Volume 2009, Article ID 141589, 15 pagesdoi:10.1155/2009/141589 Research Article Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion Syste
Trang 1Volume 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,liouj2008@126.com
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 the time scaleT 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
Copyrightq 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 Ginzburg5 first proposed the following ratio-dependent predator-prey model:
·
x xa − bx − cxy
my x ,
·
y y
my x
Trang 22 Advances in Difference Equations
which incorporates mutual interference by predators, where gx cx/my x is a
Michaelis-Menten type functional response function Equation 1.1 has been studied by many authors and seen great progresse.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:
x1t x1t
a1− a11x1t − a13x3t
m13x3t x1t−
a14x4t
m14x4t x1t
D1tx2t − x1t,
x2t x2ta2− a22x2t D2tx1t − x2t,
x3t x3t
− a3 a31x1t − τ1
m13x3t − τ1 x1t − τ1
,
x4t x4t
− a4 a41x1t − τ2
m14x4t − τ2 x1t − τ2
.
1.2
In view of periodicity of the actual environment, Huo and Li12 investigated a more general delayed ratio-dependent predator-prey model with periodic coefficients of the form
·
x1t x1t
a1t − a11tx1t − τ11 −a12tx2t
m1 x1t
,
·
x2t x2t
− a2t a21tx1t − τ21
m1 x1t − τ21 − a22tx2t − τ22 −a23tx3t
m2 x2t
,
·
x3t x3t
− a3t a32tx2t − τ32
m2 x2t − τ32 − a33tx3t − τ33
.
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:
zΔ1t b1t − a1t exp{z1t} − c1t exp{z3t}
m1t exp{z3t} exp{z1t}
D1texp{z2t − z1t} − 1,
zΔ2t b2t − a2t exp{z2t} D2texp{z1t − z2t} − 1,
Trang 3zΔ3t −r1t − a3t exp{z3t − τ11} d1t exp{z1t − τ12}
m1t exp{z3t − τ12} exp{z1t − τ12}
− c2t exp{z4t}
m2t exp{z4t} exp{z3t} ,
zΔ4t −r2t − a4t exp{z4t − τ21} d2t exp{z3t − τ22}
m2t exp{z4t − τ22} exp{z3t − τ22}
1.4 with the initial conditions
z i s ϕ i s ≥ 0, s ∈ −τ, 0 ∩ T, ϕ i 0 > 0, ϕ i s ∈ Crd−τ, 0 ∩ T, R, i 1, 2, 3, 4,
1.5
where τ max{τ ij , i, j 1, 2} In 1.4, z i t represents the prey population in the ith patch
i 1, 2, and z i t i 3, 4 represents the predator population z1t is the prey for z3t, and
z3t is the prey for z4t 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 system1.4:
H a i t ∈ CrdT, R i 1, 2, 3, 4, b i t, c i t, d i t, r i t, m i t, D i t ∈ CrdT, R i
1, 2 are all rd-continuous positive periodic functions with period ω > 0; τ ij i, j 1, 2 are
nonnegative constants
In1.4, set x i t exp{z i t}, y j t exp{z j2t}, 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
x1t x1t
b1t − a1tx1t − c1ty1t
m1ty1t x1t
D1tx2t − x1t,
x2t x2tb2t − a2tx2t D2tx1t − x2t,
y1t y1t
− r1t − a3ty1t − τ11 d1tx1t − τ12
m1ty1t − τ12 x1t − τ12 −
c2ty2t
m2ty2t y1t
,
y2t y2t
− r2t − a4ty2t − τ21 d2ty1t − τ22
m2ty2t − τ22 y1t − τ22
.
1.6
IfT Z, then 1.4 is reformulated as
x1k 1 x1k exp
b1k − a1kx1k − c1ky1k
m1ky1k x1k D1k
x2k
x1k− 1
,
x2k 1 x2k exp
b2k − a2kx2k D2k
x1k
x2k− 1
,
Trang 44 Advances in Difference Equations
y1k 1 y1k exp
− r1k − a3ky1k − τ11 d1kx1k − τ12
m1ky1k − τ12 x1k − τ12
m2ky2k − τ12 y1k − τ12
,
y2k 1 y2k exp
− r2k − a4ky2k − τ21 d2ky1k − τ22
m2ky2k − τ22 y1k − τ22
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 of1.6
2 Preliminaries
A time scaleT is an arbitrary nonempty closed subset of the real numbers R Throughout the paper, we assume the time scaleT is unbounded above and below, such as R, Z and
∪
k∈Z2k, 2k 1 The following definitions and lemmas can be found in 13
Definition 2.1 The forward jump operator σ : T → T, the backward jump operator ρ : T →
T, and the graininess μ : T → R 0, ∞ are defined, respectively, by
σ t inf{s ∈ T | s > t}, ρt sup{s ∈ T | s < t}, μt σt − t for t ∈ T. 2.1
If σt t, then t is called right-dense otherwise: right-scattered, and if ρt t, then t is
called left-denseotherwise: left-scattered
IfT has a left-scattered maximum m, then T k T \ {m}; otherwise T k T If T has a
right-scattered minimum m, thenTk T \ {m}; otherwise T k T
Definition 2.2 Assume f : T → R is a function and let t ∈ T k Then one defines fΔt to be the
numberprovided it exists with the property that given any ε > 0, there is a neighborhood
U of t such that
f σt − fs − fΔtσt − s ≤ ε|σt − s| ∀s ∈ U. 2.2
In this case, fΔt is called the delta or Hilger derivative of f at t Moreover, f is said to be
delta or Hilger differentiable on T if fΔt exists for all t ∈ T k A function F :T → R is called
an antiderivative of f : T → R provided FΔt ft for all t ∈ T k Then one defines
s
r
Definition 2.3 A function f : T → R is said to be rd-continuous if it is continuous at right-dense points inT and its left-sided limits exists finite at left-dense points in T The set of
rd-continuous functions f : T → R will be denoted by CrdT, R.
Trang 5Definition 2.4 If a ∈ T, inf T −∞, and f is rd-continuous on −∞, a, then one defines the
improper integral by
a
−∞f tΔt lim
T→ −∞
a
T
provided this limit exists, and one says that the improper integral converges in this case
Definition 2.5see 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 2.6see 14 Let T / 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 ω ft for all t ∈ T and ω is the smallest number such that ft ω ft.
IfT R, one says that f is periodic with period ω > 0 if ω is the smallest positive number such that ft ω ft 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 ∈ CrdT, R, then
a b
a αft βgtΔt α b
a f tΔt β b
a g tΔt;
b if ft ≥ 0 for all a ≤ t < b, then b
a f tΔt ≥ 0;
c if |ft| ≤ gt on a, b : {t ∈ T | a ≤ t < b}, then | b
a f tΔt| ≤ b
a g tΔt.
Lemma 2.10 If fΔt ≥ 0, then f is nondecreasing.
Notation 1 To facilitate the discussion below, we now introduce some notation to be used
throughout this paper LetT 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
f sΔs 1
ω
κ ω
κ
f sΔs, f M sup
t∈Tf t, f L inf
t∈Tf t, 2.5 where f ∈ CrdT, R is an ω-periodic function, that is, ft ω ft for all t ∈ T, t ω ∈ T Notation 2 Let X, Z be two Banach spaces, let L : Dom L ⊂ 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 : Im Q → 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
Trang 66 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 t1, t2∈ I ω If g : T → R is ω-periodic, then
g t ≤ gt1
κ ω
κ
|gΔs|Δs, g t ≥ gt2 −
κ ω
κ
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 b1− D1>c1
m1
,
iii d1> r1c2
m2
,
iv d2> r2,
then the system1.4 has at least one ω-periodic solution.
Proof Consider vector equation
zΔt Yt, where z z1, z2, z3, z4T
, zΔzΔ1, zΔ2, zΔ3, zΔ4T
, Y Y1, Y2, Y3, Y4T
,
Y1 b1t − D1t − a1t exp{z1t} − c1t exp{z3t}
m1t exp{z3t} exp{z1t}
D1t exp{z2t − z1t},
Y2 b2t − D2t − a2t exp{z2t} D2t exp{z1t − z2t},
Y3 −r1t − a3t exp{z3t − τ11} d1t exp{z1t − τ12}
m1t exp{z3t − τ12} exp{z1t − τ12}
− c2t exp{z4t}
m2t exp{z4t} exp{z3t} ,
Y4 −r2t − a4exp{z4t − τ21} d2t exp{z3t − τ22}
m2t exp{z4t − τ22} exp{z3t − τ22}.
3.1
Trang 7X Z z ∈ CrdT, R4|z i t ω z i t, i 1, 2, 3, 4, ∀t ∈ T ,
||z|| z1, z2, z3, z4T 4
i1
max
t ∈I ω
where| · | is the Euclidean norm Then X and Z are both Banach spaces with the above norm
|| · || Let Nzt Y, Lzt zΔt, Pzt Qzt z, z ∈ X Then
Ker L R4, Im L
z ∈ Z
κ ω
κ
z i tΔt 0, i 1, 2, 3, 4, for t ∈ T
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 : Im L →
Ker P ∩ Dom L exists and is given by K P z t
κ z sΔs − 1/ω κ ω
κ
t
κ z sΔs Δt, thus
QNz 1
ω
κ ω
κ
Y tΔt,
K P I − QNz
t
κ
Y sΔs − 1
ω
κ ω
κ
t
κ
Y sΔs Δt −
t − κ − 1 ω
κ ω
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 setΩ ⊂ X Moreover, QNΩ is bounded, thus, N is L-compact on Ω for any
open bounded setΩ ⊂ X Corresponding to the operator equation Lz λNz, λ ∈ 0, 1, we
have
Suppose that z ∈ X is a solution of 3.5 for certain λ ∈ 0, 1 Integrating on both sides
of3.5 from κ to κ ω with respect to t, we have
κ ω
κ
b1t − D1tΔt
κ ω
κ
D1t exp{z2t − z1t}Δt
κ ω
κ
a1t exp{z1t}Δt
κ ω
κ
c1t exp{z3t}
m1t exp{z3t} exp{z1t} Δt,
3.6
κ ω
κ
b2t − D2tΔt
κ ω
κ
D2t exp{z1t − z2t}Δt
κ ω
κ
a2t exp{z2t}Δt,
3.7
Trang 88 Advances in Difference Equations
κ ω
κ
d1t exp{z1t − τ12}
m1t exp{z3t − τ12} exp{z1t − τ12}Δt
r1ω
κ ω
κ
a3t exp{z3t − τ11}Δt
κ ω
κ
c2t exp{z4t}
m2t exp{z4t} exp{z3t} Δt,
3.8
r2ω
κ ω
κ
a4t exp{z4t − τ21}Δt
κ ω
κ
d2t exp{z3t − τ22}
m2t exp{z4t − τ22} exp{z3t − τ22}Δt
3.9
It follows from3.5 to 3.9 that
κ ω
κ
zΔ
1tΔt ≤ 2 κ ω
κ
a1t exp{z1t}Δt 2
κ ω
κ
c1t exp{z3t}
m1t exp{z3t} exp{z1t} Δt
< 2a M1
κ ω
κ
exp{z1t}Δt 2c1
m1
ω,
3.10
κ ω
κ
|zΔ
2t|Δt ≤ 2a M
2
κ ω
κ
κ ω
κ
zΔ
3tΔt ≤ 2 κ ω
κ
d1t exp{z1t − τ12}
m1t exp{z3t − τ12} exp{z1t − τ12}Δt
< 2d1ω : l3,
3.12
κ ω
κ
zΔ
4tΔt ≤ 2 κ ω
κ
d2t exp{z3t − τ22}
m2t exp{z4t − τ22} exp{z3t − τ22}Δt
< 2d2ω : l4.
3.13
Multiplying3.6 by exp{z1t} and integrating over κ, κ ω gives
κ ω
κ
a1t exp{2z1t}Δt <
κ ω
κ
b1t − D1t exp{z1t}Δt
κ ω
κ
D1t exp{z2t}Δt,
3.14 which yields
a L
1
κ ω
κ
exp{2z1t}Δt < b1− D1M
κ ω
κ
exp{z1t}Δt D M
1
κ ω
κ
exp{z2t}Δt. 3.15
Trang 9By using the inequality κ ω
κ exp{z1t}Δt2≤ ω κ ω
κ exp{2z1t}Δt, we have
a L
1
ω
κ ω
κ
exp{z1t}Δt
2
< b1− D1M
κ ω
κ
exp{z1t}Δt D M
1
κ ω
κ
exp{z2t}Δt.
3.16 Then
2a L1
ω
κ ω
κ
exp{z1t}Δt
< b1− D1M
b1− D1M24a L1D M
1
ω
κ ω
κ
exp{z2t}Δt
1/2
.
3.17
By using the inequalitya b 1/2 < a 1/2 b 1/2 , a > 0, b > 0, we derive from3.17 that
a L
1
ω
κ ω
κ
exp{z1t}Δt < b1− D1M
a L
1D M
1
ω
κ ω
κ
exp{z2t}Δt
1/2
Similarly, multiplying3.7 by exp{z2t} and integrating over κ, κ ω, then synthesize the
above, we obtain
a L2
ω
κ ω
κ
exp{z2t}Δt < b2− D2M
a L2D2M ω
κ ω
κ
exp{z1t}Δt
1/2
It follows from3.18 and 3.19 that
a L1
a L
2
κ ω
κ
exp{z1t}Δt
< ω
a L
2b1− D1M
ωa L1D1M
ω b2− D2M
ωa L2D M2
κ ω
κ
exp{z1t}Δt
1/4
,
3.20
so, there exists a positive constant ρ1such that
κ ω
κ
which together with3.19, there also exists a positive constant ρ2such that
κ ω
κ
Trang 1010 Advances in Difference Equations This, together with3.11, 3.12, and 3.21, leads to
κ ω
κ
zΔ
1tΔt < 2a M
1 ρ1 2
c1
m1
ω : l1,
κ ω
κ
zΔ
2tΔt < 2a M
2 ρ2: l2.
3.23
Sincez1t, z2t, z3t, z4t 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 from3.21 and 3.22 that
z i ξ i < ln ρ i
From3.8 and 3.9, we obtain that
z3ξ3 < ln d1− r1
a3 : L3, z4ξ4 < ln d2− r2
This, together with3.12, 3.13, and 3.26, deduces
z i t ≤ z i ξ i
κ ω
κ
zΔ
i tΔt < L i l i , i 1, 2, 3, 4. 3.27
From3.6 and 3.24, we have
z1η1 ≥ lnb1− D1−
c1
m1
From3.7 and 3.24, it yields that
z2η2 > ln b2− D2
... Trang 9By using the inequality κ ω
κ exp{z1t}Δt2≤... exp{z2t}Δt,
3.7
Trang 88 Advances in Difference Equations
κ... is L-compact on Ω with Ω is open bounded in X
Trang 66 Advances in Difference Equations
Furthermore