Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 52, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
PERIODIC SOLUTIONS FOR A FOOD CHAIN SYSTEM WITH
MONOD-HALDANE FUNCTIONAL RESPONSE ON
TIME SCALES
KEJUN ZHUANG
Abstract. In this article, we study a three species food chain model on time
scales, with Monod-Haldane functional response and time delay. With the help
of coincidence degree theory, we establish the existence of periodic solutions.
1. Introduction
Research on food chain system has been a hot spot in population dynamics.
Dynamical behavior of these models governed by differential equations and difference equations has been extensively studied in [5, 10, 12, 13, 15]. Hsu, Hwang and
Kuang [9] considered the ratio-dependent food chain model
x
1 m1 xy
ẋ(t) = rx(1 − ) −
,
K
η1 a 1 y + x
1 m2 yz
m1 xy
− d1 y −
,
ẏ(t) =
a1 y + x
η2 a 2 z + y
m2 yz
ż(t) =
− d2 z,
a2 z + y
where x, y and z sand for the population densities of prey, predator and top predator, respectively. The boundness, extinction and periodicity were studied.
Xu, Chaplain and Davidson [14] studied the delayed three-species Lotka-Volterra
food chain system
ẋ1 (t) = x1 (t)[r1 (t) − a11 (t)x1 (t − τ11 ) − a12 (t)x2 (t)],
ẋ2 (t) = x2 (t)[−r2 (t) + a21 (t)x1 (t − τ21 ) − a22 (t)x2 (t − τ22 ) − a23 (t)x3 (t)],
ẋ3 (t) = x3 (t)[−r3 (t) + a32 (t)x2 (t − τ32 ) − a33 x3 (t − τ33 )].
The existence, uniqueness and global stability of positive periodic solutions of the
system were studied. In population dynamics, the relationship between predator
and prey can be represented as the functional response which refers to the change
in the density of prey attached per unit time per predator as the prey density
changes. Holling [8] gave three different kinds of functional responses, which are
monotonic in the first quadrant. But some experiments and observations indicate
2000 Mathematics Subject Classification. 92D25, 34C25.
Key words and phrases. Food chain system; periodic solution; time scales; coincidence degree.
c
2014
Texas State University - San Marcos.
Submitted July 7, 2012. Published February 21, 2014.
1
2
K. ZHUANG
EJDE-2014/52
that nonmonotonic response is more realistic [1]. To model such an inhibitory effect,
Andrews suggested the so-called Monod-Haldane function proposed in [1]. From
above all, we can get the following non-autonomous food chain model with time
delays,
m12 (t)u2 (t)
],
a1 (t) + b1 (t)u1 (t) + u21 (t)
m21 u1 (t − τ )
u̇2 (t) = u2 (t)[−r2 (t) +
− d2 (t)u2 (t)
a1 (t) + b1 (t)u1 (t − τ ) + u21 (t − τ )
m23 (t)u3 (t)
],
−
a2 (t) + b2 (t)u2 (t) + u22 (t)
m32 (t)u2 (t − σ)
u̇3 (t) = u3 (t)[−r3 (t) +
− d3 (t)u3 (t)],
a2 (t) + b2 (t)u2 (t − σ) + u22 (t − σ)
u̇1 (t) = u1 (t)[r1 (t) − d1 (t)u1 (t) −
(1.1)
where u1 (t), u2 (t) and u3 (t) stand for the population density of prey, predator
and top–predator at time t, respectively. All coefficients are positive continuous
functions. mi,i+1 (t) is the capture rate of the predator, mi+1,i (t) is a measure of
the food quality that the prey provided for conversion into predator birth, where
i = 1, 2.
On the other hand, if the populations have non-overlapping generations, the
discrete model governed difference equations is more appropriate
u1 (n + 1) = u1 (n) exp[r1 (n) − d1 (n)u1 (n) −
m12 (n)u2 (n)
],
a1 (n) + b1 (n)u1 (n) + u21 (n)
m21 u1 (n − τ )
− r2 (n) − d2 (n)u2 (n)
a1 (n) + b1 (n)u1 (n − τ ) + u21 (n − τ )
m23 (n)u3 (n)
−
],
a2 (n) + b2 (n)u2 (n) + u22 (n)
m32 (n)u2 (n − σ)
u3 (n + 1) = u3 (n) exp[
− r3 (n) − d3 (n)u3 (n)],
a2 (n) + b2 (n)u2 (n − σ) + u22 (n − σ)
(1.2)
where all the coefficients are positive periodic sequences.
To explore the periodic solutions of differential equation and difference equation
models, coincidence degree theory is a common method. However, for these two
types of systems, the methods and results are significantly similar. Enlightened by
the idea of Stefan Hilger [7], to unify the continuous and discrete dynamic systems,
we consider the following dynamic system on time scales,
u2 (n + 1) = u2 (n) exp[
x∆ (t) = r1 (t) − d1 (t)ex(t) −
y ∆ (t) = −r2 (t) +
−
m12 (t)ey(t)
,
a1 (t) + b1 (t)ex(t) + e2x(t)
m21 (t)ex(t−τ )
− d2 (t)ey(t)
a1 (t) + b1 ex(t−τ ) + e2x(t−τ )
m23 (t)ez(t)
,
a2 (t) + b2 (t)ey(t) + e2y(t)
z ∆ (t) = −r3 (t) +
m32 (t)ey(t−σ)
− d3 (t)ez(t) ,
a2 (t) + b2 (t)ey(t−σ) + e2y(t−σ)
(1.3)
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PERIODIC SOLUTIONS FOR A FOOD CHAIN MODEL
3
where t ∈ T and T is a time scale that is unbounded above. x∆ (t) is the delta derivative of x at t, which is defined in [4]. All the coefficients are positive ω−periodic
functions. Set u1 (t) = ex(t) , u2 (t) = ey(t) , u3 (t) = ez(t) , then (1.3) can be reduced
to (1.1) and (1.2) when T = R and T = Z, respectively.
The purpose of this paper is to study the periodicity of three-species food chain
system on time scales and this model has not been investigated before. We would
like to mention that there are several papers on periodicity in dynamic systems on
time scales by using the coincidence degree theory, see [2, 3, 11, 17]. The remainder
of the paper is organized as follows. In the following section, some preliminary
results about calculus on time scales and the continuation theorem are stated.
Next, the sufficient conditions for the existence of periodic solutions are explored.
2. Preliminaries
For convenience, we first present the useful lemma about time scales and the
continuation theorem of the coincidence degree theory; more details can be found
in [16, 6].
A time scale T is an arbitrary nonempty closed subset of real numbers R.
Throughout this paper, we assume that the time scale T is unbounded above and
below, such as R, Z and ∪k∈Z [2k, 2k + 1]. The following definitions and lemmas
about time scales are from [16, 6].
Lemma 2.1 ([16]). Let t1 , t2 ∈ Iω and t ∈ T. If g : T → R ∈ Crd (T) is ω-periodic,
then
Z
1 k+ω ∆
|g (s)|∆s,
g(t) ≤ g(t1 ) +
2 k
Z
1 k+ω ∆
g(t) ≥ g(t2 ) −
|g (s)|∆s,
2 k
where the constant factor 1/2 is the best possible.
For simplicity, we use the following notation throughout this paper. Let T be
ω-periodic; that is, t ∈ T implies t + ω ∈ T,
k = min{R+ ∩ T},
g M = sup g(t),
t∈T
Iω = [k, k + ω] ∩ T, g L = inf g(t),
t∈T
Z
Z
1
1 k+ω
ḡ =
g(s)∆s =
g(s)∆s,
ω Iω
ω k
where g ∈ Crd (T) is an ω-periodic real function; i.e., g(t + ω) = g(t) for all t ∈ T.
Next, we state the Mawhin’s continuation theorem, which is a main tool in the
proof of our theorem.
Lemma 2.2 ([6]). Let L be a Fredholm mapping of index zero and N be L-compact
on Ω̄. Suppose
(a) for each λ ∈ (0, 1), every solution u of Lu = λN u is such that u ∈
/ ∂Ω;
(b) QN u 6= 0 for each u ∈ ∂Ω ∩ ker L and the Brouwer degree deg{JQN, Ω ∩
ker L, 0} =
6 0.
Then the operator equation Lu = N u has at least one solution lying in Dom L ∩ Ω̄.
4
K. ZHUANG
EJDE-2014/52
3. Main Results
L2
M M2
Theorem 3.1. If mL
> r3M (aM
+ e2M2 ) holds, where
32 e
2 + b2 e
L2 = ln
L
aL
mM
2 r3
21 ω
−
M
m32
bL
1
and
M2 = ln
r1M (aM
1 +
M
bM
1 r1
dL
1
mL
12
+
2r1M
dL
1
)
+
mM
21 ω
,
bL
1
then (1.3) has at least one ω-periodic solution.
Proof. Let
n
X = Z = (x, y, z)T ∈ C(T, R3 ) : x(t + ω) = x(t), y(t + ω) = y(t),
o
z(t + ω) = z(t), ∀t ∈ T ,
k(x, y, z)T k = max |x(t)| + max |y(t)| + max |z(t)|,
t∈Iω
t∈Iω
t∈Iω
(x, y, z)T ∈ X
(or in Z).
Then X and Z are both Banach spaces when they are endowed with the above
norm k · k. Let
   
x
N1
N y  = N2  ,
z
N3
where
N1 = r1 (t) − d1 (t)ex(t) −
N2 = −r2 (t) +
−
m12 (t)ey(t)
,
a1 (t) + b1 (t)ex(t) + e2x(t)
m21 (t)ex(t−τ )
− d2 (t)ey(t)
a1 (t) + b1 ex(t−τ ) + e2x(t−τ )
m23 (t)ez(t)
,
a2 (t) + b2 (t)ey(t) + e2y(t)
m32 (t)ey(t−σ)
− d3 (t)ez(t) .
a2 (t) + b2 (t)ey(t−σ) + e2y(t−σ)

   ∆
 
   1 R k+ω
x(t)∆t
x
x
x
x
ω Rk


L y  = y ∆  , P y  = Q y  =  ω1 kk+ω t(t)∆t  .
R
∆
1 k+ω
z
z
z
z
z(t)∆t
ω k
Obviously, ker L = R3 , Im L = (x, y, z)T ∈ Z : x̄ = ȳ = z̄ = 0, t ∈ T , dim ker L =
3 = codim Im L. Since Im L is closed in Z, then L is a Fredholm mapping of
index zero. It is easy to show that P and Q are continuous projections such that
Im P = ker L and Im L = ker Q = Im(I − Q). Furthermore, the generalized inverse
(of L) KP : Im L → ker P ∩ Dom L exists and is given by

R k+ω R t
  R t
x(s)∆s − ω1 k
x(s)∆s∆t
x
k
k
R
R
R

KP y  =  kt y(s)∆s − ω1 kk+ω kt y(s)∆s∆t  .
R
R
R
t
k+ω t
z
z(s)∆s − ω1 k
z(s)∆s∆t
k
k
N3 = −r3 (t) +
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PERIODIC SOLUTIONS FOR A FOOD CHAIN MODEL
5
Thus

m12 (t)ey(t)
(r1 (t) − d1 (t)ex(t) − a1 (t)+b
x(t) +e2x(t) )∆t
(t)e
1

R
m21 (t)ex(t−τ )
1 k+ω

(−r
(t)
+
2
x(t−τ
) +e2x(t−τ )

ω k
a1 (t)+b1 e

z(t)
m23 (t)e
,
−d2 (t)ey(t) − a2 (t)+b
)∆t
y(t) +e2y(t)

2 (t)e
R
y(t−σ)

m32 (t)e
1 k+ω

(−r
(t)
+
3
y(t−σ)
2y(t−σ)
ω k
 R k+ω
1
ω
  

x


QN y  = 

z


k
a2 (t)+b2 (t)e
+e
−d3 (t)ez(t) )∆t
and
Rt

k
x(s)∆s − ω1
R k+ω R t
x(s)∆s∆t

kR

k+ω
− t − k − ω1 k (t − k)∆t x̄

Rt
R
R

1 k+ω t
y(s)∆s
−
y(s)∆s∆t
k
kR
ω k

.
k+ω
− t − k − ω1 k (t − k)∆t ȳ 


R
Rt
R
1 k+ω t

z(s)∆s
−
z(s)∆s∆t−
k
kR

ω k
1 k+ω
t − k − ω k (t − k)∆t z̄


  

x

KP (I − Q)N y  = 


z



k
Clearly, QN and KP (I − Q)N are continuous. According to the Arzela-Ascoli
theorem, it is not difficulty to show that KP (I − Q)N (Ω̄) is compact for any open
bounded set Ω ⊂ X and QN (Ω̄) is bounded. Thus, N is L-compact on Ω̄.
Now, we shall search an appropriate open bounded subset Ω for the application
of the continuation theorem, Lemma 2.2. For the operator equation Lu = λN u,
where λ ∈ (0, 1), we have
x(t)
u∆
−
1 (t) = λ(r1 (t) − d1 (t)e
u∆
2 (t) = λ(−r2 (t) +
m12 (t)ey(t)
),
a1 (t) + b1 (t)ex(t) + e2x(t)
m21 (t)ex(t−τ )
− d2 (t)ey(t)
a1 (t) + b1 ex(t−τ ) + e2x(t−τ )
m23 (t)ez(t)
),
−
a2 (t) + b2 (t)ey(t) + e2y(t)
u∆
3 (t) = λ(−r3 (t) +
(3.1)
m32 (t)ey(t−σ)
− d3 (t)ez(t) ).
a2 (t) + b2 (t)ey(t−σ) + e2y(t−σ)
Assume that (u1 , u2 , u3 )T ∈ X is a solution of system (3.1) for a certain λ ∈ (0, 1).
Integrating (3.1) on both sides from k to k + ω, we obtain
k+ω
Z
[d1 (t)ex(t) +
k
Z
m12 (t)ey(t)
]∆t = r̄1 ω,
a1 (t) + b1 (t)ex(t) + e2x(t)
k+ω
[r2 (t) + d2 (t)ey(t) +
k
k+ω
m23 (t)ez(t)
]∆t
a2 (t) + b2 (t)ey(t) + e2y(t)
m21 (t)ex(t−τ )
=
∆t,
a1 (t) + b1 ex(t−τ ) + e2x(t−τ )
k
Z k+ω
Z k+ω
m32 (t)ey(t−σ)
z(t)
[r3 (t) + d3 (t)e ]∆t =
∆t.
a2 (t) + b2 (t)ey(t−σ) + e2y(t−σ)
k
k
Z
(3.2)
6
K. ZHUANG
EJDE-2014/52
Since (x, y, z)T ∈ X, there exist ξi , ηi ∈ Iω , i = 1, 2, 3, such that
x(ξ1 ) = min{x(t)},
x(η1 ) = max{x(t)},
y(ξ2 ) = min{y(t)},
y(η2 ) = max{y(t)},
z(ξ3 ) = min{z(t)},
z(η3 ) = max{z(t)}.
t∈Iω
t∈Iω
t∈Iω
t∈Iω
From (3.1) and (3.2), we have
Z
t∈Iω
k+ω
|x∆ (t)|∆t ≤ 2r̄1 ω,
k
k+ω
Z
|y ∆ (t)|∆t ≤ 2
mM
21 ω
,
bL
1
|z ∆ (t)|∆t ≤ 2
mM
32 ω
.
bL
2
k
k+ω
Z
(3.3)
t∈Iω
k
By the first equation of (3.2) and (3.3),
d1 (ξ1 )ex(ξ1 ) < r1 (ξ1 );
that is,
x(ξ1 ) < ln
r1M
.
dL
1
From the second equation of (3.2), we have
r2 (η2 ) <
m21 (η2 )ex(η2 −τ )
a1 (η2 )
and
x(η1 ) ≥ x(η2 − τ ) > ln
r2L aL
1
.
mM
21
According to Lemma 2.1, we have
Z
rM
1 k+ω ∆
|x (t)|∆t ≤ ln 1L + r̄1 ω := M1 ,
x(t) ≤ x(ξ1 ) +
2 k
d1
Z k+ω
1
rL aL
x(t) ≥ x(η1 ) −
|x∆ (t)|∆t ≥ ln 2 M1 − r̄1 ω := L1 .
2 k
m21
From the first equation of (3.2) and (3.3), we obtain
m12 (ξ1 )ey(ξ1 )
< r1 (ξ1 ),
a(ξ1 ) + b1 (ξ1 )ex(ξ1 ) + e2x(ξ1 )
y(ξ2 ) < y(ξ1 ) < ln
r1M (aM
1 +
M
bM
1 r1
dL
1
mL
12
+
2r1M
dL
1
)
.
Then
1
y(t) ≤ y(ξ2 ) +
2
Z
k+ω
∆
|y (t)|∆t < ln
r1M (aM
1 +
k
M
bM
1 r1
dL
1
mL
12
From the third equation of (3.2), we have
m32 (ξ3 ) y(ξ3 −σ)
e
> r3 (t),
a2 (ξ3 )
+
2r1M
dL
1
)
+
mM
21 ω
:= M2 .
bL
1
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PERIODIC SOLUTIONS FOR A FOOD CHAIN MODEL
this reduces to
y(η2 ) ≥ y(ξ3 − σ) > ln
7
L
aL
2 r3
.
M
m32
Then
Z
aL rL
1 k+ω ∆ mM ω
y (t) ∆t ≥ ln 2 M3 − 21
:= L2 .
2 k
m32
bL
1
According to the first equation of (3.2), we have
y(t) ≥ y(η2 ) −
m32 (ξ3 )
> d3 (ξ3 )ez(ξ3 ) .
b2 (ξ3 )
Then
1
z(t) ≤ z(ξ3 ) +
2
Also we have
k+ω
Z
|z ∆ (t)|∆t ≤ ln
k
z(η3 )
dM
>
3 e
aM
2
mM
mM
32
32 ω
+
:= M3 .
L
L
b2 d3
bL
2
L2
mL
32 e
− r3M .
M2 + e2M2
+ bM
2 e
Then
1
z(t) ≥ z(η3 ) −
2
k+ω
Z
∆
|z (t)|∆t ≥ ln
k
L2
mL
32 e
M eM2 +e2M2
+b
aM
2
2
dM
3
− r3M
−
mM
32 ω
:= L3 .
bL
2
Therefore, we have
max
t∈[k,k+ω]
max
t∈[k,k+ω]
max
t∈[k,k+ω]
|x(t)| ≤ max{|M1 |, |L1 |} := R1 ,
|y(t)| ≤ max{|M2 |, |L2 |} := R2 ,
|z(t)| ≤ max{|M3 |, |L3 |} := R3 .
Clearly, R1 , R2 and R3 are independent of λ. Let R = R1 + R2 + R3 + R0 , where
R0 is taken sufficiently large such that for the algebraic equations
Z
1 κ+ω
m12 (t)ey
x
¯
r̄1 − d1 e −
∆t = 0,
ω κ
a1 (t) + b1 (t)ex + e2x
Z
1 κ+ω
m21 (t)ex
− r̄2 +
∆t − d¯2 ey
ω κ
a1 (t) + b1 (t)ex + e2x
(3.4)
Z
1 κ+ω
m̄23 ez
−
∆t
=
0,
ω κ
a2 (t) + b2 (t)ey + e2y
Z κ+ω
1
m32 (t)ey
−r̄3 +
∆t − d¯3 ez = 0,
ω κ
a2 (t) + b2 (t)ey + e2y
every solution (x∗ , y ∗ , z ∗ )T of (3.4) satisfies k(x∗ , y ∗ , z ∗ )T k < R. Now, we define
Ω = {(x, y, z)T ∈ X : k(x, y, z)T k < R}. Then it is clear that Ω satisfies the
requirement (a) of Lemma 2.2. If (u1 , u2 , u3 )T ∈ ∂Ω ∩ ker L = ∂Ω ∩ R3 , then
(x, y, z)T is a constant vector in R3 with k(x, y, z)T k = |x| + |y| + |z| = R, so we
have
   
x
0
QN y  6= 0 .
z
0
By the assumption in Theorem 3.1 and the definition of topological degree, a
direct calculation yields deg(JQN, Ω ∩ ker L, 0) 6= 0. We have verified that Ω
8
K. ZHUANG
EJDE-2014/52
satisfies all requirements of Lemma 2.2; therefore, system (1.3) has at least one
ω-periodic solution in Dom L ∩ Ω̄. This completes the proof.
Conclusion. In this article, a three species food chain model on time scales is
proposed. This model not only unifies the food chain system with Monod-Haldane
functional response and time delay governed by differential equations and their discrete analogues in form of difference equations, but also extends the results to more
general time scales. By using the Mawhin’s continuation theorem of coincidence degree theory, the existence of periodic solutions is established, which means that we
do not have to investigate the same problem in systems (1.1) and (1.2) repeatedly.
Moreover, based on the sharp inequalities in [16], the priori estimates of periodic
solutions are better than previous work.
Acknowledgments. The author would like to acknowledge the support from National Natural Science Foundation of China (11301001), Anhui Provincial Natural
Science Funds (1208085QA11), Excellent Youth Scholars Foundation and the Natural Science Foundation of Anhui Province (2013SQRL030ZD) and Humanities and
Social Science Research Projects of Anhui Universities (SK2013B018).
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Kejun Zhuang
Institute of Applied Mathematics, Anhui University of Finance and Economics, Bengbu
233030, China
E-mail address: zhkj123@163.com