Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
25(1) (2009), 45–54
www.emis.de/journals
ISSN 1786-0091
DYNAMICS OF SPECIES IN A NONAUTONOMOUS
LOTKA-VOLTERRA SYSTEM
TA VIET TON
Abstract. In this paper, we study a Lotka-Volterra model with two predators and one prey.
The explorations involve the permance, extinction, the existence, uniqueness and global asymptotic stability of a positive solution.
1. Introduction
In this paper, we consider the Lotka-Volterra model with BeddingtonDeAngelis functional response of two predators and one prey.

cxy
0

x = x [a − bx] −
− mxz



α + βx + γy

 0
f xy
y = −dy +
− nyz
(1)
α + βx + γy



hz [mx + ny]
0


z = −gz +
ξ + ηz
Here x(t), y(t) and z(t) represent the population density of the two predator species, and of the prey at times t, respectively. a(t), b(t), c(t), d(t),
m(t), f (t), n(t), g(t), h(t), β(t), γ(t), η(t) are continuous and bounded
above and below by positive constants; α(t), ξ(t) are continuous and nonnegative.
This paper is organized as follows. Section 2 provides some definitions and
notations. In section 3 we state our main result of this paper for problem (1).
2. Definition and notation
In this section we summarize the basic definitions and facts which are used
later (cf. [7]). Let R3+ := {(x, y, z) ∈ R3 |x > 0, y > 0, z > 0}. For a bounded
2000 Mathematics Subject Classification. 34D05.
Key words and phrases. Beddington-DeAngelis Predator-Prey system, permanence,
extinction.
45
46
TA VIET TON
continuous function g(t) on R, we use the following notation:
g u := sup g(t),
g l := inf g(t).
t∈R
t∈R
Similarly to Lemma 1 in [10], one can easily prove that
Lemma 1. Both the nonnegative and positive cones of R3 are positively invariant for (1).
In the remainder of this paper, for biological reasons, we only consider the
solutions (x(t), y(t), z(t)) with positive initial values, i.e., x(to ) > 0, y(to ) > 0
and z(to ) > 0.
Definition 1. System (1) is said to be permanent if there exist positive constants δ, ∆ with 0 < δ < ∆ such that
min{lim inf x(t),
lim inf y(t),
lim inf z(t)} > δ,
max{lim sup x(t),
lim sup y(t),
lim sup z(t)} 6 ∆
t→∞
t→∞
t→∞
t→∞
t→∞
t→∞
for all solutions of (1) with positive initial values. System (1) is said to be
nonpersistent if there is a positive solution (x(t), y(t), z(t)) of (1) satisfying
min{lim inf x(t), lim inf y(t), lim inf z(t)} = 0.
t→∞
t→∞
t→∞
Definition 2. A set A is called to be an ultimately bounded region of system
(1) if for any solution (x(t), y(t), z(t)) of (1) with positive initial values, there
exists T1 > 0 such that (x(t), y(t), z(t)) ∈ A for all t > t0 + T1 .
Definition 3. A bounded nonnegative solution (x̂(t), ŷ(t), ẑ(t)) of (1) is said
to be globally asymptotically stable (or globally attractive) if any other solution (x(t), y(t), z(t)) of (1) with positive initial values satisfies
lim (|x(t) − x̂(t)| + |y(t) − ŷ(t)| + |z(t) − ẑ(t)|) = 0.
t→∞
Lemma 2. [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
lim f (t) = 0.
t→∞
3. Main Results
Theorem 1. If mε1 > 0, mε2 > 0 and mε3 > 0, then set Γε defined by
(2)
Γε = {(x, y, z) ∈ R3 | mε1 6 x 6 M1ε , mε2 6 y 6 M2ε , mε3 6 z 6 M3ε }
is positively invariant with respect to system (1), where
au
+ ε,
bl
(f u − dl β l )M1ε
,
M2ε :=
dl γ l
M1ε :=
al γ l − cu mu M3ε
−
,
bu γ l
bu
f l mε1 − (du + nu M3ε )(αu + β u mε1 )
mε2 :=
,
γ u (du + nu M3ε )
mε1 :=
LOTKA-VOLTERRA SYSTEMS . . .
47
hu (mu M1ε + nu M2ε ) − g l ξ l
hl (mu mε1 + nu m2ε ) − g u ξ u
ε
,
m
:=
3
gl ηl
guηu
and ε > 0 is constant.
M3ε :=
Proof. We know that the logistic equation
X 0 (t) = A(t)X(t)[B − X(t)]
(B 6= 0)
has a unique solution
Rt
X(t) =
BX0 e
Rt
X0 [e
t0
t0
A(s)Bds
A(s)Bds
− 1] + B
where X0 := X(t0 ). By Lemma 1, we have x(t) > 0, y(t) > 0 and z(t) > 0 for
all t ≥ t0 . Because M1ε > x0 > 0 and (1), we have
x0 (t) 6 x(t)[a(t) − b(t)x(t)] 6 x(t)[al − bu x(t)] = bu x(t)(M10 − x).
Thus, by using a standard comparison argument, we obtain that
u
(3)
u
x0 M10 ea (t−t0 )
x0 M1ε ea (t−t0 )
x(t) 6
6
< M1ε ,
u
x0 [ea (t−t0 ) − 1] + M10
x0 [eau (t−t0 ) − 1] + M1ε
t > t0 .
Similarly, because
0
y 6 −dl y +
f u M1ε y
f u xy
l
6
−d
y
+
6
αl + β l x + γ l y
αl + β l M1ε + γ l y
(4)
6
[(f u − dl β l )M1ε − dl γ l y]y
dl γ l
=
y[M2ε − y],
ε
ε
l
l
l
l
l
l
α + β M1 + γ y
α + β M1 + γ y
t > t0 ,
and 0 < y0 < M2ε , we have
Rt
(5)
y(t) 6
y0 e
M2ε
Rt
y0 [e
t0
t0
M2ε dl γ l
ds
αl + β l M1ε + γ l y(s)
M2ε dl γ l
ds
αl + β l M1ε + γ l y(s) − 1] + M ε
< M2ε ,
2
And because 0 < z0 <
M3ε
and
hu z(mu x + nu y)
ξ l + ηl z
hu z(mu M1ε + nu M2ε )
6 −g l z +
ξ l + ηl z
gl ηl
z(M3ε − z),
= l
ξ + ηl z
z 0 (t) 6 −g l z +
we also get that
(6)
z(t) < M3ε ,
t > t0 .
t > t0 .
48
TA VIET TON
Now, from (1), (3), (5) and (6), we have
µ
¶
cu y
0
l
u
u
x (t) > x a − b x − l
−m z
α + γly
µ
¶
cu M2ε
l
u
u
ε
>x a −b x− l
− m M3
α + γ l M2ε
= bu x(mε1 − x)
and x0 > mε1 . Thus,
u mε (t−t )
0
1
mε1 x0 eb
x(t) >
u mε (t−t )
0
1
x0 [eb
− 1] + mε1
> mε1 ,
for all
t > t0 .
Similarly, we have
f l mε1
y − nu M3ε y
αu + β u mε1 + γ u y
γ u (du + nu M3ε )
= u
y[mε2 − y],
ε
u
u
α + β m1 + γ y
y 0 (t) > −du y +
for which follows that y(t) > mε2 , for all t > t0 , and
z 0 (t) > −g u z +
hl z(mu mε1 + nu mε2 )
ξ u + ηuz
guηu
= u
z(mε3 − z),
u
ξ +η z
for which follows that z(t) > mε3 , for all t > t0 . So the proof is complete.
¤
Corollary 1. If mε1 > 0, mε2 > 0 and mε3 > 0, then we have
lim inf x(t) > mε1 ,
lim sup x(t) 6 M1ε ,
lim inf y(t) >
mε2 ,
lim sup y(t) 6 M2ε ,
lim inf z(t) >
mε3 ,
lim sup z(t) 6 M3ε .
t→∞
t→∞
t→∞
t→∞
t→∞
t→∞
Proof. From (3) we have lim supt→∞ x(t) < M1ε . Thus there exists t1 > t0 such
that
x(t) 6 M1ε , t > t1 .
Similarly to (4) and (5) we obtain that
Rt
(7)
y(t) 6 M2ε
y1 e
ε dl γ l
M2
t1 αl +β l M ε +γ l y(s) ds
1
Rt
y1 [e
ε dl γ l
M2
t1 αl +β l M ε +γ l y(s) ds
1
.
− 1] + M2ε
Thus, 0 < y(t) 6 max{M2ε , y1 } for all t > t1 , where y1 := y(t1 ). Therefore,
from (7) we get that
lim sup y(t) 6 M2ε .
t→∞
LOTKA-VOLTERRA SYSTEMS . . .
49
As a consequence, there exists t2 > t1 such that y(t) 6 M3ε for all t > t2 .
Similarly, we claim that
lim sup z(t) 6 M3ε ,
lim inf x(t) > mε1 ,
lim inf y(t) > mε2 ,
lim inf z(t) > mε3 .
t→∞
t→∞
t→∞
t→∞
The proof is complete.
¤
Corollary 2. If mε1 > 0, mε2 > 0 and mε3 > 0, then system (1) is permanent
and set Γε with ε > 0 defined by (2) is an ultimately bounded region of system
(1).
Using Corollary 1, it is easy to verify the statement of the above lemma.
Theorem 2. If M20 < 0 or M30 < 0, then lim y(t) = 0 or lim z(t) = 0
t→∞
t→∞
respectively.
Proof. We see that if M2o < 0 or M3o < 0 then M2ε < 0 or M3ε < 0, respectively,
with ε is sufficiently small. Therefore, similarly to the proof of Theorem 1 we
get that
(8)
y 0 (t) 6
dl γ l
y[M2ε − y]
αl + β l M1ε + γ l y
and
gl ηl
z[M3ε − z].
ξ l + ηl z
If M2ε < 0 then y 0 (t) < 0. Thus, 0 < y(t) 6 y(t0 ), for all t > t0 and then there
exists C1 > 0 such that
lim y(t) = C1 .
z 0 (t) 6
t→∞
If C1 > 0 then from system (1), (8) and 0 < C1 6 y(t) 6 y(t0 ), t > t0 ,
we have that there exists µ > 0 such that y 0 (t) < −µ for all t > t0 . Thus
y(t) < −µ(t − t0 ) + y0 . Therefore, lim y(t) = −∞. So we have a contraction
t→∞
to the fact that y(t) > 0 for all t > t0 . Hence,
lim y(t) = 0.
t→∞
Similarly, if M3ε < 0 then lim z(t) = 0. The proof is complete.
t→∞
¤
Theorem 3. Let (x∗ (t), y ∗ (t), z ∗ (t)) be bounded positive solutions of system
(1). If mε1 > 0, mε2 > 0 and mε3 > 0 and the following conditions hold:
½
α(t)f (t)
[β(t)c(t) + f (t)γ(t)]M2ε
inf b(t) −
−
t>t0
u(t, mε1 , mε2 )
u(t, mε1 , mε2 )
¾
m(t)h(t)ξ(t) m(t)h(t)η(t)M3ε
−
−
> 0,
v(t, mε3 )
v(t, M3ε )
50
½
inf
t>t0
(9)
inf {
t>t0
TA VIET TON
f (t)γ(t)mε1
α(t)c(t)
h(t)ξ(t)n(t)
−
−
ε
ε
ε
ε
u(t, m1 , M2 ) u(t, m1 , m2 )
v(t, mε3 )
¾
β(t)c(t)M1ε
h(t)n(t)η(t)M3ε
−
−
> 0,
u(t, M1ε , mε2 )
v(t, M3ε )
m(t)h(t)η(t)mε1 + n(t)h(t)η(t)mε2
− m(t) − n(t)} > 0,
v(t, M3ε )
where
u(t, x, y) := [α(t) + β(t)x∗ + γ(t)y ∗ (t)][α(t) + β(t)x(t) + γ(t)y(t)],
v(t, z) := [ξ(t) + η(t)z ∗ ][ξ(t) + η(t)z]
then (x∗ (t), y ∗ (t), z ∗ (t)) is globally asymptotically stable.
Proof. Let (x(t), y(t), z(t)) be any solution of (1) with positive initial value.
Since Γε is an ultimately bounded region of (1), there exists T1 > 0 such that
(x(t), y(t), z(t)) ∈ Γε and
(x∗ (t), y ∗ (t), z ∗ (t)) ∈ Γε for all t > t0 + T1 .
Considering a Liapunov function defined by
V (t) = | ln(x(t))−ln(x∗ (t))|+| ln(y(t))−ln(y ∗ (t))|+| ln(z(t))−ln(z ∗ (t))|, t > t0
a direct calculation of the right derivative D+ V (t) of V (t) along the solution
of (1) produces
LOTKA-VOLTERRA SYSTEMS . . .
(10)
µ
0
0
x
x∗
D V (t) = sgn(x − x )
− ∗
x
x
µ 0
¶
∗0
z
z
+ sgn(z − z ∗ )
− ∗
z
z
+
∗
¶
µ
∗
+ sgn(y − y )
0
51
0
y
y∗
− ∗
y
y
¶
·
cy
cy ∗
= sgn(x − x ) a − bx −
+ mz ∗
− mz − a + bx∗ +
∗
∗
α + βx + γy
α + βx + γy
¸
∗
·
+ sgn(y − y ) −d +
∗
fx
f x∗
− nz + d −
+ nz ∗
α + βx + γy
α + βx∗ + γy ∗
·
h(mx + ny) h(mx∗ + ny ∗ )
+ sgn(z − z )
−
ξ + ηz
ξ + ηz ∗
¸
¸
∗
·
µ
∗
∗
∗
= sgn(x − x ) −b(x − x ) − m(z − z ) − c
− n sgn(y − y ∗ )(z − z ∗ ) + f sgn(y − y ∗ )
+ h sgn(z − z ∗ )
α(x − x∗ ) + γ(xy ∗ − x∗ y)
u(t, x, y)
(mx + ny)(ξ + ηz ∗ ) − (mx∗ + ny ∗ )(ξ + ηz)
v(t, z)
6 −b|x − x∗ | + m|z − z ∗ | +
+ n|z − z ∗ | +
y
y∗
−
α + βx + γy α + βx∗ + γy ∗
αc|y − y ∗ |
βc(xy ∗ − x∗ y)
+ sgn(x − x∗ )
u(t, x, y)
u(t, x, y)
f γ(xy ∗ − x∗ y)
αf |x − x∗ |
+ sgn(y − y ∗ )
u(t, x, y)
u(t, x, y)
+ h sgn(z − z ∗ )
(mx + ny)(ξ + ηz ∗ ) − (mx∗ + ny ∗ )(ξ + ηz)
.
v(t, z)
We have
(mx + ny)(ξ + ηz ∗ ) − (mx∗ + ny ∗ )(ξ + ηz) =
= mξ(x − x∗ ) + nξ(y − y ∗ ) + mη(xz ∗ − x∗ z) + nη(yz ∗ − y ∗ z)
¶¸
52
TA VIET TON
and
Thus,
xy ∗ − x∗ y = x(y ∗ − y) + y(x − x∗ ),
xz ∗ − x∗ z = x(z ∗ − z) + z(x − x∗ ),
yz ∗ − y ∗ z = y(z ∗ − z) + z(y − y ∗ ).
¸
·
hmξ
αf
βcy + f γy
hmηz
|x − x∗ |+
D V (t) 6 −b +
+
+
+
v(t, z) u(t, x, y)
u(t, x, y)
v(t, z)
+
·
¸
αc
βcx − f γx hξn + hnηz
+
+
+
|y − y ∗ |+
u(t, x, y)
u(t, x, y)
v(t, z)
(11)
·
¸
hmηx + hnηy
+ m+n−
|z − z ∗ |.
v(t, z)
Because (x(t), y(t), z(t)) ∈ Γε , for all t > t0 + T1 , we get that
(12)
·
α(t)f (t)
β(t)c(t) + f (t)γ(t) ε
+
D V (t) 6 − b(t) −
−
M2 −
ε
ε
u(t, m1 , m2 )
u(t, mε1 , M2ε )
¸
m(t)h(t)ξ(t) m(t)h(t)η(t)M3ε
−
−
|x − x∗ |−
v(t, mε3 )
v(t, M3ε )
·
−
f (t)γ(t)mε1
α(t)c(t)
h(t)ξ(t)n(t)
−
−
−
ε
ε
ε
ε
u(t, m1 , M2 ) u(t, m1 , m2 )
v(t, mε3 )
¸
β(t)c(t)M1ε
h(t)n(t)η(t)M3ε
−
−
|y − y ∗ |−
u(t, M1ε , mε2 )
v(t, M3ε )
·
¸
m(t)h(t)η(t)mε1 + h(t)n(t)η(t)mε2
−
− n(t) − m(t) |z − z ∗ |,
v(t, M3ε )
for all t > t0 + T1 . ¿From (9) follows that there exists a positive constant µ > 0
such that
(13)
D+ V (t) 6 −µ[|x(t) − x∗ (t)| + |y(t) − y ∗ (t)| + |z(t) − z ∗ (t)|],
for all t > t0 + T1 .
Integrating on both sides of (13) from t0 + T1 to t produces
Z t
V (t)+µ
[|x(s)−x∗ (s)|+|y(s)−y ∗ (s)|+|z(s)−z ∗ (s)|]ds 6 V (t0 +T1 ) < +∞,
t0 +T1
for all t > t0 + T1 .
LOTKA-VOLTERRA SYSTEMS . . .
Z
53
Then
t
[|x(s) − x∗ (s)| + |y(s) − y ∗ (s)| + |z(s) − z ∗ (s)|]ds 6 µ−1 V (t0 + T1 ) < +∞,
t0 +T1
for all t > t0 + T1 ,
Hence, |x − x∗ | + |y − y ∗ | + |z − z ∗ | ∈ L1 ([t0 + T1 , +∞)).
The boundedness of x∗ , y ∗ , z ∗ and the ultimate boundedness of x(t), y(t),
z(t) imply that x(t), y(t), z(t), x∗ (t), y ∗ (t) and z ∗ (t) all have bounded derivatives for t > t0 + T1 (from the equations satisfied by them). As a consequence
|x(s) − x∗ (s)| + |y(s) − y ∗ (s)| + |z(s) − z ∗ (s)|
is uniformly continuous on [t0 + T1 , +∞).
By Lemma 2 we have
lim (|x(t) − x∗ (t)| + |y(t) − y ∗ (t)| + |z(t) − z ∗ (t)|) = 0
t→∞
which completes the proof.
¤
Acknowledgment
The author would like to thank professor Nguyen Huu Du for his interest,
encouragement, fruitful discussions and helpful comments.
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54
TA VIET TON
Received September 19, 2007.
Faculty of Mathematics, Mechanics and Informatics,
College of Science,
Vietnam National University,
Hanoi,
Vietnam.
E-mail address: tontv@vnu.edu.vn