Applied Mathematics and Computation 218 (2012) 5310–5318
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Study of the asymptotic behavior of the solutions of three systems
of difference equations of exponential form
G. Papaschinopoulos a,⇑, M. Radin b, C.J. Schinas a
a
b
School of Engineering, Democritus University of Thrace, Xanthi 67100, Greece
Rochester Institute of Technology, School of Mathematical Sciences, College of Science Rochester, NY 14623, USA
a r t i c l e
i n f o
Keywords:
Difference equations
Boundedness
Persistence
Asymptotic behavior
a b s t r a c t
In this paper we study the boundedness, the persistence and the asymptotic behavior of
the positive solutions of the following systems of two difference equations of exponential
form:
a þ beyn
d þ exn
; ynþ1 ¼
;
c þ yn1
f þ xn1
yn
xn
a þ be
d þ e
xnþ1 ¼
; ynþ1 ¼
;
c þ xn1
f þ yn1
xn
yn
a þ be
d þ e
; ynþ1 ¼
;
xnþ1 ¼
c þ yn1
f þ xn1
xnþ1 ¼
where a, b, c, d, , f are positive constants and the initial values x1, x0, y1, y0 are positive
constants.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
In [26] the authors studied the boundedness, the asymptotic behavior, the periodicity and the stability of the positive
solutions of the difference equation
ynþ1 ¼
a þ beyn
;
c þ yn1
where a, b, c are positive constants and the initial values x1, x0 are positive numbers.
Motivated by the above paper, we will investigate the boundedness, the persistence and the asymptotic behavior of the
positive solutions of the following systems of difference equations of exponential form:
a þ beyn
d þ exn
; ynþ1 ¼
;
c þ yn1
f þ xn1
a þ beyn
d þ exn
xnþ1 ¼
; ynþ1 ¼
;
c þ xn1
f þ yn1
a þ bexn
d þ eyn
xnþ1 ¼
; ynþ1 ¼
c þ yn1
f þ xn1
xnþ1 ¼
⇑ Corresponding author.
E-mail addresses: [email protected] (G. Papaschinopoulos), [email protected] (M. Radin), [email protected] (C.J. Schinas).
0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2011.11.014
ð1:1Þ
ð1:2Þ
ð1:3Þ
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
5311
where a, b, c, d, , f are positive constants and the initial values x1, x0, y1, y0 are positive constants. Also for applications in
Mathematical Biology or Population Dynamics, we consider b to be the growth or reproduction rate of species xn and to be
the growth or reproduction rate of species yn.
Difference equations and systems of difference equations of exponential form can be found in the following papers:
[23,25,26,29–31,33,39,43]. Moreover, as difference equations have many applications in applied sciences there are many papers and books concerning theory and applications of difference equations, (for partial review of the theory of difference
equations, systems of difference equations and their applications see [1–43] and the references sited therein).
2. Global behavior of solutions of three general systems
In this section we consider the following three general systems of two difference equations
xnþ1 ¼ f ðyn ; yn1 Þ;
ynþ1 ¼ gðxn ; xn1 Þ;
ð2:1Þ
xnþ1 ¼ f ðxn1 ; yn Þ;
ynþ1 ¼ gðxn ; yn1 Þ;
ð2:2Þ
xnþ1 ¼ f ðxn ; yn1 Þ;
ynþ1 ¼ gðxn1 ; yn Þ;
ð2:3Þ
and
where f, g are continuous functions and the initial values x1, x0, y1, y0 are positive numbers.
Working in a similar fashion as in relative theorems included in [10,11,17,24] we can state the following theorem which is
useful for the study of our systems Eqs. (1.1)–(1.3).
Theorem 2.1. Let f, g, f : R+ R+ ? R+, g : R+ R+ ? R+ be continuous functions, R+ = (0, 1). Then the following statements are
true
(i) Let a1, b1, a2, b2 be positive numbers such that a1 < b1, a2 < b2 and
f : ½a2 ; b2 ½a2 ; b2 ! ½a1 ; b1 ;
g : ½a1 ; b1 ½a1 ; b1 ! ½a2 ; b2 :
Suppose that the function f(u, v) is a decreasing function with respect to u (resp. v) for all v (resp. u) and g(z, w) is a decreasing
function with respect to z (resp. w) for every w (resp. z). Finally suppose that if m, M, r, R are real numbers such that if
M ¼ f ðr; rÞ;
m ¼ f ðR; RÞ;
R ¼ gðm; mÞ;
r ¼ gðM; MÞ
Þ and every positive solution of
then m = M and r = R. Then the system of difference Eq. (2.1) has a unique positive equilibrium ð
x; y
the system Eq. (2.1) which satisfies
xn0 2 ½a1 ; b1 ;
xn0 þ1 2 ½a1 ; b1 ;
yn0 2 ½a2 ; b2 ;
yn0 þ1 2 ½a2 ; b2 ;
n0 2 N
ð2:4Þ
tends to the unique positive equilibrium of Eq. (2.1).
(ii) Let
f : ½a1 ; b1 ½a2 ; b2 ! ½a1 ; b1 ;
g : ½a1 ; b1 ½a2 ; b2 ! ½a2 ; b2 :
Suppose that the functions f(x, y) and g(x, y) are decreasing functions with respect to x (resp. y) for all y (resp. x). Finally suppose
that if m, M, r, R are real numbers such that if
M ¼ f ðm; rÞ;
m ¼ f ðM; RÞ;
R ¼ gðm; rÞ;
r ¼ gðM; RÞ
ð2:5Þ
Þ and every pothen m = M and r = R. Then the systems of difference Eqs. (2.2) and (2.3) have a unique positive equilibrium ð
x; y
sitive solution of the system Eqs. (2.2) (resp. (2.3)) satisfying Eq. (2.4) tends to the unique positive equilibrium of Eqs. (2.2) (resp.
(2.3)).
Proof. (i) System Eq. (2.1) is equivalent to the system of separated equations
xnþ1 ¼ f ðgðxn1 ; xn2 Þ; gðxn2 ; xn3 ÞÞ ¼ Fðxn1 ; xn2 ; xn3 Þ;
ynþ1 ¼ gðf ðyn1 ; yn2 Þ; f ðyn2 ; yn3 ÞÞ ¼ Gðyn1 ; yn2 ; yn3 Þ;
n P 2:
We consider the equation
xnþ1 ¼ Fðxn1 ; xn2 ; xn3 Þ:
ð2:6Þ
From the conditions of f, g we have that F is a function from [a1, b1] [a1, b1] [a1, b1] into [a1, b1] and F(x, y, z) is increasing in
x for all y, z, increasing in y for all x, z and increasing in z for all x, y. Let now that M, m be positive numbers such that
M ¼ FðM; M; MÞ ¼ f ðgðM; MÞ; gðM; MÞÞ;
m ¼ Fðm; m; mÞ ¼ f ðgðm; mÞ; gðm; mÞÞ:
5312
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
By setting r = g(M, M), R = g(m, m) we have that relations Eq. (2.5) are satisfied. Then from hypothesis of Theorem 2.1 we have
that m = M. Therefore from Theorem 1.15 of [17] we have that Eq. (2.6) has a unique positive equilibrium x and every positive
solution of Eq. (2.6) tends to the unique positive equilibrium x. Similarly we can prove that equation
ynþ1 ¼ gðf ðyn1 ; yn2 Þ; f ðyn2 ; yn3 ÞÞ ¼ Gðyn1 ; yn2 ; yn3 Þ
ð2:7Þ
and every positive solution of Eq. (2.7) tends to the unique positive equilibrium y
. This
has a unique positive equilibrium y
completes the proof of the statement (i).
The proof of the statement (ii) is a modification of similar results included in [10,11,17,24]. h
3. Global behavior of solutions of system Eq. (1.1)
In the first lemma we study the boundedness and persistence of the positive solutions of Eq. (1.1).
Lemma 3.1. Every positive solution of Eq. (1.1) is bounded and persists.
Proof. Let (xn, yn) be an arbitrary solution of Eq. (1.1). From Eq. (1.1) we can see that
xn 6
aþb
dþ
; yn 6
; n ¼ 1; 2; . . . :
c
f
ð3:1Þ
In addition, from Eqs. (1.1) and (3.1) we get
aþb
dþ
xn P
a þ be f
d þ e c
; yn P
; n ¼ 3; 4; . . .
dþ
cþ f
f þ aþb
c
ð3:2Þ
Therefore, from Eqs. (3.1) and (3.2) the proof of the lemma is complete.
h
In the next proposition we will study the asymptotic behavior of the positive solutions of Eq. (1.1).
Proposition 3.1. Consider system Eq. (1.1). Suppose that the following relation holds true:
< c;
ð3:3Þ
b < f:
Þ and every positive solution of Eq. (1.1) tends to the unique positive
Then system Eq. (1.1) has a unique positive equilibrium ð
x; y
equilibrium of Eq. (1.1) as n ? 1.
Proof. We consider the functions
f ðu; v Þ ¼
a þ beu
d þ ez
; gðz; wÞ ¼
;
cþv
fþw
ð3:4Þ
where
"
dþ
#
"
aþb
#
a þ be f a þ b
d þ e c d þ z; w 2 I1 ¼
;
;
:
; u; v 2 I2 ¼
dþ
c
f
cþ f
f þ aþb
c
From Eqs. (3.4) and (3.5), we get the following relations for u,
f ðu; v Þ 2 I1 ;
ð3:5Þ
v 2 I2, z, w 2 I1
gðz; wÞ 2 I2
and so f:I2 I2 ? I1, g : I1 I1 ? I2. Let (xn, yn) be an arbitrary solution of Eq. (1.1). Therefore from Lemma 3.1 for n P 3 we
have:
xn 2 I 1 ;
yn 2 I2 :
Now let m, M, r, R be positive numbers such that
M¼
a þ ber
a þ beR
d þ em
d þ eM
; m¼
; R¼
; r¼
:
cþr
cþR
fþm
fþM
ð3:6Þ
Then we consider the functions
FðxÞ ¼
a þ behðxÞ
d þ ex
x; hðxÞ ¼
; x 2 I1 :
c þ hðxÞ
fþx
ð3:7Þ
Note that F maps the interval I1 into itself. We claim that equation F(x) = 0 has a unique solution in I1. From Eq. (3.7) we get
5313
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
0
F 0 ðxÞ ¼ h ðxÞ be
0
h ðxÞ ¼ hðxÞ ðcþhðxÞÞþaþbehðxÞ
ðcþhðxÞÞ2
ex ðfþxÞþdþex
ð3:8Þ
x 2 I1 :
;
ðfþxÞ2
1;
Let x; x 2 I1 be a solution of equation F(x) = 0. Then from Eq. (3.7) we have
xðc þ hðxÞÞ ¼ a þ behðxÞ ;
hðxÞðf þ xÞ ¼ d þ ex :
ð3:9Þ
Now observe that relations Eqs. (3.8) and (3.9) imply that
0
h ðxÞ ¼ ex þ hðxÞ
f þ x
behðxÞ ðc þ hðxÞÞ þ a þ behðxÞ behðxÞ þ x
¼
:
c þ hðxÞ
ðc þ hðxÞÞ2
;
ð3:10Þ
Then from Eqs. (3.3), (3.8) and (3.10), we get
F 0 ðxÞ ¼
ex þ hðxÞ
c þ hðxÞ
behðxÞ þ x
1 < 0:
f þ x
ð3:11Þ
Therefore, from Eq. (3.11) we see that equation F(x) = 0 has a unique solution in I1. In addition, relations Eq. (3.6) imply that
m, M are roots of F(x) = 0. Hence we get m = M. Therefore from Eq. (3.6) it follows that r = R. Thus from statement (i) of TheÞ and every positive solution of system Eq. (1.1) tends to the
orem 2.1, system Eq. (1.1) has a unique positive equilibrium ð
x; y
unique positive equilibrium as n ? 1. This completes the proof of the proposition. h
In the next proposition of this section we will study the global asymptotic stability of the positive equilibrium of Eq. (1.1).
Proposition 3.2. Consider system Eq. (1.1) where the condition Eq. (3.3) holds true. Also suppose that
b þ ðb þ Þe1 ða þ bÞðd þ Þ
< 1:
þ
cf
c2 f2
ð3:12Þ
Þ of Eq. (1.1) is globally asymptotically stable.
Then the unique positive equilibrium ð
x; y
Þ is locally asymptotically stable. The linearized system of Eq. (1.1) about ð
Þ is
Proof. First we will prove that ð
x; y
x; y
y
y
x
x
aþbe
xnþ1 ¼ be
cþy yn ðcþyÞ2 yn1
ð3:13Þ
e
dþe
ynþ1 ¼ fþ
x xn ðfþxÞ2 xn1 :
Observe that system Eq. (3.13) is equivalent to the following system
0
wnþ1 ¼ Awn ;
0
Bc 0
B
A¼B
@1 0
0
a¼
bey
;
c þ y
a
b¼
0 b
1
0C
C
C;
0 0A
d
1 0 0
0
xn
1
By C
B n C
wn ¼ B
C;
@ xn1 A
yn1
a þ bey
ex
d þ ex
; c¼
:
; d¼
2
f þ x
Þ
ðc þ y
ðf þ xÞ2
The characteristic equation of A is
k4 ack2 ðad þ bcÞk bd ¼ 0:
ð3:14Þ
Þ is the positive equilibrium of Eq. (1.1), then we have
Since ð
x; y
x ¼
a þ bey d þ ex
; y¼
:
c þ y
f þ x
ð3:15Þ
Hence from Eqs. (3.12) and (3.15) and since xex < e1, x > 0 we get
jacj þ jadj þ jbcj þ jbdj ¼
bexy
bey ðd þ ex Þ ex ða þ bey Þ ðd þ ex Þða þ bey Þ
þ
þ
þ
Þðf þ xÞ ðc þ y
ðc þ y
Þðf þ xÞ2 ðf þ xÞðc þ y
Þ2
Þ2
ðf þ xÞ2 ðc þ y
¼
ey ðf þ xÞ
bexy
by
xex ðc þ yÞ ðd þ ex Þða þ bey Þ
þ
þ
þ
2
Þðf þ xÞ ðc þ y
ðc þ y
Þðf þ xÞ
Þ2
Þ2
ðf þ xÞðc þ y
ðf þ xÞ2 ðc þ y
<
b þ ðb þ Þe1 ða þ bÞðd þ Þ
< 1:
þ
cf
c2 f2
ð3:16Þ
Therefore, from Eq. (3.16) and from Remark 1.3.1 of [23], all the roots of Eq. (3.14) are of modulus less than 1 which implies
Þ is locally asymptotically stable. Using Proposition 3.1, we see that ð
Þ is globally asymptotically stable. This
that ð
x; y
x; y
completes the proof of the proposition. h
5314
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
4. Global Character of solutions of system Eq. (1.2)
In the following lemma we study the boundedness and persistence of system Eq. (1.2).
Lemma 4.1. Every positive solution of Eq. (1.2) is bounded and persists.
Proof. Let (xn, yn) be an arbitrary solution of Eq. (1.2). Similarly as in lemma 3.1, for n = 3, 4, . . . by induction we get
"
"
#
dþ
aþb
#
a þ be f a þ b
d þ e c d þ xn 2 I 3 ¼
;
2
I
¼
;
;
;
y
4
n
c
f
f þ dþf c þ aþb
c
ð4:1Þ
and so the proof of the lemma is complete. h
In the next proposition we study the asymptotic behavior of the positive solutions of Eq. (1.2).
Proposition 4.1. Consider system Eq. (1.2). Suppose that the following relation holds true:
b < cf:
ð4:2Þ
Þ and every positive solution of Eq. (1.2) tends to the unique positive
Then system Eq. (1.2) has a unique positive equilibrium ð
x; y
equilibrium of Eq. (1.2) as n ? 1.
Proof. We consider the functions
f ðx; yÞ ¼
a þ bey
d þ ex
; gðx; yÞ ¼
;
cþx
fþy
ð4:3Þ
x 2 I3 ;
y 2 I4 ;
ð4:4Þ
where
and I3, I4 are defined in Eq. (4.1). From (4.3) and (4.4), we see that for x 2 I3, y 2 I4
f ðx; yÞ 2 I3 ;
gðx; yÞ 2 I4
and so f : I3 I4 ? I3, g : I3 I4 ? I4.
Let m, M, r, R be positive numbers such that
M¼
a þ ber
a þ beR
d þ em
d þ eM
; m¼
; R¼
; r¼
:
cþm
cþM
fþr
fþR
ð4:5Þ
From Eq. (4.5) we get
a
er ¼ MðcþmÞ
;
b
;
em ¼ RðfþrÞd
a
eR ¼ mðcþMÞ
;
b
eM ¼ rðfþRÞd
;
which imply that
M m ¼ bc ðer eR Þ ¼ bc erR ðeR er Þ;
R r ¼ f ðem eM Þ ¼ f emM ðeM em Þ:
ð4:6Þ
Moreover, we get
eR er ¼ en ðR rÞ;
minfR; rg 6 n 6 maxfR; rg;
eM em ¼ eh ðM mÞ;
minfM; mg 6 h 6 maxfM; mg:
ð4:7Þ
Then relations Eqs. (4.6) and (4.7) imply that
b
M m ¼ erRþn ðR rÞ;
c
R r ¼ emMþh ðM mÞ
f
and so
b
jM mj 6 jR rj;
c
jR rj 6 jM mj:
f
In addition, observe that relations Eqs. (4.2) and (4.8) imply that
b
1
jM mj 6 0;
cf
b
1
jR rj 6 0
cf
ð4:8Þ
5315
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
from which we see that M = m and R = r. Therefore from Eq. (4.1) and statement (ii) of Theorem 2.1 system Eq. (1.2) has a
Þ and every positive solution of system Eq. (1.2) tends to the unique positive equilibrium
unique positive equilibrium ð
x; y
as n ? 1. This completes the proof of the proposition. h
In the next proposition of this section we will study the global asymptotic stability of the positive equilibrium of Eq. (1.2).
Proposition 4.2. Consider system Eqs. (1.2) where (4.2) holds true. Also suppose that
a þ b d þ b ða þ bÞðd þ Þ
þ 2 þ þ
< 1:
c2
cf
f
c2 f2
ð4:9Þ
Þ of Eq. (1.2) is globally asymptotically stable.
Then the unique positive equilibrium ð
x; y
Þ is locally asymptotically stable. The linearized system of Eq. (1.2) about ð
Þ is
Proof. First we will prove that ð
x; y
x; y
y
y
x
x
aþbe
xnþ1 ¼ be
cþx yn ðcþxÞ2 xn1
ð4:10Þ
e
dþe
ynþ1 ¼ fþ
y :
xn ðfþy
y
Þ2 n1
Notice that system Eq. (4.10) is equivalent to the following system
1
0 a b 0
C
B
Bc 0 0 dC
C;
B¼B
C
B
@1 0 0 0A
0
wnþ1 ¼ Bwn ;
0
a¼
bey
;
c þ x
b¼
1
xn
C
B
B yn C
C;
wn ¼ B
C
B
@ xn1 A
0
yn1
1 0 0
a þ bey
ex
d þ ex
; c¼
:
; d¼
2
fþy
Þ2
ðc þ xÞ
ðf þ y
Then the characteristic equation of B is
k4 ðb þ d þ acÞk2 þ bd ¼ 0:
ð4:11Þ
From Eq. (4.9) we get
a þ bey d þ ex
bexy
ðd þ ex Þða þ bey Þ
þ
þ
þ
Þ
Þ2 ðc þ xÞ2
Þ2 ðc þ xÞðf þ y
ðf þ y
ðc þ xÞ2
ðf þ y
a þ b d þ b ða þ bÞðd þ Þ
< 2 þ 2 þ þ
< 1:
c
cf
f
c2 f2
jbj þ jdj þ jacj þ jbdj ¼
ð4:12Þ
Therefore, from Eq. (4.12) and from Remark 1.3.1 of [23] all the roots of Eq. (4.11) are of modulus less than 1 which implies
Þ is locally asymptotically stable. Using Proposition 4.1, we see that ð
Þ is globally asymptotically stable. This
that ð
x; y
x; y
completes the proof of the proposition. h
5. Global character of solutions of system Eq. (1.3)
In the following lemma we study the boundedness and persistence of system Eq. (1.3).
Lemma 5.1. Every positive solution of Eq. (1.3) is bounded and persists.
Proof. Let (xn, yn) be an arbitrary solution of Eq. (1.3). Similarly as in lemma 3.1, for n = 3, 4, . . . we get
"
aþb
#
"
dþ
#
a þ be c a þ b
d þ e f d þ xn 2 I 5 ¼
;
;
; yn 2 I 6 ¼
;
dþ
c
f
cþ f
f þ aþb
c
and so the proof of the lemma is complete. h
In the next proposition we study the asymptotic behavior of the positive solutions of Eq. (1.3).
ð5:1Þ
5316
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
Proposition 5.1. Consider system Eq. (1.3). Suppose that the following relation holds true:
< f:
b < c;
ð5:2Þ
Þ and every positive solution of Eq. (1.3) tends to the unique positive
Then system Eq. (1.3) has a unique positive equilibrium ð
x; y
equilibrium of Eq. (1.3) as n ? 1.
Proof. We consider the functions
f ðx; yÞ ¼
a þ bex
d þ ey
; gðx; yÞ ¼
;
cþy
fþx
ð5:3Þ
x 2 I5 ;
y 2 I6 ;
ð5:4Þ
where
where I5, I6 are defined in Eq. (5.1). From Eq. (5.3) and Eq. (5.4), we have for x 2 I5, y 2 I6
f ðx; yÞ 2 I5 ;
gðx; yÞ 2 I6
and so f : I5 I6 ? I5, g : I5 I6 ? I6.
Let m, M, r, R be positive numbers such that
M¼
a þ bem
a þ beM
d þ er
d þ eR
; m¼
; R¼
; r¼
:
cþr
cþR
fþm
fþM
ð5:5Þ
Moreover arguing as in the proof of Theorem 1.16 of [17], it suffices to assume that
m 6 M;
r 6 R:
ð5:6Þ
Observe that relations Eq. (5.5) imply:
a
em ¼ MðcþrÞ
;
b
er ¼
RðfþmÞd
;
a
eM ¼ mðcþRÞ
;
b
eR ¼ rðfþMÞd
from which we see that
ðer eR Þ ¼ fðR rÞ þ Rm rM;
bðem eM Þ ¼ cðM mÞ þ Mr mR:
ð5:7Þ
Then by adding the two relations Eq. (5.7) we get
ðer eR Þ þ bðem eM Þ ¼ fðR rÞ þ cðM mÞ:
Thus from Eq. (4.7) we get
erRþn ðR rÞ þ bemMþh ðM mÞ ¼ fðR rÞ þ cðM mÞ
ð5:8Þ
where h, n are defined in Eq. (4.7). Therefore from Eq. (5.8) we have
b
fðR rÞ 1 erRþn þ cðM mÞ 1 emMþh ¼ 0:
f
c
ð5:9Þ
Then using Eqs. (5.2), (5.6) and (5.9), gives us m = M and r = R. Hence from Eq. (5.1), statement (ii) of Theorem 2.1 system Eq.
Þ and every positive solution of system Eq. (1.3) tends to the unique positive
(1.3) has a unique positive equilibrium ð
x; y
equilibrium as n ? 1. This completes the proof of the proposition. h
In the next proposition of this paper, we study the global asymptotic stability of the positive equilibrium of Eq. (1.3).
Proposition 5.2. Consider system Eqs. (1.3) where (5.2) hold true. Also suppose that
b
c
b ða þ bÞðd þ Þ
þ þ þ
< 1:
f cf
c2 f2
ð5:10Þ
Þ of Eq. (1.3) is globally asymptotically stable.
Then the unique positive equilibrium ð
x; y
Þ is locally asymptotically stable. The linearized system of Eq. (1.3) about ð
Þ is
Proof. First we will prove that ð
x; y
x; y
x
x
y
y
aþbe
xnþ1 ¼ be
cþy xn ðcþyÞ2 yn1 ;
e
dþe
ynþ1 ¼ fþ
x yn ðfþxÞ2 xn1 :
Clearly we see that system Eq. (5.11) is equivalent to the system
ð5:11Þ
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
0
wnþ1 ¼ Cwn ;
a 0
B0 c
B
C¼B
@1 0
0
a¼
bex
;
c þ y
b¼
1
0 b
d 0C
C
C;
0 0A
1 0 0
0
xn
5317
1
By C
B n C
wn ¼ B
C;
@ xn1 A
yn1
a þ bex
ey
d þ ey
; c¼
:
; d¼
2
fþx
Þ
ðc þ y
ðf þ xÞ2
Then the characteristic equation of C is
k4 ða þ cÞk3 þ ack2 bd ¼ 0:
ð5:12Þ
From Eq. (5.10) we get
jaj þ jcj þ jacj þ jbdj ¼
bex
ey
bexy
ðd þ ey Þða þ bex Þ b b ða þ bÞðd þ Þ
< þ þ þ
< 1:
þ
þ
þ
c þ y f þ x ðc þ yÞðf þ xÞ
c f cf
Þ2
c2 f2
ðf þ xÞ2 ðc þ y
ð5:13Þ
Therefore, from Eq. (5.13) and from Remark 1.3.1 of [23] we see that all the roots of Eq. (5.12) are of modulus less than 1,
Þ is locally asymptotically stable. Using Proposition 5.1, ð
Þ is globally asymptotically stable. This
which implies that ð
x; y
x; y
completes the proof of the proposition. h
Acknowledgement
The authors thank the referees for their helpful suggestions.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.
A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part 1, Int. J. Difference Eq. 3 (1) (2008).
A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part 2, Int. J. Difference Eq., to appear.
J.R. Beddington, C.A. Free, J.H. Lawton, Dynamic complexity in predator prey models framed in difference equations, Nature 255 (1975) 58–60.
D. Benest, C. Froeschle, Analysis and Modelling of Discrete Dynamical Systems, Cordon and Breach Science Publishers, The Netherlands, 1998.
L. Berg, On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen 21 (No. 4) (2002) 1061–1074.
L. Berg, Inclusion theorems for non-linear difference equations, J. Difference Eq. Appl. 10 (4) (2004) 399–408.
L. Berg, On the asymptotics of difference equation xn3 = xn(1 + xn1xn2), J. Difference Eq. Appl. 14 (1) (2008) 105–108.
E. Camouzis, M.R.S. Kulenovic, G. Ladas, O. Merino, Rational systems in the plain, J. Difference Eq. Appl. (2008).
E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton,
London, 2008.
þb2 xn þc2 yn
n
E. Camouzis, C.M. Kent, G. Ladas, C.D. Lynd, On the Global Character of solutions of the system xnþ1 ¼ a1xþy
and ynþ1 ¼ a2AþBx
: Open problems and
n
n þCyn
conjectures, J. Difference Eq. Appl., to appear.
D. Clark, M.R.S. Kulenovic, A coupled system of rational difference equations, Comput. Math. Appl 43 (2002) 849–867.
D. Clark, M.R.S. Kulenovic, J.F. Selgrade, Global asymptotic behavior of a two-dimensional difference equation modelling competition, Nonlinear Anal.
52 (2003) 1765–1776.
J.M. Cushing, Periodically forced nonlinear systems of difference equations, J. Difference Eq. Appl. 3 (1998) 547–561.
L. Edelstein-Keshet, Mathematical Models in Biology, Birkhauser Mathematical Series, New York, 1988.
S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996.
E.A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations Chapman & Hall/CRC, 2005.
L. Gutnik, S. Stević, On the behaviour of the solutions of a second order difference equation, Discrete Dyn. Nat. Soc. Vol. 2007, Article ID 27562, (2007),
14 pages.
B. Iricanin, S. Stević, On a class of third order nonlinear difference equations, Appl. Math. Comput. 213 (2009) 479–483.
B. Iricanin, S. Stević, Eventually constant solutions of a rational difference equations, Appl. Math. Comput. 215 (2009) 854–856.
C.M. Kent, Converegence of solutions in a nonhyperbolic case, Nonlinear Anal. 47 (2001) 4651–4665.
C.M. Kent, W. Kosmala, S. Stević, On the Difference Equation xn+1 = xnxn2 1, Abstr. Appl. Anal., Vol. 2011, Article ID 815285, 15 pages.
V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers., Dordrecht,
1993.
M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, 2002.
E. El-Metwally, E.A. Grove, G. Ladas, R. Levins, M. Radin, On the difference equation, xnþ1 ¼ a þ bxn1 exn , Nonlinear Anal. 47 (2001) 4623–4634.
yn
I. Ozturk, F. Bozkurt, S. Ozen, On the difference equation ynþ1 ¼ acþbe
þyn1 , Appl. Math. Comput. 181 (2006) 1387–1393.
G. Papaschinopoulos, G. Stefanidou, K.B. Papadopoulos, On a modification of a discrete epidemic model, Comput. Math. Appl. 59 (11) (2010) 3559–
3569.
xp
G. Papaschinopoulos, C.J. Schinas, G. Stefanidou, On the nonautonomous difference equation xnþ1 ¼ An þ n1
, Appl. Math. Comput. 217 (2011) 5573–
xqn
5580.
Y. Saito, W. Ma, T. Hara, A necessary and sufficient condition for permanence of a Lotka-Voltera discrete system with delays, J. Math. Anal. Appl. 256
(2001) 162–174.
G. Stefanidou, G. Papaschinopoulos, C.J. Schinas, On a system of two exponential type difference equations, Comm. Appl. Nonlinear Anal. 17 (2) (2010)
113.
S. Stević, Asymptotic behaviour of a sequence defined by iteration with applications, Colloq. Math. 93 (2) (2002) 267–276.
S. Stević, On the recursive sequence xn+1 = xn1/g(xn), Taiwanese J. Math. 6 (3) (2002) 405–414.
S. Stević, Asymptotic behaviour of a nonlinear difference equation, Indian J. Pure Appl. Math. 34 (12) (2003) 1681–1687.
S. Stević, Global stability and asymptotics of somes classes of difference equations, J. Math. Anal. Appl. 316 (2006) 60–68.
S. Stević, On momotone solutions of some classes of difference equations, Discrete Dyn. Nat. Soc. vol. 2006, Article ID 53890, 2006, 9 pages.
S. Stević, On positive solutions of a (k + 1)th order difference equation, Appl. Math. Lett. 19 (5) (2006) 427–431.
5318
G. Papaschinopoulos et al. / Applied Mathematics and Computation 218 (2012) 5310–5318
[37] S. Stević, Asymptotics of some classes of higher order difference equations, Discrete Dyn. Nat. Soc. vol. 2007, Article ID 56813, 2007, 20 pages.
[38] S. Stević, Existence of nontrivial solutions of a rational difference equations, Appl. Math. Lett. 20 (2007) 28–31.
[39] S. Stević, On a Discrete Epidemic Model, Discrete Dynamics in Nature and Society, vol. 2007, Article ID 87519, 10 pages, 2007. doi:10.1155/2007/
87519.
[40] S. Stević, Nontrivial solutions of a higher-order rational
difference
equation, Mat. Zameki 84 (5) (2008) 772–780.
[41] S. Stević, On the recursive sequence xnþ1 ¼ max c; xpn =xpn1 , Appl. Math. Lett. 21 (8) (2008) 791–796.
[42] S. Stević, On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl. 376 (2011) 317–328.
[43] D.C. Zhang, B. Shi, Oscillation and global asymptotic stability in a discrete epidemic model, J. Math. Anal. Appl. 278 (2003) 194–202.