Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 281052, 15 pages
doi:10.1155/2012/281052
Research Article
Dynamic Inequalities on
Time Scales with Applications in Permanence of
Predator-Prey System
Meng Hu and Lili Wang
Department of Mathematics, Anyang Normal University, Anyang, Henan 455000, China
Correspondence should be addressed to Meng Hu, humeng2001@126.com
Received 15 February 2012; Revised 1 April 2012; Accepted 9 April 2012
Academic Editor: Vimal Singh
Copyright q 2012 M. Hu and L. Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
By using the theory of calculus on time scales and some mathematical methods, several dynamic
inequalities on time scales are established. Based on these results, we derive some sufficient
conditions for permanence of predator-prey system incorporating a prey refuge on time scales.
Finally, examples and numerical simulations are presented to illustrate the feasibility and
effectiveness of the results.
1. Introduction
An important and ubiquitous problem in predator-prey theory and related topics in
mathematical ecology concerns the long-term coexistence of species. In the past few years,
permanence of different classes of continuous or discrete ecosystem has been studied wildly
both in theories and applications; we refer the readers to 1–6 and the references therein.
However, in the natural world, there are many species whose developing processes
are both continuous and discrete. Hence, using the only differential equation or difference
equation cannot accurately describe the law of their developments. Therefore, there is a need
to establish correspondent dynamic models on new time scales.
The theory of calculus on time scales see 7 and references cited therein was initiated
by Stefan Hilger in his Ph.D. thesis in 1988 8 in order to unify continuous and discrete
analysis, and it has a tremendous potential for applications and has recently received much
attention since his foundational work; one may see 9–15. Therefore, it is practicable to study
that on time scales which can unify the continuous and discrete situations. However, to the
best of the authors’ knowledge, there are few papers considered permanence of predator-prey
system on time scales.
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Discrete Dynamics in Nature and Society
Motivated by the previous, in this paper, we first establish some dynamic inequalities
on time scales by using the theory of calculus on time scales and some mathematical
methods, then, based on these results, as an application, we will study the permanence of the
following delayed predator-prey system incorporating a prey refuge with Michaelis-Menten
and Beddington-DeAngelis functional response on time scales:
xΔ t xt bt − ctxδ− τ1 , t −
dtyt1 − m
atyt xt1 − m
htzt1 − m
−
,
kt ptxt1 − m ntzt
ftxδ− τ2 , t1 − m
yΔ t yt −d1 t ,
atyδ− τ2 , t xδ− τ2 , t1 − m
gtxδ− τ3 , t1 − m
Δ
z t zt −d2 t ,
kt ptxδ− τ3 , t1 − m ntzδ− τ3 , t
1.1
where t ∈ T, T is a time scale. xt denotes the density of prey specie and yt and zt denote
the density of two predators species. at, bt, ct, dt, ft, gt, ht, kt, pt, nt, d1 t,
d2 t are continuous, positive, and bounded functions, m ∈ 0, 1 is a constant, and m denotes
the prey refuge parameter. δ− τi , t, i 1, 2, 3, are delay functions with t ∈ T and τi ∈ 0, ∞T ,
i 1, 2, 3, where δ− be a backward shift operator on the set T∗ and T∗ is a nonempty subset of
the time scale T. For the ecological justification of 1.1, one can refer to 2, 12.
The initial conditions of 1.1 are of the form
xt ϕ1 t,
yt ϕ2 t,
zt ϕ3 t,
t ∈ δ− τ, 0, 0T ,
ϕi 0 > 0,
i 1, 2, 3,
1.2
where τ max{τi }, τi ≥ 0, i 1, 2, 3.
For convenience, we introduce the notation
f u sup ft ,
f l inf ft ,
t∈T
1.3
t∈T
where f is a positive and bounded function.
2. Dynamic Inequalities on Time Scales
Let T be a nonempty closed subset time scale of R. The forward and backward jump
operators σ, ρ : T → T, and the graininess μ: T → R
are defined, respectively, by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
μt σt − t.
2.1
A point t ∈ T is called left dense if t > inf T and ρt t, left scattered if ρt < t,
right dense if t < sup T and σt t, and right scattered if σt > t. If T has a left scattered
Discrete Dynamics in Nature and Society
3
maximum m, then Tk T \ {m}; otherwise Tk T. If T has a right scattered minimum m,
then Tk T \ {m}; otherwise Tk T.
The basic theories of calculus on time scales, one can see 7.
A function p : T → R is called regressive provided 1 μtpt / 0 for all t ∈ Tk . The
set of all regressive and rd-continuous functions p : T → R will be denoted by R RT RT, R. We define the set R
R
T, R {p ∈ R : 1 μtpt > 0, for all t ∈ T}.
If r is a regressive function, then the generalized exponential function er is defined by
t
er t, s exp
2.2
ξμτ rτΔτ
s
for all s, t ∈ T, with the cylinder transformation
⎧
⎪
⎨ Log1 hz ,
h
ξh z ⎪
⎩z,
if h /
0,
2.3
if h 0.
Let p, q : T → R be two regressive functions and define
p ⊕ q p q μpq,
p −
p
,
1 μp
p q p ⊕ q .
2.4
Lemma 2.1 See 7. Assume that p, q : T → R are two regressive functions, then
i e0 t, s ≡ 1 and ep t, t ≡ 1;
ii ep σt, s 1 μtptep t, s;
iii ep t, s 1/ep s, t ep s, t;
iv ep t, sep s, r ep t, r;
v ep t, sΔ ptep t, s.
Lemma 2.2. Assume that a > 0, b > 0 and −a ∈ R
. Then
yΔ t ≥ ≤b − ayt,
yt > 0, t ∈ t0 , ∞T
2.5
implies
ayt0 b
yt ≥ ≤ 1 − 1 e−a t, t0 ,
a
b
t ∈ t0 , ∞T .
2.6
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Discrete Dynamics in Nature and Society
Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For
Δ
ye−a ·, t0 t yΔ te−a σt, t0 yt−ae−a t, t0 yΔ te−a σt, t0 yt
−a
e−a σt, t0 1 μt−a
yΔ t − −ayt e−a σt, t0 2.7
yΔ t − −ayt e−a σt, t0 ,
then, integrate both side from t0 to t to conclude
t yte−a t, t0 − yt0 yΔ s − −ays e−a σs, t0 Δs
t0
≥
t
2.8
be−a σs, t0 Δs
t0
b
t
e−a t0 , σsΔs,
t0
that is, yt ≥ yt0 e−a t, t0 b
t
t0
e−a t, σsΔs. So
yt ≥ yt0 e−a t, t0 b
t
e−a t, σsΔs
t0
yt0 e−a t, t0 −
b
a
t
e−a t, σs−aΔs
t0
b
yt0 e−a t, t0 − e−a t, t0 − 1
a
b
b
e−a t, t0 yt0 −
,
a
a
2.9
that is, yt ≥ b/a1 ayt0 /b − 1e−a t, t0 . This completes the proof.
Lemma 2.3. Assume that a > 0, b > 0. Then
yΔ t ≥ ≤b − ayσt,
yt > 0, t ∈ t0 , ∞T
2.10
implies
ayt0 b
yt ≥ ≤ 1 − 1 ea t, t0 ,
a
b
t ∈ t0 , ∞T .
2.11
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5
Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For
yea ·, t0 Δ
t yΔ tea t, t0 aea t, t0 yσt ea t, t0 yΔ t ayσt ,
2.12
then, integrate both side from t0 to t to conclude
ytea t, t0 − yt0 t
ea s, t0 yΔ s ayσs Δs
t0
≥b
2.13
t
ea s, t0 Δs,
t0
then
yt ≥ ea t, t0 yt0 b
t
ea t, sΔs
t0
ea t, t0 yt0 b
t
1 μsa ea s, t
t0
ea t, t0 yt0 b
t
ea σs, t
t0
ea t, t0 yt0 b
t
1
Δs
1 μsa
ea t, σs
t0
b
ea t, t0 yt0 −
a
t
1
Δs
1 μsa
1
Δs
1 μsa
2.14
ea t, σsaΔs
t0
b
ea t, t0 − 1
a
b
b
,
ea t, t0 yt0 −
a
a
ea t, t0 yt0 −
that is, yt ≥ b/a1 ayt0 /b − 1ea t, t0 . This completes the proof.
Lemma 2.4. Assume that a > 0, b > 0 and −b ∈ R
. Then
yΔ t ≤ ≥yσt b − ayt ,
yt > 0, t ∈ t0 , ∞T
2.15
implies
yt ≤ ≥
−1
b
b
1
− 1 e−b t, t0 ,
a
ayt0 t ∈ t0 , ∞T .
Proof. We only prove the “≤” case; the proof of the “≥” case is similar.
2.16
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Discrete Dynamics in Nature and Society
Let yt 1/xt, then 1/xtΔ ≤ 1/xσtb − a/xt, that is, −xΔ t/xt
xσt ≤ 1/xσtb − a/xt, so, xΔ t ≥ a − bxt. By Lemma 2.2, we have xt ≥
a/b1 bxt0 /a − 1e−b t, t0 . Therefore, yt ≤ b/a1 b/ayt0 − 1e−b t, t0 −1 .
This completes the proof.
Lemma 2.5. Assume that a > 0, b > 0. Then
yΔ t ≤ ≥yt b − ayσt ,
yt > 0, t ∈ t0 , ∞T
2.17
implies
yt ≤ ≥
b
b
1
− 1 eb t, t0 ,
a
ayt0 t ∈ t0 , ∞T .
2.18
Proof. We only prove the “≤” case; the proof of the “≥” case is similar.
Let yt 1/xt, then 1/xtΔ ≤ 1/xtb − a/xσt, that is, −xΔ t/xtx
σt ≤ 1/xtb − a/xσt, so, xΔ t ≥ a − bxσt. By Lemma 2.3, we have xt ≥
a/b1 bxt0 /a − 1eb t, t0 . Therefore, yt ≤ b/a1 b/ayt0 − 1eb t, t0 −1 . This
completes the proof.
Definition 2.6 see 16. Let T∗ be a nonempty subset of the time scale T and t0 ∈ T∗ a
fixed number. The operator δ− associated with t0 ∈ T∗ called the initial point is said to
be backward shift operator on the set T∗ . The variable s ∈ t0 , ∞T in δ− s, t is called the shift
size. The value δ− s, t in T∗ indicate s units translation of the term t ∈ T∗ to the left.
Now, we state some different time scales with their corresponding backward shift
operators: let T R and t0 0; then δ− s, t t − s; let T Z and t0 0; then δ− s,
√ t t − s;
let T hZ and t0 0; then δ− s, t t − s; let T N1/2 and t0 0; then δ− s, t t2 − s2 ; let
T 2N and t0 1; then δ− s, t t/s; and so on.
Lemma 2.7. If a ∈ R
and yΔ t ≤ ≥atyt, t ∈ t0 , ∞T , then
yδ− s, t ≥ ≤ea σt, δ− s, tyσt,
2.19
where δ− s, t be defined in Definition 2.6 and s, t ∈ t0 , ∞T × s, ∞T .
Proof. We only prove the “≤” case; the proof of the “≥” case is similar. For
Δ
yea ·, t0 t yΔ tea σt, t0 ytatea t, t0 at
ea σt, t0 1 μtat
yΔ t − atyt ea σt, t0 yΔ tea σt, t0 yt
yΔ t − atyt ea σt, t0 ≤ 0,
2.20
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7
then, integrate both side from δ− s, t to σt to conclude
yσtea σt, t0 ≤ yδ− s, tea δ− s, t, t0 ,
2.21
yδ− s, t ≥ ea σt, δ− s, tyσt.
2.22
then
This completes the proof.
3. Permanence
As an application, based on the results obtained in Section 2, we will establish a permanent
result for system 1.1.
Definition 3.1. System 1.1 is said to be permanent if there exists a compact region D ⊆ Int R3
,
such that for any positive solution xt, yt, zt of system 1.1 with initial condition 1.2
eventually enters and remains in region D.
In this section, we consider the time scale T that satisfies ea σt, δ− τ, t to be a
constant on 0, ∞T , where a ∈ R
, τ ∈ 0, ∞T are constants. For example, let t0 0 the
initial point, when T R, then ea σt, δ− τ, t eaτ ; when T Z, then ea σt, δ− τ, t 1 a1
τ ; when T hZ, then ea σt, δ− τ, t 1 ah1
τ/h , and so on.
For convenience, we introduce the following notations:
M1 M2 bu ebu σt, δ− τ1 , t
ε,
cl
f u − d1l 1 − mM1 ef u −dl σt, δ− τ2 , t
1
M3 m1 m2 m3 al d1l
ε,
g u − pl d2l /pl − kl d2l / pu 1 − mM1 pu M1 1 − meg u /pl −d2l σt, δ− τ3 , t
d2l nl
l
b − du 1 − m/al − hu 1 − m/nl eβl σt, δ− τ1 , t
cu
m1 /2 f l − d1u 1 − me−d1u σt, δ− τ2 , t
au d1u
− ε,
− ε,
m1 /2pl 1 − m g l − d2u − ku d2u e−d2u σt, δ− τ3 , t
nu d2u
where βl bl − du 1 − m/al − hu 1 − m/nl − cu M1 .
ε,
− ε,
3.1
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Hereafter, we assume that
H1 ft − d1 t ∈ R
;
H2 f u − d1l > 0;
H3 gt/pt − d2 t ∈ R
;
H4 g u − pl d2l /pl − kl d2l /pu 1 − mM1 > 0;
H5 bt − dt1 − m/at − ht1 − m/nt − ctM1 ∈ R
;
H6 bl − du 1 − m/al − hu 1 − m/nl > 0;
H7 −d1 t ∈ R
;
H8 f l − d1u > 0;
H9 −d2 t ∈ R
;
H10 m1 /21 − mg l − d2u pu − ku d2u > 0.
Proposition 3.2. Assume that xt, yt, zt is any positive solution of system 1.1 with initial
condition 1.2. If (H1)–(H4 ) hold, then
xt ≤ M1 ,
yt ≤ M2 ,
zt ≤ M3 .
3.2
Proof. Assume that xt, yt, zt is any positive solution of system 1.1 with initial
condition 1.2. From the first equation of system 1.1, for t ∈ τ1 , ∞T , we have
xΔ t ≤ xtbt − ctxδ− τ1 , t.
3.3
From 3.3, we can see xΔ t ≤ btxt; then by Lemma 2.7, we can get
xδ− τ1 , t ≥ eb σt, δ− τ1 , txσt.
3.4
Together with 3.3 and 3.4, we have
xΔ t ≤ xtbt − cteb σt, δ− τ1 , txσt
≤ xt bu − cl ebu σt, δ− τ1 , txσt .
3.5
By Lemma 2.5, for arbitrary small positive constant ε, there exists T1 > 0 such that
xt ≤
bu
ε
cl ebu σt, δ− τ1 , t
bu ebu σt, δ− τ1 , t
ε : M1 ,
cl
3.6
t ∈ T1 , ∞T .
Discrete Dynamics in Nature and Society
9
Again, from the second equation of system 1.1 and 3.6, for t ∈ T1 τ2 , ∞T , we
have
yΔ t ≤ yt −d1 t ft1 − mM1
.
atyδ− τ2 , t 1 − mM1
3.7
From 3.7, we can see yΔ t ≤ −d1 t ftyt; then by H1 and Lemma 2.7, we can
get
yδ− τ2 , t ≥ eα σt, δ− τ2 , tyσt,
3.8
where αt ft − d1 t.
Together with 3.7 and 3.8, we have
ft1 − mM1
y t ≤ yt −d1 t ateα σt, δ− τ2 , tyσt 1 − mM1
Δ
≤ yt
1 − mM1 ft − d1 t − atd1 teα σt, δ− τ2 , tyσt
ateα σt, δ− τ2 , tyσt 1 − mM1
⎛
≤ yt⎝f u − d1l −
al d1l ef u −dl σt, δ− τ2 , t
1
1 − mM1
3.9
⎞
yσt⎠.
By H2 and Lemma 2.5, for arbitrary small positive constant ε, there exists T2 > T1 τ2
such that
yt ≤
f u − d1l 1 − mM1
ε
al d1l ef u −dl σt, δ− τ2 , t
1
f u − d1l 1 − mM1 ef u −dl σt, δ− τ2 , t
1
al d1l
3.10
ε : M2 ,
t ∈ T2 , ∞T .
Similarly, under conditions H3 -H4 , by Lemmas 2.5 and 2.7, we can get that, for
arbitrary small positive constant ε, there exists T3 > T2 τ3 such that
zt ≤
g u − pl d2l /pl − kl d2l / pu 1 − mM1 pu M1 1 − meg u /pl −d2l σt, δ− τ3 , t
d2l nl
ε : M3 ,
t ∈ T3 , ∞T .
The conclusion of Proposition 3.2 follows. This completes the proof.
3.11
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Proposition 3.3. Assume that xt, yt, zt is any positive solution of system 1.1 with initial
condition 1.2. If (H1 )–(H10 ) hold, then
xt ≥ m1 ,
yt ≥ m2 ,
zt ≥ m3 .
3.12
Proof. Assume that xt, yt, zt is any positive solution of system 1.1 with initial
condition 1.2. From Theorem 3.4, there exists a T1 > 0, such that xt ≤ M1 , t ∈ T1 , ∞T . By
the first equation of system 1.1, for t ∈ T1 τ1 , ∞T , we have
dt1 − m ht1 − m
−
− ctxδ− τ1 , t ,
xΔ t ≥ xt bt −
at
nt
3.13
dt1 − m ht1 − m
−
− ctM1 .
xΔ t ≥ xt bt −
at
nt
3.14
then
By 3.14, H5 , and Lemma 2.7, we can get
xδ− τ1 , t ≤ eβ σt, δ− τ1 , txσt,
3.15
where β bt − dt1 − m/at − ht1 − m/nt − ctM1 .
Together with 3.13 and 3.15, we have
dt1 − m ht1 − m
−
− cteβ σt, δ− τ1 , txσt
x t ≥ xt bt −
at
nt
du 1 − m hu 1 − m
l
u
≥ xt b −
−
− c eβl σt, δ− τ1 , txσt ,
al
nl
Δ
3.16
where βl bl − du 1 − m/al − hu 1 − m/nl − cu M1 .
By H6 and Lemma 2.5, for arbitrary small positive constant ε, there exists T4 > T1 τ1
such that
xt ≥
bl − du 1 − m/al − hu 1 − m/nl
−ε
cu eβl σt, δ− τ1 , t
l
b − du 1 − m/al − hu 1 − m/nl eβl σt, δ− τ1 , t
cu
− ε : m1 ,
t ∈ T4 , ∞T .
3.17
Again, from the second equation of system 1.1 and 3.17, for t ∈ T4 τ2 , ∞T , we
have
y t ≥ yt −d1 t Δ
m1 /2ft1 − m
.
atyδ− τ2 , t m1 /21 − m
3.18
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11
From 3.18, we can see yΔ t ≥ −d1 tyt; then by H7 and Lemma 2.7, we can get
yδ− τ2 , t ≤ e−d1 σt, δ− τ2 , tyσt.
3.19
Together with 3.18 and 3.19, we have
yΔ t ≥ yt −d1 t ≥ yt
m1 /2ft1 − m
ate−d1 σt, δ− τ2 , tyσt m1 /21 − m
3.20
m1 /2 f l − d1u 1 − m − au d1u e−d1u σt, δ− τ2 , tyσt
au e−d1u σt, δ− τ2 , tM2 m1 /21 − m
.
By H8 and Lemma 2.5, for arbitrary small positive constant ε, there exists T5 > T4 τ2
such that
m1 /2 f l − d1u 1 − m
yt ≥ u u
−ε
a d1 e−d1u σt, δ− τ2 , t
m1 /2 f l − d1u 1 − me−d1u σt, δ− τ2 , t
− ε : m2 ,
au d1u
3.21
t ∈ T5 , ∞T .
Similarly, under conditions H9 and H10 , by Lemmas 2.5 and 2.7, we can get that,
for arbitrary small positive constant ε, there exists T6 > T5 τ3 such that
zt ≥
m1 /2pl 1 − m g l − d2u − ku d2u e−d2u σt, δ− τ3 , t
nu d2u
− ε : m3 ,
t ∈ T6 , ∞T .
The conclusion of Proposition 3.3 follows. This completes the proof.
Together with Propositions 3.2 and 3.3, we can obtain the following theorem.
Theorem 3.4. Assume that (H1 )–(H10 ) hold; then system 1.1 is permanent.
3.22
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x(t)
0.35
0.3
0.25
0
20
40
0
20
40
t
60
80
100
60
80
100
60
80
100
y(t)
0.4
0.2
0
t
z(t)
0.4
0.3
0.2
0
20
40
x1
Figure 1: T R, m 0.6. Dynamics behavior of system 4.1 with initial condition x0 0.3, y0 0.3,
z0 0.3.
4. Examples and Simulations
Consider the following system on time scales with m 0.6:
0.05yt1 − m
π xΔ t xt 0.08 0.05 sin t − 0.2xδ− τ1 , t −
2
2yt xt1 − m
0.05zt1 − m
−
,
0.005 xt1 − m 2zt
2xδ− τ2 , t1 − m
5π
yΔ t yt − 0.7 − 0.25 cos
t ,
2
2yδ− τ2 , t xδ− τ2 , t1 − m
4xδ− τ3 , t1 − m
3π
zΔ t zt − 0.75 0.2 sin
t .
2
0.005 xδ− τ3 , t1 − m 2zδ− τ3 , t
4.1
Let T R; then μt 0. Obviously, H1 , H3 , H5 , H7 , and H9 hold. Taking
τi 1, i 1, 2, 3, by a direct calculation, we can get
H2 f u − d1l 1.5500 > 0;
H4 g u − pl d2l /pl − kl d2l /pu 1 − mM1 3.4407 > 0;
H6 bl − du 1 − m/al − hu 1 − m/nl 0.0100 > 0;
H8 f l − d1u 1.0500 > 0;
H10 m1 /21 − mg l − d2u pu − ku d2u 0.0218 > 0.
From the above results, we can see that all conditions of Theorem 3.4 hold. So, system
4.1 is permanent, see Figure 1.
Let T Z; then μt 1. It is easy to check H1 , H3 , H5 , H7 , and H9 hold.
Taking τi 1, i 1, 2, 3, by a direct calculation, we can get
Discrete Dynamics in Nature and Society
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x(t)
0.4
0.35
0
10
20
30
40
50
30
40
50
30
40
50
t
y(t)
0.2
0.1
0
0
10
20
t
z(t)
0.4
0.3
0.2
0
10
20
t
Figure 2: T Z, m 0.6. Dynamics behavior of system 4.1 with initial condition x0 0.35, y0 0.2,
z0 0.4.
x(t)
0.4
0.2
0
0
5
10
15
20
t
25
30
35
40
0
5
10
15
20
t
25
30
35
40
0
5
10
15
20
t
25
30
35
40
y(t)
0.4
0.2
0
z(t)
0.4
0.3
0.2
0.1
Figure 3: T ∞
k0 2k, 2k 1, m 0.6. Dynamics behavior of system 4.1 with initial condition x0 0.3,
y0 0.2, z0 0.3.
H2 f u − d1l 1.5500 > 0;
H4 g u − pl d2l /pl − kl d2l /pu 1 − mM1 3.4086 > 0;
H6 bl − du 1 − m/al − hu 1 − m/nl 0.0100 > 0;
H8 f l − d1u 1.0500 > 0;
H10 m1 /21 − mg l − d2u pu − ku d2u 0.0244 > 0.
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From the above results, we can see that all conditions of Theorem 3.4 hold. So, system
4.1 is permanent, see Figure 2.
0, for t ∈ ∞
2k,2k
1,
k0
It is easy to check H1 , H3 ,
Let T ∞
k0 2k, 2k 1; then μt 1, for t ∈ ∞
k0 {2k
1}.
H5 , H7 , and H9 hold. Taking τi 2, i 1, 2, 3, by a direct calculation, we can get
H2 f u − d1l 1.5500 > 0;
H6 bl − du 1 − m/al − hu 1 − m/nl 0.0100 > 0;
H8 f l − d1u 1.0500 > 0.
Furthermore, if t ∈ ∞
k0 2k, 2k 1, then
H4 g u − pl d2l /pl − kl d2l /pu 1 − mM1 3.4418 > 0;
H10 m1 /21 − mg l − d2u pu − ku d2u 0.0175 > 0;
if t ∈ ∞
k0 {2k 1}, then
H4 g u − pl d2l /pl − kl d2l /pu 1 − mM1 3.4133 > 0;
H10 m1 /21 − mg l − d2u pu − ku d2u 0.0233 > 0.
From the above results, we can see that all conditions of Theorem 3.4 hold. So, system
4.1 is permanent, see Figure 3.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of China under Grant
61073065.
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