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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 141589, 15 pages
doi:10.1155/2009/141589
Research Article
Existence of Periodic Solutions for a Delayed
Ratio-Dependent Three-Species Predator-Prey
Diffusion System on Time Scales
Zhenjie Liu
School of Mathematics and Computer, Harbin University, Harbin, Heilongjiang 150086, China
Correspondence should be addressed to Zhenjie Liu,
Received 3 September 2008; Accepted 21 January 2009
Recommended by Binggen Zhang
This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey
diffusion system with Michaelis-Menten functional responses and time delays in a two-patch
environment on time scales. By using a continuation theorem based on coincidence degree theory,
we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when
thetimescale
T is chosen as R or Z, the existence of the periodic solutions of the corresponding
continuous and discrete models follows. Therefore, the methods are unified to provide the
existence of the desired solutions for the continuous differential equations and discrete difference
equations.
Copyright q 2009 Zhenjie Liu. 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
The traditional predator-prey model has received great attention from both theoretical and
mathematical biologists and has been studied extensively e.g., see 1–4 and references
therein. Based on growing biological and physiological evidences, some biologists have
argued that in many situations, especially when predators have to search for food and
therefore, have to share or compete for food, the functional response in a prey-predator
model should be ratio-dependent, 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. Starting from
this argument and the traditional prey-dependent-only mode, Arditi and Ginzburg 5 first
proposed the following ratio-dependent predator-prey model:
·
x
xa − bx −
cxy
my x
,
·
y
y
− d
fx
my x
1.1
2 Advances in Difference Equations
which incorporates mutual interference by predators, where gxcx/my x is a
Michaelis-Menten type functional response function. Equation 1.1 has been studied by
many authors and seen great progress e.g., see 6–11.
Xu and Chen 11 studied a delayed two-predator-one-prey model in two patches
which is described by the following differential equations:
x
1
tx
1
t
a
1
− a
11
x
1
t −
a
13
x
3
t
m
13
x
3
tx
1
t
−
a
14
x
4
t
m
14
x
4
tx
1
t
D
1
tx
2
t − x
1
t,
x
2
tx
2
ta
2
− a
22
x
2
t D
2
tx
1
t − x
2
t,
x
3
tx
3
t
− a
3
a
31
x
1
t − τ
1
m
13
x
3
t − τ
1
x
1
t − τ
1
,
x
4
tx
4
t
− a
4
a
41
x
1
t − τ
2
m
14
x
4
t − τ
2
x
1
t − τ
2
.
1.2
In view of periodicity of the actual environment, Huo and Li 12 investigated a more general
delayed ratio-dependent predator-prey model with periodic coefficients of the form
·
x
1
tx
1
t
a
1
t − a
11
tx
1
t − τ
11
−
a
12
tx
2
t
m
1
x
1
t
,
·
x
2
tx
2
t
− a
2
t
a
21
tx
1
t − τ
21
m
1
x
1
t − τ
21
− a
22
tx
2
t − τ
22
−
a
23
tx
3
t
m
2
x
2
t
,
·
x
3
tx
3
t
− a
3
t
a
32
tx
2
t − τ
32
m
2
x
2
t − τ
32
− a
33
tx
3
t − τ
33
.
1.3
In order to consider periodic variations of the environment and the density regulation
of the predators though taking into account delay effect and diffusion between patches,
more realistic and interesting models of population interactions should take into account
comprehensively other than one or two aspects. On the other hand, in order to unify the
study of differential and difference equations, people have done a lot of research about
dynamic equations on time scales. The principle aim of this paper is to systematically unify
the existence of periodic solutions for a delayed ratio-dependent predator-prey system with
functional response and diffusion modeled by ordinary differential equations and their
discrete analogues in form of difference equations and to extend these results to more
general time scales. The approach is based on Gaines and Mawhin’s continuation theorem
of coincidence degree theory, which has been widely applied to deal with the existence of
periodic solutions of differential equations and difference equations.
Therefore, it is interesting and important to study the following model on time
scales T:
z
Δ
1
tb
1
t − a
1
t exp{z
1
t}−
c
1
t exp{z
3
t}
m
1
t exp{z
3
t} exp{z
1
t}
D
1
t
exp{z
2
t − z
1
t}−1
,
z
Δ
2
tb
2
t − a
2
t exp{z
2
t} D
2
t
exp{z
1
t − z
2
t}−1
,
Advances in Difference Equations 3
z
Δ
3
t−r
1
t − a
3
t exp{z
3
t − τ
11
}
d
1
t exp{z
1
t − τ
12
}
m
1
t exp{z
3
t − τ
12
} exp{z
1
t − τ
12
}
−
c
2
t exp{z
4
t}
m
2
t exp{z
4
t} exp{z
3
t}
,
z
Δ
4
t−r
2
t − a
4
t exp{z
4
t − τ
21
}
d
2
t exp{z
3
t − τ
22
}
m
2
t exp{z
4
t − τ
22
} exp{z
3
t − τ
22
}
1.4
with the initial conditions
z
i
sϕ
i
s ≥ 0,s∈ −τ, 0 ∩ T,ϕ
i
0 > 0,ϕ
i
s ∈ C
rd
−τ,0 ∩ T, R
,i 1, 2, 3, 4,
1.5
where τ max{τ
ij
,i,j 1, 2}.In1.4, z
i
t represents the prey population in the ith patch
i 1, 2,andz
i
ti 3, 4 represents the predator population. z
1
t is the prey for z
3
t,and
z
3
t is the prey for z
4
t so that they form a food chain. D
i
t denotes the dispersal rate of the
prey in the ith patch i 1, 2. For the sake of generality and convenience, we always make
the following fundamental assumptions for system 1.4:
H a
i
t ∈ C
rd
T, R
i 1, 2, 3, 4,b
i
t,c
i
t,d
i
t,r
i
t,m
i
t,D
i
t ∈ C
rd
T, R
i
1, 2 are all rd-continuous positive periodic functions with period ω>0; τ
ij
i, j 1, 2 are
nonnegative constants.
In 1.4,setx
i
texp{z
i
t},y
j
texp{z
j2
t},i 1, 2,j 1, 2. If T R, then
1.4 reduces to the ratio-dependent predator-prey diffusive system of three species with time
delays governed by the ordinary differential equations
x
1
tx
1
t
b
1
t − a
1
tx
1
t −
c
1
ty
1
t
m
1
ty
1
tx
1
t
D
1
tx
2
t − x
1
t,
x
2
tx
2
tb
2
t − a
2
tx
2
t D
2
tx
1
t − x
2
t,
y
1
ty
1
t
− r
1
t − a
3
ty
1
t − τ
11
d
1
tx
1
t − τ
12
m
1
ty
1
t − τ
12
x
1
t − τ
12
−
c
2
ty
2
t
m
2
ty
2
ty
1
t
,
y
2
ty
2
t
− r
2
t − a
4
ty
2
t − τ
21
d
2
ty
1
t − τ
22
m
2
ty
2
t − τ
22
y
1
t − τ
22
.
1.6
If T Z, then 1.4 is reformulated as
x
1
k 1x
1
k exp
b
1
k − a
1
kx
1
k −
c
1
ky
1
k
m
1
ky
1
kx
1
k
D
1
k
x
2
k
x
1
k
− 1
,
x
2
k 1x
2
k exp
b
2
k − a
2
kx
2
kD
2
k
x
1
k
x
2
k
− 1
,
4 Advances in Difference Equations
y
1
k 1y
1
k exp
− r
1
k − a
3
ky
1
k − τ
11
d
1
kx
1
k − τ
12
m
1
ky
1
k − τ
12
x
1
k − τ
12
−
c
2
ky
2
k
m
2
ky
2
k − τ
12
y
1
k − τ
12
,
y
2
k 1y
2
k exp
− r
2
k − a
4
ky
2
k − τ
21
d
2
ky
1
k − τ
22
m
2
ky
2
k − τ
22
y
1
k − τ
22
1.7
which is the discrete time ratio-dependent predator-prey diffusive system of three species
with time delays and is also a discrete analogue of 1.6.
2. Preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers R. Throughout
the paper, we assume the time scale T is unbounded above and below, such as R, Z and
∪
k∈Z
2k, 2k 1. The following definitions and lemmas can be found in 13.
Definition 2.1. The forward jump operator σ : T → T, t he backward jump operator ρ : T →
T, and the graininess μ : T → R
0, ∞ are defined, respectively, by
σtinf{s ∈ T | s>t},ρtsup{s ∈ T | s<t},μtσt − t for t ∈ T. 2.1
If σtt, then t is called right-dense otherwise: right-scattered,andifρtt, then t is
called left-dense otherwise: left-scattered.
If T has a left-scattered maximum m, then T
k
T \{m}; otherwise T
k
T.IfT has a
right-scattered minimum m, then T
k
T \{m}; otherwise T
k
T.
Definition 2.2. Assume f : T → R is a function and let t ∈ T
k
. Then one defines f
Δ
t to be the
number provided it exists with the property that given any ε>0, there is a neighborhood
U of t such that
fσt − fs − f
Δ
tσt − s
≤ ε|σt − s|∀s ∈ U. 2.2
In this case, f
Δ
t is called the delta or Hilger derivative of f at t. Moreover, f is said to be
delta or Hilger differentiable on T if f
Δ
t exists for all t ∈ T
k
.AfunctionF : T → R is called
an antiderivative of f : T → R provided F
Δ
tft for all t ∈ T
k
. Then one defines
s
r
ftΔt Fs − Fr for r, s ∈ T. 2.3
Definition 2.3. A function f : T → R is said to be rd-continuous if it is continuous at right-
dense points in T and its left-sided limits exists finite at left-dense points in T.Thesetof
rd-continuous functions f : T → R will be denoted by C
rd
T, R.
Advances in Difference Equations 5
Definition 2.4. If a ∈ T,infT −∞,andf is rd-continuous on −∞,a, then one defines the
improper integral by
a
−∞
ftΔt lim
T →−∞
a
T
ftΔt 2.4
provided this limit exists, and one says that the improper integral converges in this case.
Definition 2.5 see 14. One says that a time scale T is periodic if there exists p>0 such that
if t ∈ T, then t ± p ∈ T. For T
/
R, the smallest positive p is called the period of the time scale.
Definition 2.6 see 14.LetT
/
R be a periodic time scale with period p. One says that the
function f : T → R is periodic with period ω if there exists a natural number n such that
ω np, ft ωft for all t ∈ T and ω is the smallest number such that ft ωft.
If T R, one says that f
is periodic with period ω>0ifω is the smallest positive
number such that ft ωft for all t ∈ T.
Lemma 2.7. Every rd-continuous function has an antiderivative.
Lemma 2.8. Every continuous function is rd-continuous.
Lemma 2.9. If a, b ∈ T,α,β∈ R and f, g ∈ C
rd
T, R,then
a
b
a
αftβgtΔt α
b
a
ftΔt β
b
a
gtΔt;
b if ft ≥ 0 for all a ≤ t<b,then
b
a
ftΔt ≥ 0;
c if |ft|≤gt on a, b : {t ∈ T | a ≤ t<b},then|
b
a
ftΔt|≤
b
a
gtΔt.
Lemma 2.10. If f
Δ
t ≥ 0,thenf is nondecreasing.
Notation 1. To facilitate the discussion below, we now introduce some notation to be used
throughout this paper. Let T be ω-periodic, that is, t ∈ T implies t ω ∈ T,
κ min
0, ∞ ∩ T
,I
ω
κ, κ ω ∩ T,
f
1
ω
I
ω
fsΔs
1
ω
κω
κ
fsΔs, f
M
sup
t∈T
ft,f
L
inf
t∈T
ft,
2.5
where f ∈ C
rd
T, R is an ω-periodic function, that is, ft ωft for all t ∈ T, t ω ∈ T.
Notation 2. Let X, Z be two Banach spaces, let L :DomL ⊂ X → Z be a linear mapping,
and let N : X → Z be a continuous mapping. If L is a Fredholm mapping of index zero
and there exist continuous projectors P : X → X and Q : Z → Z such that Im P Ker L,
Ker Q Im L ImI − Q, then the restriction L|
Dom L∩ Ker P
: I − PX → Im L is invertible.
Denote the inverse of that map by K
P
.IfΩ is an open bounded subset of X, the mapping N
will be called L-compact on
Ω if QNΩ is bounded and K
P
I − QN : Ω → X is compact.
Since Im Q is isomorphic to Ker L, there exists an isomorphism J :ImQ → Ker L.
Lemma 2.11 Continuation theorem 15. Let X, Z be two Banach spaces, and let L be a Fredholm
mapping of index zero. Assume that N :
Ω → Z is L-compact on Ω with Ω is open bounded in X .
6 Advances in Difference Equations
Furthermore assume the following:
a for each λ ∈ 0, 1,x∈ ∂Ω ∩ Dom L, Lx
/
λNx;
b for each x ∈ ∂Ω ∩ Ker L, QNx
/
0;
c deg{JQN, Ω ∩ Ker L, 0}
/
0.
Then the operator equation Lx Nx has at least one solution in Dom L ∩
Ω.
Lemma 2.12 see 16. Let t
1
,t
2
∈ I
ω
.Ifg : T → R is ω-periodic, then
gt ≤ gt
1
κω
κ
|g
Δ
s|Δs, gt ≥ gt
2
−
κω
κ
|g
Δ
s|Δs. 2.6
3. Existence of Periodic Solutions
The fundamental theorem in this paper is stated as follows about the existence of an ω-
periodic solution.
Theorem 3.1. Suppose that (H) holds. Furthermore assume the following:
i b
i
t >D
i
t,t∈ T,i 1, 2,
ii
b
1
− D
1
>
c
1
m
1
,
iii
d
1
> r
1
c
2
m
2
,
iv
d
2
> r
2
,
then the system 1.4 has at least one ω-periodic solution.
Proof. Consider vector equation
z
Δ
tYt, where z
z
1
,z
2
,z
3
,z
4
T
,z
Δ
z
Δ
1
,z
Δ
2
,z
Δ
3
,z
Δ
4
T
,Y
Y
1
,Y
2
,Y
3
,Y
4
T
,
Y
1
b
1
t − D
1
t − a
1
t exp{z
1
t}−
c
1
t exp{z
3
t}
m
1
t exp{z
3
t} exp{z
1
t}
D
1
t exp{z
2
t − z
1
t},
Y
2
b
2
t − D
2
t − a
2
t exp{z
2
t} D
2
t exp{z
1
t − z
2
t},
Y
3
−r
1
t − a
3
t exp{z
3
t − τ
11
}
d
1
t exp{z
1
t − τ
12
}
m
1
t exp{z
3
t − τ
12
} exp{z
1
t − τ
12
}
−
c
2
t exp{z
4
t}
m
2
t exp{z
4
t} exp{z
3
t}
,
Y
4
−r
2
t − a
4
exp{z
4
t − τ
21
}
d
2
t exp{z
3
t − τ
22
}
m
2
t exp{z
4
t − τ
22
} exp{z
3
t − τ
22
}
.
3.1
Advances in Difference Equations 7
Define
X Z
z ∈ C
rd
T, R
4
|z
i
t ωz
i
t,i 1, 2, 3, 4, ∀t ∈ T
,
||z|| z
1
,z
2
,z
3
,z
4
T
4
i1
max
t∈I
ω
|z
i
t|,z∈ X or Z,
3.2
where |·|is the Euclidean norm. Then X and Z are both Banach spaces with the above norm
|| · ||.LetNztY, Lztz
Δ
t,PztQztz, z ∈ X. Then
Ker L R
4
, Im L
z ∈ Z
κω
κ
z
i
tΔt 0,i 1, 2, 3, 4, for t ∈ T
, 3.3
and dim KerL codim Im L 4. Since Im L is closed in X, then L is a Fredholm mapping
of index zero. It is easy to show that P, Q are continuous projectors such that Im P
Ker L, Ker Q Im L ImI − Q. Furthermore, the generalized inverse to L K
P
:ImL →
Ker P ∩ Dom L exists and is given by K
P
z
t
κ
zsΔs − 1/ω
κω
κ
t
κ
zsΔs Δt,thus
QNz
1
ω
κω
κ
YtΔt,
K
P
I − QNz
t
κ
YsΔs −
1
ω
κω
κ
t
κ
YsΔs Δt −
t − κ −
1
ω
κω
κ
t − κΔt
Y.
3.4
Obviously, QN : X → Z, K
P
I − QN : X → X are continuous. Since X is a Banach space,
using the Arzela-Ascoli theorem, it is easy to show that
K
P
I − QNΩ is compact for any
open bounded set Ω ⊂ X. Moreover, QN
Ω is bounded, thus, N is L-compact on Ω for any
open bounded set Ω ⊂ X. Corresponding to the operator equation Lz λNz, λ ∈ 0, 1,we
have
z
Δ
i
tλY
i
t,i 1, 2, 3, 4. 3.5
Suppose that z ∈ X is a solution of 3.5 for certain λ ∈ 0, 1. Integrating on both sides
of 3.5 from κ to κ ω with respect to t, we have
κω
κ
b
1
t − D
1
tΔt
κω
κ
D
1
t exp{z
2
t − z
1
t}Δt
κω
κ
a
1
t exp{z
1
t}Δt
κω
κ
c
1
t exp{z
3
t}
m
1
t exp{z
3
t} exp{z
1
t}
Δt,
3.6
κω
κ
b
2
t − D
2
tΔt
κω
κ
D
2
t exp{z
1
t − z
2
t}Δt
κω
κ
a
2
t exp{z
2
t}Δt,
3.7
8 Advances in Difference Equations
κω
κ
d
1
t exp{z
1
t − τ
12
}
m
1
t exp{z
3
t − τ
12
} exp{z
1
t − τ
12
}
Δt
r
1
ω
κω
κ
a
3
t exp{z
3
t − τ
11
}Δt
κω
κ
c
2
t exp{z
4
t}
m
2
t exp{z
4
t} exp{z
3
t}
Δt,
3.8
r
2
ω
κω
κ
a
4
t exp{z
4
t − τ
21
}Δt
κω
κ
d
2
t exp{z
3
t − τ
22
}
m
2
t exp{z
4
t − τ
22
} exp{z
3
t − τ
22
}
Δt.
3.9
It follows from 3.5 to 3.9 that
κω
κ
z
Δ
1
t
Δt ≤ 2
κω
κ
a
1
t exp{z
1
t}Δt 2
κω
κ
c
1
t exp{z
3
t}
m
1
t exp{z
3
t} exp{z
1
t}
Δt
< 2a
M
1
κω
κ
exp{z
1
t}Δt 2
c
1
m
1
ω,
3.10
κω
κ
|z
Δ
2
t|Δt ≤ 2a
M
2
κω
κ
exp{z
2
t}Δt,
3.11
κω
κ
z
Δ
3
t
Δt ≤ 2
κω
κ
d
1
t exp{z
1
t − τ
12
}
m
1
t exp{z
3
t − τ
12
} exp{z
1
t − τ
12
}
Δt
< 2
d
1
ω : l
3
,
3.12
κω
κ
z
Δ
4
t
Δt ≤ 2
κω
κ
d
2
t exp{z
3
t − τ
22
}
m
2
t exp{z
4
t − τ
22
} exp{z
3
t − τ
22
}
Δt
< 2
d
2
ω : l
4
.
3.13
Multiplying 3.6 by exp{z
1
t} and integrating over κ, κ ω gives
κω
κ
a
1
t exp{2z
1
t}Δt<
κω
κ
b
1
t − D
1
t exp{z
1
t}Δt
κω
κ
D
1
t exp{z
2
t}Δt,
3.14
which yields
a
L
1
κω
κ
exp{2z
1
t}Δt<b
1
− D
1
M
κω
κ
exp{z
1
t}Δt D
M
1
κω
κ
exp{z
2
t}Δt.
3.15
Advances in Difference Equations 9
By using the inequality
κω
κ
exp{z
1
t}Δt
2
≤ ω
κω
κ
exp{2z
1
t}Δt, we have
a
L
1
ω
κω
κ
exp{z
1
t}Δt
2
< b
1
− D
1
M
κω
κ
exp{z
1
t}Δt D
M
1
κω
κ
exp{z
2
t}Δt.
3.16
Then
2a
L
1
ω
κω
κ
exp{z
1
t}Δt
< b
1
− D
1
M
b
1
− D
1
M
2
4a
L
1
D
M
1
ω
κω
κ
exp{z
2
t}Δt
1/2
.
3.17
By using the inequality a b
1/2
<a
1/2
b
1/2
,a>0,b>0, we derive from 3.17 that
a
L
1
ω
κω
κ
exp{z
1
t}Δt<b
1
− D
1
M
a
L
1
D
M
1
ω
κω
κ
exp{z
2
t}Δt
1/2
. 3.18
Similarly, multiplying 3.7 by exp{z
2
t} and integrating over κ, κ ω, then synthesize the
above, we obtain
a
L
2
ω
κω
κ
exp{z
2
t}Δt<b
2
− D
2
M
a
L
2
D
M
2
ω
κω
κ
exp{z
1
t}Δt
1/2
. 3.19
It follows from 3.18 and 3.19 that
a
L
1
a
L
2
κω
κ
exp{z
1
t}Δt
<ω
a
L
2
b
1
− D
1
M
ωa
L
1
D
M
1
ωb
2
− D
2
M
ωa
L
2
D
M
2
κω
κ
exp{z
1
t}Δt
1/4
,
3.20
so, there exists a positive constant ρ
1
such that
κω
κ
exp{z
1
t}Δt<ρ
1
, 3.21
which together with 3.19, there also exists a positive constant ρ
2
such that
κω
κ
exp{z
2
t}Δt<ρ
2
. 3.22
10 Advances in Difference Equations
This, together with 3.11, 3.12,and3.21,leadsto
κω
κ
z
Δ
1
t
Δt<2a
M
1
ρ
1
2
c
1
m
1
ω : l
1
,
κω
κ
z
Δ
2
t
Δt<2a
M
2
ρ
2
: l
2
.
3.23
Since z
1
t,z
2
t,z
3
t,z
4
t
T
∈ X, there exist some points ξ
i
,η
i
∈ I
ω
,i 1, 2, 3, 4,
such that
z
i
ξ
i
min
t∈I
ω
{z
i
t},z
i
η
i
max
t∈I
ω
{z
i
t},i 1, 2, 3, 4. 3.24
It follows from 3.21 and 3.22 that
z
i
ξ
i
< ln
ρ
i
ω
: L
i
,i 1, 2. 3.25
From 3.8 and 3.9,weobtainthat
z
3
ξ
3
< ln
d
1
− r
1
a
3
: L
3
,z
4
ξ
4
< ln
d
2
− r
2
a
4
: L
4
. 3.26
This, together with 3.12, 3.13,and3.26, deduces
z
i
t ≤ z
i
ξ
i
κω
κ
z
Δ
i
t
Δt<L
i
l
i
,i 1, 2, 3, 4. 3.27
From 3.6 and 3.24, we have
z
1
η
1
≥ ln
b
1
− D
1
−
c
1
m
1
a
1
: δ
1
. 3.28
From 3.7 and 3.24, it yields that
z
2
η
2
> ln
b
2
− D
2
a
2
: δ
2
. 3.29
Advances in Difference Equations 11
Noticing that
κω
κ
exp{zt − τ
1
}Δt
κω
κ
exp{zt − τ
2
}Δt,from3.8 and 3.9, deduces
κω
κ
d
1
t exp{z
1
t − τ
12
}
m
1
t exp{z
3
t − τ
12
} exp{z
1
t − τ
12
}
Δt
<
r
1
ω a
M
3
κω
κ
exp{z
3
t − τ
12
}Δt
c
2
m
2
ω,
κω
κ
d
2
t exp{z
3
t − τ
22
}
m
2
t exp{z
4
t − τ
22
} exp{z
3
t − τ
22
}
Δt
<
r
2
ω a
M
4
κω
κ
exp{z
4
t − τ
21
}Δt.
3.30
There exist two points t
i
∈ κ, κ ωi 1, 2 such that
d
1
t
1
τ
12
exp{z
1
t
1
}
m
1
t
1
τ
12
exp{z
3
t
1
} exp{z
1
t
1
}
<
r
1
a
3
exp{z
3
t
1
}
c
2
m
2
,
d
2
t
2
τ
22
exp{z
3
t
2
}
m
2
t
2
τ
22
exp{z
4
t
2
} exp{z
3
t
2
}
<
r
2
ω a
4
exp{z
4
t
2
}.
3.31
Hence,
z
3
t
1
> ln
1
2a
3
m
M
1
r
1
m
M
1
m
M
1
c
2
m
2
a
3
A
1
2
4a
3
m
M
1
A
1
d
1
− r
1
−
c
2
m
2
−
r
1
m
M
1
m
M
1
c
2
m
2
a
3
A
1
: δ
3
,
z
4
t
2
> ln
r
2
m
M
2
a
4
A
2
2
4a
4
m
M
2
A
2
d
2
− r
2
−
r
2
m
M
2
a
4
A
2
2a
4
m
M
2
: δ
4
,
3.32
where A
1
exp{z
1
ξ
1
},A
2
exp{z
3
ξ
3
}. Then, this, together with 3.12, 3.13, 3.23,
3.28, 3.29,and3.32, deduces
z
i
t ≥ z
i
η
i
−
κω
κ
z
Δ
i
t
Δt ≥ δ
i
− l
i
,i 1, 2, 3, 4, for any t ∈ κ, κ ω. 3.33
It follows from 3.27 to 3.33 that
max
t∈I
ω
|z
i
t|≤max{|L
i
l
i
|, |δ
i
− l
i
|} : B
i
,i 1, 2, 3, 4. 3.34
12 Advances in Difference Equations
From 3.34, we clearly know that B
i
i 1, 2, 3, 4 are independent of λ,andfromthe
representation of QNz, it is easy to know that there exist points ζ
i
∈ κ, κ ωi 1, 2, 3, 4
such that QNz Y
∗
z
1
,z
2
,z
3
,z
4
, where
Y
∗
⎛
⎜
⎜
⎝
z
1
z
2
z
3
z
4
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b
1
− D
1
− a
1
exp{z
1
}−
c
1
exp{z
3
}
m
1
ζ
1
exp{z
3
} exp{z
1
}
−
D
1
exp{z
2
− z
1
}
b
2
− D
2
− a
2
exp{z
2
} D
2
exp{z
1
− z
2
}
−
r
1
− a
3
exp{z
3
}
d
1
exp{z
1
}
m
1
ζ
2
exp{z
3
} exp{z
1
}
−
c
2
exp{z
4
}
m
2
ζ
3
exp{z
4
} exp{z
3
}
−
r
2
− a
4
exp{z
4
}
d
2
exp{z
3
}
m
2
ζ
4
exp{z
4
} exp{z
3
}
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
3.35
Take B
4
i0
B
i
, where B
0
is taken sufficiently large such that B
0
≥
4
i1
|L
i
|
4
i1
|δ
i
|,and
such that each solution u
∗
u
∗
1
,u
∗
2
,u
∗
3
,u
∗
4
T
of the system Y
∗
u
1
,u
2
,u
3
,u
4
T
0satisfies
||u
∗
||
4
i1
|u
∗
i
| <B
0
if the system 3.35 has solutions. Now take Ω{z
1
,z
2
,z
3
,z
4
T
∈
X|||z
1
,z
2
,z
3
,z
4
T
|| <B}. Then it is clear that Ω verifies the requirement a of Lemma 2.11.
When z
1
,z
2
,z
3
,z
4
T
∈ ∂Ω ∩ Ker L ∂Ω ∩ R
4
, z
1
,z
2
,z
3
,z
4
T
is a constant vector in R
4
with ||z
1
,z
2
,z
3
,z
4
T
|| B, from the definition of B, we can naturally derive QNz
/
0 whether
the system 3.35 has solutions or not. This shows that the condition b of Lemma 2.11 is
satisfied.
Finally, we will prove that the condition c of Lemma 2.11 is valid. Define the
homotopy H
μ
z
1
,z
2
,z
3
,z
4
:DomL × 0, 1 → R
4
by
H
μ
z
1
,z
2
,z
3
,z
4
μQN
z
1
,z
2
,z
3
,z
4
1 − μG
z
1
,z
2
,z
3
,z
4
, for μ ∈ 0, 1, 3.36
where
G
⎛
⎜
⎜
⎝
z
1
z
2
z
3
z
4
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b
1
− D
1
− a
1
exp{z
1
}
b
2
− D
2
− a
2
exp{z
2
}
−
r
1
− a
3
exp{z
3
}
d
1
exp{z
1
}
m
1
ζ
2
exp{z
3
} exp{z
1
}
−
r
2
− a
4
exp{z
4
}
d
2
exp{z
3
}
m
2
ζ
4
exp{z
4
} exp{z
3
}
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, 3.37
where z
1
,z
2
,z
3
,z
4
T
∈ R
4
,μ∈ 0, 1 is a parameter. From 3.37, it is easy to show that
0
/
∈ H
μ
∂Ω ∩ ker L. Moreover, one can easily show that the algebraic equation
Advances in Difference Equations 13
b
1
− D
1
− a
1
u
1
0,
b
2
− D
2
− a
2
u
2
0,
−
r
1
− a
3
u
3
d
1
u
1
m
1
ζ
2
u
3
u
1
0,
−
r
2
− a
4
u
4
d
2
u
3
m
2
ζ
4
u
4
u
3
0
3.38
has a unique positive solution u
1
,u
2
,u
3
,u
4
T
in R
4
.NotethatJ I identical mapping,
since Im Q Ker L, according to the invariance property of homotopy, direct calculation
produces
deg
JQNz
1
,z
2
,z
3
,z
4
T
, Ω ∩ Ker L, 0, 0, 0, 0
T
deg
Gz
1
,z
2
,z
3
,z
4
T
, Ω ∩ Ker L, 0, 0, 0, 0
T
sign
−
a
1
u
∗
1
00 0
0 −
a
2
u
∗
2
00
d
1
m
1
ζ
2
u
∗
3
m
1
ζ
2
u
∗
3
u
∗
1
2
0 −a
3
−
d
1
m
1
ζ
2
u
∗
1
m
1
ζ
2
u
∗
3
u
∗
1
2
0
00
d
2
m
2
ζ
4
u
∗
4
m
2
ζ
4
u
∗
4
u
∗
3
2
−a
4
−
d
2
m
2
ζ
4
u
∗
3
m
2
ζ
4
u
∗
4
u
∗
3
2
1,
3.39
where deg{·, ·, ·} is the Brouwer degree. By now we have proved that Ω verifies all
requirements of Lemma 2.11. Therefore, 1.4 has at least one ω-periodic solution in Dom L ∩
Ω. The proof is complete.
Corollary 3.2. If the conditions in Theorem 3.1 hold, then both the corresponding continuous model
1.6 and the discrete model 1.7 have at least one ω-periodic solution.
Remark 3.3. If T R and τ
11
≡ τ
21
≡ 0in1.6, then the system 1.6 reduces t o the continuous
ratio-dependence predator-prey diffusive system proposed in 17.
Remark 3.4. If we only consider the prey population in one-patch environment and ignore
the dispersal process in the system 1.4, then the classical ratio-dependence two species
predator-prey model in particular of 1.4 with Michaelis-Menten functional response and
time delay on time scales
z
Δ
1
tr
1
t − at exp{z
1
t}−
ct exp{z
2
t}
mt exp{z
2
t} exp{z
1
t}
,
z
Δ
2
t−r
2
t
dt exp{z
1
t − τ}
mt exp{z
2
t − τ} exp{z
1
t − τ}
,
3.40
14 Advances in Difference Equations
where at,ct,dt,r
i
t,mt ∈ C
rd
T, R
i 1, 2 are positive ω-periodic functions, τ is
nonnegative constant. It is easy to obtain the corresponding conclusions on time scales for
the system 3.40.
Corollary 3.5. Suppose that (i)
r
1
>
c
m
, (ii) dt >r
2
t,t∈ T hold, then 3.40 has at least one
ω-periodic solution.
Remark 3.6. The result in Corollary 3.5 is same as those for the corresponding continuous and
discrete systems.
Acknowledgments
The author is very grateful to his supervisor Prof. M. Fan and the anonymous referees for
their many valuable comments and suggestions which greatly improved the presentation
of this paper. This work is supported by the Foundation for subjects development of
Harbin University no. HXK200716 and by the Foundation for Scientific Research Projects of
Education Department of Hei-longjiang Province of China no. 11513043.
References
1 E. Beretta and Y. Kuang, “Convergence results in a well-known delayed predator-prey s ystem,”
Journal of Mathematical Analysis and Applications, vol. 204, no. 3, pp. 840–853, 1996.
2 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
3 Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics
in Science and Engineering, Academic Press, Boston, Mass, USA, 1993.
4 Y. Kuang, “Rich dynamics of Gause-type ratio-dependent predator-prey system,” in Differential
Equations with Applications to Biology (Halifax, NS, 1997), vol. 21 of Fields Institute Communications, pp.
325–337, American Mathematical Society, Providence, RI, USA, 1999.
5 R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of
Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.
6 M. Fan and K. Wang, “Periodicity in a delayed ratio-dependent predator-prey system,” Journal of
Mathematical Analysis and Applications, vol. 262, no. 1, pp. 179–190, 2001.
7 M. Fan and Q. Wang, “Periodic solutions of a class of nonautonomous discrete time semi-ratio-
dependent predator-prey systems,” Discrete and Continuous Dynamical Systems. Series B, vol. 4, no.
3, pp. 563–574, 2004.
8 S B. Hsu, T W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-
dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001.
9 C. Jost, O. Arino, and R. Arditi, “About deterministic extinction in ratio-dependent predator-prey
models,” Bulletin of Mathematical Biology, vol. 61, no. 1, pp. 19–32, 1999.
10 D. Xiao and S. Ruan, “Multiple bifurcations in a delayed predator-prey system with nonmonotonic
functional response,” Journal of Differential Equations, vol. 176, no. 2, pp. 494–510, 2001.
11 R. Xu and L. Chen, “Persistence and global stability for a three-species ratio-dependent predator-prey
system with time delays in two-patch environments,” Acta Mathematica Scientia. Series B, vol. 22, no.
4, pp. 533–541, 2002.
12 H F. Huo and W T. Li, “Periodic solution of a delayed predator-prey system with Michaelis-Menten
type functional response,” Journal of Computational and Applied Mathematics, vol. 166, no. 2, pp. 453–
463, 2004.
13 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkh
¨
auser, Boston, Mass, USA, 2001.
14 E. R. Kaufmann and Y. N. Raffoul, “Periodic solutions for a neutral nonlinear dynamical equation on
a time scale,” Journal of Mathematical Analysis and Applications, vol. 319, no. 1, pp. 315–325, 2006.
15 R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.
Advances in Difference Equations 15
16 M. Bohner, M. Fan, and J. Zhang, “Existence of periodic solutions in predator-prey and competition
dynamic systems,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1193–1204, 2006.
17 S. Wen, S. Chen, and H. Mei, “Positive periodic solution of a more realistic three-species Lotka-
Volterra model with delay and density regulation,” Chaos, Solitons and Fractals. In press.
