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ASYMPTOTIC BEHAVIOR OF SOME MATHEMATICAL MODELS
Vu Van Khuong*
Department of Mathematical Analysis, University of Transport and Communications
Hanoi City, Vietnam
Department of Mathematics, Hung Yen University of Technology and Education
Khoai Chau District, Hung Yen, Viet Nam
(Received on: 19-11-12; Revised & Accepted on: 25-12-12)
ABSTRACT
T
he objective of this paper is to investigate the asymptotic behavior of the solutions of an economic model(
)
(
)
1 1
1 2
2
,
0,1,2,....
1
2
n n n n
n
n n
x
x
x a c
bx
by
n
y
a c
bx
b
α
+
+
= +
− −
−
=
=
− −
(1)
where
α
∈
[0; 1], a, b, c1, c2 are positive constants, and of a known population model1
.
1 n,
0,1,2...
x
n n
x
+= +
α β
x
−e
−n
=
(2)where
α
is the immigration rate andβ
is the population growth rate. Mathematics Subject Classification: 39A10.Keywords: Difference equations, asymptotic behavior, Asymptotic approximation.
1 INTRODUCTION
Difference equations manifest themselves as mathematical models describing real life situation in probability theory, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical network, quanta in radiation, genetics in biology, economics, psychology, sociology, etc,... Nonlinear difference equations of order greater than one are of paramount importance in applications where the (n+1)st generation (or state) of the system depends on the previous n generations (or state). Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering and economics.
Stability criteria and periodic behavior are derived for difference equations by many authors. In this paper, we study the asymptotic behavior of all positive solutions of the following economic system:
(
)
(
)
1 1
1 2
2
,
0,1,2,....
1
2
n n n n
n
n n
x
x
x a c
bx
by
n
y
a c
bx
b
α
+
+
= +
− −
−
=
=
− −
(1)
where
α
∈
[0; 1], a; b; c1; c2 are positive constants, and of a known pop-ulation model1
.
1,
0,1, 2...
n
x
n n
x
+= +
α β
x e
− −n
=
(2) whereα
is the immigration rate andβ
is the population growth rate.Corresponding author: Vu Van Khuong*
2. ASYMPTOTIC APPROXIMATION OF THE ECONOMIC SYSTEM (1)
In [5] H. El. Metwally investigated some qualitative behavior of the solutions of the following economic model
(
)
(
)
1 1
1 2
2
,
0,1,2,....
1
2
n n n n
n
n n
x
x
x a c
bx
by
n
y
a c
bx
b
α
+
+
= +
− −
−
=
=
− −
(1)
where
α
∈
[0; 1], a, b, c1, c2 are positive constants.The author obtained some results about the boundedness, the periodicity and the global attractivity of the solutions of this higher order difference equa-tion. In this section, we study the asymptotic behavior of all positive solutions of the following system (1).
Set
β
= a-c1 and2
2
a
c
a
b
−
=
, we can write the system (1) in the form(
)
1
. 1
2
,
0,1,2,....
1
2
n n n n
n
n n
x
x
x
bx
by
n
y
a
x
α
β
+
+
= +
−
−
=
= −
(1)
Finally, the system (1) can be written in only equation
(
)
21
1
n2
n2
n 1.
n n
b
x
+= +
αβ α
−
ab x
−
α
bx
+
α
x x
− (3)The equilibrium point equation is
(
)
7
1
2
b
x
= +
αβ α
−
ab x
−
α
x
Thus, we get
(
)
1 2
2
0
0.
3
ab
x
or x
b
β
−
=
=
>
We set
(
) (
)
21 1
,
1
2
.
2
n n n n n n
b
f x x
−= +
αβ α
−
ab x
−
α
abx
+
α
x x
−(
)
(
)
2
1 2
'
'
3 4
3
3
n
n
x x
x x
ab
f
ab
f
α β
α β
−
−
−
=
−
=
Note that the linearized equation of Eq (3) about the positive equilibrium point. We now investigate the asymptotical stability of Eq. (1)
(
)
2
2
3
ab
x
b
β
−
=
is(
)
(
)
1 1.
3 4
3
n3
n n
ab
ab
y
+=
−
α β
−
y
+
α β
−
y
− (4)The characteristic polynomial associated with Eq (4) is
( )
24
(
)
3
(
)
0.
3
3
ab
ab
Since,
( )
0
(
)
0,
( )
1
(
)
0
3
ab
p
= −
α β
−
<
p
=
α β
−
ab
>
( )
4
(
)
3
( )
'
2
0
0,1
3
ab
p t
= +
t
α β
−
−
>
for t
∈
It follows that for
(
3
)
0
min 1;
4
ab
α
β
≤ ≤
−
, there is a unique positive root
t
0∈
( )
0,1
such that( )
20
0
0
0 01
p t
=
and
< < <
t
t
such that( )
2( )
0 0
0
p t
<
p t
=
. It means that( )
2(
)
(
)
0 0 0
4
3
0
3
3
ab
ab
p t
= +
t
α β
−
−
t
−
α β
−
=
(6)This fact motivated us to believe that there are solutions of Equation (5) which have the following asymptotics
( )
1 0
0
0n n
n
x
= +
x a t
+
t
(7)where
a
1∈
R
and t0 is the above mentioned root of Eq (6) we solve the open problem, showing that such a solutionexists, developing Berg's ideas in [1-4] which are based on the asymptotics. The asymptotics for solutions of difference equation have been investigated by L. Berg and S. Stevi'c, see, for example [7-9], [13,14] and the reference therein. The problem is solved by constructing appropriate sequences yn and zn
n n n
y
≤ ≤
x
z
(8) for suffiently large n. In [1-4] some methods can be found for the construction of these bounds, see, also [13, 14]. From (7) we expect that for k≥
2 such solutions have the first three members in their asymptotics in the following form2
1 0 1 0
n n
n
x a t
b t
ϕ
= +
+
(9)This is proved by developing Berg's ideas in [1-4] which are based on asymptotics. We need the following result in the proof of main theorems. The proof of the following theorem can be found in [13].
Theorem 2.1. Let f :
I
k+2→
I
be a continuous and nondecreasing function in each argument on the interval I⊂
R, and let{ }
y
n and{ }
z
n be sequences withy
n≤
z
nfor n
≥
n
0 and such that(
,
1,....,
n 1) (
,
,
1,...,
n 1)
01
n k n k n k n k
y
−≤
f n y
− +y
+f n z
− +z
+≤
z
−fo r
n n
> + −
k
(10)Then there is a solution of the following difference equation
(
,
1,...,
1)
n k n k n
x
−=
f n x
− +x
+ (11)with property (8) for n
≥
n
0Theorem 2.2. For a; b; c1; c2 are positive constants and
(
3
)
0,min 1,
4
ab
α
β
∈
−
there is a nonoscillatory solution
of Eq (3) converging to the positive equilibirium point
2
(
)
3
ab
x
as n
b
β
−
Proof: First, Eq (3) can be written in the form
1
1 1
2
2
2
.
4
2
n n n n
x
x
x
x
a
ab
b
b
β
α
−
−
=
++
+
−
−
(
)
11 1 1 1
2
2
2
,
,
.
4
2
n n n n n n n
F x
x x
x
x
x
a
x
ab
b
b
β
α
−
− +
=
++
+
−
−
−
− (12)We expect the solutions of Eq (3) have the asymptotic appropriation (9).
(
)
(
)(
) (
1)
1 2 2 2 2 1 2 2
1 1 1 1 1 1 1 1 1 1
.
,
,
4
2
2
2
2
n n n n n n n n
n
n n
F
F
c x a t
b t
x a t
b t
x a t
b t
d
x a t
b t
with c
and d
a
b
b
b
ϕ ϕ ϕ
β
α
α
−
+ + + +
− + + − −
=
=
+
+
+
+
+
+
+ −
=
=
−
−
1
1 2 2 2 2 1 2 2
1 1 1 1
1 1 1 1
1
a
nb
n1
a
nb
n4
4
n4
n n nF
c
t
t
t
t
x
a t
b t
d
x a t
b t
x
x
x
x
−
+ + − −
−
=
+
+
+
+
+
+
+
+ − −
(
)
21
1 1 1
1
2
4
2
3
2
2
2
3
n
ab
ca t
a x ca
F
a
t a t
b
b
b
ab
x
x
β
β
α
−
−
−
=
+
+
−
−
+
+
−
1 4 12 3 12 1 2 2 2
( )
21 1
2
4
2n n
cb
ca
ca
cb
t
t
b
t
b t
o t
x
x
x
x
−
+
−
+
+
−
−
+
By easy calculation we have
(
)
2
2
ab
2
a
2
2
0
ab
b
b
b
β
β
α
−
+
+
−
−
=
and we can write the function F in following form
(
)
(
)
2
8
6
12
3
3
n
ab
ab a t
F
t
t
b
xt
α β
α β
αβ
α
+
−
−
−
=
+
−
(
)
(
)
( )
2 2 2
4 2 1 2 2 2
1 1
2 2
4
n n
n
b t
t
ct
x c t
x
ca
ca t
o t
xt
x
+
+
+
−
−
+
−
+
(13)As mentioned earlier exists t0
∈
(0; 1) such that p(t0) = 0 and( )
2 2
0 0 0
0
< < <
t
t
1,
p t
<
0
Posing t = t0, we obtain( )
( )
2 2 2 2
4 2 1 0 1 2
0 0 2 0 2 0
0
4
x c
x
cb t
n1
ca t
on nF
t
t
t
o t
c
c
xt
x
+
−
=
+
−
+ −
+
( )
2( )
2 2 2( )
2 1 0 1 2
0 2 0 2 0
0
.
1
n n
o n
cb t
ca t
F
p t
t
o t
xt
x
+
=
+ −
+
Therefore, we have
( )
2( )
2 2 2( )
2 1 0 1 2 2
0 2 0 2 0 0
0
.
1
n n
o n n
cb t
ca t
F
p t
t
t
o t
xt
x
+
=
+ −
+
(
)
2( )
21 0 0
n n
Setting
( )
0
0
0t
C
H q
Bq C
q
B
=
+ = → = −
( )
( )
0
2 0 '
2 0
.
0
t
p t
c
H
q
B
xt
= =
<
We obtain that there are q1 < q0 and q2 > q0 such that
H
t0( )
q
1>
0;
H
t0(
q
2)
<
0
; q1 < q0 < q2.We assume that
21
0,
1 0 0 0n n
n
a
≠
if
ϕ
= +
x a t
+
q t
, we obtain
(
)
2( )
21
,
,
1 0 0 0n n
n n n
F
ϕ
−ϕ ϕ
+
q B C t
+
+
o t
With the notation
2
2
1 0
n
1 0
n
,
1 0
n
2 0
n
n
n
y
= +
x a t
+
q t
z
= +
x a t
+
q t
We get
(
)
(
)
2
1 1 1 0
2
2 1 2 0
,
,
,
,
,
n n
n n
n n
n n
F y
y y
q B C t
F z
z
z
q B C t
− +
− −
+
+
These relations show that inequalities (10) are satisfied for sufficiently large n, where f = F+xn-1 and F is given by
(12). Because the function
(
)
1,
n,
1n n
f x
−x x
+ is continuous and nondecreasing on
x
,
)
3
x
,
)
+∞ →
+∞
. Weeasily have
f x x x
(
, ,
)
=
x
We can apply the Theorem (2.1) withI
=
x
,
∞
)
and see that there is ann
0≥
0
and a solution of equation (3) with the asymptotics
(
02n)
n
n
x
=
ϕ
+
o t
, forn
0≥
0
whereb
=
q
0 in
ϕ
n
. In particular, the solution converges monotonically to the positive equilibrium(
)
0
2
3
ab
x
for n n
b
β
−
=
≥
Hence, the solution0
2
n n
x
+ +
is also such a solution when n≥
-2.Theorem 2.3. For a; b; c1; c2 are positive constants and
(
3
)
0,min 1,
4
ab
α
β
∈
−
there is a nonoscillatory
solution of system equation
(
)
(
)
1 1
1 2
2
,
0,1,2,....
1
2
n n n n
n
n n
x
x
x a c
bx
by
n
y
a c
bx
b
α
+
+
= +
− −
−
=
=
− −
(1)
converging to the positive equilibrium point
x y
,
where(
)
2
3
ab
x
b
β
−
=
(
)
2
2
1
2
ab
y
a c
b
b
β
−
=
− −
and
( )
( )
2 0
2
2 0
1
(
) ,
2
2
n
n n
n
n n
x
o t
y
a c
b
o t
n
b
ϕ
ϕ
=
+
3. ASYMPTOTIC APPROXIMATION OF POPULATION MODEL (2)
In [5,6] the authors studied the asymptotic behavior of some know population models. They established that every solution of the bounded below by positive constants. They also provided sufficient conditions for the global asymptotically stable of all solution of that higher order difference equations.
The study of a nonoscillatory solution of difference equation converging to the positive equilibrium point
x
is extremely useful in the behavior of mathematical models of various biological systems and other application. This is due to the fact that difference equation are appropriate models for discribing situations where the variable is assumed to take only a discrete set of values and they arise frequently in the study of biological models, in the formation and analysis of discrete - time systems, the numerical intergation of differential equation by finite - difference schemes, the study of deterministic chaos, etc. For example El - Metwally [5] investigated the asymptotic behavior of the population model1
.
1,
0,1,2,...
n
x
n n
x
+= +
α β
x
−e
−n
=
(14) whereα
is the immigration rate andβ
is the population growth rate.In final section, we study the nonoscillatory solution of the equation (14) converging to the positive equilibrium point. The equilibrium point equation is
x
= +
α β
. .
x e
−x
x
x
x
e
β
α
−
=
We pose
(
,
1)
.
1 nx
n n n
f x x
−= +
α β
x
−e
−1
' '
.
.
n
n
x
x x
x
x x
f
xe
x
x
f
e
x
β
α
α
β
−
−
−
= −
= −
−
= −
=
Note that the linearized equation of Eq (14) about the positive equilibrium point
(
)
1 1
,
0,1, 2,...
n n n
x
y
x y
y
n
x
α
α
+
=
−
+
−
−=
(15)The characteristic polynomial associated with Eq (15) is
( )
(
)
( )
( )
(
)
(
)
2
0
, 0
0,
1
0
p t
xt
x x
t
x
Since p
x
p
x
x x
x
x x
α
α
α
α α
α α
=
+
−
+ − =
= − <
= +
−
+ − =
−
+ >
p t
'
( )
=
2
xt
+
x x
(
−
α
)
>
0
for t
∈
( )
0,1
There is a unique positive roott
0∈
( )
0,1
such that( )
20
0
0
0 01
p t
=
and
< < <
t
t
such that
( )
2( )
0 0
0
p t
<
p t
=
. It means that( )
2(
)
0 0 0
0
p t
=
xt
+
x x
−
α
t
+ − =
α
x
This fact motivated us to believe that there are solutions of Equation (5) which have the following asymptotics
( )
1 0 0
n n
n
x
= +
x
a t
+
o t
(16)from (16) we expect that for k
≥
2 such solutions have the first three members in their asymptotics in the following form2 1 0 1 0
n n
n
x
a t
b t
Theorem 3.1: For
α β
,
are positive constants there is a nonoscillatory solution of Equation (14) converging to the positive equilibrium point that satisfies1
x
x
e
β
α
−
=
as n
→ ∞
Proof: From Eq (14) we can write in the form
(
1 1)
(
1)
1(
1)
11
,
,
.
xn,
n n
n n n n n n
F x
−x x
+=
β
e
x
+−
α
−
x
−=
g x x
+−
x
− (18)(
)
1
1
' '
1
1
0
1
0
n n
n n
x x
x
x n
g
e
g
e
x
β
α
β
+
+ +
=
>
=
−
>
We expect the solutions of Eq (14) have the asymptotic appropriation (16)’
(
)
2(
1 2 2) (
1 2 2)
1 1
1
,
n,
x atn bt n n n n nn n
F
=
F
ϕ ϕ ϕ
− +=
β
e
+ +x at
+
++
bt
+−
α
− +
x at
−+
bt
−( )
0
for t
0,
=
∈ ∞
(
)(
)
2 1 2 2 1 2 2 1
1
1 2 2 1 2 2
2
ln
1
1
n n
x at bt n n n n
n n n n
n n
e
x
at
bt
x
at
bt
at
bt
at
bt
x
at
bt
x
x
x
x
β
α
β
α
α
−
+ + − − + +
−
− − + +
=
+
+
+
+
−
+
+
=
+
+
+
+
−
−
2
1 2 2 1 2 2 1 2 2
ln
1
. 1
...
n n n n n n
x
at
bt
at
bt
at
bt
x
x
x
x
x
x
β
α
α
α
α
− −
+ + + +
+
=
+
+
−
−
+
−
−
−
−
−
1 2 2 2 2 2 3 3
2
2
...
ln
...
2
n n n n
at
bt
a t
abt
x
x
x
−
+
− −+
−+
+
−
+
(
)
1 2 2 2 2 2
2
ln
...
2
n n n
n n
x
at
bt
a t
x
at
bt
x
x
α
β
α
α
+ + +
−
+
= +
+
+
+
−
+
−
−
(19)From (19) we have
(
)
ln
x
x
ln
x
α
or
x
α
e
xβ
x
β
−
= +
−
=
This is trust and from (19) we obtain
1 1
n
n n
a
at
t
at
x
x
α
+
−
=
+
−
(
) (
)
(
)
(
( )
)
2 n
n
xt
t x x
x
np t at
a
at
a
t
at
t x
x
t x x
t x x
α
α
α
α
α
+
−
+
−
− −
=
=
−
−
−
As mentioned earlier exists
t
0∈
( )
0,1
such that p(t0) = 0 and( )
2 2
0 0 0
( )
(
0 0)
0 0
0 0
0
n n
p t at
a
at
a
t
t x
x
α
t x x
α
− −
=
=
−
−
From (19) it follows
(
)
2 2 2 2 2 2
2 2 2 2 2 0 0
0 2 0 0 2
2
2
n n
n n n
b
a
bt
a t
t
t
bt
x
x
x
α
x
α
+ +
−
−
−=
+
−
−
−
(
)
( )
2 2
2 2 2
0 0 0
2 2 2
0
1
2
2
n n n
b
a
b
a
F
b
t
t
o t
t
x
x
x
α
x
α
=
−
− −
−
+
−
−
(
)
( )
(
)
(
)
( )
2 2
2 2 4 4
0 0 0
2 2
0 2
2 2
0
2
.
2
n n
o
b x
xp t
a t
x t
x
t
o t
x
x
t
α
α
α
−
−
−
−
=
+
−
Setting F = (Bb+C)
t
02n+
o t
( )
02n( )
0
0o
t
C
H
q
Bq
C
q
B
=
+ = →
= −
( )
(
)
( )
(
)
0
2 0 '
2
2 2
0
2
0
2
t
b x
xp t
H
q
B
x
x
t
α
α
−
= =
<
−
We obtain that there are q1 < q0 and q2 > q0 such that Hto(q1) > 0,
( )
0 2
0,
1 0 2t
H
q
<
q
<
q
<
q
. We assume that
20 0 0
0,
n na
≠
if
ϕ
= +
x
at
+
q t
, we obtain
(
)
[
]
2( )
21
,
,
1 0 0 0n n
n n n
F
ϕ
−ϕ ϕ
+
q B
+
C t
+
o t
With the notation2 2
0 1 0
,
0 2 0n n n n
n n
y
= +
x
at
+
q t
z
= +
x
at
+
q t
We get
(
)
[
]
(
)
[
]
2
1 1 1 0
2
1 1 2 0
,
,
;
,
,
n n n n
n n n n
F y
y y
q B
C t
F z
z z
q B
C t
− +
− +
+
+
These relations show that inequalities (10) are satisfied for sufficiently large n, where
g
= +
F
x
n−1 and F is given by (18). Because the functiong x
(
n−1,
x x
n,
n+1)
is continuous and nondecreasing on)
)
3
,
,
x
x
+∞ →
+∞
. Weeasily have
g x x x
(
, ,
)
=
x
. We can apply the Theorem 2.1 with I = +∞
x
,
)
and see that there is ann
0≥
0
and a solution of equation (14) with theasymptotics
( )
2
0
,
0
0n
n n
n
x
=
ϕ
+
o t
for n
≥
where b
=
q in
ϕ
. In particular, the solution converges monotonically to the positive equilibrium point forn
0≥
0
. The proof is complete.ACKNOWLEDGMENTS
We would like to extend our thanks to the referees for their suggestions which certainly improved the exposition of our paper. This work was supported by Vietnam's National Foundation for Science and Technology Development (NAFOSTED: 101.01.58.09).
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