Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 713503, 14 pages
doi:10.1155/2012/713503
Research Article
Multiple Periodic Solutions of a Ratio-Dependent
Predator-Prey Discrete Model
Tiejun Zhou, Xiaolan Zhang, and Min Wang
College of Science, Hunan Agricultural University, Changsha, Hunan 410128, China
Correspondence should be addressed to Tiejun Zhou, hntjzhou@126.com
Received 29 September 2012; Accepted 6 December 2012
Academic Editor: Xiang Ping Yan
Copyright q 2012 Tiejun Zhou et al. 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.
A delayed ratio-dependent predator-prey discrete-time model with nonmonotone functional
response is investigated in this paper. By using the continuation theorem of Mawhins coincidence
degree theory, some new sufficient conditions are obtained for the existence of multiple positive
periodic solutions of the discrete model. An example is given to illustrate the feasibility of the
obtained result.
1. Introduction
It is known that one of important factors impacted on a predator-prey system is the functional
response. Holling proposed three types of functional response functions, namely, Holling I,
Holling II, and Holling III, which are all monotonously nondescending 1. But for some
predator-prey systems, when the prey density reaches a high level, the growth of predator
may be inhibited; that is, to say, the predator’s functional response is not monotonously
increasing. In order to describe such kind of biological phenomena, Andrews proposed the
so-called Holling IV functional response function 2
gx cx
,
m2 nx x2
1.1
which is humped and declines at high prey densities x. Recently, many authors have explored
the dynamics of predator-prey systems with Holling IV type functional responses 3–11. For
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Discrete Dynamics in Nature and Society
example, Ruan and Xiao considered the following predator-prey model 5:
cyt
dx
xt a − bxt − 2
,
dt
m x2 t
dy
hxt − τ
yt −d 2
,
dt
m x2 t − τ
1.2
where xt and yt represent predator and prey densities, respectively. In 1.2, the functional
response function gIV x cx/m2 x2 is a special case of Holling IV functional response.
The functional response functions mentioned previously only depend on the prey x.
But some biologists have argued that the functional response should be ratio dependent or
semi-ratio dependent in many situations. Based on biological and physiological evidences,
Arditi and Ginzburg first proposed the ratio-dependent predator-prey model 12
cyt
dx
xt a − bxt −
,
dt
myt xt
dy
hxt
yt −d ,
dt
myt xt
1.3
where the functional response function gr x, y cx/y/m x/y is ratio dependent.
Many researchers have putted up a great lot of works on the ratio-dependent or semi-ratiodependent predator-prey system 13–19.
Recently, some researchers incorporated the ratio-dependent theory and the inhibitory
effect on the specific growth rate into the predator-prey model 3, 7, 11, 15. Ding et
al. considered a semi-ratio-dependent predator-prey system with nonmonotonic functional
response and time delay 11; they obtained some sufficient conditions for the existence
and global stability of a positive periodic solution to this system. Hu and Xia considered
a functional response function 7, 15:
gIV
cxy
x
.
2 2
y
m y x2
1.4
With the functional response function, Xia and Han proposed the following periodic ratiodependent model with nonmonotone functional response 15:
t
cty2 t
dxt
xt at − bt
,
Kt − sxsds − 2 2
dt
m y t x2 t
−∞
htxt − τtyt − τt
dyt
yt −dt 2 2
,
dt
m y t − τt x2 t − τt
1.5
where at, bt, ct, dt, and ht are all positive periodic continuous functions with period
ω > 0, m is a positive real constant, and Ks : R → R is a delay kernel function. Based
on Mawhins coincidence degree, they obtained some sufficient conditions for the existence
of two positive periodic solutions of the ratio-dependent model 1.5.
Discrete Dynamics in Nature and Society
3
It is well known that discrete population models are more appropriate than the
continuous models when the populations do not overlap among generations. Therefore,
many scholars have studied some discrete population models 3, 4, 14, 16–19. For example,
Lu and Wang considered the following discrete semi-ratio-dependent predator-prey system
with Holling type IV functional response and time delay 3:
a12 nyn
,
xn 1 xn exp r1 n − a11 nxn − τ − 2
m x2 n
yn
.
yn 1 yn exp r2 n − a21 n
xn
1.6
They proved that the system 1.6 is permanent and globally attractive under some appropriate conditions. Furthermore, they also obtained some sufficient conditions which
guarantee the existence and global attractivity of positive periodic solution.
Motivated by the mentioned previously, this paper is to investigate the existence
of multiple periodic solutions of the following discrete ratio-dependent model with
nonmonotone functional response:
∞
xn 1 xn exp an − bn Klxn − l −
l0
cny2 n
,
m2 y2 n x2 n
hnxn − τnyn − τn
yn 1 yn exp −dn 2 2
,
m y n − τn x2 n − τn
1.7
for n ∈ Z0 , where a, d : Z0 → R, b, c, h : Z0 → R , and τ : Z0 → Z0 are all ω-periodic
sequences, ω is a positive integer, m is a positive real constant, and K : Z0 → Z0 satisfies
∞
l0 Kl 1, where Z, Z0 , Z , R, R0 , and R denote the sets of all integers, nonnegative
integers, positive integers, real numbers, nonnegative real numbers, and positive real
numbers, respectively. The model 1.7 is created from the continuous-time system 1.5 by
employing the semidiscretization technique.
The initial conditions associated with 1.7 are of the form
xn φn,
yn ψn,
n ∈ Z − Z ,
1.8
where φn ≥ 0, ψn ≥ 0 for n ∈ Z − Z0 and φ0 > 0, ψ0 > 0.
2. Preliminaries
For convenience, we will use the following notations in the discussion:
Iω {0, 1, . . . , ω − 1},
f :
1 ω−1
fk,
ω k0
Δun un 1 − un,
where f is a ω-periodic sequence of real numbers defined for k ∈ Z.
2.1
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Discrete Dynamics in Nature and Society
In the system 1.7, the time delay kernel sequence Kl satisfies
fore, if we define
Gl ∞
Kl kω,
∞
l0
Kl 1. There-
l ∈ Iω ,
2.2
k0
then Gl is uniformly convergent with respect to l ∈ Iω , and it satisfies
ω−1
l0
Gl 1.
Lemma 2.1. x∗ n, y∗ n is a positive ω-periodic solution of system 1.7 if and only if u∗1 n,
u∗2 n lnx∗ n/y∗ n, ln y∗ n is a ω-periodic solution of the following system 2.3:
Δu1 n an dn − bn
ω−1
Gl expu1 n − l u2 n − l
l0
−
m2
hn expu1 n − τn
cn
− 2
,
exp2u1 n m exp2u1 n − τn
Δu2 n −dn 2.3
hn expu1 n − τn
,
m2 exp2u1 n − τn
where an, bn, cn, dn, hn, and τn are the same as those in model 1.7.
Proof. Let u1 n, u2 n lnxn/yn, ln yn; then the system 1.7 can be rewritten as
Δu1 n an dn − bn
∞
Kl expu1 n − l u2 n − l
l0
−
m2
hn expu1 n − τn
cn
− 2
,
exp2u1 n m exp2u1 n − τn
Δu2 n −dn 2.4
hn expu1 n − τn
.
m2 exp2u1 n − τn
Therefore, x∗ n, y∗ n is a positive ω-periodic solution of system 1.7 if and only if
u∗1 n, u∗2 n lnx∗ n/y∗ n, ln y∗ n is a ω-periodic solution of the system 2.4.
Notice that
∞
Kl expu1 n − l u2 n − l
l0
∞ k1ω−1
Kl expu1 n − l u2 n − l
k0
lkω
∞ ω−1
Ks kω expu1 n − s − kω u2 n − s − kω.
k0 s0
2.5
Discrete Dynamics in Nature and Society
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If u1 n, u2 n is ω-periodic, then we have
∞
Kl expu1 n − l u2 n − l
l0
2.6
∞ ω−1
Ks kω expu1 n − s u2 n − s.
k0 s0
Because Gl ∞
k0
Kl kω is uniformly convergent with respect to l ∈ Iω , so we have
∞
Kl expu1 n − l u2 n − l
l0
ω−1
∞
Ks kω expu1 n − s u2 n − s
2.7
s0 k0
ω−1
Gs expu1 n − s u2 n − s.
s0
Therefore, u∗1 n, u∗2 n is a ω-periodic solution of the system 2.3 if and only if it is a ωperiodic solution of the system 2.4. This completes the proof.
From 1.8, the initial conditions associated with 2.3 are of the form
xn φn,
yn ψn,
n ∈ {0, −1, −2, . . . , τ0 },
2.8
where τ0 maxn∈Z−Z {ω − 1, τn}, φn ≥ 0, ψn ≥ 0 for n ∈ Z − Z0 , and φ0 > 0, ψ0 > 0.
Throughout this paper, we assume that
H1 d > 0, h > 2md exp|a| |d| a dω;
H2 m2 a > c.
Under the assumption H1, there exist the following six positive numbers:
l± v± 2
2
h exp |a| |d| a d ω ± h exp 4 a d ω − 4m2 d
,
2d
2
2
h exp − |a| |d| a d ω ± h exp −4 a d ω − 4m2 d
u± h±
,
2d
2
h − 4m2 d
2d
2
.
2.9
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Discrete Dynamics in Nature and Society
Obviously,
2.10
l− < u− < v− < v < u < l .
In this paper, we adopt coincidence degree theory to prove the existence of multiple
positive periodic solutions of 1.7. We first summarize some concepts and results from the
book by Gaines and Mawhin 20. Let X and Y be normed vector spaces. Define an abstract
equation in X,
Lx λNx,
2.11
where L : Dom L ⊂ X → Y is a linear mapping, and N : X → Y is a continuous mapping.
The mapping L is called a Fredholm mapping of index zero if dim ker L codim Im L < ∞
and Im L is closed in Y . If L is a Fredholm mapping of index zero, then there exist continuous
projectors P : X → X and Q : Y → Y such that Im P ker L and Im L ker Q ImI − Q.
It follows that L|Dom L∩ker P : I − P X → Im L is invertible, and its inverse is denoted by Kp .
If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω if QNΩ is
bounded and Kp I − QN : Ω → X is compact. Because Im Q is isomorphic to ker L, there
exists an isomorphism J : Im Q → ker L.
In our proof of the existence, we also need the following two lemmas.
Lemma 2.2 continuation theorem 20. Let L be a Fredholm mapping of index zero and let N be
L-compact on Ω. Suppose that
a for each λ ∈ 0, 1, x ∈ ∂Ω ∩ Dom L, Lx / λNx;
b for each x ∈ ∂Ω ∩ ker L, QNx /
0;
c degJQN, Ω ∩ ker L, 0 /
0.
Then the operator equation Lx Nx has at least one solution in Dom L ∩ Ω.
Lemma 2.3 see 14. If u : Z → R is a ω-periodic sequence, then for any fixed n1 ,n2 ∈ Iω , one has
un ≤ un1 ω−1
|Δuk|,
un ≥ un2 −
k0
ω−1
|Δuk|.
2.12
k0
3. Existence of Two Positive Periodic Solutions
We are ready to state and prove our main theorem.
Theorem 3.1. Suppose that (H1) and (H2) hold. Then model 1.7 has at least two positive ω-periodic
solutions.
Proof. It is easy to see that if the system 2.3 has a ω-periodic solution u∗1 n, u∗2 n, then
x∗ n, y∗ n expu∗1 n − u∗2 n, expu∗2 n is a positive ω-periodic solution to the system 1.7. Therefore, to complete the proof, it suffices to show that the system 2.3 has at
least two ω-periodic solutions.
Discrete Dynamics in Nature and Society
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We take
X Y {u1 n, u2 n | ui n ω ui n, i 1, 2, n ∈ Z}
3.1
and define the norm of X and Y
u
max|u1 n| max|u2 n|,
n∈Iω
3.2
n∈Iω
for u u1 , u2 ∈ X or Y . Then X and Y are Banach spaces when they are endowed with the
previous norm · .
For any u u1 , u2 ∈ X, because of its periodicity, it is easy to verify that
Λ1 u, n an dn − bn
ω−1
Gl expu1 n − l u2 n − l
l0
−
hn expu1 n − τn
cn
−
,
m2 exp2u1 n m2 exp2u1 n − τn
Λ2 u, n −dn 3.3
hn expu1 n − τn
m2 exp2u1 n − τn
are ω-periodic with respect to n.
Set
L : Dom L ∩ X −→ Y,
N : X −→ Y,
Lun Lu1 , u2 n Δu1 n, Δu2 n,
Nun Nu1 , u2 n Λ1 u, n, Λ2 u, n.
3.4
Obviously, ker L R2 , Im L {u1 , u2 ∈ Y : ω−1
n0 ui n 0, i 1, 2} is closed in Y , and
dim ker L codim Im L 2. Therefore, L is a Fredholm mapping of index zero.
Define two mappings P and Q as
1 ω−1
1 ω−1
u1 n,
u2 n ,
Pu ω n0
ω n0
1 ω−1
1 ω−1
Qv v1 n,
v2 n ,
ω n0
ω n0
P : X −→ X,
Q : Y −→ Y,
u u1 , u2 ∈ X,
3.5
v v1 , v2 ∈ Y.
It is easy to prove that P and Q are two projectors such that Im P ker L and Im L ker Q ImI − Q. Furthermore, by a simple computation, we find that the inverse Kp of Lp : Im L →
Dom L ∩ ker P has the form
Kp u1 , u2 n−1
1 ω−1
1 ω−1
u1 k −
ω − ku1 k, u2 k −
ω − ku2 k .
ω k0
ω k0
k0
k0
n−1
3.6
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Discrete Dynamics in Nature and Society
Evidently,
QNu1 , u2 1 ω−1
1 ω−1
Λ1 u, n,
Λ2 u, n
ω n0
ω n0
3.7
and Kp I − QN are continuous by the Lebesgues convergence theorem. Moreover, by Arzela
Ascolis theorem, QNΩ and Kp I − QNΩ are relatively compact for the open bounded
set Ω ⊂ X. Therefore, N is L-compact on Ω for the open bounded set Ω ⊂ X.
Corresponding to the operator equation 2.11, we get the following system:
Δu1 n λΛ1 u, n,
Δu2 n λΛ2 u, n,
3.8
where λ ∈ 0, 1. Suppose that u1 n, u2 n ∈ X is an arbitrary solution of system 3.8 for a
constant λ ∈ 0, 1. Summing 3.8 over Iω , we obtain
aω ω−1
n0
cn
bn Gl expu1 n − l u2 n − l 2
,
m exp2u1 n
l0
ω−1
dω ω−1
hn expu1 n − τn
.
2 exp2u n − τn
m
1
n0
3.9
3.10
From system 3.8, we have
ω−1
ω−1
n0
n0
|Δu1 n| <
|an| |dn|
ω−1
bn
n0
ω−1
Gl expu1 n − l u2 n − l
l0
hn expu1 n − τn
cn
2
,
m exp2u1 n m2 exp2u1 n − τn
ω−1
ω−1
n0
n0
|Δu2 n| <
|dn| 3.11
ω−1
hn expu1 n − τn
.
2
n0 m exp2u1 n − τn
By using 3.9 and 3.10, we obtain
|Δu1 n| < |a| |d| a d ω,
3.12
|Δu2 n| < |d| d ω.
3.13
ω−1
n0
ω−1
n0
Discrete Dynamics in Nature and Society
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Obviously, there exist ξi ,ηi ∈ Iω , such that
ui ηi max ui n,
ui ξi min ui n,
n∈Iω
n∈Iω
i 1, 2.
3.14
From 3.10, it follows that
exp u1 η1
,
dω ≤ hω 2
m exp2u1 ξ1 3.15
d 2
u1 η1 ≥ ln
m exp2u1 ξ1 .
h
3.16
therefore
By using Lemma 2.3 and 3.12, we obtain
ω−1
d 2
u1 n ≥ u1 η1 −
m exp2u1 ξ1 − |a| |d| a d ω.
|Δu1 s| > ln
h
s0
3.17
In particular, we have
u1 ξ1 > ln
d
h
m exp2u1 ξ1 2
− |a| |d| a d ω,
3.18
or
d exp2u1 ξ1 − h exp |a| |d| a d ω expu1 ξ1 m2 d < 0.
3.19
The assumption H1 implies that h exp|a| |d| a dω > 2md. So we have
ln l− < u1 ξ1 < ln l .
3.20
From 3.10, we also have
dω ≥ hω
expu1 ξ1 ,
m2 exp 2u1 η1
3.21
it follows that
u1 ξ1 ≤ ln
d
h
.
m exp 2u1 η1
2
3.22
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Discrete Dynamics in Nature and Society
By using Lemma 2.3 and 3.12 again, we have
u1 n ≤ u1 ξ1 ω−1
|Δu1 s| < ln
d
h
s0
|a| |d| a d ω.
m exp 2u1 η1
2
3.23
In particular, we have
d 2
u1 η1 < ln
|a| |d| a d ω,
m exp 2u1 η1
h
3.24
or
d exp 2u1 η1 − h exp − |a| |d| a d ω exp u1 η1 m2 d > 0.
3.25
Therefore,
u1 η1 < ln v−
or u1 η1 > ln v .
3.26
From 3.12 and 3.20, we have
u1 n ≤ u1 ξ1 |Δu1 s| < ln l |a| |d| a d ω : B11 .
ω−1
3.27
s0
Similarly, from 3.12 and 3.26, we have
ω−1
u1 n ≥ u1 η1 −
|Δu1 s| > ln v − |a| |d| a d ω : B12 .
3.28
s0
By using 3.14, 3.27, and 3.28, it follows from 3.9 that
aω ≥ bω expu2 ξ2 B12 ,
3.29
cω
aω ≤ bω exp u2 η2 B11 2 .
m
3.30
From 3.29, we have
u2 ξ2 ≤ ln
a
b
− B12 .
3.31
In view of 3.12, we obtain
u2 n ≤ u2 ξ2 ω−1
a
s0
b
|Δu2 s| < ln
− B12 |d| d ω : B21 .
3.32
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Under the assumption H2, it follows from 3.30 that
a − c/m2
− B11 .
u2 η2 ≥ ln
b
3.33
By using 3.12, we obtain again
a − c/m2
ω−1
u2 n ≥ u2 η2 −
− B11 − |d| d ω : B22 .
|Δu2 s| > ln
b
s0
3.34
It follows from 3.32 and 3.34 that
max|u2 n| < max{|B21 |, |B22 |} : B2 .
3.35
n∈Iω
Notice that
c h expu1 h expu1 QNu1 , u2 a d − b expu1 u2 − 2
, −d 2
m exp2u1 m exp2u1 3.36
for u u1 , u2 ∈ R2 . Under the conditions H1 and H2, we can obtain two distinct solutions of QNu1 , u2 0
a m2 u2− − c
u ln u− , ln
,
bu− m2 u2−
a m2 u2 − c
u u1 , u2 ln u , ln
.
bu m2 u2
−
u−1 , u−2
3.37
After choosing a constant C > 0 such that
am2 u2 − c am2 u2 − c −
C > max ln
, ln
,
bu− m2 u2 bu m2 u2 3.38
−
we can define two bounded open subsets of X as follows:
Ω1 u u1 , u2 ∈ X | u1 ∈ ln l− , ln v− , max|u2 | < B2 C ,
n∈Iω
Ω2 u u1 , u2 ∈ X | min u1 ∈ ln l− , ln l , max u1 ∈ ln v , B11 , max|u2 | < B2 C, .
n∈Iω
n∈Iω
n∈Iω
3.39
It follows from 2.10 and 3.38 that u− ∈ Ω1 and u ∈ Ω2 . Because of ln v− < ln v , it is easy to
see that Ω1 ∩Ω2 is empty, and Ωi satisfies the condition a in Lemma 2.2 for i 1, 2. Moreover,
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Discrete Dynamics in Nature and Society
QNu /
0 for u ∈ ∂Ωi ker L ∂Ωi R2 . This shows that the condition b in Lemma 2.2 is
satisfied.
Because Im Q ker L, we can take the isomorphic J as the identity mapping, then we
have
degJQNu1 , u2 , Ωi ∩ ker L, 0, 0 degQNu1 , u2 , Ωi ∩ ker L, 0, 0.
3.40
From 3.37, QNu1 , u2 0 has two solutions u− u−1 , u−2 ∈ Ω1 ∩ Ker L and u u1 , u2 ∈
Ω2 ∩ Ker L. Therefore we have
degQNu1 , u2 , Ω1 ∩ ker L, 0, 0
−
−
h exp u−1 m2 − exp 2u−1
−
− − b exp u1 u2 −b exp u1 u2 −
2
2
m exp2u−1 sign
−
−
2
h exp u1 m − exp 2u1
0
− 2
2
m exp 2u
1
b h exp 2u−1 u−2 m2 − exp 2u−1
sign
sign m − exp u−1
− 2
2
m exp 2u1
⎞
⎛
e − 2md
e 2md − e − 2md
⎟
⎜
⎟
sign ⎜
⎠
⎝
2d
3.41
1/
0.
Similarly, we can obtain that
degQNu1 , u2 , Ω2 ∩ ker L, 0, 0
⎞
⎛ e 2md e − 2md
e − 2md
⎜
⎟
⎟ −1 sign m − exp u1 sign⎜
/ 0.
⎝−
⎠
2d
3.42
So the condition c in Lemma 2.2 is also satisfied.
By now we know that Ωi i 1, 2 satisfies all the requirements of Lemma 2.2. Hence
the system 2.3 has at least two ω-periodic solutions. This completes the proof.
4. An Example
In the system 1.7, let an 0.5 0.25 cos2/3πn, let bn 1.1 cos2/3πn, let cn 0.11 0.1 cos2/3πn, let dn 0.011 0.01 sin2/3πn, let hn 1 0.5 cos2/3πn,
and let τn 2. Obviously, they are positive periodic sequences with period ω 3. The time
Discrete Dynamics in Nature and Society
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delay kernel sequence Kn 1 − exp−1 exp−n, which satisfies ∞
n0 Kn 1. It is easy
to obtain that d 0.011 > 0, h − 2md exp|a| |d| a dω ≈ 0.0559 > 0, m2 a − c 1.89 > 0.
Therefore, the conditions H1 and H2 are satisfied. From Theorem 3.1, the system 1.7 has
at least two 3-periodic solutions.
5. Conclusion
In 3, Lu and Wang investigated a discrete time semi-ratio-dependent predator-prey system
1.6 with Holling type IV functional response and time delay. They established sufficient
conditions which guarantee the existence and global attractivity of a positive periodic
solution of the system. In this paper, a ratio-dependent predator-prey discrete-time model
with discrete distributed delays and nonmonotone functional response is investigated. By
using the continuation theorem of Mawhins coincidence degree theory, we prove that the
system 1.7 has at least two positive periodic solutions under conditions H1 and H2. As
3, we would like to know the local stability of the two positive periodic solutions of system
1.7, which is our future work.
Acknowledgment
This work was supported by the Project of Hunan Provincial Department of Finance
201051.
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