DSpace at VNU: Dynamics of species in a model with two predators and one prey

Nonlinear Analysis 74 (2011) 4868–4881

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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na

Dynamics of species in a model with two predators and one prey
Ta Viet Ton a,∗ , Nguyen Trong Hieu b
a

Department of Applied Physics, Graduate School of Engineering, Osaka University, Suita Osaka 565-0871, Japan

b

Faculty of Mathematics, Mechanics and Informatics, Hanoi National University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

article

abstract

info

Article history:
Received 2 May 2010
Accepted 27 April 2011
Communicated by Ravi Agarwal

In this paper, we study a predator–prey model which has one prey and two predators
with Beddington–DeAngelis functional responses. Firstly, we establish a set of sufficient
conditions for the permanence and extinction of species. Secondly, the periodicity of

positive solutions is studied. Thirdly, by using Liapunov functions and the continuation
theorem in coincidence degree theory, we show the global asymptotic stability of such
solutions. Finally, we give some numerical examples to illustrate the behavior of the
model.
© 2011 Elsevier Ltd. All rights reserved.

MSC:
34C27
34D05
Keywords:
Predator–prey system
Beddington–DeAngelis functional response
Permanence
Extinction
Periodic solution
Asymptotic stability
Liapunov function

1. Introduction
The dynamical relationship between predators and prey has been studied by several authors for a long time. In those
researches, to represent the average number of prey killed per individual predator per unit of time, a functional, called
the functional response, was introduced. The functional response can depend on only the prey’s density or both the prey’s
and the predator’s densities. However, some biologists have argued that in many situations, especially when predators
have to search for food, the functional response should depend on both the prey’s and the predator’s densities [1–6]. One
of the most popular functional responses is the fractional one as in the following prey–predator model. It is called the
Beddington–DeAngelis functional response:



x′1 = x1 (a1 − b1 x1 ) −

c1 x1 x2

α + β x1 + γ x2
c2 x1 x2
′

.
x2 = −a2 x2 +
α + β x1 + γ x2

,

In this model, xi (t ) represents the population density of species Xi at time t (i ≥ 1); X1 is the prey and X2 is the predator.
At time t , a1 (t ) is the intrinsic growth rate of X1 and ai (t ) is the death rate of X2 ; b1 (t ) measures the inhibiting effect of
the environment on X1 . This model was originally proposed by Beddington [7] and DeAngelis et al. [8] independently. Since
the appearance of these two investigations, there have been many other ones for analogous systems with diffusion in a

∗

Corresponding author. Tel.: +81 6 6879 4249.
E-mail addresses: (T.V. Ton), (N.T. Hieu).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2011.04.061


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

4869

constant environment [9–14]. However, a constant environment is rarely the case in real life. Most natural environments are
physically highly variable, i.e., the coefficients in those models should depend on time [15–18]. In order to continue studying
such models, in this paper, we consider a predator–prey model of one prey and two predators with Beddington–DeAngelis
functional responses:


c3 (t )x1 x3
c2 (t )x1 x2


−
,
x′1 = x1 [a1 (t ) − b1 (t )x1 ] −


α(
t
)
+
β(
t
)
x
+
γ
(
t
)
x

α(
t
)
+
β(t )x1 + γ (t )x3

1
2
[
]


d2 (t )x1
x′2 = x2 −a2 (t ) +
− b2 (t )x3 ,
(1.1)
α(t ) + β(t )x1 + γ (t )x2


[
]


d3 (t )x1


x′3 = x3 −a3 (t ) +
− b3 (t )x2 .
α(t ) + β(t )x1 + γ (t )x3
Here xi (t ) represents the population density of species Xi at time t (i ≥ 1), X1 is the prey and X2 , X3 are the predators. Two

predators share one prey and it is assumed that there are two types of competition between the two predators. The first type
is direct interference where individuals of each predator species act with aggression against individuals of the other predator
species. In our model, this type of competition is described by the coefficients b2 (t ) and b3 (t ). The second type of competition
is interference competition that occurs during hunting because predators spend time interacting with each other rather than
seeking prey. Here we assume that there is no competition of that type between individuals of the two different predator
d (t )x
species. Therefore, the Beddington–DeAngelis functional responses are of the form α(t )+β(it )x 1+γ (t )x (i = 2, 3). We use the
i
1
same coefficients α, β, γ in the functional responses of both predators, since it is assumed that both predators take the same
time to handle a prey once they encounter it and that individuals of each predator species interfere with each other when
hunting by exactly the same amount in both species. This assumption is somewhat restrictive from the biological viewpoint,
but it could be removed without greatly changing the analysis of system (1.1).
Throughout this paper, it is assumed that the functions ai (t ), bij (t ), ci (t ), di (t ), α(t ), β(t ), γ (t ) (1 ≤ i, j ≤ 3) are
continuous on R and bounded above and below by some positive constants.
This article is organized as follows. Section 2 provides some definitions and notation. In Section 3, we state some results
on invariant sets, and the permanence and extinction of system (1.1). Then, the asymptotic stability of solution is proved
by using a Liapunov function. In Section 4, we continue using other Liapunov functions and the continuation theorem
in coincidence degree theory to show the existence and global stability of a positive periodic solution. The final section
illustrates the behavior of system (1.1) by some computational results and gives our conclusion.
2. Definitions and notation
In this section we introduce some basic definitions and facts which will be used throughout this paper. Let R3+ =
{(x1 , x2 , x3 ) ∈ R3 | xi > 0 (i ≥ 1)}. Denote by x(t ) = (x1 (t ), x2 (t ), x3 (t )) the solution of system (1.1) with initial condition
x0 = (x01 , x02 , x03 ) = (x1 (t0 ), x2 (t0 ), x3 (t0 )), t0 ≥ 0. For biological reasons, throughout this paper, we only consider the
solutions x(t ) with positive initial values, i.e., x0 ∈ R3+ . Let g (t ) be a continuous function; for brevity, instead of writing g (t )
we write g. If g is bounded on R, we denote
g u = sup g (t ),
t ∈R

g l = inf g (t ),

t ∈R

ω

and gˆ = ω 0 g (t )dt, if g is a periodic function with period ω. The global existence and uniqueness of solution of system
(1.1) are guaranteed by the properties of the map defined by the right hand side of system (1.1) [19]. We have the following
lemma.
1

Lemma 2.1. Both the non-negative and positive cones of R3 are positively invariant for (1.1).
Proof. The solution x(t ) of (1.1) with initial value x0 satisfies

∫ t [
] 

c2 x2
c3 x3
0


x1 = x1 exp
a1 − b 1 x 1 −
−
du ,


α + β x1 + γ x2 ]α + β x1 + γ x3


∫t0t [


d2 x1
−a 2 +
x2 = x02 exp
− b2 x3 du ,

α + β x1 + γ x2
t0 [

∫
] 

t

d3 x1


x3 = x03 exp
−a 3 +
− b3 x2 du .
α + β x1 + γ x3
t0
The conclusion follows immediately for all t ∈ [t0 , ∞). The proof is complete.
Definition 2.2. System (1.1) is said to be permanent if there exist some positive δj (j = 1, 2) such that

δ1 ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ δ2 (i ≥ 1)
t →∞

for all solutions of (1.1).

t →∞


4870

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

Definition 2.3. A set A ⊂ R3+ is called an ultimately bounded region of system (1.1) if for any solution x(t ) of (1.1) with
positive initial values, there exists T1 > 0 such that x(t ) ∈ A for all t ≥ t0 + T1 .
Definition 2.4. A bounded non-negative solution x∗ (t ) of (1.1) is said to be globally asymptotically stable (or globally
∑3
attractive) if any other solution x(t ) of (1.1) with positive initial values satisfies limt →∞ i=1 |xi (t ) − x∗i (t )| = 0.
Remark 2.5. It is easy to see that if a solution of (1.1) is globally asymptotically stable, then so are all solutions. In this case,
system (1.1) is also said to be globally asymptotically stable.
3. The model with general coefficients
Let ϵ ≥ 0 be sufficiently small. Put
M1ϵ =
mϵ1 =
mϵi =

au1
bl1

Miϵ =

+ ϵ,

al1 γ l − c2u − c3u
bu1 γ l

dui M1ϵ − ali α l
ali γ l

,

− ϵ,

(3.1)

dli mϵ1 − (aui + bui Mjϵ )(β u mϵ1 + α u )

(aui + bui Mjϵ )γ u

(i, j ≥ 2, i ̸= j),

then Miϵ > mϵi (i ≥ 1). We will show that max{m0i , 0} (i ≥ 1) are the lower bounds for the limiting bounds of species Xi as
time t tends to infinity. This is obvious when m0i ≤ 0. Therefore, it is assumed that m0i > 0.
Hypothesis 3.1. m0i > 0 (i ≥ 1).
Theorem 3.2. Under Hypothesis 3.1, for any sufficiently small ϵ > 0 such that mϵi > 0 (i ≥ 1), a set Γϵ defined by
Γϵ = {(x1 , x2 , x3 ) ∈ R3 | mϵi < xi < Miϵ (i ≥ 1)} is positively invariant with respect to system (1.1).
Proof. Throughout this proof, we use the fact that the solution to the equation
X ′ (t ) = A(t , X )X (t )[B − X (t )]

(B ̸= 0)

is given by
X (t ) =

BX 0 exp



X 0 exp



t
t0



t
t0

BA(s, X (s))ds






BA(s, X (s))ds − 1 + B

,

where t0 ≥ 0 and X 0 = X (t0 ). Consider the solution of system (1.1) with an initial value x0 ∈ Γϵ . From Lemma 2.1 and from
the first equation of (1.1), we have
x′1 (t ) ≤ x1 (t )[a1 (t ) − b1 (t )x1 (t )]

≤ x1 (t )[au1 − bl1 x1 (t )]

= bl1 x1 (t )(M10 − x1 ).
Using the comparison theorem gives
x1 ( t ) ≤

≤

x01 M10 exp{au1 (t − t0 )}
x01 [exp{au1 (t − t0 )} − 1] + M10
x01 M1ϵ exp{au1 (t − t0 )}

x01

[exp{au1 (t − t0 )} − 1] + M1ϵ

< M1ϵ ,

t ≥ t0 .

It follows from the third equation of (1.1) and from (3.2) that
x′2 ≤ −al2 x2 +

≤ −al2 x2 +

du2 x1 x2

α l + β l x1 + γ l x2
du2 M1ϵ x2

α l + γ l x2

(3.2)


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

x2 (du2 M1ϵ − al2 α l ) − al2 γ l x2



=
=

4871



α l + γ l x2
al2 γ l

α + γ l x2
l

x2 (M2ϵ − x2 ).

Putting
C2 ( t ) =

al2 γ l

α l + γ l x2 (t )

,

(3.3)

and using the comparison theorem again yields



x2 (t ) ≤

M2ϵ x02 exp M2ϵ





x02 exp M2ϵ

t
t0

t
t0

C2 (s)ds







C2 (s)ds − 1 + M2ϵ

< M2ϵ ,

t ≥ t0 .

(3.4)

Similarly, x3 (t ) < M3ϵ for every t ≥ t0 .
Now, by the first equation of (1.1), it implies that
x′1 (t ) ≥ x1



al1 −

c2u + c3u

γl

− bu1 x1



= bu1 x1 (m01 − x1 ).

Since x01 > mϵ1 , by the comparison theorem, we obtain

x1 (t ) ≥

x01 m01 exp{bu1 m01 (t − t0 )}
x01 [exp{bu1 m01 (t − t0 )} − 1] + m01

> mϵ1 for all t ≥ t0 .

Similarly, for i, j ≥ 2 (i ̸= j),
x′i = −ai xi +

di x1 xi

α + β x1 + γ xi

− bi xi xj
d l m ϵ xi

i 1
≥ −(aui + bui Mjϵ )xi + u
α + β u mϵ1 + γ u xi

 l ϵ
di m1 − (aui + bui Mjϵ )(β u mϵ1 + α u ) xi − (aui + bui Mjϵ )γ u x2i
=
α u + β u mϵ1 + γ u xi
u
u ϵ
u
(ai + bi Mj )γ
= u

xi (mϵi − xi ),
α + β u mϵ1 + γ u xi

from which follows that xi (t ) > mϵi for all t ≥ t0 . We complete the proof.
In the next theorem, the permanence of system (1.1) is shown. A treatment called practical persistence to prove the
permanence of models and its application to various types of models can be seen in [20,9,21].
Theorem 3.3. Under Hypothesis 3.1, for any sufficiently small ϵ > 0 such that mϵi > 0,
mϵi ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ Miϵ
t →∞

t →∞

(i ≥ 1).

Consequently, system (1.1) is permanent.
Proof. According to the proof of Theorem 3.2 we have
x1 (t ) ≤

x01 M10 exp{au1 (t − t0 )}
x01

.

[exp{au1 (t − t0 )} − 1] + M10

Thus, lim supt →∞ x1 (t ) ≤ M10 , i.e., there exists t1 ≥ t0 such that x1 (t ) < M1ϵ for all t ≥ t1 . By the same arguments as made
for (3.4), it follows that



x2 (t ) ≤

M2ϵ x12 exp M2ϵ





x12 exp M2ϵ

t
t1

t
t1

C2 (s)ds







C2 (s)ds − 1 + M2ϵ

,

(3.5)

from which it is implied that 0 < x2 (t ) ≤ max{M2ϵ , x12 } for all t ≥ t1 , where x12 = x2 (t1 ). Then from (3.3), inft ≥t1 C2 (s) > 0.
By using (3.5), we have lim supt →∞ x2 (t ) ≤ M2ϵ . Similarly, lim supt →∞ x3 (t ) ≤ M3ϵ and lim inft →∞ xi (t ) ≥ mϵi (i ≥ 1). The
permanence follows from Definition 2.2. The proof is complete.


4872

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

Theorem 3.4. Let i ∈ {2, 3}. If Mi0 < 0 then limt →∞ xi (t ) = 0, i.e., the ith predator goes to extinction.
Proof. It follows from Mi0 < 0 that Miϵ < 0 with a sufficiently small ϵ . Similarly to the proof of Theorem 3.2 we have
x′i (t ) ≤

ali γ l

α + γ l xi
l

xi (Miϵ − xi ) < 0.

(3.6)

Thus, there exists C ≥ 0 such that limt →∞ xi (t ) = C and C ≤ xi (t ) ≤ x0i for all t ≥ t0 . If C > 0 then from (3.6) there exists
µ > 0 such that x′i (t ) < −µ for all t ≥ t0 . We therefore have xi (t ) < −µ(t − t0 ) + x0i and then limt →∞ xi (t ) = −∞, which
contradicts xi (t ) > 0 for all t ≥ t0 . Hence, limt →∞ xi (t ) = 0.
In order to consider the global asymptotic stability of system (1.1), we need the following result called Barbalat’s lemma.
Lemma 3.5 (See [22]). Let h be a real number and f be a non-negative function defined on [h, +∞) such that f is integrable and
uniformly continuous on [h, +∞). Then limt →∞ f (t ) = 0.
Theorem 3.6. Suppose that Hypothesis 3.1 holds and let ϵ > 0 be sufficiently small such that mϵi > 0 (i ≥ 1). Let x∗ be a
solution of system (1.1) satisfying

β c2 M2ϵ + d2 (α + γ M2ϵ ) β c3 M3ϵ + d3 (α + γ M3ϵ )
+
u2 (mϵ1 , M2ϵ )
u3 (mϵ1 , M3ϵ )
t →∞


ϵ
ϵ
di γ m1
ci (α + β M1 )
−
< 0,
lim sup bj +
ϵ
ϵ
ui (M1 , mi )
ui (mϵ1 , Miϵ )
t →∞




lim sup −b1 +

< 0,
(3.7)

where ui (a, b) = (α + β x∗1 + γ x∗i )(α + β a + γ b) (i, j ≥ 2, i ̸= j). Then x∗ is globally asymptotically stable.

Proof. Let x be the other solution of (1.1). From Theorem 3.3, Γϵ is an ultimately bounded region of (1.1). Then there exists
∑3
∗
T1 > 0 such that x, x∗ ∈ Γϵ for all t ≥ t0 + T1 . Consider a Liapunov function defined by V (t ) =
i=1 | ln xi − ln xi |, t ≥ t0 .
+
A direct calculation of the right derivative D V (t ) of V (t ) along the solution of (1.1) gives
D+ V (t ) =

3
−

sgn(xi − x∗i )



i=1


= sgn(x1 − x∗1 ) −c2


x′i
xi



−

x∗i ′



x∗i
x2

α + β x1 + γ x2

x3

x∗3

−

x∗2



α + β x∗1 + γ x∗2




−
− b1 (x1 − x1 )
α + β x1 + γ x3
α + β x∗1 + γ x∗3
[
]
x∗1

x1
b2
∗
+ d2 sgn(x2 − x∗2 )
−
(
x
−
x
)
−
3
3
α + β x1 + γ x2
α + β x∗1 + γ x∗2
d2
[
]
b3
x1
x∗1
∗
+ d3 sgn(x3 − x∗3 )
−
−
(
x
−
x
)

2
2
α + β x1 + γ x3
α + β x∗1 + γ x∗3
d3
∗
∗
∗
α(x2 − x2 ) + β(x1 x2 − x1 x2 )
≤ −b1 |x1 − x∗1 | − c2 sgn(x1 − x∗1 )
u2 (x1 , x2 )
∗
∗
∗
∗ α(x3 − x3 ) + β(x1 x3 − x1 x3 )
− c3 sgn(x1 − x1 )
u3 (x1 , x3 )
∗
∗
∗
α(
x
−
x
1
1 ) + γ (x1 x2 − x2 x1 )
+ d2 sgn(x2 − x∗2 )
+ b2 |x3 − x∗3 |
u2 (x1 , x2 )
α(x1 − x∗1 ) + γ (x1 x∗3 − x3 x∗1 )

+ d3 sgn(x3 − x∗3 )
+ b3 |x2 − x∗2 |.
u3 (x1 , x3 )
− c3

∗

It follows from x, x∗ ∈ Γϵ for t ≥ t0 + T1 and x1 x∗i − x∗1 xi = x1 (x∗i − xi ) + xi (x1 − x∗1 ) (i = 2, 3) that

(α + β x1 )(x2 − x∗2 ) − β x2 (x1 − x∗1 )
u2 (x1 , x2 )
∗
(α
+
β
x
)(
x
−
x
)
−
β
x
(
x1 − x∗1 )
1
3
3
3

− c3 sgn(x1 − x∗1 )
u3 (x1 , x3 )
(α + γ x2 )(x1 − x∗1 ) − γ x1 (x2 − x∗2 )
+ d2 sgn(x2 − x∗2 )
+ b2 |x3 − x∗3 |
u2 (x1 , x2 )

D+ V (t ) ≤ −b1 |x1 − x∗1 | − c2 sgn(x1 − x∗1 )


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

(α + γ x3 )(x1 − x1 ) − γ x1 (x3 − x3 )
+ d3 sgn(x3 − x∗3 )
+ b3 |x2 − x∗2 |
u2 (x1 , x3 )
]
[
β c2 M2ϵ + d2 (α + γ M2ϵ ) β c3 M3ϵ + d3 (α + γ M3ϵ )
+
|x1 − x∗1 |
≤ −b 1 +
u2 (mϵ1 , M2ϵ )
u3 (mϵ1 , M3ϵ )
[
]
c2 (α + β M1ϵ )
d2 γ mϵ1
+ b3 +
−

|x2 − x∗2 |
u2 (M1ϵ , mϵ2 )
u2 (mϵ1 , M2ϵ )
[
]
c3 (α + β M1ϵ )
d3 γ mϵ1
+ b2 +
−
|x3 − x∗3 | for t ≥ t0 + T1 .
u3 (M1ϵ , mϵ3 )
u3 (mϵ1 , M3ϵ )
∗

4873

∗

(3.8)

Combining (3.7) and (3.8) gives the existence of a positive number µ > 0 and of T2 ≥ t0 + T1 such that
D+ V (t ) ≤ −µ

3
−

|xi − x∗i | for every t ≥ T2 .

(3.9)

i =1

Integrating both sides of (3.9) from T2 to t yields
V (t ) + µ

∫ t −
3
T2


Then

t
T2


|xi − xi | ds ≤ V (T2 ) < ∞ for every t ≥ T2 .
∗

i=1


∑3

i =1

|xi − x∗i | ds ≤ µ−1 V (T2 ) < ∞ for every t ≥ T2 . Hence,

∑3

i=1

|xi − x∗i | ∈ L1 ([T2 , ∞)).

On the other hand, it follows from x, x∗ ∈ Γϵ for all t ≥ t0 + T1 and from the equations of (1.1) that the derivatives
∑3
∗
of xi (t ), x∗i (t )(i ≥ 1) are bounded on [T2 , ∞). As a consequence
i=1 |xi − xi | is uniformly continuous on [T2 , ∞). By
Lemma 3.5 we have limt →∞

∑3

i=1

|xi − x∗i | = 0, which completes the proof.

4. The model with periodic coefficients
In this section, we assume that the coefficients in system (1.1) are ω-periodic in t and bounded above and below by some
positive constants. We study the existence and stability of a periodic solution of this system. To do this, we will employ
an alternative approach to establish some criteria in terms of the average of the related functions over an interval of the
common period. That is continuation theorem in coincidence degree theory, which has been successfully used to establish
criteria for the existence of positive periodic solutions of some mathematical models of predator–prey type; we refer the
reader to [23–26]. To this end, we shall summarize in the following a few concepts and results from [27] that will be basic
for this section.
Let X and Y be two Banach spaces, let L:Dom L ⊂ X → Y be a linear mapping, and let N :X → Y be a continuous mapping.
The mapping L will be called a Fredholm mapping of index zero if the following conditions hold:
(i) Im L is closed;
(ii) dim Ker L = codim Im L < ∞.
If L is a Fredholm mapping of index zero and there exist continuous projections P :X → X and Q :Y → Y such that

Im P = Ker L, Im L = Ker Q = Im(I − Q ), it follows that
Lp = L|Dom L∩Ker P :(I − P )X → Im L
is invertible. We denote by Kp the inverse of that map. If Ω is an open bounded subset of X, the mapping N will be called
¯ if the mapping QN : Ω
¯ → Y is continuous and bounded, and Kp (I − Q )N : Ω
¯ → X is compact, i.e.,
L-compact on Ω
¯ ) is relatively compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism
it is continuous and Kp (I − Q )N (Ω
J :Im Q → Ker L. The following continuation theorem is from [27].
Lemma 4.1 (Continuation Theorem). Let X and Y be two Banach spaces and L a Fredholm mapping of index zero. Assume that
¯ → Y is L-compact on Ω
¯ with Ω is open and bounded in X. Furthermore, assume that
N :Ω
(a) for each λ ∈ (0, 1), x ∈ ∂ Ω ∩ Dom L, Lx ̸= λNx;
(b) for each x ∈ ∂ Ω ∩ Ker L, QNx ̸= 0;
(c) deg{QNx, Ω ∩ Ker L, 0} ̸= 0;

¯.
then the operator equation Lx = Nx has at least one solution in Dom L ∩ Ω


4874

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

We now put
aˆ 1

L11 = ln

L12

bˆ
1

,

H11 = ln

aˆ 1
bˆ 1

+ 2aˆ 1 ω,

  
c2 + c3
= ln aˆ 1 −
,
γ

Li1 = ln

aˆ 1 (dˆ i − aˆ i β l ) exp{2aˆ 1 ω} − aˆ i bˆ 1 α l
aˆ i bˆ 1 γ l




di

Hi1 = 2
Li2 = ln

H12 = L12 − 2aˆ 1 ω,

β

,

ω + Li1 ,

di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 })

γ u (ˆai + bˆ i exp{Hj1 })




Hi2 = Li2 − 2

di

β

,

ω (i, j ≥ 2, i ̸= j).

The convention here is that ln x = −∞ if x ≤ 0. In the next theorem, a sufficient condition for existence of an ω-periodic

solution of (1.1) is presented.
Theorem 4.2. If Li2 > −∞ (i ≥ 1) then system (1.1) has at least one positive ω-periodic solution.
Proof. Put xi (t ) = exp{ui (t )} (i ≥ 1), then system (1.1) becomes


c3 exp{u3 }
c2 exp{u2 }


u′1 = a1 − b1 exp{u1 } −
−
,


α
+
β
exp{u1 } + γ exp{u2 }
α
+
β
exp{u1 } + γ exp{u3 }


d2 exp{u1 }
− b2 exp{u3 },
u′2 = −a2 +
(4.1)

α + β exp{u1 } + γ exp{u2 }



 ′
d3 exp{u1 }

u3 = −a3 +
− b3 exp{u2 }.
α + β exp{u1 } + γ exp{u3 }
∑3
Let X = Y = {u = (u1 , u2 , u3 )T ∈ C1 (R, R3 ) | ui (t ) = ui (t + ω) (i ≥ 1)} with ‖u‖ =
i=1 maxt ∈[0,T ] |ui (t )|, u ∈ X. Then
X, Y are both Banach spaces with the above norm ‖ · ‖. Let
 


u1
N1 (t )
N u2 = N2 (t )
u3
N (t )
 3

c2 exp{u2 }
c3 exp{u3 }
a1 − b1 exp{u1 } −
−

α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 } 



d2 exp{u1 }


− b2 exp{u3 }
−a 2 +
= 
,


α + β exp{u1 } + γ exp{u2 }


d3 exp{u1 }
− b3 exp{u2 }
−a 3 +
α + β exp{u1 } + γ exp{u3 }

 ∫ ω
1
u1 (t )dt



 
 
 
   ω ∫0

1 ω
u′1

u1
u1
u1
u1


′
u2 ∈ X .
u2 (t )dt  ,
L u2 = u2  ,
P u2 = Q u2 = 
 ω ∫0

u3
u3
u3
u3
u′3

1 ω
u3 (t )dt
ω 0
ω


T
3
Then Ker L = {u ∈ X | u = (h1 , h2 , h3 ) ∈ R }, Im L = u ∈ Y | 0 ui (t )dt = 0 (i ≥ 1) , and dim Ker L = 3 = codim Im L.
Since Im L is closed in Y, L is a Fredholm mapping of index zero. It is easy to show that P , Q are continuous projections such
that Im P = Ker L, Im L = Ker Q = Im (I − Q ). Furthermore, the generalized inverse (to L) KP :Im L → Dom L ∩ Ker P exists

and is given by
t

∫
u1
u2
u3

 
KP

u1 (s)ds −

∫0
 t

=  u2 (s)ds −
∫0
 t
u3 (s)ds −
0

1

ω

∫

∫

t

T ∫0 ∫0
ω
t
1

u1 (s)dsdt






u2 (s)dsdt  .

T ∫0 ∫0

ω
t
1
u3 (s)dsdt
T

0

0


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

4875

¯ with any open bounded
Obviously, QN and KP (I − Q )N are continuous. It is easy to see that N is L-compact on Ω
set Ω ⊂ X.
Now we will find an appropriate open, bounded subset Ω for application of the continuation theorem. Corresponding to
the operator equation Lu = λNu, λ ∈ (0, 1), we have

u1 = λ a1 − b1 exp{u1 } −

c2 exp{u2 }

′


u 2 = λ −a 2 +

d2 exp{u1 }

′

α + β exp{u1 } + γ exp{u2 }


u 3 = λ −a 3 +

α + β exp{u1 } + γ exp{u2 }

−

α + β exp{u1 } + γ exp{u3 }


,

− b2 exp{u3 } ,

(4.2)



d3 exp{u1 }

′



c3 exp{u3 }

α + β exp{u1 } + γ exp{u3 }

− b3 exp{u2 } .

Suppose that (u1 , u2 , u3 ) ∈ X is an arbitrary solution of system (4.2) for a certain λ ∈ (0, 1). Integrating both sides of (4.2)
over the interval [0, ω], we obtain
aˆ 1 ω =

∫ ω

c3 exp{u3 }

c2 exp{u2 }



+
dt ,
α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 }
∫ ω
∫ ω
di exp{u1 }dt
aˆ i ω +
bi exp{uj }dt =
α + β exp{u1 } + γ exp{ui }
0
0


∫ ω

di
di
dt =
ω (i, j ≥ 2, i ̸= j).
≤
β
β
0
b1 exp{u1 } +

0

(4.3)

It follows from (4.2) and (4.3) that for i, j ≥ 2 (i ̸= j),
ω

∫

∫

|u1 (t ) |dt ≤ λ
′

ω

∫
a1 dt +

∫
+
0

0

ω

ω

∫
b1 exp{u1 }dt +

0

0

∫

ω

0

c2 exp{u2 }
dt
α + β exp{u1 } + γ exp{u2 }



c3 exp{u3 }
dt
α + β exp{u1 } + γ exp{u3 }

< 2aˆ 1 ω,


ω

di
|ui (t )′ |dt < 2

ω.
β
0

Since u ∈ X, there exist ξi , ηi ∈ [0, ω] such that
ui (ξi ) = min ui (t ),

ui (ηi ) = max ui (t )

t ∈[0,ω]

t ∈[0,ω]

(i ≥ 1).

From the first equation of (4.3) and (4.4), we obtain aˆ 1 ω ≥
u1 (ξ1 ) < L11 . Hence
u1 (t ) ≤ u1 (ξ1 ) +

ω

∫

(4.4)

ω
0

b1 exp{u1 (ξ1 )}dt = bˆ 1 ω exp{u1 (ξ1 )}, from which follows

|u′1 (t )|dt < L11 + 2aˆ 1 ω = H11 for all t ≥ 0.

0

On the other hand, from the first equation of (4.3) and (4.4), we also have
aˆ 1 ω ≤

ω

∫

b1 exp{u1 (η1 )}dt +

0

ω

∫

c2 (t ) + c3 (t )

0


= bˆ 1 exp{u1 (η1 )} +

  
c2 + c3
ω.
γ

Then for any t ≥ 0,
u1 (t ) ≥ u1 (η1 ) −

ω

∫

|u′1 (t )|dt

  
c2 + c3
≥ ln aˆ 1 −
− 2aˆ 1 ω
γ
0

= H12 .

γ (t )

dt


4876

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

From the arguments above, we have H12 ≤ u1 (t ) ≤ H11 for all t ∈ [0, ω]. It then follows from the second equation of (4.3)
and (4.4) that

ω

∫

di exp{u1 }dt
α + β exp{u1 } + γ exp{ui }
∫ ω
di exp{H11 }dt
≤
l
l
α + β exp{H11 } + γ l exp{ui (ξi )}
0
ωdˆ i exp{H11 }
= l
,
l
α + β exp{H11 } + γ l exp{ui (ξi )}

aˆ i ω ≤

0

from which it is implied that
ui (ξi ) ≤ ln

(dˆ i − aˆ i β l ) exp{H11 } − aˆ i α l
aˆ i γ l
aˆ 1 (dˆ i − aˆ i β l ) exp{2aˆ 1 ω} − aˆ i bˆ 1 α l

= ln

aˆ i bˆ 1 γ l

,

and then
ui (t ) ≤ ui (ξi ) +

ω

∫

|u′i (t )|dt

0

≤ ln

aˆ 1 (dˆ i − aˆ i β l ) exp{2aˆ 1 ω} − aˆ i bˆ 1 α l
aˆ i bˆ 1 γ l




+2

di

β

ω

= Hi1 (i ≥ 2).
Similarly, for i, j ≥ 2 (i ̸= j) and t ≥ 0, we have
aˆ i ω =

ω

∫

[

di exp{u1 }

]
− bi exp{uj } dt

α + β exp{u1 } + γ exp{ui }
]
di exp{H12 }
≥
− bi exp{Hj1 } dt
α u + β u exp{H12 } + γ u exp{ui (ηi )}
0


dˆ i exp{H12 }
=
− bˆ i exp{Hj1 } ω,

α u + β u exp{H12 } + γ u exp{ui (ηi )}
0

∫

ω

[

from which it is implied that
ui (ηi ) ≥ ln

di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 })

γ u (ˆai + bˆ i exp{Hj1 })

,

and
ui (t ) ≥ ui (ηi ) −

ω

∫

|u′i (t )|dt

0

≥ ln

di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 })

γ u (ˆai + bˆ i exp{Hj1 })




di
−2
ω
β

= Hi2 .
Put Bi = max{|Hi1 |, |Hi2 |} (i ≥ 1), then maxt ∈[0,ω] |ui | ≤ Bi . Thus, for any solution u ∈ X of (4.2), we have ‖u‖ ≤

and, clearly, Bi (i ≥ 1) are independent of λ. Taking B =

∑4

∑3

i=1

Bi ,

i=1 Bi where B4 is taken sufficiently large such that B4 ≥

i=1
j=1 |Lij | and letting Ω = {u ∈ X| ‖u‖ < B}, then Ω satisfies the condition (a) of Lemma 4.1. To compute the

Brouwer degree, let us consider the homotopy

∑3 ∑2

Hµ (u) = µQN (u) + (1 − µ)G(u),

µ ∈ [0, 1],


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

4877

where G : R3 → R3 ,

ˆ
aˆ 1 −
∫ bω1 exp{u1 }
d2 (t ) exp{u1 }dt
1




−ˆa2 − bˆ 2 exp{u3 } +

G(u) = 




ω

1
−ˆa3 − bˆ 3 exp{u2 } +
ω

∫0 ω
0





α + β exp{u2 } + γ exp{u2 } 
.

d3 (t ) exp{u1 }dt
α + β exp{u3 } + γ exp{u3 }

We have

] 
∫ ω[
µc2 exp{u2 }
µc3 exp{u3 }
1
ˆ
+
dt

aˆ − b1 exp{u1 } −
 1

ω 0 α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 }


∫
ω


d
exp
{
u
}
dt
1
2
1

.
ˆ
Hµ (u) = 
−ˆa2 − b2 exp{u3 } +

ω 0 α + β exp{u1 } + γ exp{u2 }


∫ ω



1
d3 exp{u1 }dt
−ˆa3 − bˆ 3 exp{u2 } +
ω 0 α + β exp{u1 } + γ exp{u3 }


By carrying out similar arguments to those above, one can easily show that any solution u∗ of the equation Hµ (u) = 0 ∈ R3
with µ ∈ [0, 1] satisfies Li1 ≤ u∗i ≤ Li2 (i ≥ 1). Thus, 0 ̸∈ Hµ (∂ Ω ∩ Ker L) for µ ∈ [0, 1], and then QN (∂ Ω ∩ Ker L) ̸= 0.
Note that the isomorphism J can be the identity mapping I; since Im P = Ker L, by the invariance property of homotopy, we
have
deg(JQN , Ω ∩ Ker L, 0) = deg(QN , Ω ∩ Ker L, 0)

= deg(QN , Ω ∩ R3 , 0)
= deg(G, Ω ∩ R3 , 0)
 

−bˆ 1 exp{u1 }
0
0





∂ f2 (u1 , u2 )
  ∂ f2 (u1 , u2 )


ˆ


−
b2 exp{u3 }
= sgn det 

∂
u
∂
u
2

 ∂ f (u 1, u )


∂ f3 (u1 , u3 ) 
3
1
3




−bˆ 3 exp{u2 }
∂ u1
∂ u3



∂ f2 (u1 , u2 ) ∂ f3 (u1 , u3 ) ˆ ˆ
= −sgn bˆ 1 exp{u1 }

+ b2 b3 exp{u2 + u3 } ,
∂ u2
∂ u3

(4.5)

where deg(·, ·, ·) is the Brouwer degree [28] and
f 2 ( u1 , u2 ) =
f 3 ( u1 , u3 ) =

1

ω

ω

∫

1

∫0 ω

ω

0

d2 exp{u1 }dt
,
α + β exp{u1 } + γ exp{u2 }
d3 exp{u1 }dt

.
α + β exp{u1 } + γ exp{u3 }

It is easy to see that the functions fi (u1 , ui ) are decreasing in ui ∈ R (i ≥ 2). Then

∂ f2 (u1 , u2 ) ∂ f3 (u1 , u3 )
> 0.
∂ u2
∂ u3

(4.6)

Combining (4.5) and (4.6) gives deg(JQN , Ω ∩ Ker L, 0) = −1 ̸= 0. By now we have proved that Ω verifies all requirements
¯ , i.e., (4.1) has at least one ω-periodic solution u∗ in
of Lemma 4.1, then Lu = Nu has at least one solution in Dom L ∩ Ω
¯ . Set x∗i = exp{u∗i }(i ≥ 1), then x∗ is an ω-periodic solution of system (1.1) with strictly positive components. We
Dom L ∩ Ω
complete the proof.
Corollary 4.3. If the ω-periodic solution x∗ in Theorem 4.2 satisfies the assumptions in Theorem 3.6, then x∗ is globally
asymptotically stable.
Proof. The proof of this corollary is derived directly from Theorems 3.6 and 4.2.
5. Numerical examples and conclusion
In this section, we present some numerical examples. As a first example, we consider the case a1 = 4.7 + sin(π t ), b1 =
2.4 − cos(2.7t ), c2 = 10.1 + 2.2 sin(1.4t ), c3 = 9.3 + cos(2.8t ), a2 = (1.1 − cos(2π t ))/2.5, b2 = 1.4 + sin(0.8t ), d2 = 9.9 −
0.4 sin(0.6t ), a3 = (2.2 − cos(1.7t ))/6, b3 = (2.3 + 1.3 sin(3.2t ))/2, d3 = 8.5 + sin(0.9π t ), α = (1.2 − cos(2t ))/4, β =
(2.3 + cos(1.2t ))/5, γ = 5.5 − 0.5 sin(t ). By (3.1), Mi0 > 0, m0i > 0 (i ≥ 1) then Hypothesis 3.1 holds and system (1.1)


4878

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

x3

x1

x2
Fig. 1. Orbit of globally asymptotically stable system.

t-x1

1.5

1.26

x1

1.02

0.78

0.54

0.3

0

6

12

16

24

30

t
Fig. 2. Population sizes X1 with respect to time.

t-x2

1.6
1.28

x2

0.96
0.64
0.32
0

0

6

12

18

24

30

t
Fig. 3. Population sizes X2 with respect to time.

has an invariant set. Fig. 1 is the orbit of solution with initial value x0 = (1.3, 1.4, 1.1); it seems to be very chaotic but it is
permanent. According to Theorem 3.6, it is not only permanent but also globally asymptotically stable. In spite of different
initial values,
x0 = (1.3, 1.4, 1.1) and x¯ 0 = (0.6, 1, 1.5), solutions
xi and x¯ i (i ≥ 1) still tend to one trajectory (see Figs. 2–4).
In the next example, Theorem 3.4 will be illustrated by system (1.1) with coefficients a1 = 3 + 1.2 sin(2.4t ), b1 =
2.4 + 2 cos(π t ), c2 = 5.1 − 0.9 sin(1.7π t ), c3 = 4.4 − 1.2 cos(π t ), a2 = 1.1 − cos(1.9t ), b2 = (2.6 + sin(3π t ))/4, d2 =
4.3 − 1.7 sin(0.5π t ), a3 = 2.1 − 2.5 cos(1.4π t ), b3 = 0.8 − 1.3 sin(1.6t ), d3 = 2.9 + 2.3 sin(0.4t ), α = (1.8 − cos(5.7t ))/3,


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

4879

t-x3

1.56

1.32

x3

1.08

0.84

0.6

0.36

0

6

12

18

24

30

t
Fig. 4. Population sizes X3 with respect to time.

x3

x2

x1
Fig. 5. Orbit of non-permanent system.

3

X1
X2
X3

2.4

1.8

1.2

0.6

0

0

4

8

12

16

20

t
Fig. 6. Orbit of non-permanent system with respect to time.

β = 1 − 0.2 cos(0.2π t ), γ = 3.4 − 1.6 sin(1.8t ) and initial condition x0 = (1.3, 2.1, 2.4). Since M30 < 0 then the density
of species X3 becomes extinct (see Figs. 5–6). Therefore, the predator X3 vanishes and system (1.1) is not permanent.
For the model with periodic coefficients, we consider the last example concerning the numerical solutions of system
(1.1) where a1 = 3 + 1.3 sin(2t ), b1 = 2.2 + 1.9 cos(2t ), c2 = 2.8(2.2 − sin(2t )), c3 = (3.5 − 2 cos(2t ))/2, a2 =
1 − 0.6 cos(2t ), b2 = 1.2 + 0.5 sin(2t ), d2 = 4 + 1.8 sin(2t ), a3 = (1.2 − cos(2t ))/3, b3 = 1.4 + 1.1 sin(2t ), d3 =
3.1 − 2.3 sin(2t ), α = 0.1(3.2 − 2 cos(2t )), β = (2.1 + 1.8 cos(2t ))/5, γ = 3.3 − sin(2t ) and the initial value
x0 = (1.1, 1.9, 1.4). Under π -periodic perturbation satisfying Theorem 4.2, system (1.1) has positive π -periodic solution
(see Figs. 7–8). Moreover, as the hypothesis of Theorem 3.6 also holds then it is globally asymptotically stable.
In conclusion, this work provides some results about the asymptotic behavior of a model of one prey and two
predators with Beddington–DeAngelis functional responses. The mathematical analysis presented in this model shows


4880

T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881

x3

x1

x2

Fig. 7. Orbit of periodic system.

2

X1
X2
X3

1.6

1.2

0.8

0.4

0
0

3.6

7.2

10.8

14.4

18

t
Fig. 8. Orbit of periodic system with respect to time.

that according to the values of the coefficients, one can make suitable predictions about the asymptotic behavior of the
overall predator–prey system including the permanence, the periodicity, the global asymptotic stability and especially the
extinction of species. Those conclusions warn us to make timely decisions to protect species in our ecological system. Further,
the given conditions on coefficients can be easily numerically computed.
Acknowledgments
The authors would like to thank the anonymous referees for their very helpful suggestions which improved the

manuscript.
References
[1] R. Arditi, H.R. Akcakaya, Underestimation of mutual interference of predators, Oecologia 83 (1990) 358–361.
[2] R. Arditi, N. Perrin, H. Saiah, Functional response and heterogeneities: an experimental test with cladocerans, Oikos 60 (1991) 69–75.
[3] P.M. Dolman, The intensity of interference varies with resource density: evidence from a field study with snow buntings, Plectroph. Niv. Oecologia
102 (1995) 511–514.
[4] C. Jost, R. Arditi, From pattern to process: identifying predator–prey interactions, Popul. Ecol. 43 (2001) 229–243.
[5] C. Jost, S. Ellner, Testing for predator dependence in predator–prey dynamics: a nonparametric approach, Proc. R. Soc. Lond. Ser. B 267 (2000)
1611–1620.
[6] G.T. Skalski, J.F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology 82 (2001)
3083–3092.
[7] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol. 44 (1975) 331–340.
[8] D.L. DeAngelis, R.A. Goldstein, R.V. O’Neill, A model for trophic interaction, Ecology 56 (1975) 881–892.
[9] R.S. Cantrell, C. Cosner, Effects of domain size on the persistence of populations in a diffusive food chain model with DeAngelis–Beddington functional
response, Nat. Resour. Model. 14 (2001) 335–367.
[10] R.S. Cantrell, C. Cosner, On the dynamics of predator–prey models with the Beddington–DeAngelis functional response, J. Math. Anal. Appl. 257 (2001)
206–222.


T.V. Ton, N.T. Hieu / Nonlinear Analysis 74 (2011) 4868–4881
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]

[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]

4881

R.S. Cantrell, C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, Wiley, Chichester, 2003.
C. Cosner, D.L. DeAngelis, J.S. Ault, D.B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol. 56 (1999) 65–75.
T. Hwang, Global analysis of the predator–prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl. 281 (2003) 395–401.
T. Hwang, Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl. 290 (2004)
113–122.
J. Cui, Y. Takeuchi, Permanence, extinction and periodic solution of predator–prey system with Beddington–DeAngelis functional response, J. Math.
Anal. Appl. 317 (2006) 464–474.
M. Fan, Y. Kuang, Dynamics of a non-autonomous predator–prey system with the Beddington–DeAngelis functional response, J. Math. Anal. Appl. 295
(2004) 15–39.
T.V. Ton, Dynamics of species in a non-autonomous Lotka–Volterra system, Acta Math. Acad. Paedagog. Nyházi. 25 (2009) 45–54.
Z. Zeng, M. Fan, Study on a non-autonomous predator prey system with Beddington–DeAngelis functional response, Math. Comput. Modelling 48
(2008) 1755–1764.
D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, in: Pitman Monographs and Surveys in Pure and
Applied Mathematics, 1993.
R.S. Cantrell, C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 247–272.
C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol. 58 (1996) 207–246.
I. Barbălat, Systèmes dèquations diffèrentielles dosillations non linéaires, Rev. Roumaine Math. Pures Appl. 4 (1975) 267–270.
Y. Li, Positive periodic solution for neutral delay model, Acta Math. Sinica 39 (1996) 790–795.
Y. Li, Periodic solutions of a periodic neutral delay equations, J. Math. Anal. Appl. 214 (1997) 11–21.

Y. Li, Periodic solution of a periodic delay predator–prey system, Proc. Amer. Math. Soc. 127 (1999) 1331–1335.
S.H. Saker, Oscillation and global attractivity of hematopoiesis model with delay time, Appl. Math. Comput. 136 (2003) 27–36.
R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.
W. Krawcewicz, J. Wu, Theory of Degrees, with Applications to Bifurcations and Differential Equations, John Wiley, New York, 1997.