Proceedings of the 4th International Conference on Research and Education in Mathematics
BIRKHOFF'S THEOREM FOR TRIPLE STOCHASTIC MATRICES RELATED TO QUADRATIC STOCHASTIC OPERATORS 1 Mansoor 1 Department
Saburov, 2 Farruh Shahidi
of Computational and Theoretical Sciences, Faculty of Science,
International Islamic University Malaysia, 25200 Bandar Indera Mallkota, Kuantan, Pahang, MALAYSIA. 2 Department
of Mechanics and Mathematics,
National University of Uzbekistan, 100174, Vuzgorodok, Tashkent, UZBEKISTAN. email:
1
[email protected], 2 fa:
[email protected]
Abstract In this paper we introduce a. concept ofT- quadratic stochastic operator. We study geometrical structures of the set of such a class operators based on an analogues BirkhofPs theorem for triple stochastic cubic matrices. Keywords: Quadratic stochastic operator, triple stochastic cubic matrix.
1
Introduction
In recent years penetration of mathematical models in problems of biology is steadily increasing. In a simple case an investigation of an evolution of biological system reduces to studying of asymptotical behavior of trajectory of so-called quadratic stochastic operators. It is well known (see (1}) that an evolution operator of free population is described by a quadratic stochastic operator (in short QSO) v: sm-l ~ sm-l given by m
E
(Vx)k
k=lm
PijkXiXj,
'
i,j=l
(1.1)
here 771
Pijk
~ 0,
E
Pijk
= 1,
(1.2)
k=l
and the simplex m sm-l
=
X= (Xt,X2, ...
,x111) E JRffi: Exk = l,xk ~ 0 . k=l
From biological point of vie\v ( 1.1) and (1. 2) mean the following: assume that every individual in a population belongs to precisely one of the species 1, 2, · · · , m. Then Pij,k is a probability that individuals in the i-th and j-th species interbreed to produce an individual of k-th specie. It is obvious that Pii,k ~ 0 and
m
E Pii,k = 1. Assume that the population is so large that frequency fluctuations can be k=l
neglected. Then the species of the population can be described by the tuple x = (x1 , x 2 , ••. , xm) of species probabilities, that is, Xi is the fraction of the species i in the population. Thus, each element X E sm-l is a probability distribution on I= {1, 2, ... , m}. In the case of panmixia, the parent pairs i and j arise for a fixed state x = (x 1,x2, ... ,xm) with probability XiXj · Hence (1.1) is the total probability of the species k in the first generation of direct descendants. In this setting an evolution of the system is described by the QSO acting on the simplex. One of the main problems in mathematical biology consists of the study the asymptotical behavior of the trajectory of QSO. This problem was solved for a Volterra type QSO (see [2]) which is defined by (1.1), (1.2) with an additional assumption Pij,k = 0 if k f/ {i, j}. The Volterra type QSO has simple biological meaning: in each generation an individual can inherit only species of its parents. 478
Proceedings of the 4th International Conference on Research and Education in Mathematics In this paper we consider another class of QSO which is defined by triple stochastic cubic matrices. We study geometrical structures of the set of such a class operators by means of an analogues Birkhoff's theorem for triple stochastic cubic matrices.
2
Preliminaries
In this section we consider one class of QSO which is defined by triple stochastic cubic matrices and study its basic properties. Definition 2.1. A cubic matrix P = {Pijk}i,'i,k= 1 is said to be triple stochastic if the following conditions are satisfied: m
LPijk = 1, i=1
By
~(!;
m
1'11
LPijk = 1, j=1
LPiik k=1
= 1,
Piik
> 0,
Yi,j, k
= 1, m.
we denote the set all of triple stochastic cubic matrices.
Definition 2.2. An operatorV: sm- 1 --> sm- 1 given by (1.1) is calledT-quadratic stochastic operator (in short T-QSO) if the corresponding cubic matrix P = {Pijk}i,j,k= 1, is triple stochastic. By 'I'.Q we denote the set all of the T - QSO . Example 2.3. Let V : 8 2
-->
8 2 be given by
V:
(Vx)l = (Vx)z =
(Vxh
X~+ X~+ X~
+ X1X3 + XzX3 = X1X2 + X1X3 + X2X3
(2.3)
X1X2
0
One can see that the operator V given by (2.3) is T-QSO. Let Pk = {p;jk}iJ=l• for each k = l,m. Then QSO given by (1.1) has the following form: Vx
= ((P1x,x), (P2 x,x), · · ·, (Pmx,x)),
here (·, ·) stands for the usual scalar product on Rm. Further, we use the notations (P1IP2J· .. IPm) and (P1,P2, ... ,Pm) for the QSO and for the cubic matrix 'P, respectively. Let 'I
