DSpace at VNU: Survival of three species in a nonautonomous Lotka-Volterra system

J. Math. Anal. Appl. 362 (2010) 427–437

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa

Survival of three species in a nonautonomous Lotka–Volterra system
Ta Viet Ton
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Viet Nam

a r t i c l e

i n f o

a b s t r a c t

Article history:
Received 10 December 2008
Available online 4 August 2009
Submitted by C.V. Pao

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra
system of three species with constant interaction coefficients. In this paper, we study a
nonautonomous Lotka–Volterra model with one predator and two preys. The explorations
involve the persistence, extinction and global asymptotic stability of a positive solution.
© 2009 Elsevier Inc. All rights reserved.

Keywords:
Predator–prey model

Survival
Extinction
Persistence
Asymptotic stability
Liapunov function

1. Introduction
We consider a Lotka–Volterra model of one predator and two preys

⎧
⎪
⎨ x1 (t ) = x1 (t ) a1 (t ) − b11 (t )x1 (t ) − b12 (t )x2 (t ) − b13 (t )x3 (t ) ,
x2 (t ) = x2 (t ) a2 (t ) − b21 (t )x1 (t ) − b22 (t )x2 (t ) − b23 (t )x3 (t ) ,
⎪
⎩
x3 (t ) = x3 (t ) −a3 (t ) + b31 (t )x1 (t ) + b32 (t )x2 (t ) − b33 (t )x3 (t ) .

(1.1)

Here xi (t ) represents the population density of species X i at time t (i = 1, 2, 3), x1 (t ), x2 (t ) are the two preys and they
interact other and x3 (t ) is the predator. ai (t ), b i j (t ) (i , j = 1, 2, 3) are continuous on R and bounded above and below
function by positive constants. At time t, ai (t ) is the intrinsic growth rate of prey species X i (i = 1, 2), a3 (t ) is the death
rate of the predator species X 3 , b i j (t ) measures the amount of competition between the prey X i and X j (i = j, i , j = 1, 2),
b3i (t )
b i3 (t )

denotes the coefficient in conversing prey species X i into new individual of predator species X 3 (i = 1, 2) and b ii (t )

(i = 1, 2, 3) measures the inhibiting effect of environment on the ith population.
This paper is organized as follows. Section 2 provides some definitions and notations. In Section 3 we state some results

about invariant set and asymptotic stability for problem (1.1). Section 4 is special case of Section 3 when the coefficient
b i j (t ) is constant and Section 5 is special case of Section 4 when the coefficient ai (t ) is constant (i , j = 1, 2, 3).
2. Definition and notation
In this section we summarize the basic definitions and facts which are used later. Let R3+ := {(x1 , x2 , x3 ) ∈ R3 | xi
i = 1, 2, 3}. For a bounded continuous function g (t ) on R, we use the following notation:

g u := sup g (t ),
t ∈R

g l := inf g (t ).
t ∈R

E-mail address:
0022-247X/$ – see front matter
doi:10.1016/j.jmaa.2009.07.053

©

2009 Elsevier Inc. All rights reserved.

0,


428

T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

The global existence and uniqueness of the solutions of system (1.1) can be found in [3]. From the uniqueness theorem, it
is easy to prove that
Lemma 2.1. Both the nonnegative and positive cones of R3 are positively invariant for (1.1).

In the remainder of this paper, for biological reasons, we only consider the solutions (x1 (t ), x2 (t ), x3 (t )) with positive
initial values, i.e, xi (t 0 ) > 0, i = 1, 2, 3.
Definition 2.2. System (1.1) is said to be permanent if there exist positive constants δ ,
with 0 < δ <
such that for
all i = 1, 2, 3, lim inft →∞ xi (t ) δ , lim supt →∞ xi (t )
for all solutions of (1.1) with positive initial values. System (1.1) is
called persistent if for all i = 1, 2, 3, lim supt →∞ xi (t ) > 0, ∀i = 1, 2, 3 and strongly persistent if lim inft →∞ xi (t ) > 0 for all
solutions with positive initial values.
Definition 2.3. A set A is called to be an ultimately bounded region of system (1.1) if for any solution (x1 (t ), x2 (t ), x3 (t ))
of (1.1) with positive initial values, there exists T 1 > 0 such that (x1 (t ), x2 (t ), x3 (t )) ∈ A for all t t 0 + T 1 .
Definition 2.4. A bounded nonnegative solution (x∗1 (t ), x∗2 (t ), x∗3 (t )) of (1.1) is said to be globally asymptotically stable (or
globally attractive) if any other solution (x1 (t ), x2 (t ), x3 (t )) of (1.1) with positive initial values satisfies
3

xi (t ) − x∗i (t ) = 0.

lim

t →∞

i =1

Remark 2.5. It is easy to see that if system (1.1) has a solution is globally asymptotically stable, then any solution of (1.1) is
also globally asymptotically stable.
Lemma 2.6. (See [2].) Let h be a real number and f be a nonnegative function defined on [h, +∞) such that f is integrable on [h, +∞)
and is uniformly continuous on [h, +∞), then limt →∞ f (t ) = 0.
3. The model with general coefficients
Theorem 3.1. If mi > 0, i = 1, 2, 3, then set Γ defined by

Γ = (x1 , x2 , x3 ) ∈ R3 mi

M i , i = 1, 2, 3

x

(3.1)

is positively invariant with respect to system (1.1), where

a1u

M 1 :=

bl11
−al3

M 3 :=
m2 :=
and

+ ,

M 2 :=

a2u
bl22

u
u

+ b31
M 1 + b32
M2

bl33
u
u
al2 − b21
M 1 − b23
M3
u
b22

+ ,

,

m1 :=
m3 :=

,

u
u
al1 − b12
M 2 − b13
M3
u
b11

,

−a3u + bl31m1 + bl32m2
u
b33

(3.2)

0 is constant.

Proof. First, we know that the logistic equation

X (t ) = A (t ) X (t ) B − X (t )

( B = 0)

has a unique solution

X (t ) =

B X 0 exp B
X 0 exp B

t
t0

t
t0

A (s) ds

A (s) ds − 1 + B

(3.3)

where X 0 := X (t 0 ).
Next, we consider the solution of system (1.1) with the initial values (x10 , x20 , x30 ) ∈ Γ . By Lemma 2.1, we have xi (t ) > 0
for all t t 0 and i = 1, 2, 3. We have

x1 (t )

x1 (t ) a1 (t ) − b11 (t )x1 (t )

x1 (t ) a1u − bl11 x1 (t ) = bl11 x1 (t ) M 10 − x1 (t ) .


T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

429

Using the comparison theorem, we obtain that

x1 (t )
Because x10

x3 (t )

x10 M 10 exp{a1u (t − t 0 )}

x10 M 1 exp{a1u (t − t 0 )}

x10 [exp{a1u (t − t 0 )} − 1] + M 10

x10 [exp{a1u (t − t 0 )} − 1] + M 1

M 1 , we have x1 (t )

M 3 for all t

M 2 for all t

t 0 and because

t 0 . Now, from above results, we have

u
u
u
u
x1 (t ) al1 − b12
M 2 − b13
M 3 − b11
x1 (t ) = b11
x1 (t ) m1 − x1 (t ) .

mi , i = 1, 2, 3, we get that

From the comparison theorem and from xi0

x1 (t )

t 0 . Similarly, we get that x2 (t )

u
u
x3 (t ) −al3 + b31
M 1 + b32
M 2 − bl33 x3 (t ) = bl33 x3 (t ) M 3 − x3 (t ) ,

we also get that x3 (t )

x1 (t )

M 1 for all t

(3.4)

.

u
m1 x10 exp{b11
m1 (t − t 0 )}
u
x10 [exp{b11
m1 (t − t 0 )} − 1] + m1

Similarly, we obtain that x2 (t )

m2 , x3 (t )

m1

m3 for all t

for all t

t0 .

t 0 . The proof is complete.

✷

Theorem 3.2. If mi > 0, i = 1, 2, 3, then the set Γ is an ultimately bounded region, i.e., system (1.1) is permanent.
Proof. From (3.4) we have

lim sup x1 (t )
t →∞

M1 .

Similarly,

lim sup x2 (t )
t →∞

M2 .

Thus there exists t 1 t 0 such that xi (t ) M i , i = 1, 2 for all t t 1 . By the same argument in Theorem 3.1, we also get that
lim supt →∞ x3 (t ) M 3 . Similarly, we claim that lim inft →∞ xi (t ) mi . Then Γ is an ultimately bounded region. ✷
Theorem 3.3. If M 30 < 0 then limt →∞ x3 (t ) = 0.

Proof. We see that if M 30 < 0 then M 3 < 0 with

x3 (t )

is sufficiently small. Similar to the proof of Theorem 3.1 we get that

bl33 x3 (t ) M 3 − x3 (t ) < 0.

(3.5)

Therefore, 0 < x3 (t ) x3 (t 0 ) for t t 0 and there exists c 0 such that limt →∞ x3 (t ) = c . If c > 0 then 0 < c
x3 (t )
x3 (t 0 ), t t 0 . From (3.5), there exists ν > 0 such that x3 (t ) < −ν for all t t 0 . It follows x3 (t ) < −ν (t − t 0 ) + x3 (t 0 ) and
limt →∞ x3 (t ) = −∞ which contradicts our result that x3 (t ) > 0 for all t t 0 . Hence, limt →∞ x3 (t ) = 0. ✷
Theorem 3.4. Let (x∗1 (t ), x∗2 (t ), x∗3 (t )) be a solution of system (1.1). If mi > 0, i = 1, 2, 3 and the following conditions hold

⎧
lim inf 2m1 b11 (t ) + m2 b12 (t ) + m3 b13 (t ) − M 2 b21 (t ) − M 3 b31 (t ) − a1 (t ) > 0,
⎪
⎪
⎪ t →∞
⎨
lim inf 2m2 b22 (t ) + m1 b21 (t ) + m3 b23 (t ) − M 1 b12 (t ) − M 3 b32 (t ) − a2 (t ) > 0,
t →∞
⎪
⎪
⎪
⎩ lim inf 2m b33 (t ) − M b13 (t ) + b31 (t ) − M b23 (t ) + b32 (t ) + a3 (t ) > 0,
3
1

2

(3.6)

t →∞

then (x∗1 (t ), x∗2 (t ), x∗3 (t )) is globally asymptotically stable.
Proof. From (3.6), there exists t 1 > t 0 such that (3.6) holds when we replace lim inft →∞ in (3.6) by inft t1 . Let
(x1 (t ), x2 (t ), x3 (t )) be any solution of (1.1) with positive initial value. Since Γ is an ultimately bounded region, there exists
T 1 > t 1 such that (x1 (t ), x2 (t ), x3 (t )) ∈ Γ and (x∗1 (t ), x∗2 (t ), x∗3 (t )) ∈ Γ for all t T 1 .
3
T 1 . For brevity, we denote xi (t ), x∗i (t ), ai (t )
Considering a Liapunov function defined by V (t ) = i =1 |xi (t ) − x∗i (t )|, t
∗
and b i j (t ) by xi , xi , ai and b i j , respectively. A direct calculation of the right derivative D + V (t ) of V (t ) along the solution of

system (1.1) produces


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T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

3

D + V (t ) =

sgn xi − x∗i xi − x∗i

i =1

2

3

=

xi a i −
i =1

bi j x j

− x∗i ai −

j =1

+ sgn x3 − x∗3

3

b i j x∗j

sgn xi − x∗i

j =1
2

x3 −a3 +

b3 j x j − b33 x3
j =1

2

ai − b ii xi + x∗i

=

2

− x∗3 −a3 +

b3 j x∗j − b33 x∗3

j =1
3

xi − x∗i − sgn xi − x∗i

i =1

b i j xi x j − x∗i x∗j

− a3 + b33 x3 + x∗3

x3 − x∗3

j =1 , j = i

+ sgn x3 − x∗3

2

b3 j x3 x j − x∗3 x∗j

j =1
2

3

ai − b ii xi + x∗i −

=
i =1

bi j x j

xi − x∗i − sgn xi − x∗i

j =1 , j = i

3

b i j x∗i x j − x∗j

j =1 , j = i

− a3 + b33 x3 + x∗3 − b31 x1 − b32 x2 x3 − x∗3 + sgn x3 − x∗3

2

b3 j x∗3 x j − x∗j .

j =1

Then
2

D + V (t )

3

ai − b ii xi + x∗i −

i =1

bi j x j

3

xi − x∗i +

j =1 , j = i

b i j x∗i x j − x∗j

j =1 , j = i
2

− a3 + b33 x3 + x∗3 − b31 x1 − b32 x2 x3 − x∗3 +

b3 j x∗3 x j − x∗j

j =1
2

3

ai − 2b ii mi −
i =1

bi j m j

3

xi − x∗i + M i

j =1 , j = i

b i j x j − x∗j

j =1 , j = i

− a3 + 2b33m3 − b31 M 1 − b32 M 2 x3 − x∗3 + M 3

2

b3 j x j − x∗j

j =1

= M 2 b21 + M 3 b31 + a1 − 2m1 b11 − m2 b12 − m3 b13 x1 − x∗1
+ M 1 b12 + M 3 b32 + a2 − 2m2 b22 − m1 b21 − m3 b23 x2 − x∗2
+ M 1 (b13 + b31 ) + M 2 (b23 + b32 ) − 2m3 b33 − a3 x3 − x∗3 .
From (3.6) it follows that there exists a positive constant
3

D + V (t )

−μ

xi (t ) − x∗i (t )

for all t

μ > 0 such that

T 1.

(3.8)

i =1

Integrating on both sides of (3.8) from T 1 to t produces
t

3

V (t ) + μ
T1

xi (t ) − x∗i (t )

dt

V ( T 1 ) < +∞ for all t

i =1

Then
t

T1

Hence,

3

xi (t ) − x∗i (t )

i =1
3
i =1 |xi

dt

1

μ

− x∗i | ∈ L 1 ([ T 1 , +∞)).

V ( T 1 ) < +∞

(3.7)

for all t

T 1.

T 1.


T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

431

On the other hand, the ultimate boundedness of xi (t ) and x∗i imply that xi (t ) and x∗i , i = 1, 2, 3 all have bounded
derivatives for t T 1 (from the equations satisfied by them). As a consequence
on [ T 1 , +∞). By Lemma 2.6 we have
3

is uniformly continuous

xi (t ) − x∗i (t ) = 0

lim

t →∞

3

∗
i =1 |xi (t ) − xi (t )|

i =1

✷

which completes the proof.

4. The model with constant effects
In this section, we assume that the coefficients b i j , 1
shall assume that

M [ai ] = lim

3 in system (1.1) are positive constants. Furthermore, we

t0 +T

1

T →∞

i, j

ai (t ) dt

T

(4.1)

t0

exists uniformly with respect to t 0 in (−∞, ∞).
First, we consider a predator–prey system

x1 (t ) = x1 (t ) a1 (t ) − b11 x1 (t ) − b13 x3 (t ) ,
x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) − b33 x3 (t ) .

(4.2)

Put

Z i (T ) =

1

t0 +T

zi (t ) dt ,

T
t0

we have the following theorem.
Theorem 4.1. Assume that

b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0.
Then inft

t0

x1 (t ) > 0. Furthermore,

i) If

M [a3 ] <
then inft

t0

b31
b11

M [a1 ]

x3 (t ) > 0 and

lim X 1 ( T ) =

T →∞

lim X 3 ( T ) =

b33 M [a1 ] + b13 M [a3 ]
b13 b31 + b11 b33
b31 M [a1 ] − b11 M [a3 ]
b13 b31 + b11 b33

T →∞

,
.

ii) If

M [a3 ] >

b31
b11

M [a1 ]

then

lim X 1 ( T ) =

M [a1 ]

T →∞

b11

,

lim X 3 ( T ) = 0.

T →∞

Proof. To proof the first statement, we use the same proof as in Theorem 3.1. Let > 0 be a sufficient small constant. From

the comparison theorem and from x1 (t ) x1 (t )[a1u − b11 x1 (t )], it is easy to get that

lim sup x1 (t )
t →∞

a1u
b11

.


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T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

Then there exists T 1 > t 0 such that x1 (t ) < P 1 :=

x3 (t ) < x3 (t ) −al3 + b31 P 1 − b33 x3 (t )

a1u
b11

+

for t

for all t

T 1 . Thus

T 1.

(4.3)

Consider two cases.
Case 1: There exists

> 0 such that −al3 + b31 P 1 < 0.
From (4.3), it follows that limt →∞ x3 (t ) = 0. Therefore, there exists T 2 > T 1 such that a1 (t ) − b13 x3 (t ) > 12 al1 . It follows

from the first equation of system (4.2) that

x1 (t )

1

x1 (t )

2

al1 − b11 x1 (t )

for t

T 2.

Using the comparison theorem, we obtain

al1

lim inf x1 (t )
t →∞

2b11

Case 2: −al3 + b31 P 10

.
0.

It follows from (4.3) that

P 3 :=

lim sup x3 (t )
t →∞

−al3 + b31 P 1
b33

.

Then, we can choose a sufficient positive small
equation of system (4.2), we have

x1 (t ) al1 − b13 P 3 − b11 x1 (t )

x1 (t )

and T 3 > T 1 such that x1 (t )

for t

P 1 , x3 (t )

P 3 for all t

T 3.

Because of our assumption b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0, there exists a sufficient positive small

b11 b13 al3 + b11 b33 al1 − b13 b31 a1u

al1 − b13 P 3 =

b11 b33

−

b13 b31
b33

Then lim inft →∞ x1 (t ) > 0.
From the conclusions of two above cases, we obtain that inft

c 1 < x1 (t ) < d1

for all t

T 3 . From the first

such that

> 0.

t0

x1 (t ) > 0. Then there exists c 1 > 0 such that

t0 .

(4.4)

To prove Part i), first, we show that it is impossible to have

lim x3 (t ) = 0.

(4.5)

t →∞

Assume the contrary, it follows from (4.4) and (4.5) that

lim

T →∞

lim

T →∞

1
T
1

ln

x1 (t 0 + T )
x1 (t 0 )

= 0,

t0 +T

x3 (s) ds = 0.

T
t0

Then, we have from the first equation of (4.2) that

lim

T →∞

1

t0 +T

b11 x1 (s) ds = lim

T

T →∞

t0

Since (4.5) implies that

1
T

ln

x3 (t 0 + T )
x3 (t 0 )

<0

for large values of T , by (4.6),

1

t0 +T

t0 +T

a1 (s) ds −

T

t0

b13 x3 (s) ds − ln
t0

x1 (t 0 + T )
x1 (t 0 )

= M [a1 ].

(4.6)


T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

− M [a3 ] + b31

M [a1 ]

= lim

T →∞

b11

= lim

T →∞

t0 +T

1
T

t0 +T

a3 (s) ds + b31

−
t0

1

ln

T

433

x1 (s) ds
t0
t0 +T

x3 (t 0 + T )

+ b33

x3 (t 0 )

x3 (s) ds

0

t0

which contradicts our assumption. This contradiction proves that

lim sup x3 (t ) = d > 0.
t →∞

If, contrary to the assertion of the theorem, inft t0 x3 (t ) = 0, then there exists a sequence of numbers {sn }∞
1 such that
sn t 0 , sn → ∞ as n → ∞ and x3 (sn ) → 0 as n → ∞. Put

1

c=

2

t0 +T

1

lim inf
T →∞

x3 (t ) dt .

T

t0

Since x3 (t ) > c for arbitrarily large values of t and since sn → ∞ and x3 (sn ) → 0 as n → ∞, there exist sequences { pn }∞
1 ,
∞
{qn }∞
1, t 0 < pn < τn < qn < pn+1 , x3 ( pn ) = x3 (qn ) = c and
1 and {τn }1 such that for all n

0 < x3 (τn ) <

c
n

exp{−b31 d1n}.

∗ ∞
From this we see that there exist sequences {tn }∞
1 and {tn }1 such that for n

1

tn < τn < tn∗ ,
c
x3 (tn ) = x3 tn∗ = ,
n
c
for t ∈ tn , tn∗ .
x3 (t )
n

(4.7)

Thus
tn∗

1

0< ∗
tn − tn

c

x3 (t ) dt

n

→ 0 as n → ∞.

(4.8)

tn

We declare the following inequalities hold:

tn∗ − tn > tn∗ − τn

n

for n

1.

(4.9)

In fact,

x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) − b33 x3 (t ) < b31 d1 x3 (t )
for all t

τn

t 0 , then for t

t

−a3 (s) + b31 x1 (s) − b33 x3 (s) ds

x3 (t ) = x3 (τn ) exp
τn

c

=

n
c
n

exp{−b31 d1n} exp b31 d1 (t − τn )

exp b31 d1 (t − τn − n) .

From (4.10) and (4.7), we obtain that tn∗ − τn

(4.10)
n. It follows (4.9) that

t∗

M [ai ] = lim

∗
n→∞ tn

n

1

− tn

ai (t ) (i = 1, 3).
tn

From the first equation of system (4.2) we get that

1
tn∗ − tn

ln

x1 (tn∗ )
x1 (tn )

=

1
tn∗ − tn

tn∗

tn∗

a1 (t ) dt − b11
tn

tn∗

x1 (t ) dt − b13
tn

x3 (t ) dt .
tn


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T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

Then, it follows from (4.4), (4.8) and (4.9) that

lim ∗
n→∞ tn

tn∗

1

− tn

x1 (t ) dt =

M [a1 ]

tn

b11

(4.11)

.

Similarly, from the second equation of system (4.2) we have

1
tn∗ − tn

ln

x3 (tn∗ )

=

x3 (tn )

tn∗

1

a3 (t ) dt + b31

−

tn∗ − tn

tn∗

tn

tn∗

x1 (t ) dt − b33
tn

x3 (t ) dt .
tn

From this and from (4.7), (4.8) and (4.11), we get that

− M [a3 ] +

b31
b11

M [a1 ] = 0.

Since this contradicts our assumption, we obtain that inft

c 3 < x3 (t ) < d3

for all t

t0

x3 (t ) > 0. Therefore, there exists c 3 > 0 such that

t0 .

(4.12)

Now, from system (4.2), for all T > 0, we have

⎧
x1 (t 0 + T )
1
⎪
⎪
= A 1 ( T ) − b11 X 1 ( T ) − b13 X 3 ( T ),
⎨ ln
T
x1 (t 0 )

⎪
1
⎪
⎩ ln x3 (t0 +T ) = − A ( T ) + b X ( T ) − b X ( T ).
3

x3 (t 0 )

T

31

1

33

3

x1 (t 0 + T )
] + b13 [ T1
x1 (t 0 )

ln

Then

X1 (T ) =
X3 (T ) =

b33 [ A 1 ( T ) −

1
T

ln

x3 (t 0 + T )
x3 (t 0 )

+ A 3 ( T )]

x3 (t 0 + T )
x3 (t 0 )

+ A 3 ( T )]

b13 b31 + b11 b33
b31 [ A 1 ( T ) −

1
T

ln

x1 (t 0 + T )
] − b11 [ T1
x1 (t 0 )

ln

b13 b31 + b11 b33

,
.

(4.13)

It follows from (4.4) and (4.12) that

lim

T →∞

1
T

ln

xi (t 0 + T )
xi (t 0 )

= 0 (i = 1, 3),

then

lim X 1 ( T ) =

T →∞

lim X 3 ( T ) =

T →∞

b33 M [a1 ] + b13 M [a3 ]
b13 b31 + b11 b33
b31 M [a1 ] − b11 M [a3 ]
b13 b31 + b11 b33

,
.

To prove Part ii), first, we show that limt →∞ x3 (t ) = 0. Assume the contrary, then there exist δ > 0 and a sequence of
numbers { T n }∞
1 , T n > 0, T n → ∞ (n → ∞) such that δ < x3 (t 0 + T n ) < d3 for all n. Then, from the second equation of (4.13),
we get that

lim X 3 ( T n ) =

n→∞

b31 M [a1 ] − b11 M [a3 ]
b13 b31 + b11 b33

<0

which contradicts X 3 ( T ) 0 for all T > 0. This contradiction follows that limt →∞ x3 (t ) = 0 and then lim T →∞ X 3 ( T ) = 0. It
follows from the first equation of (4.13) that limT →∞ X 1 ( T ) = Mb[a1 ] . ✷
11

Now, we consider the following system

⎧
⎪
⎨ x1 (t ) = x1 (t ) a1 (t ) − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t ) ,
x2 (t ) = x2 (t ) a2 (t ) − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t ) ,
⎪
⎩
x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) + b32 x2 (t ) − b33 x3 (t ) .

(4.14)


T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

435

Proposition 4.2. If the following conditions hold

M [a1 ] >
M [a3 ] <

b12
b22

M [a2 ],

M [a2 ] >

b21
b11

M [a1 ]

and

(b21 b32 − b31 b22 ) M [a1 ] + (b12 b31 − b11 b32 ) M [a2 ]
b12 b21 − b11 b22

(4.15)

then lim supt →∞ x3 (t ) > 0.
Proof. Assume the contrary, then limt →∞ x3 (t ) = 0. Thus

lim X 3 ( T ) = 0.

(4.16)

T →∞

By replacing t 0 by a larger number, if necessary, we may assume that ai (t ) − b i3 x3 (t ) > 0 for t
for i = 1, 2,

⎧
t t0 ,
⎨ ai (t ) − b i3 x3 (t ),
ai (t ) = ai (t ) − (t − t 0 + 1)b i3 x3 (t ), t 0 − 1 t < t 0 ,
⎩
ai (t ),
t < t 0 − 1,
∗

t 0 − 1 and i = 1, 2. We put,

(4.17)

then a∗i is continuous on R, a∗i l > 0, a∗i u < ∞. Moreover, since limt →∞ x3 (t ) = 0, the limit
∗

M [ai ] = lim

T →∞

1

t∗ +T

∗

ai (t ) dt = lim

T
t∗

T →∞

1

t∗ +T

ai (t ) dt = M [ai ]

T
t∗

exists uniformly with respect to t ∗ ∈ R and i = 1, 2. Then for t
system

t 0 , (x1 (t ), x2 (t )) is a solution of the following competitive

x1 (t ) = x1 (t ) a∗1 (t ) − b11 x1 (t ) − b12 x2 (t ) ,
x2 (t ) = x2 (t ) a∗2 (t ) − b21 x1 (t ) − b22 x2 (t ) .

This system has been studied in [1] (see Theorem 2.1). By condition (4.15), we have

−b22 M [a1 ] + b12 M [a2 ]
,
T →∞
b12 b21 − b11 b22
b21 M [a1 ] − b11 M [a2 ]
lim X 2 ( T ) =
.
T →∞
b12 b21 − b11 b22
lim X 1 ( T ) =

(4.18)

From the third equation of system (4.14) we have

1

T

ln

x3 (t 0 + T )
x3 (t 0 )

= − A 3 ( T ) + b31 X 1 ( T ) + b32 X 2 ( T ) − b33 X 3 ( T ).

Then

− A 3 ( T ) + b31 X 1 ( T ) + b32 X 2 ( T ) − b33 X 3 ( T ) < 0
for T sufficiently large. Let T → ∞ and using (4.16) and (4.18) we obtain that

− M [a3 ] +

(b21 b32 − b31 b22 ) M [a1 ] + (b12 b31 − b11 b32 ) M [a2 ]
b12 b21 − b11 b22

which contradicts (4.15). This proves the proposition.

0

✷

Proposition 4.3. If one of the following conditions holds

⎧
b31
⎪

M [a3 ] <
M [a1 ],
⎪
⎪
⎪
b11
⎨
(b21 b33 + b23 b31 ) M [a1 ] + (b21 b13 − b11 b23 ) M [a3 ]
M [a2 ] >
,
⎪
⎪
b13 b31 + b11 b33
⎪
⎪
⎩
b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0,

(4.19)


436

T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

⎧
b31
⎪
M [a3 ] >
M [a1 ],

⎪
⎪
⎪
b11
⎨
b21

(4.20)

M [a2 ] >
M [a1 ],
⎪
⎪
b11
⎪
⎪
⎩
b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0
then lim supt →∞ x2 (t ) > 0.
Proof. Similarly to Proposition 4.2, we assume the contrary, then
t0 +T

lim x2 (t ) = 0,

x2 (t ) dt = 0

lim

t →∞

T →∞

(4.21)

t0

and (x1 (t ), x3 (t )) is a solution of a predator–prey system

x1 (t ) = x1 (t ) a∗1 (t ) − b11 x1 (t ) − b13 x3 (t ) ,

x3 (t ) = x3 (t ) −a∗3 (t ) + b31 x1 (t ) − b33 x3 (t )
where a∗1 (t ), a∗3 (t ) are defined as in (4.17) by replacing x3 (t ) by x2 (t ) and b i3 by b i2 .
First, if the condition (4.19) holds. From Part i) of Theorem 4.1 and from (4.21) and

1

t0 +T

3

a2 (t ) dt −

T

i =1

t0

1

t0 +T

b2i xi (t ) dt =

T

1
T

ln

t0

x2 (t 0 + T )
x2 (t 0 )

<0

(4.22)

for T sufficiently large, it is easy to get that

M [a2 ] −

(b21 b33 + b23 b31 ) M [a1 ] + (b21 b13 − b11 b23 ) M [a3 ]
<0
b13 b31 + b11 b33

which contradicts (4.19).
Second, if the condition (4.20) holds. From Part ii) of Theorem 4.1, (4.21) and (4.22), we have

M [a2 ] −

b21
b11

M [a1 ] < 0

which contradicts (4.20). Those contradictions prove the theorem.

✷

Similarly to Proposition 4.3, we have
Proposition 4.4. If one of the following conditions holds

⎧
b32
⎪
M [a3 ] <
M [a2 ],
⎪
⎪
⎪
b22
⎨
(b12 b33 + b13 b32 ) M [a2 ] + (b12 b23 − b22 b13 ) M [a3 ]
M [a1 ] >
,
⎪
⎪

b23 b32 + b22 b33
⎪
⎪
⎩
b22 b23 al3 + b22 b33 al2 − b23 b32 a2u > 0,

(4.23)

⎧
b32
⎪
M [a3 ] >
M [a2 ],
⎪
⎪
⎪
b22
⎨
b12

M [a1 ] >
M [a2 ],
⎪
⎪
b22
⎪
⎪
⎩
b22 b23 al3 + b22 b33 al2 − b23 b32 a2u > 0
then lim supt →∞ x1 (t ) > 0.

From Propositions 4.2–4.4, we obtain the main theorem in this section
Theorem 4.5. If one of the following conditions holds
A1 : (4.15), (4.20) and (4.23) hold,

(4.24)


T.V. Ton / J. Math. Anal. Appl. 362 (2010) 427–437

437

A2 : (4.15), (4.20) and (4.24) hold,
A3 : (4.15), (4.21) and (4.23) hold,
A4 : (4.15), (4.21) and (4.24) hold
then system (4.14) is persistent.
5. The model with the constant intrinsic growth rates
In this section, we consider system (1.1) under the condition ai , b i j , 1

⎧
⎪
⎨ x1 (t ) = x1 (t ) a1 − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t ) ,
x2 (t ) = x2 (t ) a2 − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t ) ,
⎪
⎩
x3 (t ) = x3 (t ) −a3 + b31 x1 (t ) + b32 x2 (t ) − b33 x3 (t ) .

i, j

3 are constants

(5.1)

Put

x∗1 =
x∗3 =

a1 b33 + a3 b13
b13 b31 + b11 b33
a1 b31 − a3 b11
b13 b31 + b11 b33

,
.

Theorem 5.1. If

⎧
⎨ a < b31 a ,
3
1
b11
⎩
a2 − b21 x∗1 − b23 x∗3 < 0

(5.2)

then the constant solution (x∗1 , 0, x∗3 ) of system (5.1) is locally asymptotically stable. It means that if (x1 (t ), x2 (t ), x3 (t )) is any solution of (5.1) such that (x1 (t 0 ), x3 (t 0 )) is close to (x∗1 , x∗3 ) and x2 (t 0 ) is sufficiently small and positive, then limt →∞ x1 (t ) = x∗1 ,
limt →∞ x3 (t ) = x∗3 , limt →∞ x2 (t ) = 0.
Proof. It is easy to see that x∗1 > 0, x∗3 > 0 and (x∗1 , 0, x∗3 ) is the constant solution of system (5.1). Put

f 1 (x1 , x2 , x3 ) = x1 (a1 − b11 x1 − b12 x2 − b13 x3 ),
f 2 (x1 , x2 , x3 ) = x2 (a2 − b21 x1 − b22 x2 − b23 x3 ),
f 3 (x1 , x2 , x3 ) = x3 (−a3 + b31 x1 + b32 x2 − b33 x3 ),
then system (5.1) becomes

xi = f i (x1 , x2 , x3 ),
and f i (x∗1 , 0, x∗3 ) = 0, i

⎡ ∂ f1

⎢
A=⎣

∂ x1
∂ f2
∂ x1
∂ f3
∂ x1

∂ f1
∂ x2
∂ f2
∂ x2
∂ f3
∂ x2

i = 1, 2, 3,

= 1, 2, 3. Consider

∂ f1
∂ x3
∂ f2
∂ x3
∂ f3
∂ x3

⎤

⎤
⎡
−b12 x∗1
−b13 x∗1
−b11 x∗1
⎥ ∗
∗
∗
∗
⎦.
a2 − b21 x1 − b23 x3
0
⎦ x1 , 0, x3 = ⎣ 0
∗
∗
∗
b32 x3
−b33 x3
b31 x3

It is easy to see that

det( A − λ I ) = a2 − b21 x∗1 − b23 x∗3 − λ λ2 + b11 x∗1 + b33 x∗3 λ + (b11 b33 + b13 b31 )x∗1 x∗3
and all eigenvalues of A are less than zero. Therefore (x∗1 , 0, x∗3 ) is locally asymptotically stable.

✷

References
[1] S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. 5 (2004) 219–229.
[2] I. Barb˘alat, Systèmes dèquations diffèrentielles dosillations non linéaires, Rev. Roumaine Math. Pures Appl. 4 (1975) 267–270.
[3] Y. Xia, F. Chen, A. Chen, J. Cao, Existence and global attractivity of an almost periodic ecological model, Appl. Math. Comput. 157 (2004) 449–475.