Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 653289, 22 pages
doi:10.1155/2009/653289
Research Article
On a Generalized Discrete Ratio-Dependent
Predator-Prey System
Yong-Hong Fan1, 2 and Lin-Lin Wang2, 3
1
Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China
School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China
3
Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China
2
Correspondence should be addressed to Lin-Lin Wang, wangll 1994@sina.com
Received 28 May 2009; Accepted 19 August 2009
Recommended by Elena Braverman
Verifiable criteria are established for the permanence and existence of positive periodic solutions
of a delayed discrete predator-prey model with monotonic functional response. It is shown that
the conditions that ensure the permanence of this system are similar to those of its corresponding
continuous system. And the investigations generalize some well-known results. In particular, a
more acceptant method is given to study the bounded discrete systems rather than the comparison
theorem.
Copyright q 2009 Y.-H. Fan and L.-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.
1. Introduction
Since the end of the 19th century, many biological models have been established to illustrate
the evolutionary of species, among them, predator-prey models attracted more and more
attention of biologists and mathematicians. There are many different kinds of predator-prey
models in the literature. And since 1990s, the so-called ratio-dependent predator-prey models
play an important role in the investigations on predator-prey models, which can be roughly
stated as that the per capita predator growth rate should be a function of the ratio of prey to
predator abundance. Under some simple assumptions, a general form of a ratio-dependent
model is
x
x
−ϕ
y,
x rx 1 −
K
y
x
−D .
y y μϕ
y
1.1
2
Discrete Dynamics in Nature and Society
Here the predator-prey interactions are described by ϕx/y; this function replaces the
functional response function ϕx in the traditional prey-dependent model. For the study of
ratio-dependent predator-prey models, most works have been done on the Michaelis-Menten
type model
αxy
x
−
,
x rx 1 −
K
my x
fx
y y −d ,
my x
1.2
αtxy
,
myt x
ftx
y y −dt ,
my x
1.3
or its periodic type
x xat − btx −
see 1–7 and references therein. It is easy to see that here the functional response function is
ϕu cu/m u, u x/y, as we know, this functional response function was first used by
Holling 8, and later biologists call it Holling type II functional response function, it usually
describes the uptake of substrate by the microorganisms in microbial dynamics or chemical
kinetics 9. And in the present paper, we will concentrate on the general form of the ratiodependent predator-prey model.
For the sake of convenience, we introduce some notations and definitions. Denote Z, R,
and R as the sets of all integers, real numbers, and nonnegative real numbers, respectively.
Let C denote the set of all bounded sequences f : Z → R, C is the set of all f ∈ C such that
f > 0, and Cω {f ∈ C | fk ω fk, k ∈ Z}, Iω {0, 1, . . . , ω − 1}. We define
f M sup fk,
f L inf fk,
1.4
k∈Z
k∈Z
for any f ∈ C. Obviously, if f is an ω-periodic sequence, then
f M max fk,
k∈Iω
f L min fk.
1.5
k∈Iω
We also define
f
1 ω−1
fk,
ω k0
1.6
if f is an ω-periodic sequence. And denote
1
f
ω
ω
ftdt,
0
f L min ft,
t∈0,ω
when f is a periodic continuous function with period ω.
f M max ft,
t∈0,ω
1.7
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3
In view of the periodicity of the actual environment, we begin with the following
periodic continuous ratio-dependent predator-prey system:
xt
yt,
x t xtbt − atxt − τ1 − ctg
yt
xt − τ2 y t yt etg
− dt ,
yt − τ2 1.8
where xt and yt represent the densities of the prey population and predator population
at time t respectively; τ1 ≥ 0, τ2 ≥ 0 are real constants; b, d : R → R and
ω a, c, e : R → R
ω
are continuous periodic functions with period ω > 0 and 0 btdt > 0, 0 dtdt > 0, a, c, e
is not always zero; and gut here ut xt/yt denotes the ratio-dependent response
function, which reflects the capture ability of the predator. Here we assume that gu satisfies
the following monotonic condition, for short, we call it M:
i g ∈ C1 0, ∞, g0 ≥ 0;
ii g u ≥ 0 for u ∈ 0, ∞;
iii limu → ∞ gu α /
0.
In 10, we gave a sufficient condition for the permanence of the continuous model
xt
x t xtat − btxt − ctg
yt,
yt
xt − τ
y t yt etg
− dt .
yt − τ
1.9
In which at, bt, ct, dt, and et are all positive periodic continuous functions with
period ω > 0; τ is a positive constant. In addition to condition M, the functional response
function g also satisfies the following.
iv There exists a positive constant h such that u2 g u ≤ h.
Without loss of generality, in this paper, we always assume that α 1 if α / 1, let
c∗ t αct and still denote c∗ t as ct.
Some special cases of system 1.8 have been studied, see 11, 12 and so forth. In
those papers, the authors mainly concentrated on the existence of periodic solutions and
permanence for systems they considered.
Set
gu
,
u∈0,∞ u
m sup
1.10
from ii and iii, we can easily obtain 0 < m < ∞.
Through the above assumptions, we can see that, one of the main results in 10 can
be given as follows.
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Discrete Dynamics in Nature and Society
Theorem 1.1. If
H1 b > mc
and
H2 e > d
hold, then system 1.9 is permanent.
Remark 1.2. By similar methods proposed in 10, we can show that under conditions H1
and H2, system 1.8 is also permanent.
We also need to mention that conditions H1 and H2 are sufficient to assure the
existence of positive periodic solutions of 1.8; this problem has been solved in 13.
However, when the size of the population is rarely small or the population has
nonoverlapping generations 14, 15, a more realistic model should be considered, that is,
the discrete time model. Just as pointed out in 16, even if the coefficients are constants,
the asymptotic behavior of the discrete system is rather complex and “chaotic” than the
continuous one, see 16 for more details. Similar to the arguments of 17, we can obtain
a discrete time analogue of 1.8:
N1 k
,
N1 k 1 N1 k exp bk − akN1 k − τ1 − ckh
N2 k
N1 k − τ2 N2 k 1 N2 k exp −dk ekg
,
N2 k − τ2 1.11
where t denotes the integer part of t t > 0 and hN1 k/N2 k N2 kgN1 k/
N2 k/N1 k guk/uk, uk N1 k/N2 k. Correspondingly, the basic assumptions of 1.11 is the same as that in 1.8, of cause, here b, d : Z → R and a, c, e : Z → R are
ω−1
periodic sequences with period ω > 0 and ω−1
k0 bk > 0,
k0 dk > 0, and g satisfies M.
To the best of our knowledge, a few investigations have been carried out for the permanence
on delayed discrete ecological systems, since the dynamics of these systems are usually more
complicated than the continuous ones.
The exponential form of system 1.11 assures that, for any initial condition N0 > 0,
Nk remains positive. In the remainder of this paper, for biological reasons, we only consider
solutions Nk with
Ni −k ≥ 0,
k 1, 2, . . . , max{τ1 , τ2 },
Ni 0 > 0,
i 1, 2.
1.12
System 1.11 includes many biological models as its special cases, which have been
studied by many authors; see 17–20 and so forth. Among them, Fan and Wang see 17
Discrete Dynamics in Nature and Society
5
considered the existence of positive periodic solutions for delayed periodic Michaelis-Menten
type ratio-dependent predator-prey system
N1 k 1 N1 k exp b1 k − a1 kN1 k −
N2 k 1 N2 k exp −b2 k α1 kN2 k
,
N1 k mkN2 k
α2 kN1 k
,
N1 k mkN2 k
1.13
and obtained the following theorem.
Theorem 1.3. Assume that the following conditions hold:
A1 b1 > α1 k/mk,
A2 α2 > b2 .
Then 1.13 has at least one positive ω-periodic solution.
Later in 20, we proved that under conditions A1 and A2, system 1.13 is also
permanent, so by the main result in 21, we can also obtain Theorem 1.3, which gives another
method to prove the existence of periodic solutions.
From the works above, it is not difficult to find: that for the continuous time model
1.3 and the discrete time model 1.13, conditions that assure the existence of positive
periodic solutions are exactly the same. In addition, when we comparing the work in 11
with that in 20, we found amazedly that conditions that assure the permanence of the
discrete models are also the same as those of the continuous models. This motivated us to
consider the permanence of system 1.11 only under conditions H1 and H2, since we
have already obtained the permanence of system 1.9.
Until very recently, Yang 22 studied the permanence of system 1.11 when τ1 0
and obtained the following conclusion.
Theorem 1.4. Assume that
B1
eM > dL ,
1.14
bL > mcM ,
1.15
B2
B3
m1
e g
M2
L
dM ,
1.16
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Discrete Dynamics in Nature and Society
hold, where
bL − mcM
L
M
M
,
exp
b
−
mc
−
a
M
1
aM
1
M2 exp 2 eM − dL .
M1 L exp bM − 1 ,
a
m1 1.17
Then 1.11 is permanent.
Remark 1.5. In Theorem 1.4, condition B3 implies condition B1; and B3 is an equality, it
is too strong to satisfy.
As pointed out in 23, if we use the method of comparison theorem, then the
additional condition like B3, to some extent, is necessary. But for system itself, this
condition may be not necessary. In this paper, our aim is to improve the above results. One
of the main results in this paper is given below, furthermore, we can conclude Corollary 3.5
similarly, from which we could show that condition B3 can be deleted. Now we list the
main result in the following.
Theorem 1.6. Assume that (H1) and (H2) hold. Then system 1.11 is permanent.
Remark 1.7. Condition H2 is a necessary condition.
Corollary 1.8. Assume that (H1) and (H2) hold, then system 1.11 has at least one positive ωperiodic solution.
Clearly, Theorem 1.6 extends and improves 19, Theorem 3.1, 20, Theorem 1.4;
Theorem 1.6 also extends and improves Theorem 1.4 by weaker conditions H1 and H2
instead of B1–B3 when the coefficients are all periodic. In particular, our investigation gives
a more acceptant method to study the bounded discrete systems, which is better than the
comparison theorem.
For the permanence of biology systems, one can refer to 24–33 and the references
cited therein.
The tree of this paper is arranged as follows. In the next section, we give some useful
lemmas which are essential to prove our conclusions. And in the third section, we give a
proof to the main result.
2. Preliminary
In this section, we list the definition of permanence and establish some useful lemmas.
Definition 2.1. System 1.11 is said to be permanent if there exist two positive constants λ1 , λ2
such that
λ1 ≤ lim inf Ni k ≤ lim sup Ni k ≤ λ2 ,
k→∞
for any solution N1 k, N2 k of 1.11.
k→∞
i 1, 2,
2.1
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7
Lemma 2.2 20. The problem
xk 1 xk exp{ak − bkxk},
x0 x0 > 0,
2.2
has at least one periodic solution U if b ∈ Cω , a ∈ C and a is an ω-periodic sequence with a > 0,
moreover, the following properties hold.
a U is positive ω-periodic.
b U has the following estimations for it’s boundary:
a
exp − |a| a ω ≤ Uk ≤ exp |a| a ω ,
b
b
a
2.3
especially,
a
b
exp{−aω} ≤ Uk ≤
a
b
exp{aω},
2.4
if a ∈ Cω .
Lemma 2.3. For any positive constant K, the problem
K
xk 1 xk exp −dk ekg
,
xk
2.5
has at least one periodic solution U if d ∈ C and d is an ω-periodic sequence provided that (H2)
holds. Moreover, the following properties hold.
a U is positive ω-periodic.
b U has the following estimations for its boundary:
K
K
exp |d| d ω ,
exp − |d| d ω ≤ Uk ≤
g −1 d/e
g −1 d/e
2.6
K
K
exp −dω ≤ Uk ≤
exp dω ,
g −1 d/e
g −1 d/e
2.7
especially,
if d ∈ Cω , where g −1 represents the inverse of g.
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Discrete Dynamics in Nature and Society
Proof. We only prove that 2.6 holds, for the rest of the proof, one can refer to 17. Let xn
be any possible ω-periodic positive solution of 2.5, then
ln xn 1 − ln xn −dn eng
K
,
xn
2.8
therefore
ω−1
ω−1
n0
n0
ln xn 1 − ln xn K
−dn eng
xn
0,
2.9
this leads to
ω−1
eng
n0
K
xn
2.10
dω.
We claim that there exist some n1 and n2 such that
n1 , n2 ∈ Iω ,
xn1 ≤
g −1
K
,
d/e
xn2 ≥
g −1
K
.
d/e
2.11
If this is not true, then either
xn <
K
,
d/e
2.12
K
,
g −1 d/e
2.13
g −1
or
xn2 >
for any n ∈ Iω , in any case, we can obtain
that our claim is true.
Note that for any n ∈ Iω ,
ω−1
n0
engK/xn /
dω, this contradiction shows
K ln xn − ln xn1 ≤
−dn eng xn ,
n0
ω−1
K ln xn − ln xn2 ≥ − −dn eng
,
xn n0
ω−1
then by virtue of equality 2.10, we complete the proof.
2.14
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9
Lemma 2.4. Consider the inequality problem
xk 1 ≤ xk exp{ak − bkxk}.
2.15
If b ∈ Cω , a ∈ C, and a is an ω-periodic sequence with a > 0, then any positive solutions xk of
2.15 satisfy
lim sup xk ≤ H1 exp bM H1 ,
k→∞
2.16
where
H1 a
b
exp a |a| ω .
2.17
Moreover, if a ∈ Cω , then
lim sup xk ≤ H2 exp bM H2 ,
k→∞
2.18
where
H2 a
b
exp{aω}.
2.19
Proof. Consider the following auxiliary equation:
zk 1 zk exp{ak − bkzk},
2.20
by Lemma 2.2, 2.20 has at least one positive ω-periodic solution, denote it as z∗ k, then
z∗ k ≤ H1 .
2.21
Let
xk exp{u1 k},
z∗ k exp{u2 k},
2.22
then
u1 k 1 − u1 k ≤ ak − bk exp{u1 k},
u2 k 1 − u2 k ak − bk exp{u2 k}.
2.23
Make the transformation uk u1 k − u2 k, we can obtain
uk 1 − uk ≤ −bkz∗ k exp{uk} − 1 .
2.24
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Discrete Dynamics in Nature and Society
Now we divide the proof into two cases according to the oscillating property of uk. First we
assume that uk does not oscillate about zero, then uk will be either eventually positive or
eventually negative. If the latter holds, that is, u1 k < u2 k, we have
xk < z∗ k ≤ H1
if a ∈ Cω , then xk ≤ H2 .
2.25
Either if the former holds, then by 2.24, we know uk 1 < uk, which means that uk is
eventually decreasing, also in terms of its positivity, we know that limk → ∞ uk exists. Then
2.24 implies limk → ∞ uk 0, which leads to
if a ∈ Cω , then lim sup xk ≤ H2 .
lim sup xk ≤ H1
k→∞
k→∞
2.26
Now we assume that uk oscillates about zero, by 2.24, we know that uk >
0 implies uk1 ≤ uk. Thus, if we let {ukl } be a subsequence of {uk}, where ukl is the
first element of the lth positive semicycle of {uk}, then lim supk → ∞ uk lim supl → ∞ ukl .
For the definition of semicycle and other related concepts, we refer to 34. Notice that
ukl ≤ ukl − 1 − bkl − 1z∗ kl − 1 exp{ukl − 1} − 1
2.27
and ukl − 1 < 0, then we know
ukl ≤ bkl − 1z∗ kl − 1 1 − exp{ukl − 1}
≤ bkl − 1z∗ kl − 1 ≤ bkl − 1z∗ kl − 1M .
2.28
Therefore
lim sup ukl ≤ bkl − 1z∗ kl − 1M .
l→∞
2.29
By the medium of 2.22, 2.25, and 2.26, we have
lim sup xk ≤ H1 exp bM H1
k→∞
M
if a ∈ Cω , then lim sup xk ≤ H2 exp b H2
.
k→∞
2.30
Corollary 2.5. Any positive solution of the inequality problem 2.15 satisfies
aM
1
M
M
lim sup xk ≤ min L exp a − 1 , L exp a
,
b
b
k→∞
where a ∈ C, aM > 0 and b ∈ C .
2.31
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11
Proof. Define the function
fx x exp{a − bx},
x > 0, a ∈ R, b > 0,
2.32
it is easy to see
fx ≤
1
exp{a − 1},
b
2.33
which immediately leads to
1
1
exp{ak − 1} ≤ L exp aM − 1 .
b
k → ∞ bk
lim sup xk ≤ lim sup
k→∞
2.34
From 2.15, we have
xk 1 ≤ xk exp{ak − bkxk} ≤ xk exp aM − bL xk .
2.35
By Lemma 2.4, for any ω > 0,
lim sup xk ≤
k→∞
aM
M
M
M
exp
a
ω
exp
a
exp
a
ω
,
bL
2.36
aM
exp
aM ,
bL
2.37
let ω → 0, we can obtain
lim sup xk ≤
k→∞
by 2.34 and 2.37, we complete the proof.
Remark 2.6. Note that when aM ≤ 1/e, aM /bL exp{aM } ≤ 1/bL exp{aM − 1}.
Similarly, we can obtain the following result.
Lemma 2.7. If any positive solution xk of the inequality problem
xk 1 ≥ xk exp{ak − bkxk}
2.38
lim sup xk ≤ H,
2.39
satisfies
k→∞
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Discrete Dynamics in Nature and Society
here H is a positive constant. Then if b ∈ Cω , a ∈ C, and a is an ω-periodic sequence with a > 0, one
has
lim inf xk ≥ H3 exp −HbM ,
k→∞
H3 a
b
exp − |a| a ω .
2.40
Moreover, if a ∈ Cω , then
lim inf xk ≥ H4 exp −HbM ,
k→∞
2.41
where
H4 a
b
exp{−aω}.
2.42
The proof is similar to that of Lemma 2.4.
Corollary 2.8. If any positive solution xk of the inequality problem
xk 1 ≤ xk exp{ak − bkxk}
2.43
lim sup xk ≤ H,
2.44
satisfies
k→∞
here H is a positive constant, then
lim inf xk ≥
k→∞
aL
L
M
exp
a
−
b
H
,
bM
2.45
where a ∈ C and b ∈ C .
3. Proof of the Main Result
For the rest of this paper, we only consider the solution of 1.11 with initial conditions 1.12.
To prove Theorem 1.6, we need the following several propositions.
Proposition 3.1. There exists a positive constant K1 such that lim supk → ∞ N1 k ≤ K1 .
Proof. Given any positive solution N1 k, N2 k of 1.11, from the first equation of 1.11,
we have
N1 k 1 ≤ N1 k exp{bk − akN1 k − τ1 }.
3.1
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13
Set
N1 k exp{u1 k},
3.2
u1 k 1 − u1 k ≤ bk − ak exp{u1 k − τ1 },
3.3
then
thus
k−1
u1 i 1 − u1 i ≤
ik−τ1 k−1
3.4
bi,
ik−τ1 which is equivalent to
k−1
u1 k −
bi ≤ u1 k − τ1 ,
3.5
ik−τ1 hence
⎫
⎬
N1 k − τ1 exp{u1 k − τ1 } ≥ exp u1 k −
bi
⎭
⎩
ik−τ ⎧
⎨
k−1
1
⎫
k−1
⎬
bi .
N1 k exp −
⎭
⎩ ik−τ ⎧
⎨
3.6
1
Therefore
⎫
⎫
⎬
⎬
N1 k 1 ≤ N1 k exp bk − ak exp −
bi N1 k .
⎭
⎩
⎩ ik−τ ⎭
⎧
⎨
⎧
⎨
k−1
3.7
1
By Lemma 2.4, we have
lim sup N1 k ≤ K1 ,
k → ∞
3.8
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Discrete Dynamics in Nature and Society
where
⎧
M ⎫
⎨
⎬
k−1
ak exp −
K1 G1 exp
bi
G1 ,
⎩
⎭
ik−τ1 b
ω .
exp
b
G1 |b|
ak exp − k−1
bi
ik−τ1 3.9
Proposition 3.2. Under condition (H1), there exists a positive constant k1 such that
lim infk → ∞ N1 k ≥ k1 .
Proof. Given any positive solution N1 k, N2 k of 1.11, from the first equation of 1.11,
we have
N1 k 1 ≥ N1 k exp{bk − mck − akN1 k − τ1 }.
3.10
Set N1 k exp{u1 k}, then
u1 k 1 − u1 k ≥ bk − mck − ak exp{u1 k − τ1 },
3.11
which yields
k−1
u1 i 1 − u1 i ≥
ik−τ1 k−1
bi − mci − aiK1 ,
3.12
ik−τ1 that is,
k−1
u1 k − τ1 ≤ u1 k −
bi − mci − aiK1 ,
3.13
ik−τ1 thus
N1 k − τ1 exp{u1 k − τ1 }
⎧
⎫
k−1
⎨
⎬
≤ exp u1 k −
bi − mci − aiK1 ⎩
⎭
ik−τ 1
⎧
⎨
⎫
k−1
⎬
N1 k exp −
bi − mci − aiK1 .
⎩ ik−τ ⎭
1
3.14
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15
Therefore
N1 k 1 ≥ N1 k
⎧
⎧
⎫
⎫
k−1
⎨
⎨ ⎬
⎬
× exp bk − mck − ak exp −
bi − mci − aiK1 N1 k .
⎩
⎩ ik−τ ⎭
⎭
1
3.15
Since H1 holds, then by Lemma 2.7 and Proposition 3.1, we have
⎧
⎨
k−1
lim inf xk ≥ G2 exp −K1 ak exp −
bi − mci − aiK1 ⎩
k→∞
ik−τ1 G2 M ⎫
⎬
,
⎭
b − mc
exp − |b − mc| b − mc ω .
ak exp − k−1
ik−τ1 bi − mi − aiK1 3.16
Proposition 3.3. If (H2) holds, then there exists a positive constant K2 such that
lim sup N2 k ≤ K2 .
k→∞
3.17
Proof. Given any positive solution N1 k, N2 k of 1.11. Set N2 k exp{u1 k}, from the
second equation of 1.11 and notice that conditions ii and iii on g imply that gk, uk ≤
1, then
u1 k 1 − u1 k ≤ ek − dk,
3.18
thus
k−1
u1 i 1 − u1 i ≤
ik−τ2 k−1
ei − di,
3.19
ik−τ2 which is equivalent to
u1 k −
k−1
ik−τ2 ei − di ≤ u1 k − τ2 ,
3.20
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Discrete Dynamics in Nature and Society
hence
⎧
⎨
⎫
⎬
k−1
N2 k − τ2 exp{u1 k − τ2 } ≥ exp u1 k −
ei − di
⎩
⎭
ik−τ 2
⎧
⎨
⎫⎞L
k−1
⎬
N2 k⎝exp −
ei − di ⎠ .
⎩ ik−τ ⎭
⎛
3.21
2
Therefore for any given ε > 0, we have
N1 k − τ2 N2 k 1 N2 k exp −dk ekg
N2 k − τ2 ⎧
⎛
⎞⎫
⎪
⎪
⎨
⎬
K1 ε
⎜
⎟
≤ N2 k exp −dk ekg ⎝
,
⎠
L
⎪
⎪
⎩
⎭
N2 k exp − k−1
−
di
ei
ik−τ2 3.22
for sufficiently large k. Here we use the monotonicity of the function gu.
Consider the following auxiliary equation:
⎧
⎪
⎨
⎞⎫
⎪
⎬
K1 ε
⎜
⎟
zk 1 zk exp −dk ekg ⎝
.
⎠
L
⎪
⎪
⎩
⎭
zk exp − k−1
ei − di
⎛
3.23
ik−τ2 By Lemma 2.3 and condition H2, we can obtain that 3.23 has at least one positive ωperiodic solution, denote it as z∗1 k and
z∗1 k ≤
K
exp |d| d ω : G3 ,
g −1 d/e
3.24
K1
L .
exp − k−1
−
di
ei
ik−τ2 3.25
where
K
Let
z∗1 k exp{u2 k},
3.26
Discrete Dynamics in Nature and Society
17
then
K
,
exp{u1 k}
K
u2 k 1 − u2 k −dk ekg
.
exp{u2 k}
u1 k 1 − u1 k ≤ −dk ekg
3.27
Denote uk u1 k − u2 k, we have
uk 1 − uk ≤ ek g
K
exp{u1 k}
−g
K
exp{u2 k}
.
3.28
First we assume that uk does not oscillate about zero, then uk will be either eventually
positive or eventually negative. If the latter holds, that is, u1 k < u2 k, we have
N2 k < z∗1 k ≤ G3 .
3.29
Either if the former holds, then by 3.28, we have uk 1 < uk, which means that uk is
eventually decreasing, also in terms of its positivity, we obtain that limk → ∞ uk exists. Then
3.28 leads to limk → ∞ uk 0, this implies
lim sup N2 k ≤ G3 .
3.30
k→∞
Now we assume that uk oscillates about zero, in view of 3.28, we know that uk >
0 implies uk1 ≤ uk. Thus, if we let {ukl } be a subsequence of {uk} where ukl is the
first element of the lth positive semicycle of {uk}, then lim supk → ∞ uk lim supl → ∞ ukl .
Also, from
ukl ≤ ukl − 1 ekl − 1 g
K
exp{u1 kl − 1}
−g
K
exp{u2 kl − 1}
3.31
and ukl − 1 < 0, we know
ukl ≤ ekl − 1.
3.32
lim supukl ≤ ekl − 1M .
3.33
lim sup N2 k ≤ K2 ,
3.34
Therefore
l→∞
Thus we have
k→∞
18
Discrete Dynamics in Nature and Society
where
K2 G3 exp eM .
3.35
Proposition 3.4. Under conditions (H1) and (H2), there exists a positive constant k2 such that
lim infk → ∞ N2 k ≥ k2 .
Proof. Given any positive solution N1 k, N2 k of 1.11, from the second equation of
1.11, we have
N2 k 1 ≥ N2 k exp{−dk},
3.36
then
k−1
k−1
ln N2 i 1 − ln N2 i ≥ −
ik−τ2 di,
3.37
ik−τ2 hence
N2 k − τ2 ≤ N2 k exp
⎫
⎬
⎧
k−1
⎨ ⎩ik−τ 2
3.38
di .
⎭
Therefore from the second equation of 1.11, we have
⎧
⎪
⎨
⎛
k1
⎜
N2 k 1 ≥ N2 k exp −dk ekg ⎝
⎪
k−1
⎩
N2 k exp
ik−τ2 ⎞⎫
⎪
⎬
⎟
M ⎠⎪.
⎭
di
3.39
Consider the auxiliary equation
⎧
⎪
⎨
⎛
k1
⎜
zk 1 zk exp −dk ekg ⎝
⎪
k−1
⎩
zk exp
ik−τ2 ⎞⎫
⎪
⎬
⎟
M ⎠⎪,
⎭
di
3.40
by Lemma 2.3 and H2, 3.40 has at least one positive ω-periodic solution, denoted it as
z∗2 k, then
z∗2 k ≥
G
exp − |d| d ω : G4 .
g −1 d/e
3.41
Discrete Dynamics in Nature and Society
19
Where
G exp
k1
k−1
ik−τ2 di
M .
3.42
If we set
z∗2 k exp{u2 k},
N2 k exp{u1 k},
3.43
then
G
,
u1 k 1 − u1 k ≥ −dk ekg
exp{u1 k}
G
u2 k 1 − u2 k −dk ekg
.
exp{u2 k}
3.44
And let uk u1 k − u2 k, we have
uk 1 − uk ≥ ek g
G
exp{u1 k}
−g
G
exp{u2 k}
.
3.45
If uk does not oscillate about zero, then by a similar analysis as that in Proposition 3.1, we
have
'L
&
lim inf N2 k ≥ z∗2 k ≥ G4 .
3.46
k→∞
Otherwise, if uk oscillates about zero, by 3.45, we know that uk < 0 implies uk 1 ≥
uk. Thus, if we denote {ukl } as a subsequence of {uk} where ukl is the first element
of the lth negative semicycle of {uk}, then lim infk → ∞ uk lim infl → ∞ ukl . On the other
hand, from
ukl ≥ ukl − 1 ekl − 1 g
G
exp{u1 kl − 1}
−g
G
exp{u2 kl − 1}
3.47
and ukl − 1 > 0 we can obtain
G
ukl ≥ −ekl − 1g
exp{u2 kl − 1}
3.48
≥ −ekl − 1.
Therefore
lim inf ukl ≥ −ekl − 1L .
l→∞
3.49
20
Discrete Dynamics in Nature and Society
By the medium of 3.43, we have
'L
&
lim inf N1 k ≥ z∗2 k exp −ekl − 1L .
k→∞
3.50
Hence lim infk → ∞ N1 k ≥ k2 , where
k2 G4 exp −eM .
3.51
Proof of Theorem 1.6. From the Propositions 3.1–3.4, we can easily know that system 1.11 is
permanent. The proof is complete.
By a similar process as above, we can obtain the following result.
Corollary 3.5. Assume that b, d ∈ C and a, c, e ∈ C . If
C1
b − mcL > 0,
3.52
e − dL > 0,
3.53
C2
then system 1.11 is permanent.
Obviously, B2 includes C2, B1, and B3 include C1. Thus, Corollary 3.5
generalizes and improves Theorem 1.4.
Acknowledgments
This work was supported by NNSF of China 10771032, the Natural Science Foundation of
Ludong University 24070301, 24070302, 24200301, Program for Innovative Research Team
in Ludong University, and China Postdoctoral Science Foundation funded project.
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