Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 165-176
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2012
Asymptotic behavior of some population models
Vu Van Khuong1 and Tran Hong Thai2
Abstract
We study the asymptotic behavior of the difference equations
xn+1 = (αxn + βxn−1 )e−xn , n = 0, 1, 2, ...
(1)
xn+1 = α + βxn−1 e−xn , n = 0, 1, 2, ...
(2)
and
where α, β ∈ (0, ∞), which are interesting in their own rights, but which
may also be viewed as describing some population models.
Mathematics Subject Classification: 39A10
Keywords: Difference equations, asymptotic, positive solution, equilibrium,
nonoscillatory solution, converges
1
Introduction
Equation (1) is derived from a two lifestage model where the young nature
into adults, and adults produce young. Such systems have been well studied
1
2
Department of Mathematics, Hung Yen University of Technology and Education,
Hung Yen Province, Vietnam, e-mail: vuvankhuong@gmail.com
Department of Mathematics, Hung Yen University of Technology and Education,
Hung Yen Province, Vietnam, e-mail: hongthai78@gmail.com
Article Info: Received : October 6, 2012. Revised : November 12, 2012
Published online : December 30, 2012
166
Asymptotic behavior of some population models
when the rates of survival, maturation and reproduction are assumed to be
contact. However, they may not be contact. Adults may increase the mortality
of the young through cannibalism, exclusion from resources, or transmission of
disease. They may inhibit the maturation of the young by shading seedlings
if they are trees, or by bihavioral means as in some fish population.
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
asymptotic stability 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 model
xn+1 = α + βxn−1 e−xn , n = 0, 1, 2, ...
where α is the immigration rate and β is the population growth rate.
In this paper, we study the asymptotic behavior of the difference equations
xn+1 = (αxn + βxn−1 )e−xn , n = 0, 1, 2, ...
(1)
xn+1 = α + βxn−1 e−xn , n = 0, 1, 2, ...
(2)
and
where α, β ∈ (0, ∞).
2
Main Results
2.1
Asymptotic aproximation of population model (1)
In this section, we study the nonoscillatory solution of the equation (1)
Vu Van Khuong and Tran Hong Thai
167
converging to the positive equilibrium point.
The equilibrium point equation is x = (α + β)xe−x ⇒ x = ln(α + β) with
α + β > 1.
We pose
f (xn , xn−1 ) = (αxn + β.xn−1 )e−xn
α
fx0 n x = e−x (α − α.x − β.x) = (α + β)−1 α − (α + β).ln(α + β) =
−
α+β
ln(α + β)
β
fx0 n−1 x = β.e−x =
α+β
Note that the linearized equation of Eq (1) about the positive equilibrium
point
yn+1 =
β
α
− ln(α + β) yn +
yn−1 , n = 0, 1, 2, ...
α+β
α+β
(3)
The characteristic polynomial associated with Eq (3) is
p(t) = t2 −
Since,
p(0) = −
α
β
− ln(α + β) t −
=0
α+β
α+β
β
α
β
< 0, p(1) = 1 −
+ ln(α + β) −
= ln(α + β) > 0.
α+β
α+β
α+β
p0 (t) = 2t −
α
α
− ln(α + β) = 2t + ln(α + β) −
> 0 for t ∈ (0, 1)
α+β
α+β
It follows that for α + β > 1, there is a unique positive root t0 ∈ (0, 1) such
that p(t0 ) = 0 and 0 < t20 < t0 < 1 such that p(t20 ) < p(t0 ) = 0. It means that
p(t0 ) = t20 −
β
α
− ln(α + β) t0 −
=0
α+β
α+β
⇔ (α + β)t20 − [α − (α + β).ln(α + β)]t0 − β = 0
(4)
This fact motivated us to believe that there are solutions of Eq (3) which have
the following asymptotics
xn = x + a1 tn0 + o(tn0 )
(5)
where a1 ∈ R and t0 is the above mentioned root of Eq (4) we solve the open
problem, showing that such a solution exists, developing Berg’s ideas in [1-4]
which are based on the asymptotics. The asymptotics for solutions of difference
168
Asymptotic behavior of some population models
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
yn ≤ xn ≤ z n
(6)
for sufficiently large n. In [1-4] some methods can be found for the construction
of these bounds, see, also [13, 14].
From (5) we expect that for k ≥ 2 such solutions have the first three
members in their asymptotics in the following form
ϕn = x + a1 tn0 + b1 t2n
0
(7)
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 {yn } and {zn } be sequences
with yn ≤ zn for n ≥ n0 and such that
yn−k ≤ f (n, yn−k+1 , ..., yn+1 ), f (n, zn−k+1 , ..., zn+1 ) ≤ zn−k for n > n0 + k − 1
(8)
Then there is a solution of the following difference equation
xn−k = f (n, xn−k+1 , ..., xn+1 )
(9)
with property (6) for n ≥ n0 .
Theorem 2.2. For α, β are positive constants and α + β > 1 there is a
nonoscillatory solution of Eq (1) converging to the positive equilibrium point
x = ln(α + β) as n → ∞.
Proof. From Eq (1) we can write in the form
F (xn−1 , xn , xn+1 ) =
1
α
xn+1 exn − xn − xn−1 = g(xn , xn+1 ) − xn−1
β
β
(10)
169
Vu Van Khuong and Tran Hong Thai
gx0 n+1 =
1 xn
e >0
β
α
α
1
xn+1 exn − > 0 with xn+1 > α > xn
β
β
e
we expect the solutions of Eq (1) have the asymptotic appropriation (5)
gx0 n =
F = F (ϕn−1 , ϕn , ϕn+1 ) =
−
1 x+atn +bt2n
(x + atn+1 + bt2n+2 )
e
β
α
(x + atn + bt2n ) − (x + atn−1 + bt2n−2 ) = 0,
β
for t ∈ (0, ∞).
n
2n
ex+at +bt = [α(x + atn + bt2n ) + β(x + atn−1 + bt2n−2 )](x + atn+1 + bt2n+2 )−1
x + atn + bt2n = ln (α + β)x + αatn + βatn−1 + αbt2n + βbt2n−2
− ln(x + atn+1 + bt2n+2 )
h
αatn + βatn−1 + αbt2n + βbt2n−2 i
= ln(α + β)x + ln 1 +
(α + β)x
n+1
2n+2 at
+ bt
− lnx − ln 1 +
x
n
at + bt
2n
αatn + βatn−1 + αbt2n + βbt2n−2
=
(α + β)x
n
(αat + βatn−1 + αbt2n + βbt2n−2 )2
−
2(α + β)2 x2
(αatn + βatn−1 + αbt2n + βbt2n−2 )3
+
− ...
3(α + β)3 x3
atn+1 + bt2n+2 (atn+1 + bt2n+2 )2
+
− ...
−
x
2x2
From (11) we have
βa
at n
αa
+
−
t ⇔
(α + β)x (α + β)xt
x
−a(α + β)t2 + αat + βa n
atn ≡
.t ⇒
(α + β)xt
−a(α + β)t2 + αat + βa
⇔
a=
(α + β)xt
(α + β)xt = −(α + β)t2 + αt + β ⇔
atn ≡
(α + β)t2 + [(α + β)x − α]t − β = 0
170
Asymptotic behavior of some population models
Posing t = t0 in (11) with t0 above mentioned, we have
p(t0 ) = (α + β)t20 − [α − (α + β)ln(α + β)]t0 − β = 0.
From (11) it follows
2n−2
αbt2n
α2 a2 t2n
β 2 a2 t02n−2
2αβa2 t02n−1
b 2n+2
0 + βbt0
0
−
−
−
−
t
2
2
2
(α + β)x
2(α + β)2 x
2(α + β)2 x
2(α + β)2 x
x 0
a2 2n+2
+ 2 t0
2x
α 2 a2
a2 2
β 2 a2 t−2
αβa2 t−1
b 2
αb + βbt−2
0
0
0
−
t
+
−
−
−
t ⇒
b≡
(α + β)x
2(α + β)2 x2 2(α + β)2 x2 (α + β)2 x2 x 0 2x2 0
bt2n
0 =
αbt20 + βb
α2 a2 t20
β 2 a2
αβa2 t0
b 4 a2 4
−
−
−
−
t + t ⇒
(α + β)x
2(α + β)2 x2 2(α + β)2 x2 (α + β)2 x2 x 0 2x 0
i
h
t20 2
β
α
−
t +
b −1+
(α + β)x
x 0 (α + β)x
h t4
i
α2 t20
αβt0
β2
+ a2 02 −
−
−
=0 ⇒
2x
2(α + β)2 x2 (α + β)2 x2 2(α + β)2 x2
bt20 =
h −(α + β)x + α
t20 −
i
t40
β
b+
x (α + β)x
(α + β)x
h
i
a2
2 4
2 2
2
(α + β) t0 − α t0 − 2αβt0 − β = 0 ⇒
+ 2
2x (α + β)2
− (α + β)t40 + [α − (α + β)x]t20 + β
+
b
(α + β)x
a2
(α + β)2 t40 − α2 t20 − 2αβt0 − β 2 = 0 ⇔
2
2
2x (α + β)
p(t20 ).b
a2
2 4
2 2
2
+ 2
=0
(α
+
β)
t
−
α
t
−
2αβt
−
β
0
0
0
(α + β)x 2x (α + β)2
Finally, we have
F =
io
h
n p(t2 ).b
a2
0
2n
2 4
2 2
2
t2n
(α
+
β)
t
−
α
t
−
2αβt
−
β
+ 2
0
0 + o(t0 )
0
0
(α + β)x 2x (α + β)2
171
Vu Van Khuong and Tran Hong Thai
Setting
2n
F = (Bb + C)t2n
0 + o(t0 ),
C
Ht0 (q) = Bq + C = 0 → q0 = − ,
B
2
p(t0 )
Ht00 (q) = B =
<0
(α + β)x
we obtain that there are q1 < q0 and q2 > q0 such that
Ht0 (q1 ) > 0,
Ht0 (q2 ) < 0,
q1 < q0 < q2 .
We assume that a 6= 0, if ϕ
bn = x + atn0 + q0 t2n
0 , we obtain
2n
F (ϕ
bn−1 , ϕ
bn , ϕ
bn+1 ) ∼ [q0 B + C]t2n
0 + o(t0 )
With the notation
n
2n
yn = x + atn0 + q1 t2n
0 , zn = x + at0 + q2 t0
We get
F (yn−1 , yn , yn+1 ) ∼ [q1 B + C]t2n
0 ,
F (zn−2 , zn−1 , zn ) ∼ [q2 B + C]t2n
0 .
These relations show that inequalities (8) are satisfied for sufficiently large
n, where g = F + xn−1 and F is given by (10).
Because the function g(xn−1 , xn , xn+1 ) is continuous and nondecreasing on
[x, +∞)3 → [x, +∞). We easily have g(x, x, x) = x. We can apply the Theorem (2.1) with I = [x, ∞) and see that there is an n0 ≥ 0 and a solution of
equation (1) with the asymptotics xn = ϕ
bn + o(t2n
0 ), for n ≥ n0 where b = q0
in ϕ
bn . In particular, the solution converges monotonically to the positive equilibrium point for n ≥ n0 . The proof is complete.
2.2
Asymptotic aproximation of population model (2)
In final section, we study the nonoscillatory solution of the equation (2)
converging to the positive equilibrium point.
The equilibrium point equation is x = α + β.xe−x
x−β
x
=α
ex
172
Asymptotic behavior of some population models
We pose
f (xn , xn−1 ) = α + β.xn−1 e−xn
fx0 n x = −β.xe−x = α − x
x−α
fx0 n−1 x = β.e−x =
x
Note that the linearized equation of Eq (2) about the positive equilibrium
point
yn+1 = (α − x)yn +
x−α
yn−1 , n = 0, 1, 2, ...
x
(12)
The characteristic polynomial associated with Eq (12) is
p(t) = xt2 + x(x − α)t + α − x = 0.
Since,
p(0) = α − x < 0, p(1) = x + x(x − α) + α − x = x(x − α) + α > 0
p0 (t) = 2xt + x(x − α) > 0, for t ∈ (0, 1).
There is a unique positive root t0 ∈ (0, 1) such that p(t0 ) = 0 and
0 < t20 < t0 < 1 such that
p(t20 ) < p(t0 ) = 0.
It means that
p(t0 ) = xt20 + x(x − α)t0 + α − x = 0.
This fact motivated us to believe that there are solutions of Eq (12) which
have the following asymptotics
xn = x + a1 tn0 + o(tn0 )
(13)
from (13) we expect that for k ≥ 2 such solutions have the first three members
in their asymptotics in the following form
ϕn = x + a1 tn0 + b1 t2n
0 .
173
Vu Van Khuong and Tran Hong Thai
Theorem 2.3. For α, β are positive constants there is a nonoscillatory
solution of Eq (2) converging to the positive equilibrium point that satisfies
x 1−
as n → ∞.
β
=α
ex
Proof. From Eq (2) we can write in the form
F (xn−1 , xn , xn+1 ) =
gx0 n+1 =
1 xn
.e (xn+1 − α) − xn−1 = g(xn , xn+1 ) − xn−1
β
(14)
1 xn
e >0
β
1 xn
e (xn+1 − α) > 0
β
we expect the solutions of Eq (2) have the asymptotic appropriation (13)
gx0 n =
1 x+atn +bt2n
(x + atn+1 + bt2n+2 − α)
e
β
− (x + atn−1 + bt2n−2 ) = 0,
F = F (ϕn−1 , ϕn , ϕn+1 ) =
for t ∈ (0, ∞)
n
2n
ex+at +bt = β(x + atn−1 + bt2n−2 )(x + atn+1 + bt2n+2 − α)−1
n
x + at + bt
2n
h
atn−1 bt2n−2 bt2n+2 −1 i
atn+1
= ln β 1 +
+
+
1+
x
x
x−α x−α
atn−1 bt2n−2 βx +
1+
.
= ln
x−α
x
x
i
h
bt2n+2 atn+1 + bt2n+2 2
atn+1
−
+
− ...
. 1−
x−α x−α
x−α
atn−1 + bt2n−2 a2 t2n−2 + 2abt3n−3 + ...
−
+ ...
x
2x2
a2 t2n+2
x − α atn+1 + bt2n+2
= x + atn + bt2n + ln
+
−
+ ...
β
x−α
2(x − α)
lnx +
From (15) we have
lnx = x + ln
x−α
or (x − α)ex = βx
β
This is trust and from (15) we obtain
(15)
174
Asymptotic behavior of some population models
a n−1
atn+1
t
= atn +
x
x−α
a
at n [xt2 + tx(x − α) + (α − x)] n
p(t)atn
−a−
at =
t =
tx
x−α
tx(x − α)
tx(x − α)
As mentioned earlier exists t0 ∈ (0, 1) such that p(t0 ) = 0 and
0 < t20 < t0 < 1, p(t20 ) < 0. Posing t = t0 , we get
a
at0 n
p(t0 )atn0
−a−
=0
t0 =
t0 x
x−α
t0 x(x − α)
From (15) it follows
F =
=
(
b 2n−2
a2
bt2n+2
a2 t2n+2
0
0
t0
− 2 t02n−2 = bt2n
+
−
0
x
2x
x − α 2(x − α)2
)
i
h b
a2 1
a2
2n
t2n t2n
−
−
−b−
0 + o(t0 )
x 2x2 t20
x − α 2(x − α)2 0
b
2b(x − α)xp(t20 ) − a2 t40 [x2 t40 − (x − α)2 ] 2n
.t0 + o(t2n
0 )
2x2 (x − α)2 t20
2n
Setting F = (Bb1 + C)t2n
0 + o(t0 )
Ht0 (q) = Bq + C = 0 → q0 = −
C
B
2b(x − α)xp(t20 )
< 0.
2x2 (x − α)2 t20
We obtain that there are q1 < q0 and q2 > q0 such that
Ht00 (q) = B =
Ht0 (q1 ) > 0, Ht0 (q2 ) < 0, q1 < q0 < q2 .
We assume that a 6= 0, if ϕ
bn = x + atn0 + q0 t2n
0 , we obtain
with the notation
2n
F (ϕ
bn−1 , ϕ
bn , ϕ
bn+1 ) ∼ [q0 B + C]t2n
0 + o(t0 )
yn = x + atn0 + q1 t2n
0 ,
zn = x + atn0 + q2 t2n
0 .
We get
F (yn−1 , yn , yn+1 ) ∼ [q1 B + C]t2n
0 ,
F (zn−2 , zn−1 , zn ) ∼ [q2 B + C]t2n
0 .
These relations show that inequalities (8) are satisfied for sufficiently large
n, where g = F + xn−1 and F is given by (14).
Because the function g(xn−1 , xn , xn+1 ) is continuous and nondecreasing on
175
Vu Van Khuong and Tran Hong Thai
[x, +∞)3 → [x, +∞). We easily have g(x, x, x) = x. We can apply the Theorem (2.1) with I = [x, ∞) and see that there is an n0 ≥ 0 and a solution of
equation (2) with the asymptotics xn = ϕ
bn + o(t2n
0 ), for n ≥ n0 where b = q0
in ϕ
bn . In particular, the solution converges monotonically to the positive equilibrium point for n ≥ n0 . The proof is complete.
Acknowledgements. We would like to extend our thanks to the referees
for their suggestions which certainly improved the exposition of our paper.
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