Research ArticleOn Poisson Nonlinear Transformations

Nasir Ganikhodjaev and Nur Zatul Akmar Hamzah

Department of Computational andTheoretical Sciences Faculty of Science International Islamic University Malaysia25710 Kuantan Malaysia

Correspondence should be addressed to Nur Zatul Akmar Hamzah nz akmaryahoocom

Received 1 April 2014 Accepted 4 July 2014 Published 17 July 2014

Academic Editor Ishak Altun

Copyright copy 2014 N Ganikhodjaev and N Z A Hamzah This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We construct the family of Poisson nonlinear transformations defined on the countable sample space of nonnegative integers andinvestigate their trajectory behavior We have proved that these nonlinear transformations are regular

1 Introduction

Let (119883F) be a measurable space where 119883 is a state spaceandF is120590-algebra on119883 and 119878(119883F) the set of all probabilitymeasures on (119883F)

Let 119875(119909 119910 119860) 119909 119910 isin 119883119860 isin F be a family of functionson 119883 times 119883 times F such that for any fixed 119909 119910 isin 119883119875(119909 119910 sdot) isin

119878(119883F) 119875(119909 119910 119860) regarded as a function of two variables 119909and 119910 with fixed 119860 isin F is a measurable function on (119883 times

119883F otimes F) and 119875(119909 119910 119860) = 119875(119910 119909 119860) for any 119909 119910 isin 119883 and119860 isin F

We consider a nonlinear transformation called quadraticstochastic operator (qso) 119881 119878(119883F) rarr 119878(119883F) which isdefined by

(119881120582) (119860) = ∬119883

119875 (119909 119910 119860) 119889120582 (119909) 119889120582 (119910) (1)

where 119860 isin F is an arbitrary measurable setIf a state space 119883 = 1 2 119898 is a finite set and the

corresponding 120590-algebra is the power set P(119883) that is theset of all subsets of119883 then the set of all probability measureson (119883F) has the following form

119878119898minus1

= 119909 = (1199091 1199092 119909

119898) isin 119877119898

119909119894ge 0

for any 119894 and119898

sum119894=1

119909119894= 1

(2)

that is called a (119898 minus 1)-dimensional simplex

In this case the probabilisticmeasure119875(119894 119895 sdot) is a discretemeasure with sum

119898

119896=1119875(119894119895 119896) = 1 where 119875(119894119895 119896) equiv 119875

119894119895119896for

any 119894 119895 isin 119883 In addition the corresponding qso V has thefollowing form

(119881119909)119896 =

119898

sum119894119895=1

119875119894119895119896

119909119894119909119895 (3)

for any 119909 isin 119878119898minus1 and the coefficients 119875119894119895119896

satisfy the followingconditions

(a) 119875119894119895119896

ge 0

(b) 119875119894119895119896

= 119875119895119894119896

(c)119898

sum119896=1

119875119894119895119896

= 1 forall 119894 119895 119896 isin 1 2 119898

(4)

Such operator can be reinterpreted in terms of evolutionaryoperator of free population [1ndash10] and in this form it has a fairhistory

In this paper we consider nonlinear transformationsdefined on countable state space and investigate their limitbehavior of trajectories

2 A Poisson qso

Let 119883 = 0 1 be a countable sample space and corre-sponding 120590-algebraF a power setP(119883) that is the set of allsubsets of 119883 In order to define a probability measure 120583 on

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 832861 7 pageshttpdxdoiorg1011552014832861

2 The Scientific World Journal

countable sample space119883 it is enough to define the measure120583(119896) of each singleton 119896 119896 = 0 1 Thus we will write120583(119896) instead of 120583(119896)

Let 119875(119894 119895 119896) 119894 119895 119896 isin 119883 be a family of functions definedon119883 times 119883 timesF which satisfy the following conditions

(i) 119875(119894 119895 sdot) is a probability measure on (119883F) for anyfixed 119894 119895 isin 119883

(ii) 119875(119894 119895 119896) = 119875(119895 119894 119896) equiv 119875119894119895119896

where 119896 isin 119883 for any fixed119894 119895 isin 119883

In this case a qso (1) on measurable space (119883F) isdefined as follows

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895) (5)

where 119896 isin 119883 for arbitrary measure 120583 isin 119878(119883F)In this paper we consider a Poisson qsowhich is a Poisson

distribution 119875120582with a positive real parameter 120582 defined on119883

by the equation

119875120582 (119896) = 119890

minus120582 120582119896

119896 (6)

for any 119896 isin 119883Let 119878(119883F) be a set of all probability measures on (119883F)

and let 119875(119894 119895 sdot) be a probability measure on (119883F) for any119894 119895 isin 119883

Definition 1 A quadratic stochastic operator 119881 (5) is calleda Poisson qso if for any 119894 119895 isin 119883 the probability measure119875(119894 119895 sdot) is the Poisson distribution 119875

120582(119894119895)with positive real

parameters 120582(119894 119895) where 120582(119894 119895) = 120582(119895 119894)

Assume that 119881119899120582 119899 = 0 1 2 is the trajectory ofthe initial point 120582 isin 119878(119883 119865) where 119881119899+1120582 = 119881(119881119899120582) for all119899 = 0 1 2 with 1198810120582 = 120582

In this paper we will study limit behavior of trajectoriesof Poisson qso

3 Ergodicity and Regularity of qso

Let us consider a qso 119881 (5) defined on countable set 119883 Let119881119899120582 119899 = 0 1 2 be the trajectory of the initial point120582 isin 119878(119883F) where 119881119899+1120582 = 119881(119881119899120582) for all 119899 = 0 1 2

Definition 2 Ameasure 120583 isin 119878(119883F) is called a fixed point ofa qso 119881 if 119881120583 = 120583

Let Fix(119881) be the set of all fixed points of qso 119881

Definition 3 A qso 119881 is called regular if for any initial point120583 isin 119878(119883F) the limit

lim119899rarrinfin

119881119899(120583) (7)

exists

In measure theory there are various notions of theconvergence of measures weak convergence strong conver-gence and total variation convergence Below we considerstrong convergence

Definition 4 For (119883F) a measurable space a sequence 120583119899is

said to converge strongly to a limit 120583 if

lim119899rarrinfin

120583119899 (119860) = 120583 (119860) (8)

for every set 119860 isin F

If 119883 is a countable set then a sequence 120583119899converges

strongly to a limit 120583 if and only if

lim119899rarrinfin

120583119899 (119896) = 120583 (119896) (9)

for every singleton 119896 isin 119883In statistical mechanics the ergodic hypothesis proposes

a connection between dynamics and statistics In the classicaltheory the assumption was made that the average time spentin any region of phase space is proportional to the volume ofthe region in terms of the invariant measure More generallysuch time averages may be replaced by space averages

For nonlinear dynamical systems Ulam [11] suggestedan analogous measure-theoretic ergodicity with followingergodic hypothesis

Definition 5 A nonlinear operator 119881 defined on 119878(119883F) iscalled ergodic if the limit

lim119899rarrinfin

1

119899

119899minus1

sum119896=0

119881119896120582 (10)

exists for any 120582 isin 119878(119883 119865)

On the ground of numerical calculations for quadraticstochastic operators defined on 119878(119883 119865) with finite 119883 Ulam[11] conjectured that the ergodic theorem holds for any suchqso 119881

In 1977 Zakharevich [12] proved that this conjecture isfalse in general He considered the following operator on 1198782

1199091015840

1= 1199092

1+ 211990911199092

1199091015840

2= 1199092

2+ 211990921199093

1199091015840

3= 1199092

3+ 211990911199093

(11)

and he proved that such operator is nonergodic transforma-tion Later in [13] the sufficient condition to be nonergodictransformation was established for qso defined on 119878

2In the next section we will show that Ulamrsquos conjecture is

true for some class of Poisson qso

4 Ergodicity and Regularity of Poisson qso

Let 119881 defined in (5) be a Poisson qso We consider thefollowing cases

41 Homogenious Poisson qso We call a Poisson quadraticstochastic operator 119881 (5) homogenious if 120582(119894 119895) = 120582 for any

The Scientific World Journal 3

119894 119895 isin 119883 that is 119875119894119895119896

= 119890minus120582(120582119896119896) Then for arbitrary measure120583 isin 119878(119883F)

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895) = 119890minus120582 120582119896

119896 (12)

where 119896 isin 119883 that is 119881120583 = 119875120582

Thus 119881119899120583 = 119875120582for any 119899 = 1 2 that is Fix(119881) = 119875

120582

and we have the following statement

Proposition 6 A homogenious Poisson qso is a regulartransformation

42 Poisson qso with Two Different Parameters We considera Poisson qso such that

119875119894119895119896

=

119890minus1205821

1205821198961

119896if 119894 + 119895 is even

119890minus1205822

1205821198962

119896if 119894 + 119895 is odd

(13)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (2119899) 119861 (120583) =

infin

sum119899=0

120583 (2119899 + 1) (14)

where 119860(120583) + 119861(120583) = 1 It is easy to show that for Poissondistribution 119875

120582

119860 (119875120582) =

1 + 119890minus2120582

2 119861 (119875

120582) =

1 minus 119890minus2120582

2 (15)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

120583 (2119898) 120583 (2119899)

+1198752119898+12119899+1119896

120583 (2119898 + 1) 120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

120583 (2119898 + 1) 120583 (2119899)

+11987521198982119899+1119896

120583 (2119898) 120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 119861

2(120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

119881120583 (2119898)119881120583 (2119899)

+1198752119898+12119899+1119896

119881120583 (2119898 + 1)119881120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

119881120583 (2119898 + 1)119881120583 (2119899)

+11987521198982119899+1119896

119881120583 (2119898)119881120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861

2(119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583)]

(16)

By simple calculations we have

119860 (119881120583) =1 + 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 + 119890minus21205822

2[2119860 (120583) 119861 (120583)]

119861 (119881120583) =1 minus 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 minus 119890minus21205822

2[2119860 (120583) 119861 (120583)]

(17)

Thus by using induction on the sequence 119881119899(120583) weproduce the following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(18)

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) and119861(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) =1 + 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 + 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

119861 (119881119899+1

120583) =1 minus 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 minus 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(19)

It is obvious that the limit behavior of the recurrent equa-tion (18) is fully determined by limit behavior of recurrentequations (19)

4 The Scientific World Journal

1

08

06

04

02

0

0 02 04 06 08 1

x

(a) When 1205821 = 08 and 1205822 = 04

1

08

06

04

02

0

0 02 04 06 08 1

x

(b) When 1205821 = 04 and 1205822 = 14

Figure 1 Graph of the function (21) for some fixed values 1205821and 120582

2

Since 119860(119881119899120583) + 119861(119881119899120583) = 1 where 119860(119881119899120583) ge 0 and119861(119881119899120583) ge 0 the recurrent equations (19) are rewritten asfollows

1199091015840= 119860 (120582

1) (1199092+ 1199102) + 2119860 (120582

2) 119909119910

1199101015840= 119861 (120582

1) (1199092+ 1199102) + 2119861 (120582

2) 119909119910

(20)

with 119909 ge 0 119910 ge 0 and 119909 + 119910 = 1Solving the following quadratic equation

119909 = 119860 (1205821) (1199092+ (1 minus 119909)

2) + 2119860 (120582

2) 119909 (1 minus 119909) (21)

we have single fixed point and denoted it as (119909lowast 119910lowast) (seeFigure 1) Using simple calculus (see Figure 1) one canshow that any trajectory of the qso (20) defined on one-dimensional simplex 1198781 converges to this fixed point that isqso (20) is regular transformation so that it is ergodic

Thus for any initial measure 120583 we have

lim119899rarrinfin

119860 (119881119899120583) = 119909

lowast lim

119899rarrinfin119861 (119881119899120583) = 119910

lowast (22)

Then passing to limit in (18) for any singleton 119896 we have

lim119899rarrinfin

119881119899+1

120583 (119896)

= lim119899rarrinfin

119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

= 119890minus1205821

1205821198961

119896[119909lowast2

+ 119910lowast2

] + 119890minus1205822

1205821198962

119896[2119909lowast119910lowast]

= [119909lowast2

+ 119910lowast2

] 1198751205821(119896) + [2119909

lowast119910lowast] 1198751205822(119896)

(23)

Thus for any initial measure 120583 the strong limit ofthe sequence 119881

119899120583 exists and is equal to the convex linearcombination

lim119899rarrinfin

119881119899120583 (119896) = (119909

lowast2+ 119910lowast2

) 1198751205821(119896) + 2119909

lowast119910lowast1198751205822(119896)

(24)

of two Poisson measures 1198751205821

and 1198751205822

It is evident thatFix(119881) = (119909lowast

2+ 119910lowast2)1198751205821

(119896) + 2119909lowast119910lowast1198751205822

(119896)As corollary we have following statement

Proposition 7 A Poisson qso with two different parameters isa regular and respectively ergodic transformation with respectto strong convergence

43 A Poisson qso with Three Different Parameters We con-sider a Poisson qso such that

119875119894119895119896

=

119890minus1205821

120582119896

1

119896if 119894 + 119895 = 0 (mod 3)

119890minus1205822

1205821198962

119896if 119894 + 119895 = 1 (mod 3)

119890minus12058231205821198963

119896if 119894 + 119895 = 2 (mod 3)

(25)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (3119899) 119861 (120583) =

infin

sum119899=0

120583 (3119899 + 1)

119862 (120583) =

infin

sum119899=0

120583 (3119899 + 2)

(26)

The Scientific World Journal 5

where 119860(120583) + 119861(120583) + 119862(120583) = 1 It is easy to show that forPoisson distribution 119875

120582with parameter 120582 we have

119860 (120582) =1 + 2119890minus(32)120582 cos (radic32) 120582

3

119861 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 + 1205873)

3

119862 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 minus 1205873)

3

(27)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

120583 (3119898) 120583 (3119899)

+ 1198753119898+13119899+2119896

120583 (3119898 + 1) 120583 (3119899 + 2)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

120583 (3119898 + 1) 120583 (3119899)

+ 11987531198983119899+1119896

120583 (3119898) 120583 (3119899 + 1)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

120583 (3119898 + 2) 120583 (3119899)

+ 11987531198983119899+2119896

120583 (3119898) 120583 (3119899 + 2)

+1198753119898+13119899+1119896

120583 (3119898 + 1) 120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (120583) 119862 (120583) + 119861

2(120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

119881120583 (3119898)119881120583 (3119899)

+ 1198753119898+13119899+2119896

119881120583 (3119898 + 1)119881120583 (3119899 + 2)

+1198753119898+23119899+1119896

119881120583 (3119898 + 2)119881120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

119881120583 (3119898 + 1)119881120583 (3119899)

+ 11987531198983119899+1119896

119881120583 (3119898)119881120583 (3119899 + 1)

+1198753119898+23119899+2119896

119881120583 (3119898 + 2)119881120583 (3119899 + 2)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

119881120583 (3119898 + 2)119881120583 (3119899)

+ 11987531198983119899+2119896

119881120583 (3119898)119881120583 (3119899 + 2)

+1198753119898+13119899+1119896

119881120583 (3119898 + 1)119881120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861 (119881120583)119862 (119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583) + 119862

2(119881120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881120583)119862 (119881120583) + 119861

2(119881120583)]

(28)

By simple calculations we have

119860 (119881120583) = 119860 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119860 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119860 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119861 (119881120583) = 119861 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119861 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119861 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119862 (119881120583) = 119862 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119862 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119862 (1205823) [2119860 (120583) 119862 (120583) + 119861

2(120583)]

(29)

Thus by using induction on sequence 119881119899(120583) we producethe following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(30)

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

countable sample space119883 it is enough to define the measure120583(119896) of each singleton 119896 119896 = 0 1 Thus we will write120583(119896) instead of 120583(119896)

Let 119875(119894 119895 119896) 119894 119895 119896 isin 119883 be a family of functions definedon119883 times 119883 timesF which satisfy the following conditions

(i) 119875(119894 119895 sdot) is a probability measure on (119883F) for anyfixed 119894 119895 isin 119883

(ii) 119875(119894 119895 119896) = 119875(119895 119894 119896) equiv 119875119894119895119896

where 119896 isin 119883 for any fixed119894 119895 isin 119883

In this case a qso (1) on measurable space (119883F) isdefined as follows

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895) (5)

where 119896 isin 119883 for arbitrary measure 120583 isin 119878(119883F)In this paper we consider a Poisson qsowhich is a Poisson

distribution 119875120582with a positive real parameter 120582 defined on119883

by the equation

119875120582 (119896) = 119890

minus120582 120582119896

119896 (6)

for any 119896 isin 119883Let 119878(119883F) be a set of all probability measures on (119883F)

and let 119875(119894 119895 sdot) be a probability measure on (119883F) for any119894 119895 isin 119883

Definition 1 A quadratic stochastic operator 119881 (5) is calleda Poisson qso if for any 119894 119895 isin 119883 the probability measure119875(119894 119895 sdot) is the Poisson distribution 119875

120582(119894119895)with positive real

parameters 120582(119894 119895) where 120582(119894 119895) = 120582(119895 119894)

Assume that 119881119899120582 119899 = 0 1 2 is the trajectory ofthe initial point 120582 isin 119878(119883 119865) where 119881119899+1120582 = 119881(119881119899120582) for all119899 = 0 1 2 with 1198810120582 = 120582

In this paper we will study limit behavior of trajectoriesof Poisson qso

3 Ergodicity and Regularity of qso

Let us consider a qso 119881 (5) defined on countable set 119883 Let119881119899120582 119899 = 0 1 2 be the trajectory of the initial point120582 isin 119878(119883F) where 119881119899+1120582 = 119881(119881119899120582) for all 119899 = 0 1 2

Definition 2 Ameasure 120583 isin 119878(119883F) is called a fixed point ofa qso 119881 if 119881120583 = 120583

Let Fix(119881) be the set of all fixed points of qso 119881

Definition 3 A qso 119881 is called regular if for any initial point120583 isin 119878(119883F) the limit

lim119899rarrinfin

119881119899(120583) (7)

exists

In measure theory there are various notions of theconvergence of measures weak convergence strong conver-gence and total variation convergence Below we considerstrong convergence

Definition 4 For (119883F) a measurable space a sequence 120583119899is

said to converge strongly to a limit 120583 if

lim119899rarrinfin

120583119899 (119860) = 120583 (119860) (8)

for every set 119860 isin F

If 119883 is a countable set then a sequence 120583119899converges

strongly to a limit 120583 if and only if

lim119899rarrinfin

120583119899 (119896) = 120583 (119896) (9)

for every singleton 119896 isin 119883In statistical mechanics the ergodic hypothesis proposes

a connection between dynamics and statistics In the classicaltheory the assumption was made that the average time spentin any region of phase space is proportional to the volume ofthe region in terms of the invariant measure More generallysuch time averages may be replaced by space averages

For nonlinear dynamical systems Ulam [11] suggestedan analogous measure-theoretic ergodicity with followingergodic hypothesis

Definition 5 A nonlinear operator 119881 defined on 119878(119883F) iscalled ergodic if the limit

lim119899rarrinfin

1

119899

119899minus1

sum119896=0

119881119896120582 (10)

exists for any 120582 isin 119878(119883 119865)

On the ground of numerical calculations for quadraticstochastic operators defined on 119878(119883 119865) with finite 119883 Ulam[11] conjectured that the ergodic theorem holds for any suchqso 119881

In 1977 Zakharevich [12] proved that this conjecture isfalse in general He considered the following operator on 1198782

1199091015840

1= 1199092

1+ 211990911199092

1199091015840

2= 1199092

2+ 211990921199093

1199091015840

3= 1199092

3+ 211990911199093

(11)

and he proved that such operator is nonergodic transforma-tion Later in [13] the sufficient condition to be nonergodictransformation was established for qso defined on 119878

2In the next section we will show that Ulamrsquos conjecture is

true for some class of Poisson qso

4 Ergodicity and Regularity of Poisson qso

Let 119881 defined in (5) be a Poisson qso We consider thefollowing cases

41 Homogenious Poisson qso We call a Poisson quadraticstochastic operator 119881 (5) homogenious if 120582(119894 119895) = 120582 for any

The Scientific World Journal 3

119894 119895 isin 119883 that is 119875119894119895119896

= 119890minus120582(120582119896119896) Then for arbitrary measure120583 isin 119878(119883F)

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895) = 119890minus120582 120582119896

119896 (12)

where 119896 isin 119883 that is 119881120583 = 119875120582

Thus 119881119899120583 = 119875120582for any 119899 = 1 2 that is Fix(119881) = 119875

120582

and we have the following statement

Proposition 6 A homogenious Poisson qso is a regulartransformation

42 Poisson qso with Two Different Parameters We considera Poisson qso such that

119875119894119895119896

=

119890minus1205821

1205821198961

119896if 119894 + 119895 is even

119890minus1205822

1205821198962

119896if 119894 + 119895 is odd

(13)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (2119899) 119861 (120583) =

infin

sum119899=0

120583 (2119899 + 1) (14)

where 119860(120583) + 119861(120583) = 1 It is easy to show that for Poissondistribution 119875

120582

119860 (119875120582) =

1 + 119890minus2120582

2 119861 (119875

120582) =

1 minus 119890minus2120582

2 (15)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

120583 (2119898) 120583 (2119899)

+1198752119898+12119899+1119896

120583 (2119898 + 1) 120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

120583 (2119898 + 1) 120583 (2119899)

+11987521198982119899+1119896

120583 (2119898) 120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 119861

2(120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

119881120583 (2119898)119881120583 (2119899)

+1198752119898+12119899+1119896

119881120583 (2119898 + 1)119881120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

119881120583 (2119898 + 1)119881120583 (2119899)

+11987521198982119899+1119896

119881120583 (2119898)119881120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861

2(119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583)]

(16)

By simple calculations we have

119860 (119881120583) =1 + 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 + 119890minus21205822

2[2119860 (120583) 119861 (120583)]

119861 (119881120583) =1 minus 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 minus 119890minus21205822

2[2119860 (120583) 119861 (120583)]

(17)

Thus by using induction on the sequence 119881119899(120583) weproduce the following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(18)

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) and119861(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) =1 + 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 + 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

119861 (119881119899+1

120583) =1 minus 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 minus 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(19)

It is obvious that the limit behavior of the recurrent equa-tion (18) is fully determined by limit behavior of recurrentequations (19)

4 The Scientific World Journal

1

08

06

04

02

0

0 02 04 06 08 1

x

(a) When 1205821 = 08 and 1205822 = 04

1

08

06

04

02

0

0 02 04 06 08 1

x

(b) When 1205821 = 04 and 1205822 = 14

Figure 1 Graph of the function (21) for some fixed values 1205821and 120582

2

Since 119860(119881119899120583) + 119861(119881119899120583) = 1 where 119860(119881119899120583) ge 0 and119861(119881119899120583) ge 0 the recurrent equations (19) are rewritten asfollows

1199091015840= 119860 (120582

1) (1199092+ 1199102) + 2119860 (120582

2) 119909119910

1199101015840= 119861 (120582

1) (1199092+ 1199102) + 2119861 (120582

2) 119909119910

(20)

with 119909 ge 0 119910 ge 0 and 119909 + 119910 = 1Solving the following quadratic equation

119909 = 119860 (1205821) (1199092+ (1 minus 119909)

2) + 2119860 (120582

2) 119909 (1 minus 119909) (21)

we have single fixed point and denoted it as (119909lowast 119910lowast) (seeFigure 1) Using simple calculus (see Figure 1) one canshow that any trajectory of the qso (20) defined on one-dimensional simplex 1198781 converges to this fixed point that isqso (20) is regular transformation so that it is ergodic

Thus for any initial measure 120583 we have

lim119899rarrinfin

119860 (119881119899120583) = 119909

lowast lim

119899rarrinfin119861 (119881119899120583) = 119910

lowast (22)

Then passing to limit in (18) for any singleton 119896 we have

lim119899rarrinfin

119881119899+1

120583 (119896)

= lim119899rarrinfin

119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

= 119890minus1205821

1205821198961

119896[119909lowast2

+ 119910lowast2

] + 119890minus1205822

1205821198962

119896[2119909lowast119910lowast]

= [119909lowast2

+ 119910lowast2

] 1198751205821(119896) + [2119909

lowast119910lowast] 1198751205822(119896)

(23)

Thus for any initial measure 120583 the strong limit ofthe sequence 119881

119899120583 exists and is equal to the convex linearcombination

lim119899rarrinfin

119881119899120583 (119896) = (119909

lowast2+ 119910lowast2

) 1198751205821(119896) + 2119909

lowast119910lowast1198751205822(119896)

(24)

of two Poisson measures 1198751205821

and 1198751205822

It is evident thatFix(119881) = (119909lowast

2+ 119910lowast2)1198751205821

(119896) + 2119909lowast119910lowast1198751205822

(119896)As corollary we have following statement

Proposition 7 A Poisson qso with two different parameters isa regular and respectively ergodic transformation with respectto strong convergence

43 A Poisson qso with Three Different Parameters We con-sider a Poisson qso such that

119875119894119895119896

=

119890minus1205821

120582119896

1

119896if 119894 + 119895 = 0 (mod 3)

119890minus1205822

1205821198962

119896if 119894 + 119895 = 1 (mod 3)

119890minus12058231205821198963

119896if 119894 + 119895 = 2 (mod 3)

(25)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (3119899) 119861 (120583) =

infin

sum119899=0

120583 (3119899 + 1)

119862 (120583) =

infin

sum119899=0

120583 (3119899 + 2)

(26)

The Scientific World Journal 5

where 119860(120583) + 119861(120583) + 119862(120583) = 1 It is easy to show that forPoisson distribution 119875

120582with parameter 120582 we have

119860 (120582) =1 + 2119890minus(32)120582 cos (radic32) 120582

3

119861 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 + 1205873)

3

119862 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 minus 1205873)

3

(27)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

120583 (3119898) 120583 (3119899)

+ 1198753119898+13119899+2119896

120583 (3119898 + 1) 120583 (3119899 + 2)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

120583 (3119898 + 1) 120583 (3119899)

+ 11987531198983119899+1119896

120583 (3119898) 120583 (3119899 + 1)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

120583 (3119898 + 2) 120583 (3119899)

+ 11987531198983119899+2119896

120583 (3119898) 120583 (3119899 + 2)

+1198753119898+13119899+1119896

120583 (3119898 + 1) 120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (120583) 119862 (120583) + 119861

2(120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

119881120583 (3119898)119881120583 (3119899)

+ 1198753119898+13119899+2119896

119881120583 (3119898 + 1)119881120583 (3119899 + 2)

+1198753119898+23119899+1119896

119881120583 (3119898 + 2)119881120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

119881120583 (3119898 + 1)119881120583 (3119899)

+ 11987531198983119899+1119896

119881120583 (3119898)119881120583 (3119899 + 1)

+1198753119898+23119899+2119896

119881120583 (3119898 + 2)119881120583 (3119899 + 2)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

119881120583 (3119898 + 2)119881120583 (3119899)

+ 11987531198983119899+2119896

119881120583 (3119898)119881120583 (3119899 + 2)

+1198753119898+13119899+1119896

119881120583 (3119898 + 1)119881120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861 (119881120583)119862 (119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583) + 119862

2(119881120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881120583)119862 (119881120583) + 119861

2(119881120583)]

(28)

By simple calculations we have

119860 (119881120583) = 119860 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119860 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119860 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119861 (119881120583) = 119861 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119861 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119861 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119862 (119881120583) = 119862 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119862 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119862 (1205823) [2119860 (120583) 119862 (120583) + 119861

2(120583)]

(29)

Thus by using induction on sequence 119881119899(120583) we producethe following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(30)

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

119894 119895 isin 119883 that is 119875119894119895119896

= 119890minus120582(120582119896119896) Then for arbitrary measure120583 isin 119878(119883F)

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895) = 119890minus120582 120582119896

119896 (12)

where 119896 isin 119883 that is 119881120583 = 119875120582

Thus 119881119899120583 = 119875120582for any 119899 = 1 2 that is Fix(119881) = 119875

120582

and we have the following statement

Proposition 6 A homogenious Poisson qso is a regulartransformation

42 Poisson qso with Two Different Parameters We considera Poisson qso such that

119875119894119895119896

=

119890minus1205821

1205821198961

119896if 119894 + 119895 is even

119890minus1205822

1205821198962

119896if 119894 + 119895 is odd

(13)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (2119899) 119861 (120583) =

infin

sum119899=0

120583 (2119899 + 1) (14)

where 119860(120583) + 119861(120583) = 1 It is easy to show that for Poissondistribution 119875

120582

119860 (119875120582) =

1 + 119890minus2120582

2 119861 (119875

120582) =

1 minus 119890minus2120582

2 (15)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

120583 (2119898) 120583 (2119899)

+1198752119898+12119899+1119896

120583 (2119898 + 1) 120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

120583 (2119898 + 1) 120583 (2119899)

+11987521198982119899+1119896

120583 (2119898) 120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 119861

2(120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987521198982119899119896

119881120583 (2119898)119881120583 (2119899)

+1198752119898+12119899+1119896

119881120583 (2119898 + 1)119881120583 (2119899 + 1)]

+

infin

sum119898119899=0

[1198752119898+12119899119896

119881120583 (2119898 + 1)119881120583 (2119899)

+11987521198982119899+1119896

119881120583 (2119898)119881120583 (2119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861

2(119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583)]

(16)

By simple calculations we have

119860 (119881120583) =1 + 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 + 119890minus21205822

2[2119860 (120583) 119861 (120583)]

119861 (119881120583) =1 minus 119890minus21205821

2[1198602(120583) + 119861

2(120583)]

+1 minus 119890minus21205822

2[2119860 (120583) 119861 (120583)]

(17)

Thus by using induction on the sequence 119881119899(120583) weproduce the following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(18)

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) and119861(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) =1 + 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 + 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

119861 (119881119899+1

120583) =1 minus 119890minus21205821

2[1198602(119881119899120583) + 119861

2(119881119899120583)]

+1 minus 119890minus21205822

2[2119860 (119881

119899120583) 119861 (119881

119899120583)]

(19)

It is obvious that the limit behavior of the recurrent equa-tion (18) is fully determined by limit behavior of recurrentequations (19)

4 The Scientific World Journal

1

08

06

04

02

0

0 02 04 06 08 1

x

(a) When 1205821 = 08 and 1205822 = 04

1

08

06

04

02

0

0 02 04 06 08 1

x

(b) When 1205821 = 04 and 1205822 = 14

Figure 1 Graph of the function (21) for some fixed values 1205821and 120582

2

Since 119860(119881119899120583) + 119861(119881119899120583) = 1 where 119860(119881119899120583) ge 0 and119861(119881119899120583) ge 0 the recurrent equations (19) are rewritten asfollows

1199091015840= 119860 (120582

1) (1199092+ 1199102) + 2119860 (120582

2) 119909119910

1199101015840= 119861 (120582

1) (1199092+ 1199102) + 2119861 (120582

2) 119909119910

(20)

with 119909 ge 0 119910 ge 0 and 119909 + 119910 = 1Solving the following quadratic equation

119909 = 119860 (1205821) (1199092+ (1 minus 119909)

2) + 2119860 (120582

2) 119909 (1 minus 119909) (21)

we have single fixed point and denoted it as (119909lowast 119910lowast) (seeFigure 1) Using simple calculus (see Figure 1) one canshow that any trajectory of the qso (20) defined on one-dimensional simplex 1198781 converges to this fixed point that isqso (20) is regular transformation so that it is ergodic

Thus for any initial measure 120583 we have

lim119899rarrinfin

119860 (119881119899120583) = 119909

lowast lim

119899rarrinfin119861 (119881119899120583) = 119910

lowast (22)

Then passing to limit in (18) for any singleton 119896 we have

lim119899rarrinfin

119881119899+1

120583 (119896)

= lim119899rarrinfin

119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

= 119890minus1205821

1205821198961

119896[119909lowast2

+ 119910lowast2

] + 119890minus1205822

1205821198962

119896[2119909lowast119910lowast]

= [119909lowast2

+ 119910lowast2

] 1198751205821(119896) + [2119909

lowast119910lowast] 1198751205822(119896)

(23)

Thus for any initial measure 120583 the strong limit ofthe sequence 119881

119899120583 exists and is equal to the convex linearcombination

lim119899rarrinfin

119881119899120583 (119896) = (119909

lowast2+ 119910lowast2

) 1198751205821(119896) + 2119909

lowast119910lowast1198751205822(119896)

(24)

of two Poisson measures 1198751205821

and 1198751205822

It is evident thatFix(119881) = (119909lowast

2+ 119910lowast2)1198751205821

(119896) + 2119909lowast119910lowast1198751205822

(119896)As corollary we have following statement

Proposition 7 A Poisson qso with two different parameters isa regular and respectively ergodic transformation with respectto strong convergence

43 A Poisson qso with Three Different Parameters We con-sider a Poisson qso such that

119875119894119895119896

=

119890minus1205821

120582119896

1

119896if 119894 + 119895 = 0 (mod 3)

119890minus1205822

1205821198962

119896if 119894 + 119895 = 1 (mod 3)

119890minus12058231205821198963

119896if 119894 + 119895 = 2 (mod 3)

(25)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (3119899) 119861 (120583) =

infin

sum119899=0

120583 (3119899 + 1)

119862 (120583) =

infin

sum119899=0

120583 (3119899 + 2)

(26)

The Scientific World Journal 5

where 119860(120583) + 119861(120583) + 119862(120583) = 1 It is easy to show that forPoisson distribution 119875

120582with parameter 120582 we have

119860 (120582) =1 + 2119890minus(32)120582 cos (radic32) 120582

3

119861 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 + 1205873)

3

119862 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 minus 1205873)

3

(27)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

120583 (3119898) 120583 (3119899)

+ 1198753119898+13119899+2119896

120583 (3119898 + 1) 120583 (3119899 + 2)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

120583 (3119898 + 1) 120583 (3119899)

+ 11987531198983119899+1119896

120583 (3119898) 120583 (3119899 + 1)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

120583 (3119898 + 2) 120583 (3119899)

+ 11987531198983119899+2119896

120583 (3119898) 120583 (3119899 + 2)

+1198753119898+13119899+1119896

120583 (3119898 + 1) 120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (120583) 119862 (120583) + 119861

2(120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

119881120583 (3119898)119881120583 (3119899)

+ 1198753119898+13119899+2119896

119881120583 (3119898 + 1)119881120583 (3119899 + 2)

+1198753119898+23119899+1119896

119881120583 (3119898 + 2)119881120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

119881120583 (3119898 + 1)119881120583 (3119899)

+ 11987531198983119899+1119896

119881120583 (3119898)119881120583 (3119899 + 1)

+1198753119898+23119899+2119896

119881120583 (3119898 + 2)119881120583 (3119899 + 2)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

119881120583 (3119898 + 2)119881120583 (3119899)

+ 11987531198983119899+2119896

119881120583 (3119898)119881120583 (3119899 + 2)

+1198753119898+13119899+1119896

119881120583 (3119898 + 1)119881120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861 (119881120583)119862 (119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583) + 119862

2(119881120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881120583)119862 (119881120583) + 119861

2(119881120583)]

(28)

By simple calculations we have

119860 (119881120583) = 119860 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119860 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119860 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119861 (119881120583) = 119861 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119861 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119861 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119862 (119881120583) = 119862 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119862 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119862 (1205823) [2119860 (120583) 119862 (120583) + 119861

2(120583)]

(29)

Thus by using induction on sequence 119881119899(120583) we producethe following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(30)

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

1

08

06

04

02

0

0 02 04 06 08 1

x

(a) When 1205821 = 08 and 1205822 = 04

1

08

06

04

02

0

0 02 04 06 08 1

x

(b) When 1205821 = 04 and 1205822 = 14

Figure 1 Graph of the function (21) for some fixed values 1205821and 120582

2

Since 119860(119881119899120583) + 119861(119881119899120583) = 1 where 119860(119881119899120583) ge 0 and119861(119881119899120583) ge 0 the recurrent equations (19) are rewritten asfollows

1199091015840= 119860 (120582

1) (1199092+ 1199102) + 2119860 (120582

2) 119909119910

1199101015840= 119861 (120582

1) (1199092+ 1199102) + 2119861 (120582

2) 119909119910

(20)

with 119909 ge 0 119910 ge 0 and 119909 + 119910 = 1Solving the following quadratic equation

119909 = 119860 (1205821) (1199092+ (1 minus 119909)

2) + 2119860 (120582

2) 119909 (1 minus 119909) (21)

we have single fixed point and denoted it as (119909lowast 119910lowast) (seeFigure 1) Using simple calculus (see Figure 1) one canshow that any trajectory of the qso (20) defined on one-dimensional simplex 1198781 converges to this fixed point that isqso (20) is regular transformation so that it is ergodic

Thus for any initial measure 120583 we have

lim119899rarrinfin

119860 (119881119899120583) = 119909

lowast lim

119899rarrinfin119861 (119881119899120583) = 119910

lowast (22)

Then passing to limit in (18) for any singleton 119896 we have

lim119899rarrinfin

119881119899+1

120583 (119896)

= lim119899rarrinfin

119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 119861

2(119881119899120583)]

+119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583)]

= 119890minus1205821

1205821198961

119896[119909lowast2

+ 119910lowast2

] + 119890minus1205822

1205821198962

119896[2119909lowast119910lowast]

= [119909lowast2

+ 119910lowast2

] 1198751205821(119896) + [2119909

lowast119910lowast] 1198751205822(119896)

(23)

Thus for any initial measure 120583 the strong limit ofthe sequence 119881

119899120583 exists and is equal to the convex linearcombination

lim119899rarrinfin

119881119899120583 (119896) = (119909

lowast2+ 119910lowast2

) 1198751205821(119896) + 2119909

lowast119910lowast1198751205822(119896)

(24)

of two Poisson measures 1198751205821

and 1198751205822

It is evident thatFix(119881) = (119909lowast

2+ 119910lowast2)1198751205821

(119896) + 2119909lowast119910lowast1198751205822

(119896)As corollary we have following statement

Proposition 7 A Poisson qso with two different parameters isa regular and respectively ergodic transformation with respectto strong convergence

43 A Poisson qso with Three Different Parameters We con-sider a Poisson qso such that

119875119894119895119896

=

119890minus1205821

120582119896

1

119896if 119894 + 119895 = 0 (mod 3)

119890minus1205822

1205821198962

119896if 119894 + 119895 = 1 (mod 3)

119890minus12058231205821198963

119896if 119894 + 119895 = 2 (mod 3)

(25)

For any initial measure 120583 isin 119878(119883F) let

119860 (120583) =

infin

sum119899=0

120583 (3119899) 119861 (120583) =

infin

sum119899=0

120583 (3119899 + 1)

119862 (120583) =

infin

sum119899=0

120583 (3119899 + 2)

(26)

The Scientific World Journal 5

where 119860(120583) + 119861(120583) + 119862(120583) = 1 It is easy to show that forPoisson distribution 119875

120582with parameter 120582 we have

119860 (120582) =1 + 2119890minus(32)120582 cos (radic32) 120582

3

119861 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 + 1205873)

3

119862 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 minus 1205873)

3

(27)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

120583 (3119898) 120583 (3119899)

+ 1198753119898+13119899+2119896

120583 (3119898 + 1) 120583 (3119899 + 2)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

120583 (3119898 + 1) 120583 (3119899)

+ 11987531198983119899+1119896

120583 (3119898) 120583 (3119899 + 1)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

120583 (3119898 + 2) 120583 (3119899)

+ 11987531198983119899+2119896

120583 (3119898) 120583 (3119899 + 2)

+1198753119898+13119899+1119896

120583 (3119898 + 1) 120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (120583) 119862 (120583) + 119861

2(120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

119881120583 (3119898)119881120583 (3119899)

+ 1198753119898+13119899+2119896

119881120583 (3119898 + 1)119881120583 (3119899 + 2)

+1198753119898+23119899+1119896

119881120583 (3119898 + 2)119881120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

119881120583 (3119898 + 1)119881120583 (3119899)

+ 11987531198983119899+1119896

119881120583 (3119898)119881120583 (3119899 + 1)

+1198753119898+23119899+2119896

119881120583 (3119898 + 2)119881120583 (3119899 + 2)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

119881120583 (3119898 + 2)119881120583 (3119899)

+ 11987531198983119899+2119896

119881120583 (3119898)119881120583 (3119899 + 2)

+1198753119898+13119899+1119896

119881120583 (3119898 + 1)119881120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861 (119881120583)119862 (119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583) + 119862

2(119881120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881120583)119862 (119881120583) + 119861

2(119881120583)]

(28)

By simple calculations we have

119860 (119881120583) = 119860 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119860 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119860 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119861 (119881120583) = 119861 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119861 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119861 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119862 (119881120583) = 119862 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119862 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119862 (1205823) [2119860 (120583) 119862 (120583) + 119861

2(120583)]

(29)

Thus by using induction on sequence 119881119899(120583) we producethe following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(30)

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

where 119860(120583) + 119861(120583) + 119862(120583) = 1 It is easy to show that forPoisson distribution 119875

120582with parameter 120582 we have

119860 (120582) =1 + 2119890minus(32)120582 cos (radic32) 120582

3

119861 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 + 1205873)

3

119862 (120582) =1 minus 2119890minus(32)120582 cos ((radic32) 120582 minus 1205873)

3

(27)

Then for any measure 120583 isin 119878(119883F) we have

119881120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

120583 (119894) 120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

120583 (3119898) 120583 (3119899)

+ 1198753119898+13119899+2119896

120583 (3119898 + 1) 120583 (3119899 + 2)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

120583 (3119898 + 1) 120583 (3119899)

+ 11987531198983119899+1119896

120583 (3119898) 120583 (3119899 + 1)

+1198753119898+23119899+1119896

120583 (3119898 + 2) 120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

120583 (3119898 + 2) 120583 (3119899)

+ 11987531198983119899+2119896

120583 (3119898) 120583 (3119899 + 2)

+1198753119898+13119899+1119896

120583 (3119898 + 1) 120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (120583) 119862 (120583) + 119861

2(120583)]

1198812120583 (119896) =

infin

sum119894=0

infin

sum119895=0

119875119894119895119896

119881120583 (119894) 119881120583 (119895)

=

infin

sum119898119899=0

[11987531198983119899119896

119881120583 (3119898)119881120583 (3119899)

+ 1198753119898+13119899+2119896

119881120583 (3119898 + 1)119881120583 (3119899 + 2)

+1198753119898+23119899+1119896

119881120583 (3119898 + 2)119881120583 (3119899 + 1)]

+

infin

sum119898119899=0

[1198753119898+13119899119896

119881120583 (3119898 + 1)119881120583 (3119899)

+ 11987531198983119899+1119896

119881120583 (3119898)119881120583 (3119899 + 1)

+1198753119898+23119899+2119896

119881120583 (3119898 + 2)119881120583 (3119899 + 2)]

+

infin

sum119898119899=0

[1198753119898+23119899119896

119881120583 (3119898 + 2)119881120583 (3119899)

+ 11987531198983119899+2119896

119881120583 (3119898)119881120583 (3119899 + 2)

+1198753119898+13119899+1119896

119881120583 (3119898 + 1)119881120583 (3119899 + 1)]

= 119890minus1205821

1205821198961

119896[1198602(119881120583) + 119861 (119881120583)119862 (119881120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881120583) 119861 (119881120583) + 119862

2(119881120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881120583)119862 (119881120583) + 119861

2(119881120583)]

(28)

By simple calculations we have

119860 (119881120583) = 119860 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119860 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119860 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119861 (119881120583) = 119861 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119861 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119861 (1205823) [2119860 (120583)119862 (120583) + 119861

2(120583)]

119862 (119881120583) = 119862 (1205821) [1198602(120583) + 2119861 (120583)119862 (120583)]

+ 119862 (1205822) [2119860 (120583) 119861 (120583) + 119862

2(120583)]

+ 119862 (1205823) [2119860 (120583) 119862 (120583) + 119861

2(120583)]

(29)

Thus by using induction on sequence 119881119899(120583) we producethe following recurrent equation

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119890minus1205822

1205821198962

119896[2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119890minus1205823

1205821198963

119896[2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(30)

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(a) Diagram when 1205823 = 001

2

15

1

05

0

0 05 1 15 2

1205822

1205821

(b) Diagram when 1205823 = 100

Figure 2 Limit behavior of the dynamical system (31) 0 lt 1205821 1205822le 2 and some fixed values 120582

3

where 119899 = 0 1 Besides for parameters 119860(119881119899120583) 119861(119881119899120583)and 119862(119881119899120583) we have the following recurrent equations

119860(119881119899+1

120583) = 119860 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119860 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119860 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119861 (119881119899+1

120583) = 119861 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119861 (1205822) [2119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119861 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

119862 (119881119899+1

120583) = 119862 (1205821) [1198602(119881119899120583) + 2119861 (119881

119899120583)119862 (119881

119899120583)]

+ 119862 (1205822) [1119860 (119881

119899120583) 119861 (119881

119899120583) + 119862

2(119881119899120583)]

+ 119862 (1205823) [2119860 (119881

119899120583)119862 (119881

119899120583) + 119861

2(119881119899120583)]

(31)

It is obvious that the limit behavior of the recurrent equa-tion (30) is fully determined by limit behavior of recurrentequations (31)

Since119860(119881119899120583)+119861(119881119899120583)+119862(119881119899120583) = 1 where119860(119881119899120583) ge 0119861(119881119899120583) ge 0 and119862(119881119899120583) ge 0 the recurrent equations (31) arerewritten as follows

1199091015840= 119860 (120582

1) 1199092+ 119860 (120582

3) 1199102+ 119860 (120582

2) 1199112

+ 2119860 (1205822) 119909119910 + 119860 (120582

3) 119909119911 + 119860 (120582

1) 119910119911

1199101015840= 119861 (120582

1) 1199092+ 119861 (120582

3) 1199102+ 119861 (120582

2) 1199112

+ 2119861 (1205822) 119909119910 + 119861 (120582

3) 119909119911 + 119861 (120582

1) 119910119911

1199111015840= 119862 (120582

1) 1199092+ 119862 (120582

3) 1199102+ 119862 (120582

2) 1199112

+ 2119862 (1205822) 119909119910 + 119862 (120582

3) 119909119911 + 119862 (120582

1) 119910119911

(32)

where 119909 + 119910 + 119911 = 1

Starting from arbitrary initial data we iterate the recur-rence equations (31) and observe their behavior after a largenumber of iterations The resultant diagram in the space(1205821 1205822) with 0 lt 120582

1 1205822le 2 and some fixed 120582

3are shown

in Figure 2 In this diagram blue color corresponds to theconverges of the trajectory

One can prove that for any values of parameters 1205821 1205822

and 1205823the nonlinear transformation (31) has a single fixed

point (119909lowast 119910lowast 119911lowast) and respectively it is regular transforma-tion

If these parameters are very small for instance 1205821= 3 sdot

10minus15 1205822= 2 sdot 10minus15 and 120582

3= 1 sdot 10minus15 then any trajectory

converges to (1 0 0) But if they are rather large for instance1205821= 25 120582

2= 50 and 120582

3= 75 then any trajectory converges

to (13 13 13)As above from (31) it follows that for any singleton 119896 isin 119883

the limit of the sequence 119881119899120583(119896) exists and equals

lim119899rarrinfin

119881119899+1

120583 (119896) = 119890minus1205821

1205821198961

119896[119909lowast2

+ 2119910lowast119911lowast]

+ 119890minus1205822

1205821198962

119896[2119909lowast119910lowast+ 119911lowast2

]

+ 119890minus1205823

1205821198963

119896[2119909lowast119911lowast+ 119910lowast2

]

(33)

Thus the strong limit of the sequence 119881119899120583 exists and equalsconvex linear combination

lim119899rarrinfin

119881119899+1

120583 = (119909lowast2

+ 2119910lowast119911lowast) 1198751205821

+ (2119909lowast119910lowast+ 119911lowast2

) 1198751205822

+ (2119909lowast119911lowast+ 119910lowast2

) 1198751205823

(34)

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

of three Poisson measures 1198751205821

1198751205822

and 1198751205823

It is evident thatFix(119881) = (119909lowast

2+ 2119910lowast119911lowast)119875

1205821

+ (2119909lowast119910lowast + 119911lowast2)1198751205822

+ (2119909lowast119911lowast +

119910lowast2)1198751205823

As corollary we have following statement

Proposition 8 A Poisson qso with three different parametersis a regular and respectively ergodic transformation withrespect to strong convergence

5 Conclusion

In this paper we present a construction of Poisson quadraticstochastic operators and prove their regularity when thenumber of different parameters 120582

119894is less than or equal to

three The Poisson quadratic stochastic operators with anyfinitely many different parameters 120582

119894and countably many

different parameters 120582119894will be considered in another paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byMinistry ofHigher EducationMalaysia (MOHE) under Grant FRGS14-116-0357

References

[1] S N Bernstein ldquoThe solution of a mathematical problemrelated to the theory of heredityrdquoUchen Zapiski Nauchno-IssledKafedry Ukr Otd Matem vol 1 pp 83ndash115 1924 (Russian)

[2] R N Ganikhodzhaev ldquoQuadratic stochastic operators Lya-punov function and tournamentsrdquoRussianAcademy of SciencesSbornik Mathematics vol 76 no 2 pp 489ndash506 1993

[3] R N Ganikhodzhaev ldquoA chart of fixed points and Lyapunovfunctions for a class of discrete dynamical systemsrdquoMathemat-ical Notes vol 56 no 5-6 pp 125ndash1131 1994

[4] R N Ganikhodzhaev and D B Eshmamatova ldquoQuadraticautomorphisms of a simplex and the asymptotic behavior oftheir trajectoriesrdquo Vladikavkazskii Matematicheskii Zhurnalvol 8 no 2 pp 12ndash28 2006 (Russian)

[5] R Ganikhodzhaev F Mukhamedov and U Rozikov ldquoQuad-ratic stochastic operators and processes results and openproblemsrdquo Infinite Dimensional Analysis Quantum Probabilityand Related Topics vol 14 no 2 pp 279ndash335 2011

[6] R D Jenks ldquoQuadratic differential systems for interactivepopulation modelsrdquo Journal of Differential Equations vol 5 pp497ndash514 1969

[7] H Kesten ldquoQuadratic transformations a model for populationgrowth Irdquo Advances in Applied Probability vol 2 pp 1ndash82 1970

[8] V Losert and E Akin ldquoDynamics of games and genes discreteversus continuous timerdquo Journal of Mathematical Biology vol17 no 2 pp 241ndash251 1983

[9] Y I Lyubich Mathematical Structures in Population Geneticsvol 22 of Biomathematics Springer 1992

[10] V Volterra Variations and Fluctuations of the Number ofIndividuals in Animal Species Living Together McGrawndashHill1931

[11] SMUlamA collection ofMathematical Problems InterscienceNew-York NY USA 1960

[12] M I Zakharevich ldquoOn behavior of trajectories and the ergodichypothesis for quadratic transformations of the simplexrdquo Rus-sian Mathematical Surveys vol 33 pp 265ndash266 1978

[13] N N Ganikhodzhaev and D V Zanin ldquoOn a necessarycondition for the ergodicity of quadratic operators defined on atwo-dimensional simplexrdquo Russian Mathematical Surveys vol59 no 3 pp 571ndash572 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of