Electronic Journal of Differential Equations, Vol. 2009(2009), No. 126, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
TWIN PERIODIC SOLUTIONS OF PREDATOR-PREY
DYNAMIC SYSTEM ON TIME SCALES
RONG-HUA HE, HONG-XU LI, LIANG ZHANG
Abstract. In this article, we consider a delayed predator-prey dynamic system with type IV functional responses on time scales. Sufficient criteria for the
existence of at least two periodic solutions are established by using the wellknown continuation theorem due to Mawhin. An example is given to illustrate
the main result.
1. Introduction
In studying the interaction between predators and their prey, it is crucial to
determine what specific form of the functional response that describes the mount
of prey consumed per predator per unit of time is biological plausible provides a
sound basis for theoretical development.
In this paper, we consider the following delay predator-prey dynamic equation
with type IV functional responses on time scale T:
c(t) exp{y2 (t − γ(t))}
y1∆ (t) = b1 (t) − a1 (t) exp{y1 (t − τ1 (t))} −
,
exp{2y1 (t)}/n + exp{y1 (t)} + a
(1.1)
a2 (t) exp{y1 (t − τ2 (t))}
∆
y2 (t) = −b2 (t) +
,
exp{2y1 (t)(t − τ2 (t))}/n + exp{y1 (t − τ2 (t))} + a
where for i = 1, 2; c, γ, ai , bi , τi ∈ Crd (T) are ω-priodic functions with c(t) ≥
0, γ(t) ≥ 0, ai (t) ≥ 0, τi (t) ≥ 0, c > 0, and b̄i > 0, n and a are positive constants,
Crd (T) will be defined later.
Calculus on time scales was initiated by Stefan Hilger in 1990 with the motivation of providing a unified approach to continuous and discrete analysis. Since
then the theory of dynamic equations on time scales has become a new important mathematical branch, and it has been applied in various directions (see, eg.,
[1, 2, 3, 4, 5, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20] and the refs cited therein).
On the other hand, the Mawhin’s continuation is a powerful tool when deal with
the existence of periodic solutions for population models, and much work have been
done (see, e.g., [6, 10, 13, 17, 21] and the references cited therein). However, to the
best of our knowledge, the study on the existence of multiple periodic solutions for
population models on time scales are scarce.
2000 Mathematics Subject Classification. 92D25, 39A12.
Key words and phrases. Time scales; delayed predator-prey dynamic system;
periodic solutions; continuation theorem.
c
2009
Texas State University - San Marcos.
Submitted December 2, 2008. Published October 4, 2009.
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R.-H. HE, H.-X. LI, L. ZHANG
EJDE-2009/126
Motivated and inspired by the above excellent work, in this paper, we establish
some sufficient criteria for the existence of at least two periodic solutions for system
(1.1) by using Mawhin technique.
2. Preliminaries
We first provide without proof several definitions and results from the calculus
on time scales which are useful in the following argument. For more details, we
refer the authors to [5].
A time scale T is an arbitrary nonempty closed subset of the real numbers, and
it inherits the topology from the real numbers with the standard topology. Let
ω > 0 is a constant. Throughout this paper, the time scale we considered is always
assumed to be ω-priodic (i.e., t ∈ T implies t ± ω ∈ T) and unbounded above and
below. Set
κ = min{R+ ∩ T}, Iω = [κ, κ + ω] ∩ T.
Definition 2.1. The forward jump operator σ : T → T and the backward jump
operator ρ : T → T are defined by
σ(t) := inf{s ∈ T : s ≥ t},
ρ(t) := sup{s ∈ T : s ≤ t},
respectively, for any t ∈ T. If σ(t) = t, then t is called right-dense (otherwise: rightscattered), and if ρ(t) = t, then t is called left–dense (otherwise left-scattered).
Definition 2.2. Assume that f : T → R and fix t ∈ T. Then f is called differential at t ∈ T if there exists c ∈ R such that given any ε > 0, there is an open
neighborhood U of t satisfying
|[f (σ(t)) − f (s)] − c[σ(t) − s]| ≤ ε|σ(t) − s|,
s ∈ U.
In this case, c is called the delta (or Hilger) derivative of f at t ∈ T, and is denoted
by c = f ∆ (t).
Remark 2.3. We say that f is delta (Hilger) differential on T if f ∆ (t) exists for
all t ∈ T. A function F : T → R is called an antiderivative of f : T → R provided
that F ∆ (t) = f (t) for all t ∈ T. Then we define
Z s
f (t)∆t = F (s) − F (r), r, s ∈ T.
r
Definition 2.4. A function f : T → R is called rd-continuous if it is continuous
at right-dense points in T and its left-sided limits exist(finite) at left-dense points
in T. The set of rd-continuous functions f : T → R will be denoted by Crd =
Crd (T) = Crd [T, R).
Remark 2.5. Every rd-continuous function has an antiderivative. Every continuous function is rd-continuous.
Lemma 2.6. If a, b ∈ T, α, β ∈ R and f, g ∈ Crd (T), then
Rb
Rb
Rb
(C1) a [αf (t) + βg(t)]∆t = α a f (t)∆t + β a g(t)∆t;
Rb
(C2) if f (t) ≥ 0 for all a ≤ t ≤ b, then a f (t)∆t ≥ 0;
Rb
(C3) if |f (t)| ≤ g(t) on [a, b) := {t ∈ T : a ≤ t < b}, then | a f (t)∆t| ≤
Rb
g(t)∆t.
a
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TWIN PERIODIC SOLUTIONS
3
Lemma 2.7 ([3]). Let t1 , t2 ∈ Iω and t ∈ T. If g : T → R is ω − periodic, then
Z
Z
∆
g(t) ≤ g(t1 ) +
|g (s)|∆s and g(t) ≥ g(t2 ) −
|g ∆ (s)|∆s.
Iω
Iω
In the remainder of this section we list well known elements and result which
can be found in [8]. Let X and Z are two Banach spaces. Consider a operator
equation:
Lx = λN x,
λ ∈ (0, 1),
(2.1)
where L : Dom L ∩ X → Z is a linear operator, N : X → Z is a continuous
operator and λ is a parameter. Let P and Q denote two projectors P : X → X and
Q : Z → Z such that Im P = Ker L and Im L = Ker Q = Im(I − Q). It follows that
L|Dom L∩Ker P : (I − P )X → Im L is invertible. We denote the inverse of this map
by Kp. If Ω is a bounded open subset of X, the mapping N is called L-compact
on Ω if QN (Ω) is bounded and Kp(I − Q)N : Ω → X is compact. Because Im Q is
isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
Recall that operator L will be called a Fredholm operator of index zero if
dim Ker L = codim Im L < ∞, and Im L is closed in Z.
Lemma 2.8 (Continuation Theorem [8]). Let L be a Fredholm mapping of index
zero and let N be L − compact on Ω. Suppose
(a) for each λ ∈ (0, 1), every solution x of Lx = λN x is such that x ∈
/ ∂Ω;
(b) QN x 6= 0 for each x ∈ ∂Ω ∩ Ker L;
(c) deg{JQN, Ω, θ} =
6 0.
Then the equation Lx = N x has at least one solution lying in Dom L ∩ Ω̄.
To facilitate the discussion below, throughout this paper we adopt the following
notation
Z
Z
1
1
ḡ =
g(s)∆s, |b̄i | =
|bi (s)|∆s,
ω Iω
ω Iω
where i=1, 2; g ∈ Crd (T) is an ω-periodic real function, i.e., g(t + ω) = g(t) for all
t ∈ T. And the other symbols appearing in the sequel are denoted accordingly. Set
p
n(ā2 − b̄2 ) ± n2 (ā2 − b̄2 )2 − 4b̄22 an
u1± :=
,
2b2
1 l± :=
n[ā2 exp{(b̄1 + |b̄1 |)ω} − b̄2 ]
2b̄2
q
± n2 [ā2 exp{(b̄1 + |b̄1 |)ω} − b̄2 ]2 − 4b̄22 an ,
1
v± :=
n[ā2 − b̄2 exp{(b̄1 + |b̄1 |)ω}]
2b̄2 exp{ω(b̄1 + |b̄1 |)}
q
± n2 [ā2 − b̄2 exp{(b̄1 + |b̄1 |)ω}]2 − 4b̄22 an exp{2ω(|b̄1 | + b̄1 )} .
From the above six positive numbers, one obtains
l− < u1− < v− < v+ < u1+ < l+ .
(2.2)
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R.-H. HE, H.-X. LI, L. ZHANG
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3. Main result
In this section, our emphasis is focused on the existence of at least two periodic
solutions for (1.1). Before formulate the main result, we first embed our problem
into the frame of Lemma 2.8. Set
X = Z = {y = (y1 (t), y2 (t))T ∈ C(T, R2 )|yi (t + ω) = yi (t), i = 1, 2, t ∈ T}.
P2
Then X, Z are Banach spaces endowed with the norm kyk = i=1 maxt∈Iω |yi (t)|.
Define
N y(t) = (Φ1 (t), Φ2 (t))T , Ly(t) = (y1∆ (t), y2∆ (t))T ,
Z
T
1 Z
1
y1 (t)∆t,
y2 (t)∆t ,
P y = Qy =
ω Iω
ω Iω
where y ∈ X,
c(t) exp{y2 (t − γ(t))}
,
exp{2y1 (t)}/n + exp{y1 (t)} + a
a2 (t) exp{y1 (t − τ2 (t))}
Φ2 (t) = −b2 (t) +
.
exp{2y1 (t − τ2 (t))}/n + exp{y1 (t − τ2 (t))} + a
Φ1 (t) = b1 (t) − a1 (t) exp{y1 (t − τ1 (t))} −
Obviously,
Ker L = {y ∈ X : y = h = (h1 , h2 )T ∈ R2 , t ∈ T},
Z
Im L = y ∈ Z :
yi (t)∆t = 0, t ∈ T, i = 1, 2 ,
Iω
P, Q are continuous projectors such that
Im L = Ker Q = Im(I − Q),
Im P = Ker L,
the set Im L is closed in Z, and
dim Ker L = 2 = codim Im L.
Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse
(to L) Kp : Im L → Dom L ∩ Ker P exists and is given by
Z Z t
Z t
Z Z t
T
Z t
1
1
Kp y =
y1 (s)∆s∆t,
y2 (s)∆s −
y2 (s)∆s∆t .
y1 (s)∆s −
ω Iω κ
ω Iω κ
κ
κ
Thus
1 Z
Z
T
1
QN y =
Φ1 (s)∆s,
Φ2 (s)∆s ,
ω Iω
ω Iω
T
Kp (I − Q)N y = Θ1 (t), Θ2 (t) ,
where,
Z
t
Z Z
t
Z
1
Θ1 (t) =
Φ1 (s)∆s∆t − t − κ −
(t − κ)∆t Φ̄1 ,
ω Iω
κ
Iω κ
Z t
Z Z t
Z
1
1
Θ2 (t) =
Φ2 (s)∆s −
Φ2 (s)∆s∆t − t − κ −
(t − κ)∆t Φ̄2 .
ω Iω κ
ω Iω
κ
1
Φ1 (s)∆s −
ω
It is easy to show that QN and Kp (I − Q) are continuous. By using the ArzelaAscoli theorem, one can show that Kp (I − Q)(Ω̄) is relatively compact for any open
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TWIN PERIODIC SOLUTIONS
5
bounded set Ω ∈ X. Moreover, QN (Ω̄) is bounded. Thus, N is L-compact on Ω̄
for any open bounded set Ω ⊂ X.
After the above preparations, we now state and prove our main result.
Theorem 3.1. System (1.1) has at least two ω-periodic solutions if the following
conditions hold.
p
(i) ā2 > b̄2 (1 + 2 na ) exp{(|b̄1 | + b̄1 )ω};
(ii) b̄1 > ā1 l+ exp{(|b̄1 | + b̄1 )ω}.
Proof. Corresponding to the operator equation (2.1) we have
(y1∆ (t), y2∆ (t))T = λ(Φ1 (t), Φ2 (t))T .
(3.1)
Suppose that y ∈ X is a solution of system (3.1) for a certain λ ∈ (0, 1). Integrating
(3.1) over set Iω , we obtain
Z
c(t) exp{y2 (t − γ(t))}
∆t, (3.2)
b̄1 ω =
a1 (t) exp{y1 (t − τ1 (t))} +
exp{2y
(t)}/n
+
exp{y
(t)}
+
a
1
1
Iω
Z
a2 (t) exp y1 (t − τ2 (t))
b̄2 ω =
∆t.
(3.3)
exp{2y
(t
−
τ
(t))}/n
+
exp
y
(t
−
τ
(t))
+
a
1
2
1
2
Iω
From (3.1)–(3.3), we have
Z
Z
Z
∆
|y1 (t)|∆t ≤ λ
|b1 (t)|∆t +
[−Φ1 (t) + b1 (t)]∆t < (|b̄1 | + b̄1 )ω,
ZIω
ZIω
ZIω
|y2∆ (t)|∆t ≤ λ
|b2 (t)|∆t +
[Φ2 (t) + b2 (t)]∆t < (|b̄2 | + b̄2 )ω.
Iω
Iω
(3.4)
(3.5)
Iω
Because y = (y1 (t), y2 (t))T ∈ X, there exist ξi , ηi ∈ Iω , i = 1, 2 such that
yi (ξi ) = min{yi (t)},
t∈Iω
yi (ηi ) = max{yi (t)}.
t∈Iω
(3.6)
From (3.3) and (3.6) we get
Z
b̄2 ω ≤
a2 (t) exp{y1 (η1 )}
∆t
Iω exp{2y1 (ξ1 )}/n + exp{y1 (ξ1 )} + a
ā2 ω exp{y1 (η1 )}
=
,
exp{2y1 (ξ1 )}/n + exp{y1 (ξ1 )} + a
which implies
y1 (η1 ) ≥ ln
b̄2
(exp{2y1 (ξ1 )}/n + exp y1 (ξ1 ) + a) .
ā2
By virtue of (3.4), (3.7) and Lemma 2.7, we get
Z
y1 (t) ≥ y1 (η1 ) −
|y1∆ (t)|∆t
Iω
> ln[
b̄2
(exp{2y1 (ξ1 )}/n + exp y1 (ξ1 ) + a)] − (|b̄1 | + b̄1 )ω.
ā2
In particular,
y1 (ξ1 ) > ln
b̄2
(exp{2y1 (ξ1 )}/n + exp y1 (ξ1 ) + a) − (|b̄1 | + b̄1 )ω.
ā2
(3.7)
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R.-H. HE, H.-X. LI, L. ZHANG
EJDE-2009/126
or
b̄2
exp{2y1 (ξ1 )} − ā2 exp{(|b̄1 | + b̄1 )ω} − b̄2 exp{y1 (ξ1 )} + b̄2 a < 0.
n
According to (i), we have
ln l− < y1 (ξ1 ) < ln l+ .
(3.8)
(3.9)
From (3.3), we have
Z
a2 (t) exp{y1 (ξ1 )}
∆t
Iω exp{2y1 (η1 )}/n + exp{y1 (η1 )} + a
ā2 ω exp{y1 (ξ1 )}
=
;
exp{2y1 (η1 )}/n + exp{y1 (η1 )} + a
b̄2 ω ≥
that is,
b̄2
(exp{2y1 (η1 )}/n + exp y1 (η1 ) + a) .
ā2
Which together with (3.4) and Lemma 2.7 lead to
Z
y1 (t) ≤ y1 (ξ1 ) +
|y1∆ (t)|∆t
y1 (ξ1 ) < ln
Iω
b2
< ln
(exp{2y1 (η1 )}/n + exp y1 (η1 ) + a) + (|b̄1 | + b̄1 )ω.
a2
Hence, we have
b̄2
y1 (η1 ) < ln
(exp{2y1 (η1 )}/n + exp y1 (η1 ) + a) + (|b̄1 | + b̄1 )ω.
ā2
or
b̄2
exp{2y1 (η1 )} − ā2 exp{−(|b̄1 | + b̄1 )ω} − b̄2 exp{y1 (η1 )} + b̄2 a > 0.
n
Similarly, we can show that
y1 (η1 ) < ln v−
or y1 (η1 ) > ln v+ .
From (3.4), (3.9) and Lemma 2.7,
Z
y1 (t) ≤ y1 (ξ1 ) +
|y1∆ (t)|∆t < ln l+ + (|b̄1 | + b̄1 )ω := P1 .
(3.10)
(3.11)
Iω
On the other hand, (3.2) and (3.6) yield
c̄ω exp{y2 (ξ2 )}
,
exp{2P1 }/n + exp{P1 } + a
c̄ω exp{y2 (η2 )}
b̄1 ω ≤ ā1 ω exp{P1 } +
.
a
It follows from (3.12) that
b̄1
y2 (ξ2 ) ≤ ln
(exp{2P1 }/n + exp{P1 } + a) .
c̄
This, together with (3.5) and Lemma 2.7, yields
Z
y2 (t) ≤ y2 (ξ2 ) +
|y2∆ (t)|∆t
b̄1 ω ≥
Iω
b̄1
< ln
(exp{2P1 }/n + exp{P1 } + a) + (|b̄2 | + b̄2 )ω := P2 .
c̄
(3.12)
(3.13)
(3.14)
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TWIN PERIODIC SOLUTIONS
7
Moreover, because of (ii), it follows from (3.13) that
a
y2 (η2 ) ≥ ln (b̄1 − exp{(B̄1 + b̄1 )ωā1 l+ }) ,
c̄
which, combined with Lemma 2.7, gives
Z
y2 (t) ≥ y2 (η2 ) −
|y2∆ (t)|∆t
Iω
a
> ln[ (b̄1 − exp{(B̄1 + b̄1 )ωā1 l+ })] − (|b̄2 | + b̄2 )ω := P3 .
c̄
It follows from (3.14) and (3.15) that
max |y2 (t)| < max{|P2 |, |P3 |} := P.
t∈Iω
(3.15)
(3.16)
Obviously, ln l± , ln v± , P1 and P are independent of the choice of λ ∈ (0, 1).
Now, let’s consider QN h with h = (h1 , h2 )T ∈ R2 . Note that
!
exp{h2 }
b̄1 − ā1 exp{h1 } − exp{2h1c̄}/n+exp{h
1 }+a
QN h =
exp{h1 }
−b̄2 + exp{2hā1 2}/n+exp{h
}+a
1
In view of (i) and (ii), one can show that QN h = 0 has two distinct constant
solutions:
(b̄1 − ā1 ul− )g(ul− ) T
,
z † = ln ul− , ln
c̄
(b̄1 − ā1 ul+ )g(ul+ ) T
z ‡ = ln ul+ , ln
,
c̄
where, g(ul− ) = u2l− /n + ul− + a, and g(ul+ ) = u2l+ /n + ul+ + a. Chose M > 0
such that
(b̄1 − ā1 ul+ )g(ul+ ) (b̄1 − ā1 ul− )g(ul− )
|, | ln
| .
(3.17)
M > max | ln
c̄
c̄
And set
Ω1 = y = (y1 (t), y2 (t))T ∈ X : y1 (t) ∈ (ln l− , ln v− ), max |y2 (t)| < P + M ,
t∈Iω
T
Ω2 = y = (y1 (t), y2 (t)) ∈ X : min y1 (t) ∈ (ln l− , ln l+ ),
t∈Iω
max y1 (t) ∈ (ln v+ , P1 ), max |y2 (t)| < P + M .
t∈Iω
t∈Iω
Then both Ω1 and Ω2 are open bounded subset of X. It follows from (2.2) and
(3.17) that z † ∈ Ω1 , z ‡ ∈ Ω2 . With the help of (2.2), (3.9)–(3.11), and (3.16)–(3.17),
it is not difficult to show that Ω1 ∩ Ω2 = ∅ and Ωi verifies the requirements (a) of
Lemma 2.8 for i = 1, 2. When y ∈ ∂Ωi ∩ R2 , y is a constant vector in R2 , then
QN y 6= 0. Moreover, after direct calculation we get the Brouwer degree
deg(JQN, Ωi ∩ ker L, θ) = (−1)i+1 6= 0,
for i = 1, 2, where the isomorphism J can be chosen to be the identity mapping,
since Im Q = Ker L. Up to now, we have proved that Ωi verify all the requirements
of Lemma 2.8. Therefore, by Lemma 2.8, we derive that (1.1) has at least two
ω-periodic solutions lying in Dom L ∩ Ωi . The proof is complete.
Remark 3.2. Assume T = Z, then (1.1) becomes a discrete analogue of (1.1)
which has been discussed in [21]. Therefore, our result obtained generalized the
result in the literature.
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R.-H. HE, H.-X. LI, L. ZHANG
EJDE-2009/126
4. Example
Consider the following periodic predator-prey system with a IV functional response
1
1
y1∆ (t) = ( + sin t) − ( + cos t) exp{y1 (t − sin t)}
20
60
exp{y2 (t − cos t}
−
,
(4.1)
2 exp{y1 (t)} + 2 exp{y1 (t)} + 1
1
2(1 + cos t) exp{y1 (t − cos t)}
y2∆ (t) = −( + cos t) +
,
3
2 exp{y1 (t − cos t)} + 2 exp{y1 (t − cos t)} + 1
by choosing the 2π-periodic time scale
[
T=
[2(s − 1)π, 2sπ].
s∈Z
Then, system (4.1) has at least two 2π-periodic solutions.
Direct calculations lead to
κ = min{R+ ∩ T} = 0,
Iω = [κ, κ+ω]∩T = [0, 2π], ā1 = 1/60, ā2 = 1, b̄1 = 1/20, b̄2 = 1/3, exp{b̄1 +|b̄1 |} <
1.135, l+ < 2.302. It is straight forward to check that
p
ā2 = 1.000 > 0.757 > b̄2 1 + 2 a/n exp{(|b̄1 | + b̄1 )ω},
b̄1 = 0.050 > 0.045 > ā1 l+ exp{(|b̄1 | + b̄1 )ω},
which show that all the conditions in Theorem 3.1 are fulfilled. By Theorem 3.1
we derive that (4.1) has at least two 2π-periodic solutions.
Remark 4.1. System (4.1) models two populations (one is the predator and the
other is the prey) that are both continuous in one period of the year, die out in other
period of the year, and their offspring are renascent after incubating or dormant in
another period of the year, both of them giving rise to non-overlapping populations.
Acknowledgements. This work is supported by a grant 3777501 from the Natural
Science Development Foundation of CUIT of China.
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Rong-Hua He
Department of Mathematics, Chengdu University of Information Technology, Chengdu,
Sichuan 610103, China.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
E-mail address: ywlcd@cuit.edu.cn
Hong-Xu Li
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
E-mail address: hoxuli@sohu.com
Liang Zhang
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
E-mail address: lzhang1979@yahoo.com.cn