(t) – y∗ (t)
n
xi (t) – x∗i (t) + y(t) – y∗ (t) .
≤–
(п�ј.п�ј)
i=п›њ
For t = tk , k в€€ Z, we have
V tk+ = V tk+ + V� tk+ + V� tk+
n
λi ln xi tk+ – ln x∗i tk+ + λn+ ln y tk+ – ln y∗ tk+ + V� (tk ) + V� (tk )
=
i=п›њ
n
=
О»i ln
i=п›њ
(п›њ + hik )xi (tk ) + dik
(п›њ + hik )xв€—i (tk ) + dik
+ О»n+п›њ ln
(п›њ + hn+п›њ,k )y(tk ) + dn+п›њ,k
+ V� (tk ) + V� (tk )
(п›њ + hn+п›њ,k )yв€— (tk ) + dn+п›њ,k
= V (tk ) + V� (tk ) + V� (tk )
= V (tk ).
Therefore, V is nonincreasing. Integrating (�.�) from T� to t leads to
t
n
T�
i=п›њ
xi (s) – x∗i (s) + y(s) – y∗ (s)
V (t) +
ds ≤ V (T� ) < +∞,
∀t ≥ T� ,
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that is,
+в€ћ
T�
n
xi (s) – x∗i (s) + y(s) – y∗ (s)
ds < +в€ћ,
i=п›њ
which implies that
lim xi (s) – x∗i (s) = �,
s→+∞
lim y(s) – y∗ (s) = �,
s→+∞
i = п›њ, п�є, . . . , n.
Thus, system (п›њ.п�є) is globally asymptotically stable. This completes the proof.
Remark �. Theorem �. gives a sufficient condition for the global asymptotical stability of system (.�). Therefore, Theorem �. extends the corresponding result in [�] and
provides a possible method to study the global asymptotical stability of the models with
impulsive perturbations in biological populations.
5 Almost periodic solution
In this section, we investigate the existence and uniqueness of a globally asymptotically
stable positive almost periodic solution of system (п›њ.п�є) by using almost periodic functional
hull theory of impulsive differential equations.
Let {sn } be any integer valued sequence such that sn в†’ в€ћ as n в†’ в€ћ. Taking a subsequence if necessary, we have ri (t + sn ) в†’ riв€— (t), ai (t + sn ) в†’ aв€—i (t), bi (t + sn ) в†’ bв€—i (t),
c(t + sn ) в†’ cв€— (t), f (t + sn ) в†’ f в€— (t), О±(t + sn ) в†’ О± в€— (t), Dij (t + sn ) в†’ Dв€—ij (t), П„i (t + sn ) в†’ П„iв€— (t),
ki (t + sn ) → ki∗ (t, s), as n → ∞ for t ∈ R, s ∈ (–∞, �], i = , �, . . . , n + , j = , �, . . . , n. From
Lemma �. it follows that the set of sequences {tk – sn }, k ∈ Z is convergent to the sequence
{tks } uniformly with respect to k в€€ Z as n в†’ в€ћ.
By {kni } we denote the sequence of integers such that the subsequence {tkni } is convergent to the sequence {tks } uniformly with respect to k в€€ Z as i в†’ в€ћ.
From the almost periodicity of {hik }, it follows that there exists a subsequence of the sequence {kni } such that the sequences {hikni } are convergent uniformly to the limits denoted
by hsik , i = п›њ, п�є, . . . , n + п›њ.
Then we get hull equations of system (п›њ.п�є) as follows:
вЋ§
в€—
в€—
в€—
в€—
вЋЄ
 (t – τ (t))
⎪ x˙  (t) = x (t)[r (t) – a (t)x (t) – b (t)x
вЋЄ
в€— (t)y(t)
�
c
вЋЄ
в€—
вЋЄ
– –σ k (t, s)x (t + s) ds – +α∗ (t)x (t) ] + ni=� D∗i (t)[xi (t) – x (t)],
вЋЄ
вЋЄ
вЋЄ
вЋЄ
вЋЄ
x˙ i (t) = xi (t)[ri∗ (t) – a∗i (t)xi (t) – b∗i (t)xi (t – τi∗ (t))
вЋЄ
вЋЄ
вЋЄ
�
вЋЁ
– –σi ki∗ (t, s)xi (t + s) ds] + nj= D∗ji (t)[xj (t) – xi (t)], i = �, �, . . . , n,
f в€— (t)xп›њ (t)
в€—
в€—
в€—
в€—
вЋЄ
⎪ y˙ (t) = y(t)[–rn+ (t) + +α∗ (t)x (t) – an+ (t)y(t) – bn+ (t)y(t – τn+ (t))
вЋЄ
вЋЄ
�
вЋЄ
в€—
вЋЄ
(t, s)y(t + s) ds], t = tks ,
– –σn+ kn+
вЋЄ
вЋЄ
вЋЄ
вЋЄ
s
s
s
вЋЄ
xj (tk ) = hjk xj (tk ), j = п›њ, п�є, . . . , n,
вЋЄ
вЋЄ
вЋ©
y(tks ) = hsn+п›њ,k y(tks ), k в€€ Z.
(п�Ѕ.п›њ)
By the almost periodic theory, we can conclude that if system (.�) satisfies (H )-(H ), then
the hull equations (п�Ѕ.п›њ) of system (п›њ.п�є) also satisfy (Hп›њ )-(Hп™Ђ ).
By Lemma п�ј.п›њп�Ѕ in [п›њп™Ђ], we can easily obtain the lemma as follows.
Lemma п�Ѕ.п›њ If each hull equation of system (п›њ.п�є) has a unique strictly positive solution, then
system (п›њ.п�є) has a unique strictly positive almost periodic solution.
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By using Lemma п�Ѕ.п›њ, we obtain the following result.
Lemma �.� If system (.�) satisfies (H )-(H ), then system (.�) admits a unique strictly
positive almost periodic solution.
Proof By Lemma п�Ѕ.п›њ, in order to prove the existence of a unique strictly positive almost
periodic solution of system (п›њ.п�є), we only need to prove that each hull equation of system
(п›њ.п�є) has a unique strictly positive solution.
Firstly, we prove the existence of a strictly positive solution of any hull equations (п�Ѕ.п›њ).
According to the almost periodic hull theory of impulsive differential equations (see []),
there exists a time sequence {sn } with sn в†’ в€ћ as n в†’ +в€ћ such that ri (t + sn ) в†’ riв€— (t),
ai (t + sn ) в†’ aв€—i (t), bi (t + sn ) в†’ bв€—i (t), c(t + sn ) в†’ cв€— (t), f (t + sn ) в†’ f в€— (t), О±(t + sn ) в†’ О± в€— (t),
Dij (t + sn ) в†’ Dв€—ij (t), П„i (t + sn ) в†’ П„iв€— (t), ki (t + sn ) в†’ kiв€— (t, s), as n в†’ в€ћ for t в€€ R, t = tk ,
k ∈ Z, s ∈ (–∞, �], i = , �, . . . , n + , j = , �, . . . , n. There exists a subsequence {kn } of
{n}, kn в†’ +в€ћ, n в†’ +в€ћ such that tkn в†’ tks , hikn в†’ hsik , i = п›њ, п�є, . . . , n + п›њ. Suppose x(t) =
(x (t), x� (t), . . . , xn (t), y(t))T is any positive solution of hull equations (�.). By the proof of
Theorem �., for ∀ > �, there exists T� > � such that
Ni – ≤ xi (t) ≤ Mi + ,
Nn+ – ≤ y(t) ≤ Mn+ + ,
t ≥ T� , i = , �, . . . , n.
(п�Ѕ.п�є)
Let xn (t) = x(t + sn ) for all t ≥ –sn + T� , n = , �, . . . , such that
вЋ§
в€—
в€—
в€—
в€—
вЋЄ
вЋЄ
 (t + sn ) – a (t + sn )x (t) – b (t + sn )x (t – τ (t + sn ))
вЋЄ xЛ™ п›њ (t) = xп›њ (t)[r
в€—
вЋЄ
�
c (t+sn )y(t)
вЋЄ
вЋЄ
– –σ k∗ (t + sn , s)x (t + s) ds – +α
в€— (t+s )x (t) ]
вЋЄ
n п›њ
вЋЄ
вЋЄ
n
вЋЄ
в€—
вЋЄ
+ i=� Di (t + sn )[xi (t) – x (t)],
вЋЄ
вЋЄ
вЋЄ
вЋЄ
вЋЄ
x˙ (t) = xi (t)[ri∗ (t + sn ) – a∗i (t + sn )xi (t) – b∗i (t + sn )xi (t – τi∗ (t + sn ))
вЋЄ
вЋЄ i
вЋЄ
�
вЋЄ
– –σi ki∗ (t + sn , s)xi (t + s) ds]
вЋЄ
вЋЁ
+ nj= D∗ji (t + sn )[xj (t) – xi (t)], i = �, �, . . . , n,
вЋЄ
вЋЄ
f в€— (t+sn )xп›њ (t)
в€—
в€—
вЋЄ
(t + sn ) + п›њ+О±
y˙ (t) = y(t)[–rn+
вЋЄ
∗ (t+s )x (t) – an+ (t + sn )y(t)
вЋЄ
n п›њ
вЋЄ
вЋЄ
в€—
вЋЄ
(t + sn ))
– b∗n+ (t + sn )y(t – τn+
вЋЄ
вЋЄ
вЋЄ
�
вЋЄ
в€—
вЋЄ
–
k
(t
+
s
,
s)y(t
+ s) ds], t = tks ,
вЋЄ
n
–σn+ n+
вЋЄ
вЋЄ
вЋЄ x (t s ) = hs x (t s ), j = п›њ, п�є, . . . , n,
вЋЄ
вЋЄ
j k
jk j k
вЋЄ
вЋЄ
вЋ© y(t s ) = hs y(t s ), k в€€ Z.
k
n+п›њ,k
k
(�.�)
From the inequality (п�Ѕ.п�є), there exists a positive constant K which is independent of n
such that |˙xn | ≤ K for all t ≥ –sn + T� , n = , �, . . . . Therefore, for any positive integer
r sequence {xn (t) : n ≥ r} is uniformly bounded and equicontinuous on [–sn + T� , ∞).
According to Ascoli-Arzela theorem, one can conclude that there exists a subsequence
{sm } of {sn } such that sequence {xm (t)} not only converges on t on R, but it also converges uniformly on any compact set of R as m → +∞. Suppose limm→+∞ xm (t) = x∗ (t) =
(xв€—п›њ (t), xв€—п�є (t), . . . , xв€—n (t), y(t))T , then we have
Ni – ≤ xi (t) ≤ Mi + ,
Nn+ – ≤ y(t) ≤ Mn+ + ,
t в€€ R, i = п›њ, п�є, . . . , n.
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From differential equations (�.�) and the arbitrariness of , we can easily see that x∗ (t)
is the solution of the hull equations (�.) and Ni ≤ x∗i (t) ≤ Mi for all t ∈ R, i = , �, . . . , n.
Hence each hull equation of the almost periodic system (п›њ.п�є) has at least a strictly positive
solution.
Now we prove the uniqueness of the strictly positive solution of each hull equations
(п�Ѕ.п›њ). Suppose that the hull equations (п�Ѕ.п›њ) have two arbitrary strictly positive solutions
x(t) = (x (t), x� (t), . . . , xn (t), y(t))T and x∗ (t) = (x∗ (t), x∗� (t), . . . , x∗n (t), y∗ (t))T , which satisfy
Ni – ≤ xi (t), x∗i (t) ≤ Mi + ,
Nn+ – ≤ y(t), y∗ (t) ≤ Mn+ + ,
t в€€ R, i = п›њ, п�є, . . . , n.
Similar to Theorem �., we define a Lyapunov functional
V ∗ (t) = V∗ (t) + V�∗ (t) + V�∗ (t),
в€Ђt в€€ R,
where
n
Vп›њв€— (t) =
λi ln xi (t) – ln x∗i (t) + λn+ ln y(t) – ln y∗ (t) ,
i=п›њ
n
t
V�∗ (t) =
t–τi (t)
i=п›њ
t
+
в€— (t)
t–τn+
n
V�∗ (t) =
λi b∗i (δi∗– (s))
xi (s) – x∗i (s) ds
 – τ˙∗ i (δi∗– (s))
∗–
О»n+п›њ bв€—n+п›њ (Оґn+п›њ
(s))
y(s) – y∗ (s) ds,
∗–
в€—
Л™
 – τ n+ (δn+ (s))
�
t
–σi
t+s
О»i
i=п›њ
�
ki∗ (l – s, s) xi (l) – x∗i (l) dl ds
t
+ О»n+п›њ
–σn+
t+s
в€—
kn+п›њ
(l – s, s) y(l) – y∗ (l) dl ds,
where δj∗– is an inverse function of τj∗ , j = , �, . . . , n + . Similar to the argument in (�.�),
one has
n
D+ V ∗ (t) ≤ –
xi (t) – x∗i (t) + y(t) – y∗ (t) ,
в€Ђt в€€ R.
i=п›њ
Summing both sides of the above inequality from t to �, we have
n
�
t
xi (s) – x∗i (s) + y(s) – y∗ (s)
ds ≤ V ∗ (t) – V (�),
i=п›њ
Note that V в€— is bounded. Hence we have
�
n
–∞
i=п›њ
xi (s) – x∗i (s) + y(s) – y∗ (s)
ds < в€ћ,
∀t ≤ �.
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which implies that
n
xi (s) – x∗i (s) = lim y(s) – y∗ (s) = �,
lim
s→–∞
s→–∞
i=п›њ
For arbitrary
�
i = п›њ, п�є, . . . , n.
> �, there exists a positive constant L such that
max xi (t) – x∗i (t) , y(t) – y∗ (t)
<
�,
∀t < –L, i = , �, . . . , n.
Hence, one has
n+п›њ
V∗ (t) ≤
i=п›њ
λi �
,
Ni
n+п›њ
V�∗ (t) ≤
П„iu
i=п›њ
О»i bui
�,
 – supt∈R τ˙i (t)
n+п›њ
V�∗ (t) ≤
∀t < –L,
�
О»i
–σi
i=п›њ
(–s)kiu (s) ds � ,
∀t < –L,
∀t < –L,
which imply that there exists a positive constant ПЃ such that
V ∗ (t) < ρ � ,
∀t < –L.
So
lim V ∗ (t) = �.
t→–∞
Note that V ∗ (t) is a nonincreasing function on R, and then V ∗ (t) ≡ �. That is,
xi (t) = xв€—i (t),
y(t) = yв€— (t),
в€Ђt в€€ R, i = п›њ, п�є, . . . , n.
Therefore, each hull equation of system (п›њ.п�є) has a unique strictly positive solution.
In view of the above discussion, any hull equation of system (п›њ.п�є) has a unique strictly
positive solution. By Lemma п�Ѕ.п›њ, system (п›њ.п�є) has a unique strictly positive almost periodic
solution. The proof is completed.
By Theorem п�ј.п›њ and Lemma п�Ѕ.п�є, we obtain the following.
Theorem п�Ѕ.п›њ Suppose that (Hп›њ )-(Hп™Ђ ) hold, then system (п›њ.п�є) admits a unique strictly positive almost periodic solution, which is globally asymptotically stable.
Remark �. Theorem �. gives sufficient condition for the global asymptotical stability of
a unique positive almost periodic solution of system (п›њ.п�є). Therefore, Theorem п�Ѕ.п›њ extends
the corresponding result in [п›њп�ѕ] and provides a possible method to study the existence,
uniqueness, and stability of positive almost periodic solution of the models with impulsive
perturbations in biological populations.
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6 An example and numerical simulations
Example п�ѕ.п›њ Consider the following Lotka-Volterra type predator-prey dispersal system
with impulsive effects:
вЋ§
в€љ
вЋЄ
x˙  (t) = x (t)[� – (� + sin( �t))x (t) – �.x (t – ) – �.��y(t)
]
вЋЄ
п›њ+xп›њ (t)
вЋЄ
вЋЄ
вЋЄ
вЋЄ
+
�.�[x
(t)
–
x
(t)],
п�є
п›њ
вЋЄ
вЋЄ
в€љ
вЋЄ
�
вЋЄ
вЋЄ
�t) – �x� (t) – –�. x� (t + s) ds]
⎨ x˙ � (t) = x� (t)[ + cos(
в€љ
+ �. cos( �t)[x (t) – x� (t)],
вЋЄ
в€љ
вЋЄ
п�єxп›њ (t)
вЋЄ
вЋЄ
⎪ y˙ (t) = y(t)[–�.�| cos( �t)| + +x (t) – �y(t)], t = tk ,
вЋЄ
вЋЄ
вЋЄ
xi (tk ) = –�.�xi (tk ), i = , �,
вЋЄ
вЋЄ
вЋЄ
⎩ y(t ) = –�.�y(t ), {t : k ∈ Z} ⊂ {�k : k ∈ Z}.
k
k
k
(п�ѕ.п›њ)
Then system (п�ѕ.п›њ) is uniformly persistent and has a unique globally asymptotically stable
almost periodic solution.
Proof Corresponding to system (.�), we have ru θ – ξ l = � – �.� > � and ryu θ – ξ�l = .� ×
� – �.� > �. Then (H� ) in Proposition �. holds. By calculation, we obtain M = M� ≈ �.�,
M� ≈ .��. Further, p = � – �.� – �.�� × .�� ≥ �.� = ξu A, N ≈ �., p� = � – �. – �. ×
�.� ≥ �.� = ξ�u A, p� = –�.� + �×�.
≥ �.� = ξ�u A, which imply that (H� ) in Proposition �.�
+�.
holds. Obviously, (H )-(H� ) in Theorem �. hold and system (�.) is uniformly persistent
(see Figure п›њ).
Taking λ = λ� = , λ� = �., corresponding to system (.�), we get
inf λ a (t) –
tв€€R
–
λ b (δ– (t))
– λ
 – τ˙ (δ– (t))
О±(t)c(t)Mn+п›њ
–
[п›њ + О±(t)Nп›њ ]п�є
≥ � – �. – � –
n
j=п›њ
�
k (t – s, s) ds
–σ
О»п›њ Djп›њ (t)
О»n+п›њ f (t)
–
Nп›њ
п›њ + О±(t)Nп›њ
�.�
�.�
�.�� × .��
–
> �,
–
( + �.)� �.  + �.
Figure 1 Uniform persistence and almost periodic oscillation of system (6.1).
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Figure 2 Global asymptotical stability of state variable x1 of system (6.1).
Figure 3 Global asymptotical stability of state variable x2 of system (6.1).
inf λi ai (t) –
tв€€R
λi bi (δi– (t))
– λi
 – τ˙i (δi– (t))
≥ � – � – �. –
inf λn+ an+ (t) –
tв€€R
≥ �.� – � – � –
n
�
ki (t – s, s) ds –
–σi
j=п›њ
О»j Dij (t)
О»n+п›њ f (t)
–
Nj
п›њ + О±(t)Nп›њ
�.�
�.
–
> �,
�.  + �.
–
О»n+п›њ bn+п›њ (Оґn+п›њ
(t))
– λn+
–
 – τ˙n+ (δn+ (t))
�
kn+ (t – s, s) ds –
–σn+
c(t)
п›њ + О±(t)Nп›њ
�.��
> �.
 + �.
Hence (H ) in Theorem �. is satisfied. By Theorem �., system (�.) has a unique globally asymptotically stable almost periodic solution (see Figures �-�). This completes the
proof.
7 Conclusion
By applying the comparison theorem, the Lyapunov functional, and almost periodic functional hull theorem of the impulsive differential equations, this paper gives some new sufficient conditions for the uniform persistence, global asymptotical stability, and almost
Xu and Wu Advances in Difference Equations 2014, 2014:264
http://www.advancesindifferenceequations.com/content/2014/1/264
Figure 4 Global asymptotical stability of state variable y of system (6.1).
periodic solution to a nonautonomous dispersal competition system with impulsive effects. Theorem �. and Theorem �. indicate that the distance θ between impulse points,
the values of the impulse coefficients hik (i = , �, . . . , n, k ∈ Z), and the number A of the
impulse points in each interval of length п›њ are harmful for the uniform persistence and existence of a unique globally asymptotically stable positive almost periodic solution for the
model. The main results obtained in this paper are completely new and the method used
in this paper provides a possible method to study the uniform persistence and existence
of a unique globally asymptotically stable positive almost periodic solution of the models
with impulsive perturbations in biological populations.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors contributed to each part of this study equally and read and approved the п¬Ѓnal version of the
manuscript.
Author details
1
School of Mathematics and Computer Science, Panzhihua University, Panzhihua, Sichuan 617000, China. 2 Department
of Mathematics, Aba Teachers College, Wenchuan, Sichuan 623002, China.
Acknowledgements
The authors thank the reviewer for his constructive remarks that led to the improvement of the original manuscript.
Received: 14 February 2014 Accepted: 26 September 2014 Published: 14 Oct 2014
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Cite this article as: Xu and Wu: Dynamics of a nonautonomous Lotka-Volterra predator-prey dispersal system with
impulsive effects. Advances in Difference Equations 2014, 2014:264
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