TWO
-
DIMENSIONAL FRACTIONAL SYSTEM OF NONLINEAR
DIFFERENCE EQUATIONS IN THE MODELING COMPETITIVE
POPULATIONS
T
.
F
.
Ibrahim
1 , 21
Department of Mathematics, Faculty of Sciences and arts (S. A.) King Khalid University, Abha , Saudi Arabia
2
Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
E-mail: [email protected] & [email protected] Oct 2012
:
Abstract
In this paper we have already investigated the solutions of the two-dimensional fractional system of nonlinear difference equations in the modeling competitive populations in the form
1 1
1 1
1 1
&
n n
n n
n n n n
x
y
x
y
x
y
y
x
(1)where
and
are real numbers with the initial conditionsx
1,
x
0,
y
1,
andy
0 such thatx y
1 0
and
y x
1 0
. Moreover, we have studied the local stability, global stability, boundedness and periodicity of solutions. We will consider some special cases of (1) as applications. Finally, we give some numerical examples.---
Keywords: difference equation, solutions ,convergence ,periodicity ,eventually periodic, competitive, high orders, stability.
--- Mathematics Subject Classification: 39A10,39A11
1
Introduction
Difference equations or discrete dynamical systems [38] is diverse field which impact almost every branch of pure and applied mathematics. Every dynamical system
a
n1
f a
(
n)
determines a difference equation andvise versa .Recently, there has been great interest in studying difference equations systems. One of the reasons for this is a necessity for some techniques whose can be used in investigating equations arising in mathematical models [21] describing real life situations in population biology [17], economic, probability theory, genetics , psychology, ....etc .
The study of properties of rational difference equations [24] and systems of rational difference equations has been an area of interest in recent years, see book [20] and the references therein (see also [1] , [15] ).
1 1
, ,
n n n
n n n
x f x y
y g x y
(2)
where , n 0,1, . . . . . ,
x0,y0
R R, subset of the real plane ,
f g,
:RR, f g, are continuousfunction is competitive if f
x y,
is non-decreasing inx
and non-increasing iny
; andg x y
,
isnon-increasing in
x
and non-decreasing iny
. System (2) where the functionsf
andg
have monotonic character opposite of the monotonic character in competitive system will be called anti-competitive . It is well know that the dynamical properties of competitive populations has received great attention from both theoretical and mathematical biologists [39] due to its universal prevalence and important. Competitive and anti-competitive systems were studied by many authors (see for examples [3], [4], [11],[12], [17],[18], [22],[27], [28], [29], [39] , [40], [41]).
In a modeling setting, the two-dimensional competitive system of nonlinear rational difference
equations
1
&
1n n
n n
n n
x
y
x
y
a
y
b
x
represents the rule by which two discrete, competitive populations reproduce from one generation to the next. The phase variables
x
nandy
n denote population sizes during then
-th generation andsequence or orbit
x
n,
y
n
:
n
0,1,...
describes how the populations evolve over time. Competitivebetween the populations is reflected by the fact the transition function for each population is a decreasing function of the other population
size. For instance , In [22] M.P. Hassell, H.N. Comins studied a discrete (difference) single age-class model for two-species competition and its stability properties discussed .
There are many papers in which systems of difference equations have studied . Cinar et al. [5] has obtained the positive solution of the difference equation system
1 1
1 1
&
nn n
n n n
py
m
x
y
y
x
y
Cinar [6] has obtained the positive solution of the difference equation system
1 1
1 1
1
&
nn n
n n n
y
x
y
y
x
y
Also, Cinar [7] has obtained the positive solution of the difference equation system
1 1 1
1 1
1
1
&
n&
n n n
n n n
x
x
y
z
z
x
x
Cinar [8]-[10] has got the solutions of the following difference equations
1 1
1 1 1
1
1
1
n n
n n n n
n n
x
x
x x
x
x
x x
a x
Aloqeili [2] obtained the form of the solutions of the difference equation
1 1
1
n n
n n
x
x
a
x x
Özban [35] has investigated the positive solutions of the system of rational difference equations
1 1
1
&
nn n
n k n m n m k
y
x
y
y
x
y
In [31] , Kurbanli studied a three-dimensional system of rational difference equations
1 1 1
1 1 1
1 1 1
&
&
1
1
1
n n n
n n n
n n n n n n
x
y
z
x
y
z
y x
x y
y z
where the initial conditions are arbitrary real numbers. He expressed the solution of this system and investigated the behavior and computed for some initial values.
Elabbasy et al. [14] has obtained the solution for some particular cases of the following general system of difference equations
1 2 1 1 2 1 1 2
1 1 1
3 4 1 3 4 1 3 1 1 4 1 5
&
&
n n n n n
n n n
n n n n n n n n n n n n n
a
a y
b z
b z
c z
c z
x
y
z
a z
a x
z
b x y
b x y
c x
y
c x
y
c x y
Elsayed [16] investigated the solutions of the system of rational difference equations
1 1
1 1
1 1
&
1
1
n n
n n
n n n n
x
y
x
y
x
y
y
x
Although difference equations are sometimes very simple in their forms ,they are extremely difficult to understand thoroughly the behavior of their solutions . In book [26] V.L. Kocic, G. Ladas have studied global behavior of nonlinear difference equations of higher order . Similar nonlinear systems of rational difference equations were investigated (see [4],[39]). For some other recent papers on systems of difference equations, see, for examples, ([13] ,[24],[16],[19],[25],[30],[32],[33],[34],[36],[37],[43] ,[44]) and the related references therein.
Our goal , in this paper is to investigate the solutions of the two-dimensional fractional system of nonlinear difference equations in the modeling competitive populations in the form
1 1 1 1
1 1
&
n n
n n
n n n n
x
y
x
y
x
y
y
x
where
and
are real numbers with the initial conditionsx
1,
x
0,
y
1,
andy
0 such that1 0
x y
andy x
1 0
. Moreover, we have studied the local stability, global stability,boundedness and periodicity of solutions. We will consider some special cases of (1) as applications. Finally, we give some numerical examples.
2
Solutions for System of Nonlinear Difference Equations in (1) :
The following theorem give the solution of the system of difference equation in (1)
Theorem 2.1:Suppose that
x
n,
y
n
be a solution of equation (1) where
and
are real numbers
0
0
1 1
1
,
1,
2,
21
1
x
B
y
A
x
y
x
y
x
y
A
B
B
A
(3)
1 1 0 1 2 1 1 1 01
1
1
i j i n j n i ji i i
j
A
x
x
A
A
A
(4)
1 1 0 0 2 1 1 01
1
1
ij i i
n j n i j i i j
B
B
x
B
x
B
B
B
(5)
1 1 0 1 2 1 1 1 01
1
1
i j i n j n i ji i i
j
B
y
y
B
B
B
(6)
1 1 0 0 2 1 1 01
1
1
ij i i
n j n i j i i j
A
A
y
A
y
A
A
A
(7)where
A
x y
1 0 ,B
y x
1 0 such thatA
,1
A
,B
,B
1
,B
1
and
n
2
. ProofIt is easy , from equations (1) , to see that (3) satisfies .
We will use the mathematical induction to prove the equations(4-7) . By using equations(1) , for n 2 , we have
1 1
1 3
1 2 1 0 1 0
1
1
x
x
x
A
A
x
x y
x
y
A
x y
A
A
A
1 1 21
1
1
1
x
A
x
A
A
A
A
A
A
(8)
1 1
1 3
1 2 1 0
1 1
2
1
1
1
1
(9)
1
1
y
y
y
B
B
y
y x
y
x
B
B
B
B
B
y
B
y
B
B
B
B
B
B
By using (4) and (6) , for n 2 , we have
1 1
3 2 2
1
1
(10)
1
1
1
1
A
A
x
x
x
A
A
A
A
A
Also
1 1
3 2 2
1
1
(11)
1
1
1
1
B
B
y
y
y
B
B
B
B
B
From equations (8)-(11) , the equations (4) and (6) hold at n 2 . Now by using equations(1) , for n 3 , we have
0 2
4
2 3 0 1
2
1
1
1
1
x
B
B
x
x
x y
x
B
y
B
B
B
B
2 0
2
2 0
2 2 2
1
1
1
1
(12)
1
1
x
B
B
B
B
B
x
B
B
B
B
Also
0 2
4
2 3 0 1
2
1
1
1
1
y
A
A
y
y
y x
y
A
x
A
A
A
A
2 0
2
2 0
2 2 2
1
1
1
1
(13)
1
1
y
A
A
A
A
A
y
A
A
A
A
Also
2 0 4 2 2 0 2 21
1
1
1
1
1
(15)
1
y
A
A
A
y
A
A
A
y
A
A
A
A
A
From equations (12)-(15) , the equations (5) and (7) hold at n 2 . Now suppose that equations(4-7) hold for n k . This means that
1 1 0 1 2 1 1 1 01
16
1
1
i j i k j k i ji i i
j
A
x
x
A
A
A
1 1 0 0 2 1 1 01
1
17
1
ij i i
k j k i j i i j
B
B
x
B
x
B
B
B
1 1 0 1 2 1 1 1 01
18
1
1
i j i k j k i ji i i
j
B
y
y
B
B
B
1 1 0 0 2 1 1 01
1
19
1
ij i i
k j k i j i i j
A
A
y
A
y
A
A
A
Now we will try to prove that equations(4-7) hold at n k 1 .
2 1 2 1
2 1 1
2 1 2
k k k k k
x
x
x
x
y
2 0 4 2 2 0 2 21
1
1
1
1
1
(14)
1
x
B
B
B
x
B
B
B
x
B
B
B
B
B
1 1 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 i j i k j i ji i i
j i j i k j i j
i i i
j A x A A A A A y A x
A A A
A A
1 1 0 1 1 0 1 1 1 ij i i
k j i j i i j A A
1
1
1
1 1 0
0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 i
i j i
j i
k i j
j i
j
i i i
j i j i k j i j i i j x A A x A A A A A A A A
1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 1 k k ij i i
j i k i j i j i k i j i j i
A A A
A A
A A A
1 1 1
1 1 0 0 1 1 1 1 1 0 0 1 1 1
1 1 1
k i k i
j i j i
j j
i i
k i i
j i i j i
j j
i
x A A A A
A A A A A
1
1
1 0
1 1
1
k k i
j i
j
i i
A A A
1 1 0 1 1 0 1 1 0 1 1 01
1
1
1
1
1
k i j i j i k j k j k i j i j i i j i jA
A A
A
A
A
A
A
A
x
A
A
A
1 2 1 1 0 1 11
k k i
j i i j i i
A
1 1 0 1 1 0 11
1
1
k i j i j i k ij i i
j i
x
A
A
A
A
1 0 1 2 1 1 1 01
(20)
1
1
i j i k j k ii j i i
j
A
x
x
A
A
A
Similarly, by using equations (16-19) we can prove that
2 1 2 12 1 1
2 1 2
1 0 1 1 1 0
1
(21)
1
1
k k k k k i j i k j ii j i i
j
y
y
y
y
x
B
y
B
B
B
By using equations (16-19) and equations (20) , (21) we can prove that
2
2 2
2 2 1
1 0 0 1 1 0
1
1
1
k k k k ij i i
k j i j i i j
x
x
x
y
B
B
x
B
B
B
B
2 2 22 2 1
1 0 0 1 1 0
1
1
1
k k k k ij i i
k j i j i i j
y
y
y
x
A
A
y
A
A
A
A
Hence we have finished the proof .
Lemma 2.2:We have the following relations between the solutions in equations(4-7)
i
2 1 2 1
0
1
n n n
n j j
A
x
y
A
ii
2 1 2 1
0
1
n n n
n j j
B
y
x
B
iii
1
0
2 1 2 2 1 2
1
1
n
j j
n n n n
A
B
A
B
x
y
y
x
Proof
1
1
1 1
0 0 0
1 2 1 2
1
1 1 1
0 0
1
1
1
1
1
1
i i
j i j i i
n n
j j
n n i i
j j i
i i i i
j j
A
A
A
y A
x
x
y
A
A
A
A
A
A
1 1
0 1 0
1 1
0
1
1
i
j i
n
j i
j i
i
j
A
x y
A
A
A
10
1
n
n j
j
A A
A
A
A
A
10
1
n
n j
j
A
A
ii
As in
i
iii
By easy calculations from
i
and
ii
.
Remark 2.3: We note that
2 1 2 2 1 2
2 2 1 2 2 1
0
n n n n
n n n n
x
x
y
y
a s n
Ax
y
By x
Lemma 2.4: If
A
B
0
,We have the following relations
i
x
2n1y
2n
0
ii
x
2ny
2n1
0
Theorem 2.5: We have the following properties for the solution of system (1):
i
IfA
0
and
be a positive integer , then 2n 10
n
Lim x
ii
IfA
0
and
be a positive integer , then 2n0
nLim y
iii
IfB
0
and
be a positive integer , then 2n0
n
Lim x
iv
IfB
0
and
be a positive integer , then 2n 10
n
Lim y
3
Stability of the Solutions of Systems
Stability theory of difference equations and systems of difference equations has attracted many researchers. In recent years there has been much research activity concerning with the global asymptotic stability of system of difference equations. For these stability results, we refer, for example, to [45]. In this section we study the stability of the solutions for systems existed in the previous section and their generalizations. In the beginning, we present the basic notations and definitions concerning with the stability of equilibrium points of systems.
Consider the following two-dimensional system in the form
1
1
,
,
(22)
n n n
n n n
x
f
x
y
y
g x
y
Definition 3.1:An equilibrium point of system (22) is a point
E
x y
,
that satisfies
,
,
x
f
x y
y
g x y
Recall the Literalized Stability Theorem for two-dimensional systems in the following proposition(see[13],[28]).
Proposition 3.2:(Two-Dimensional Version of linearized Stability Theorem )
Let F
f g,
be a continuously differentiable function defined on an open set W in R .2 Let
x x
,
in W be a fixed point of F .a If all the eigenvalues of the Jacobian matrix
JF x y
,
have modulus less than one, then theequilibrium point
E
x y
,
of system(22) is asymptotically stable.bIf at least one of the eigenvalues of the Jacobian matrix
JF x y
,
has modulus greater than one,then the equilibrium point
E
x y
,
of system(22) is unstable.cThe equilibrium point
E
x y
,
of system(22) is locally asymptotically stable if every solution ofthe characteristic equation
0
f f
E E
x y
g g
E E
x y
lies inside the unit circle.
Lemma 3.3: System(1) has only one equilibrium point which is (0,0) .
Proof
For the equilibrium points of System(1) , we can write
&
x y
x y
x y
x y
Then, by solving these equations together, we have the only one equilibrium point which is 0 .
Consider ( , ) x f x y
xy
and
( , )y g x y
xy
.
The Jacobian off g
,
with respect tox y
,
isgiven by
2
2 2
2
2 2
, ,
x
xy xy
f g Jac
x y y
xy xy
At the equilibrium point
E
(0,0)
we have
10 ,
1 ,
0
f g
Jac E
x y
The corresponding eigenvalues of the equilibrium point
E
are
1
a) If
1
or
1
then the equilibrium point E of system(1) is unstable.b) If
1
or
1
then the equilibrium point E of system(1) is asymptotically stable.Now we recall some notations and previous results which will be useful in our study.
Definition 3.5:The difference equation
1
,
1,...,
,
0,1,...
(23)
n n n n k
x
F x
x
x
n
is said to be persistence if there exist numbers
m
and M with0
m
M
such that for any initial conditionsx
k,
x
k 1,...,
x
1,
x
0
0,
there exists a positive integer Nwhich depends on theinitial conditions such that
m
x
n
M
for all nN.Definition 3.6:
i
The equilibrium point of Equation(23) is locally stable if for every
0
, thereexists
0
such that for allx
k,
x
k 1,...,
x
1,
x
0
I
with1
....
0k k
x
x
x
x
x
x
we have
x
n
x
for alln
k
.
ii
The equilibrium pointx
Equation(23) is locally asymptotically stable ifx
is locally stable solution of Equation(23) and there exists
0
, such that for allx
k,
x
k 1,...,
x
1,
x
0
I
with1
....
0k k
x
x
x
x
x
x
we have n
n
Lim x
x
.The literalized equation of Equation(23) about the equilibrium
x
is the linear difference equation
1 0
, ,....,
k
n n i
i n i
F x x
x
y
y
x
.Theorem 3.7: Assume that
p q
,
R
. Then1
p
q
is a sufficient condition for the asymptotic stability of the difference equation
1 1
0
,
0,1, 2,...
n n n
x
px
qx
n
.--- ---
4
Some Special Cases:
4.1 Case 1 :The Difference Equation (24)
If we consider the one-dimensional case of the system (1) , we have the following generalized difference equation:
1 1
1
n n
n n
x
x
x
x
(24) Cinar [10] studied the difference equation1 1
1
1
n n
n n
a x
x
b x
x
If we put a b 1
