Gen. Math. Notes, Vol. 3, No. 1, March 2011, pp.46-54
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Periodic Solutions for A Predator-Prey Model with
Beddington-DeAngelis Type Functional Response
on Time Scales
Lihui Yang1 , Jianguang Yang2 and Qianhong Zhong3
1
Department of Mathematics and Computing Science, Hunan City University,
Yiyang 413000, P R China
E-mail: llhh.yang@gmail.com
2
Department of Mathematics and Computing Science, Hunan City University,
Yiyang 413000, P R China
E-mail: yangjianguang001@163.com
3
School of Mathematics and Statistics, Guizhou Key Laboratory of Economics System
Simulation, Guizhou College of Finance and Economics, Guiyang 550004, PR China
E-mail: zqianhong68@163.com
(Received: 26-11-10/ Accepted: 16-12-10)
Abstract
TIn this paper, the existence of periodic solutions of a predator-prey model
with Beddington-DeAngelis type functional response is investigated by using
the Gaines and Mawhin, s continuation theorem of coincidence degree theory
on time scales. Some conditions are obtained for the existence of periodic solutions. The approach is unified to provide the existence of the desired solutions
for the continuous differential equations and discrete difference equations.
Keywords: Beddington-DeAngelis type, coincidence degree, periodic solution, predator-prey system, time scales.
1
Introduction
From the population data of Canadian Lynx and Snowshoe hare from 1840s, we
know that interaction between a pair of predator-prey influences the population
growth of both species. The first predator-prey model was formulated by Alfred
Periodic Solutions for A Predator-Prey Model with...
47
James Lotka in 1925[12], and Vito Volterra in 1926[15]. After that, a lot of
more complicated but realistic continuous and discrete predator-prey models
have been proposed and investigated by many authors[5,6,10,14,16]. In 2010,
Baek [2] studied the stability and periodic solutions of the following predatorprey model:
i
h
(
x(t)
1 (t)x2 (t)
ẋ1 = ax1 (t) 1 − K − bxex
,
2 (t)+x1 (t)+c
(1)
ex1 (t)x2 (t)
ẋ2 = −Dx2 (t) + bx2 (t)+x1 (t)+c ,
where x1 (t), x2 (t) are the population densities of prey and the predator at time
t respectively, the positive constants K, a, m, D are the carrying capacity of
the prey, intrinsic growth rate of the prey, the conversion rate, the death rate
of the predator, respectively. The term by denotes the mutual interference
between predators. In details, one can see Ref.[2].
Since biological and environmental parameters are naturally subjected to
fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in periodically changing environment, we incorporate the varying property of the parameters into the
model and reconstruct it as follows:
h
i
(
x(t)
1 (t)x2 (t)
,
ẋ1 = a(t)x1 (t) 1 − K(t)
− b(t)xe(t)x
2 (t)+x1 (t)+c(t)
(2)
1 (t)x2 (t)
ẋ2 = −D(t)x2 (t) + b(t)xe(t)x
.
(t)+x
(t)+c(t)
2
1
It must be pointed out that although there are numerous papers investigating
the existence of positive periodic solutions of differential or difference equations by using the coincidence degree theory in mathematical ecology, one
often deal with these types of equations in a different way to prove the existence results. This motivated us to think wether we can explore such an
existence problem in an unified way. In order to unify continuous and discrete
analysis, the theory of time scales(measure chain), which has recently received
a great many attention, was introduced by Stefan Hilger in his PhD thesis
in 1998. After that, people have done a great many research about dynamic
equations on time scales. Moreover, many results on the existence of periodic solutions of dynamic equations have been reported[1,3,4,8,9]. Motivated
by Refs.[1,3,4,8,9], in this paper, we will devote our attention to investigating
the existence of periodic solutions of the following predator-prey model with
Beddington-DeAngelis type on time scales:
h
i
(
exp(x2 (t))
exp(x(t))
− b(t) exp(xe(t)
,
x∆
(t)
=
a(t)
1
−
1
K(t)
2 (t))+exp(x1 (t))+c(t)
(3)
e(t) exp(x1 (t))
x∆
2 (t) = −D(t) + b(t) exp(x2 (t))+exp(x1 (t))+c(t) ,
where x1 (t), x2 (t) are the population densities of prey and the predator at
time t respectively, K(t), a(t), m(t), D(t) are the carrying capacity of the prey,
48
Lihui Yang et al.
intrinsic growth rate of the prey, the conversion rate, the death rate of the
predator at time t respectively and they are rd-continuous positive ω-periodic
functions.
We believed that it is the first time to deal with the existence problem of
periodic solution for system (3) on time scales.
The remainder of the paper is organized as follows: in Section 2, we present
some preliminary results including some basic knowledge for dynamic systems
on time scales. In Section 3, we prove our main result.
2
Preliminary Results on Time Scales
In order to make an easy and convenient reading of this paper, we present
some foundational definitions and notations on time scales so that the paper
self-contained. For more detail, one can see[4,9].
Definition 2.1 A time scale is an arbitrary nonempty closed subset T of
R, the real numbers. The set T inherits the standard topology of R.
Definition 2.2 The forward jump operator σ : T → T, the backward jump
operator σ : T → T, and the graininess µ : T → R+ = [0, ∞) are defined,
respectively, by
σ(t) := inf{s ∈ T : s > t}, ρ(t) := sup{s ∈ T : s < t}, µ(t) = σ(t)−t for t ∈ T.
If σ(t) = t, then t is called right-dense (otherwise: right-scattered), and if
ρ(t) = t, then t is called left-dense (otherwise: left-scattered).
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-sides limits exists(finite)
1
at left-dense points in T. The set rd-continuous functions is shown by Crd
=
Crd (T) = Crd (T, R).
Definition 2.4 For f : T → R and t ∈ R, we define f ∆ (t), the deltaderivative of f at t, to be the number(provided it exists) with the property that,
given any ε > 0, there is a neighborhood U of t in T such that
|[f (σ(t)) − f (s)] − f ∆ (t)[σ(t) − s]| ≤ ε|σ(t) − s| f or all s ∈ U.
Thus f is said to be delta-differentiable if its delta-derivative exists. The set
of functions f : T → R that are delta-differentiable and whose delta-derivative
1
1
are rd-continuous functions is denoted by Crd = Crd
(T) = Crd
(T, R).
Definition 2.5 A function F : T → R is called a delta-antiderivative
of
Rs
f : T → R provided F ∆ (t) = f (t), for all t ∈ T. Then we write r f (t)∆t :=
F (s) − F (r) f or all s, t ∈ T.
Periodic Solutions for A Predator-Prey Model with...
49
For the usual time scales T = R, rd-continuous coincides with the usual
continuity in calculus. Moreover, every rd-continuous function on T has a
delta-antiderivative[11] . For more information about the above definitions and
their related concepts, one can see Refs.[1,3,4,8,9].
3
Existence of Periodic Solutions
For convenience and simplicity in the following discussion, we always use the
notations below throughout the paper. The time scale T is assumed to be
+
ω-periodic, i.e., t ∈ T implies t + ω ∈ T, κ = min{R
∩ T}, RIω = [κ, κ +
R
κ+ω
1
l
u
ω] ∩ T, g = inf t∈T g(t), g = maxt∈T g(t), ḡ = ω Iω g(s)∆s = ω1 κ g(s)∆s,
where g ∈ Crd (T) is an ω-periodic real function, i.e., g(t + ω) = g(t) for all
t ∈ T.
In order to explore the existence of positive periodic solutions of (3) and
for the reader, s convenience, we shall first summarize below a few concepts
and results without proof, borrowing from Ref.[7].
Let X, Y be normed vector spaces, L : DomL ⊂ X → Y is a linear mapping, N : X → Y is a continuous mapping. The mapping L will be called a
Fredholm mapping of index zero if dimKerL = codimImL < +∞ and ImL is
closed in Y . If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that ImP = KerL, ImL =
KerQ = Im(I − Q), It follows that L | DomL ∩ KerP : (I − P )X → ImL is
invertible. We denote the inverse of that map by KP . If Ω is an open bounded
subset of X, the mapping N will be called L−compact on Ω̄ if QN (Ω̄) is
bounded and KP (I − Q)N : Ω̄ → X is compact. Since ImQ is isomorphic to
KerL, there exist isomorphisms J : ImQ → KerL.
Lemma 3.1 ([7]Continuation Theorem ) Let L be a Fredholm mapping
of index zero and let N be L−compact on Ω̄. Suppose
(a) For each λ ∈ (0, 1), every solution
x of Lx = λN x is T
such that x ∈
/ ∂Ω;
T
(b) QN x 6= 0 for each x ∈ KerL ∂Ω, and deg{JQN, Ω ∂KerL, 0}T
=
6 0;
Then the equation Lx = N x has at least one solution lying in DomL Ω̄.
Lemma 3.2 [3] Let t1 , t2 ∈ Iω and t ∈ T. If g : T → R is ω-periodic, then
κ+ω
Z
|g ∆ (s)|∆s
g(t) ≤ g(t1 ) +
κ
and
Z
g(t) ≤ g(t2 ) −
κ
κ+ω
|g ∆ (s)|∆s.
50
Lihui Yang et al.
Lemma 3.3 The following equation:
(
¯a exp (x ) = 0,
ā − K
1
R
m(t) exp(x1 )
1 κ+ω
−D̄ + ω κ
∆t = 0
b(t) exp(x2 )+exp(x1 )+c(t)
(4)
has a unique positive solution (x∗1 , x∗2 )T .
The proofs of Lemma 3.3 are easy, so we omitted it here.
Theorem 3.1 Let S1 be defined by (12). Suppose that
(H1) ā > Ka + eb ;
(H2) mb exp(S1 ) > D̄;
(H3) ml exp(−S1 ) − D̄[exp(S1 ) + cu ] > D̄bu
hold, then (3) has at least one ω-periodic solution.
Proof. Define
X = Z = {(x1 , x2 )T ∈ C(T, R2 )|xi ∈ Crd , xi (t + ω) = xi (t), i = 1, 2},
T
||(x1 , x2 ) || =
2
X
i=1
max |xi (t)|, (x1 , x2 )T ∈ X(or Z).
t∈Iω
DomL = {x = (x1 , x2 )T ∈ X|xi ∈ Crd , i = 1, 2}.
It is easy to see that X and Z are both Banach spaces if they are endowed
with the above norm ||.||.
For (x1 , x2 )T ∈ X, we define
i
h
#
"
e(t) exp(x2 (t))
−
a(t) 1 − exp(x(t))
x1
x1 (t)
K(t)
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
,
N
(t) =
=
e(t) exp(x1 (t))
x2
x2 (t)
−D(t) +
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
∆
1 R κ+ω
x1
x1
x1
x
(t)∆t
x1
1
L
,
(t) = ∆ (t), P
(t) = Q
(t) = ω1 Rκκ+ω
x2
x2
x2
x2
x2 (t)∆t
ω κ
where (x1 , x2 )T . Then
KerL = {(x1 , x2 )T ∈ X|(x1 (t), x2 (t))T = (h1 , h2 )T ∈ R2 for t ∈ T},
Z κ+ω
Z κ+ω
T
ImL = {(x1 , x2 ) ∈ X|
x1 (t)∆t = 0,
x2 (t)∆t = 0, for t ∈ T}.
κ
κ
Then dim KerL = 2 = codim ImL. Since ImL is closed in Z, L is a Fredholm mapping of index zero, it is easy to show that P and Q are continuous
projections and ImP = KerL, ImL = KerQ = Im(I − Q). Clearly, QN and
Kp (I − Q)N are continuous. It can be shown that N is L-compact on Ω̄ for
every open bounded set, Ω ⊂ X.
51
Periodic Solutions for A Predator-Prey Model with...
Now we are at the point to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the
operator equation L(x1 , x2 )T = λN (x1 , x2 )T , λ ∈ (0, 1), we have

h
i
exp(x(t))
e(t) exp(x2 (t))
 x∆
(t)
=
λ
a(t)
1
−
−
,
1
K(t)
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
h
i
(5)
e(t) exp(x1 (t))
 x∆
2 (t) = λ −D(t) + b(t) exp(x2 (t))+exp(x1 (t))+c(t) .
Suppose that x(t) = (x1 (t), x2 (t))T ∈ X is an arbitrary solution of system (5)
for a certain λ ∈ (0, 1), Integrating (5) over the set Iω , we obtain
(
R κ+ω a(t) exp (x1 )
R κ+ω
e(t) exp(x2 (t))
āω = κ
∆t + κ
∆t,
K(t)
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
R κ+ω
(6)
e(t) exp(x1 (t))
D̄ω = κ
∆t.
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
Since (x1 , x2 )T ∈ X, there exists ξi , ηi ∈ [κ, κ + ω], i = 1, 2, 3 such that
xi (ξi ) =
min {xi (t)}, xi (ηi ) = max {xi (t)}.
t∈[κ,κ+ω]
t∈[κ,κ+ω]
It follows from (6) that
Z κ+ω
Z κ+ω
a(t) exp (x1 )
∆
|x1 (t)|∆t ≤ λ āω +
∆t
K(t)
κ
κ
Z κ+ω
e(t) exp(x2 (t))
+
∆t
b(t) exp(x2 (t)) + exp(x1 (t)) + c(t)
κ
< 2āω.
(7)
Z κ+ω
Z κ+ω
e(t) exp(x1 (t))
|x∆
∆t
2 (t)|∆t ≤ λ D̄ω +
b(t) exp(x2 (t)) + exp(x1 (t)) + c(t)
κ
κ
< 2D̄ω.
(8)
From the first equation of (6), it follows that
a
a
e
āω >
ω exp(x1 (ξ1 )), āω <
ω exp(x1 (η1 )) +
ω.
K
K
b
Then
x1 (ξ1 ) < ln
ā
e
b
:= M1 .
(9)
From (9), using the Lemma 3.2, we get
Z κ+ω
x1 (t) ≤ x1 (ξ1 ) +
|x∆
1 (t)|∆t ≤ m1 + 2āω =: A1
(10)
a
K
:= m1 , x1 (η1 ) > ln
ā −
a
K
κ
and
Z
x1 (t) ≥ x1 (η1 ) −
κ
κ+ω
|x∆
1 (t)|∆t ≥ M1 − 2D̄ω =: A2 .
(11)
52
Lihui Yang et al.
Thus
max |x1 (t)| ≤ max{|A1 |, |A2 |} := S1 .
t∈Iω
From the second equation of (6), it follows that
Z κ+ω
m
m(t) exp(x1 )
1
D̄ω <
∆t <
ωexp(S1 )
,
b(t) exp(x2 )
b
exp(x2 (ξ2 ))
κ
Z κ+ω
m(t) exp(−S1 )
∆t
D̄ω >
b(t) exp(x2 (η2 )) + exp(S1 ) + c(t)
κ
ml exp(−S1 )
ω.
≥ u
b exp(x2 (η2 )) + exp(S1 ) + cu
(12)
(13)
(14)
Then
m
b
exp(S1 )
:= m2 ,
D̄
ml exp(−S1 ) − D̄[exp(S1 ) + cu ]
x2 (η2 ) > ln
:= M2 .
D̄bu
x2 (ξ2 ) < ln
From (15),(16) and using the Lemma 3.2, we obtain
Z κ+ω
|x∆
x2 (t) ≤ x2 (ξ2 ) +
2 (t)|∆t ≤ m2 + 2āω =: A3
(15)
(16)
(17)
κ
and
Z
κ+ω
x2 (t) ≥ x2 (η2 ) −
|x∆
2 (t)|∆t ≥ M2 − 2D̄ω =: A4 .
(18)
κ
It follows from (17) and (18) that
max |x2 (t)| ≤ max{|A3 |, |A4 |} := S2 .
t∈Iω
(19)
Obviously, Si (i = 1, 2) are independent of the choice of λ ∈ (0, 1). Take
M = S1 + S2 + S0 , where S0 is taken sufficiently large such that S0 ≥ |m1 | +
|M1 | + |m2 | + |M2 |.
Next let us consider the algebraic equations
(
µe(t) exp(x2 (t))
¯a exp (x ) − 1 R κ+ω
ā − K
∆t = 0,
1
ω κ
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
R
(20)
m(t) exp(x1 )
1 κ+ω
∆t = 0
−D̄ + ω κ
b(t) exp(x2 )+exp(x1 )+c(t)
for (x, y) ∈ R2 ,where µ ∈ [0, 1] is a parameter. By carrying out similar arguments as above, it is not difficult to show that any solution (x∗1 , x∗2 ) of (20)
with µ ∈ [0, 1] satisfies
m1 < x∗1 < M1 , m2 < x∗2 < M2 .
53
Periodic Solutions for A Predator-Prey Model with...
Now we define Ω := {(x1 , x2 )T ∈ X : ||x|| < M }. T
It is clear thatTΩ verifies
T
the requirement (a) of Lemma 3.1. If (x1 , x2 ) ∈ ∂Ω KerL = ∂Ω R2 , then
(x1 , x2 )T is a constant vector in R2 with ||(x1 , x2 )T || = |x1 | + |x2 | = M . Then
# "
e(t) exp(x2 (t))
¯a exp (x ) − 1 R κ+ω
ā − K
∆t
x1
0
1
ω κ
b(t) exp(x2 (t))+exp(x1 (t))+c(t)
R κ+ω
QN
=
6=
.
m(t) exp(x1 )
1
x2
0
−D̄ + ω κ
∆t
b(t) exp(x )+exp(x )+c(t)
2
1
Now let us consider homotopy Hµ (x1 , x2 ) = µQN x + (1 − µ)Gx, µ ∈ [0, 1], x =
(x1 , x2 )T , where
"
#
ā − Ka exp (x1 )
R κ+ω
Gx =
.
m(t) exp(x1 )
−D̄ + ω1 κ
∆t
b(t) exp(x2 )+exp(x1 )+c(t)
Letting J be the identity mapping, according to Lemma 3.3 and by direct
calculation, we get
n
o
\
deg JQN (x1 , x2 )T ; ∂Ω KerL; 0
n
o
\
= deg QN (x1 , x2 )T ; ∂Ω KerL; 0
n
o
\
= deg H1 (x1 , x2 ); ∂Ω KerL; 0
n
o
\
= deg H0 (x1 , x2 ); ∂Ω KerL; 0
(
Z κ+ω
)
a
1
m(t)b(t) exp(x1 ) exp(x2 )
exp (x1 )
∆t
= sign
K
ω κ
(b(t) exp(x2 ) + exp(x1 ) + c(t))2
= 1 6= 0,
where deg(., ., ., ) is the Brower degree. Thus we have proved that Ω verifies
all requirements of Lemma 3.1, then it follows that Lx = N x has at least one
solution in DomL ∩ Ω. The proof is complete.
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