Separable Cubic Stochastic Operators

CENTRAL ASIAN JOURNAL OF MEDICAL AND NATURAL SCIENCES
Volume: 04 Issue: 06 | Nov-Dec 2023 ISSN: 2660-4159

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Separable Cubic Stochastic Operators

1. Baratov B. S. Abstract: In this paper, we study the trajectory of a
separable cubic stochastic operator on a two-
dimensional simplex, which naturally arises in the study
Received 2nd Oct 2023,
of certain problems of population biology. In the
Accepted 19th Nov 2023, simplest problem of population genetics, a biological
Online 16th Dec 2023 system of a finite set consisting of n species 1, 2, …, n is
considered.
1
Karshi State University, Uzbekistan Key words: cubic stochastic operator, separable cubic
stochastic operator.
Introduction
There are many systems which are described by nonlinear operators. A quadratic stochastic operator
(QSO) is one of the simplest nonlinear cases. A QSO has meaning of a population evolution operator
and it was first introduced by Bernstein in [1]. For more than 80 years, the theory of QSOs has been
developed and many papers were published (see e.g. [4]-[7]). In recent years it has again become of
interest in connection with its numerous applications in many branches of mathematics, biology and
physics.
Let E = 1,2,..., m be a finite set and the set of all probability distribution on E
 m

S m −1
= x = ( x1 , x2 ,..., xm )  m
: xi  0, for any i and  xi = 1 (1)
 i =1 
be the ( m − 1) -dimensional simplex. A QSO is a mapping defined as V : S
m −1
→ S m−1 of the
simplex into itself, of the form V ( x ) = x  S m−1 , where
m
xk =  Pij ,k xi x j , k  E, (2)
i , j =1
and the coefficients Pij ,k satisfy

m
Pij ,k = Pji ,k  0, P
k =1
ij ,k
= 1 for all i, j  E. (3)
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Copyright (c) 2023 Author (s). This is an open-access article distributed under the terms of Creative Commons
Attribution License (CC BY).To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/
CAJMNS Volume: 04 Issue: 06 | Nov-Dec 2023
The trajectory (orbit) x  ( n)

n0
, of V for an initial value x( 0)  S m−1 is defined by
( )
x( n+1) = V x( n) = V ( n+1) x(0) , ( ) n = 0,1,2,... .
One of the main problems in mathematical biology is to study the asymptotic behavior of the
trajectories. This problem was solved completely for the Volterra QSO.
The operator V is called Volterra QSO, if Pij ,k = 0 for any k  i, j , i, j , k  E . For the Volterra
QSO the general formula was given in [4],
 m

xk = xk  1 +  aki xi  , (4)
 i =1 
where aki = 2 Pik ,k − 1 for i  k and akk = 0 . Moreover, aki = −aik and aki  1 for all i, k  E .
In [4], the theory of Volterra QSO was developed using theory of the Lyapunov functions and
tournaments. But non-Volterra QSOs were not completely studied. Because, there is no general theory
that can be applied for study of non-Volterra operators.
In [7] Separable Quadratic Stochastic Operators (SQSOs) where introduced. Volterra QSO (4) has a
form as SQSO, but in [8] it was proved that it coincides with a SQSO if and only if it is a linear
operator.
In recent years, Cubic Stochastic Operators (CSOs) have begun to be studied, which different from
quadratic operators.
In this paper we consider another class of cubic operators which we call separable cubic stochastic
operators.
In Section 2, we recall the definition of CSOs and definitions and known results. In Section 3 for a
SCSO defined on the two-dimensional simplex, we prove that it has three fixed points and we find
conditions on parameter under which a fixed point is a repelling, attracting, or saddle point, the
boundary S is an invariant set.
2
Preliminaries and known results

Separable quadratic stochastic operator. Let us recall some necessary definition and notations. In [7]
Separable Quadratic Stochastic Operators were introduced as follows: The QSO (2), (3) with
additional condition
Pij ,k = aik b jk for all i, j , k  E (5)
where aik , b jk  entries of matrices A = ( aik ) and B = b jk ( ) such that the conditions (3) are
satisfied for the coefficients (5).
Then the QSO V corresponding to the coefficients (5) has the form.
xk = (V ( x ) )k = ( A ( x ) )k ( B ( x ) )k , (6)
where ( A ( x ) ) =  aik xi , ( B ( x ) ) =  b jk x j .
m m
k k
i =1 j =1
Definition 1. [7] The QSO (6) is called separable quadratic stochastic operator (SQSO).
m
From the conditions Pij ,k  0 and P k =1
ij , k
= 1 for all i, j it follows that the condition on matrices A
and B that aik b jk  0 , AB = 1 , where BT

T
is the transpose of B and 1 is the matrix with all
entries 1’s. If a (i ) = ( ai1 ,..., aim ) is the i -th row of the matrix A and b( j ) = ( b j1 ,..., b jm ) is the
j -th row of the matrix B , then from ABT = 1 we get

a (i )b( j ) = 1 , for all i, j = 1,..., m .
For a fixed j , the above condition implies that
( ) = (1,1,...,1) . (7)
T
A b( j )
If det ( A )  0 , then (7) gives b( j ) = b( k ) for all k, j  E , i.e., all rows of B are the same,
therefore det ( B ) = 0 . Similarly, if det ( B )  0 , then all the rows of A must be the same, so
det ( A ) = 0 .
Cubic stochastic operator. The CSO is a mapping W : S m−1 → S m−1 of the form
m
xl = P
i , j , k =1
ijk ,l
xi x j xk , l  E , (8)
where Pijk ,l are coefficients of heredity such that

m
Pijk ,l = Pkij ,l = Pikj ,l = Pkji ,l = Pjik ,l = Pjki ,l  0,  Pijk ,l = 1 i, j , k  E. (9)
l =1
and we suppose that the coefficients Pijk ,l do not change for any permutation of i, j , k .
Note that W is a non-linear operator, and its dimension increases with m . Higher-dimensional
dynamical systems are essential, but only relatively few dynamical systems have yet been analysed.
For a given x( 0)  S m−1 , the trajectory x( n)   n0

of initial point x( 0) under action of CSO (8) is
defined by ( ) , where n = 0,1, 2,... with x = x . Denote by  ( x ) the set of

x( n+1) = W x( n) (0) ( 0)
limit points of the trajectory x  . Since x   S

 
( ) ( ) n n m −1 m −1
and S is a compact set, it is
n =0 n =0
follows that  ( x )   . If  ( x ) consists of a single point, they the trajectory converges and
( 0) ( 0)
 ( x( 0 ) ) is a fixed point of the operator W. A point x  S m−1 is called a fixed of the W if
W ( x ) = x . Denote by Fix (W ) the set of all fixed points of the operator W , i.e.
Fix (W ) = x  S m−1 :W ( x ) = x .
Let DW ( x ) = ( Wi / x j ) ( x ) be a Jacobian of W at the point x .
Definition 2 ([3]): A fixed point x is called hyperbolic if its Jacobian DW ( x ) has no

eigenvalues on the unit circle in .
Definition 3 ([3]): A hyperbolic fixed point x is called:
(i) attracting if all the eigenvalues of the Jacobian DW ( x ) are in the unit disk;
(ii) repelling if all the eigenvalues of the Jacobian DW ( x ) are outside the closed unit disk;
(iii) a saddle otherwise;

Main result
In this section we consider CSO (8), (9) with additional condition
Pijk ,l = ail b jl ckl , for all i, j , k , l  E , (11)
where ail , b jl , ckl  entries of matrices A = ( ail ) , B = ( b jl ) and C = ( ckl ) such that the
conditions (9) are satisfied for the coefficients (11).
Then the CSO W corresponding to the coefficients (11) has the form
xl = (W ( x ) )l = ( A ( x ) )l ( B ( x ) )l ( C ( x ) )l (12)
( A( x )) =  a x , ( B ( x)) =  b (C ( x )) =  c
m m m
where
l il i l jl
xj and
l kl
xk .
i =1 j =1 k =1
Definition 5. The CSO (12) is called separable cubic stochastic operator (SCSO).
Lemma 1. The condition (11) is sufficient for a CSO to be product of three linear operators, but the
condition is not necessary.
The CSO with coefficients Pijk ,l can be written as the product of three matrices A = ( ail ) ,
B = ( b jl ) and C = ( ckl ) if and only if, for any i, j , k , l  E it holds
Pijk ,l + Pkij ,l + Pikj ,l + Pkji ,l + Pjik ,l + Pjki ,l = ail b jl ckl + akl bil c jl + ail bkl c jl +
Thus the
akl b jl cil + a jl bil ckl + a jl bkl cil (13)
condition (11) is a particular case of (13).
Let us consider the following matrices:
 a  
 1 1 1 +  
1 0 0 2 1 1+ a 1 
   
A =  0 1 0  , B = 1 − a 1 1 , C =  1 1 1 + a  , (18)
0 0 1  1 1− a 1   a 
     1 − 1 1 
   2 
where a  [ −1,1] .
Then corresponding SCSO W : S2 → S2 is:
   a 
 x1 = x1 ( x1 + (1 − a ) x2 + x3 )  x1 + x2 + 1 −  x3  ,
   2 

W :  x2 = x2 ( x1 + x2 + (1 − a ) x3 ) ( (1 + a ) x1 + x2 + x3 ) , (19)

 x = x  1 + a  x + x + x  ( x + (1 + a ) x + x ) .
3   1 2 3 
 3  2  
1 2 3
Using the equation x1 + x2 + x3 = 1 we rewrite the operator (19) as follows

  a 
 1 x  = x (1 − ax )
2 
1 − x3  ,
 2 
1

W :  x2 = x2 (1 − ax3 )(1 + ax1 ) , (20)

 x3 = x3 1 + a x1  (1 + ax2 ) .
  2 
Evidently, that if a =0 the SCSO (20) is the identity map. For this in the below, we consider the
case when a  0 .
Let a face of the simplex S2 be the set  = x  S 2 : xi = 0, i    1,2,3 .
Let the set int S 2 = x  S 2 : x1 x2 x3  0 and let the set S 2 = S 2 \ int S 2 be the interior
and the boundary of the simplex S2, respectively. Let e1 = (1,0,0 ) , e 2 = ( 0,1,0 ) ,
e3 = ( 0,0,1) be the vertexes of the two-dimensional simplex.
Theorem 1. For the SCSO W (20), the following assertions true:
(i) The face 1,2 , 1,3 , 2,3 of the simplex S2 are invariant sets;
(ii) Fix (W ) = e1 , e2 , e3  ;

(iii) If a   −1,0 ) , then e1 is an attracting point, e 2 is a saddle point and e 3 is a repelling point;
If a  ( 0,1 , then e1 is a repelling point, e 2 is a saddle point and e 3 is an attracting point.
Proof: (i) Obviously.
(ii) To find the fixed points we consider the equation W ( x ) = x , that is the following system of
equations
  a 
 1 x = x (1 − ax )
2 
1 − x3  ,
 2 
1

 x2 = x2 (1 − ax3 )(1 + ax1 ) , (21)

 x3 = x3 1 + a x1  (1 + ax2 ) .
  2 
(a) Let x1 = 0 . Then from the second equation of (21) it follows
x2 = x2 − ax2 x3  x2 = 0 and 1 = 1 − ax3 .
It is clear that if x2 = 0 then we have x3 = 1 . If x2  0 then from the last equation one has
x3 = 0  x2 = 1.
We take solution x2 = 1, then it follows that x3 = 0 . Consequently, if x1 = 0 , we obtain the fixed
points e 2 , e3 .
Similarly, in the case x2 = 0 and x3 = 0 , we have the fixed points e1 , e 3 and e1 , e 2 , respectively.
(b) Suppose that x1 x2 x3  0 . Then from the system (21), one has
  a 
 1 = (1 − ax )
2 
1 − x3  ,
  2 
1 = (1 − ax3 )(1 + ax1 ) , (22)

1 = 1 + a x1  (1 + ax2 ) .
  2 
If a   −1,0 ) , then from the first equation of (22) we get
 a  a 1
1 = (1 − ax2 )  1 − x3   x2 x3 = x2 + x3 .
 2  2 2
But from a   −1,0 ) and from the last equation it follows that x2 = 0 and x3 = 0 , this
contradicts to x1 x2 x3  0 .
If a  ( 0,1 , then from the third equation of (22) we have

 a  a 1 
1 = 1 + x1  (1 + ax2 )  x1 x2 = −  x1 + x2  .
 2  2 2 
But from a  ( 0,1 and from the last equation it follows that x1 = 0 and x2 = 0 , this contradicts
to x1 x2 x3  0 .
Consequently, we have that Fix (W )  int S 2 =  .
(iii) To find the type of fixed point of the SCSO (20), we rewrite it in the form
  a 
 x1 = x1 (1 − ax2 ) 1 − 2 (1 − x1 − x2 )  , (23)
  
 x = x (1 + ax ) (1 − a (1 − x − x ) ) .
 2 2 1 1 2
where ( x , x )  ( x, y ) : x, y  0, 0  x + y  1 and
1 2
x1 , x2 are the first two coordinates of
a point lying in the simplex S2.

a
The Jacobian of the operator (23) at a fixed point e1 has the following eigenvalues 1 = 1 + ,
2
2 = 1 + a , at the e 2 has the following eigenvalues 1 = 1 − a , 2 = 1 + a and at the e 3 has the
a
following eigenvalues 1 = 1 − a , 2 = 1 − . Therefore, it follows that:
2
Ifa   −1,0 ) then, the fixed point e1 is a attracting point, e 2 is a saddle fixed point and the fixed
point e 3 is a repelling point.
If a=0 then, fixed points e1 , e 2 and e 3 are non-hyperbolic points.

If a  ( 0,1 then, the fixed point e1 is a repelling point, e2 is a saddle fixed point and the fixed
point e 3 is a attracting point.
The theorem is proved.
References
1. S. Bernstein, Solution of a mathematical problem connected with the theory of heredity}, Ann.
Math. Stat. 13(1) (1942), pp. 53-61. DOI:10.1214/aoms/1177731642.
2. R. R. Davronov, U. U. Jamilov(Zhamilov), and M. Ladra, Conditional cubic stochastic operator, J.
Differ. Equ. Appl. 21(12) (2015), pp. 1163-1170. DOI:10.1080/10236198.2015.1062481.
3. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, in Studies in Nonlinearity,
Westview Press, Boulder, CO, 2003.
4. R. N. Ganikhodzhaev, Quadratic stochastic operators, Lyapunov functions and tournaments, Sb.
Math.76(2) (1993), pp. 489-506. DOI:10.1070/SM1993v076n02ABEH003423.
5. R. N. Ganikhodzhaev and D. B. Eshmamatova, Quadratic automorphisms of a simplex and the
asymptotic behavior of their trajectories, Vladikavkaz. Mat. Zh. 8(2) (2006), pp. 12-28.
6. A. Yu. Khamraev, On cubic operators of Volterra type, Uzbek. Mat. Zh. 2004(2) (2004), pp. 79-
84. (in Russian).
7. U. A. Rozikov, S. Nazir, Separable Quadratic Stochastic Operators, Lobschevskii J. Math. (3),
215(2010), pp. 215-221. DOI: 10.1134/S1995080210030030.
8. U. A. Rozikov, A. Zada, On a Class of Separable Quadratic Stochastic Operators, Lobschevskii J.
Math. (4), 32(2011), pp. 385-394. DOI: 10.1134/S1995080211040196.

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CENTRAL ASIAN JOURNAL OF MEDICAL AND NATURAL SCIENCES

Volume: 04 Issue: 06 | Nov-Dec 2023 ISSN: 2660-4159

Separable Cubic Stochastic Operators

and the coefficients Pij ,k satisfy

1040 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

The trajectory (orbit) x  ( n)

Preliminaries and known results

1041 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

and B that aik b jk  0 , AB = 1 , where BT

j -th row of the matrix B , then from ABT = 1 we get

where Pijk ,l are coefficients of heredity such that

For a given x( 0)  S m−1 , the trajectory x( n)   n0

defined by ( ) , where n = 0,1, 2,... with x = x . Denote by  ( x ) the set of

limit points of the trajectory x  . Since x   S

1042 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

 ( x( 0 ) ) is a fixed point of the operator W. A point x  S m−1 is called a fixed of the W if

Let DW ( x ) = ( Wi / x j ) ( x ) be a Jacobian of W at the point x .

Definition 2 ([3]): A fixed point x is called hyperbolic if its Jacobian DW ( x ) has no

Definition 3 ([3]): A hyperbolic fixed point x is called:

(iii) a saddle otherwise;

1043 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

Then corresponding SCSO W : S2 → S2 is:

Using the equation x1 + x2 + x3 = 1 we rewrite the operator (19) as follows

Let a face of the simplex S2 be the set  = x  S 2 : xi = 0, i    1,2,3 .

1044 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

(ii) Fix (W ) = e1 , e2 , e3  ;

1045 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

If a  ( 0,1 , then from the third equation of (22) we have

a point lying in the simplex S2.

If a=0 then, fixed points e1 , e 2 and e 3 are non-hyperbolic points.

The theorem is proved.

1046 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

1047 Published by “ CENTRAL ASIAN STUDIES" http://www.centralasianstudies.org

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