#### Abstract

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

#### 1. Introduction

Let be a measurable space, where is a state space and is -algebra on , and the set of all probability measures on .

Let be a family of functions on such that, for any fixed , regarded as a function of two variables and with fixed is a measurable function on and for any and .

We consider a nonlinear transformation called quadratic stochastic operator (qso) which is defined by where is an arbitrary measurable set.

If a state space is a finite set and the corresponding -algebra is the power set , that is, the set of all subsets of , then the set of all probability measures on has the following form: that is called a -dimensional simplex.

In this case, the probabilistic measure is a discrete measure with , where for any . In addition, the corresponding qso V has the following form: for any and the coefficients satisfy the following conditions: Such operator can be reinterpreted in terms of evolutionary operator of free population [1–10] and in this form it has a fair history.

In this paper, we consider nonlinear transformations defined on countable state space and investigate their limit behavior of trajectories.

#### 2. A Poisson qso

Let be a countable sample space and corresponding -algebra a power set , that is, the set of all subsets of . In order to define a probability measure on countable sample space , it is enough to define the measure of each singleton . Thus, we will write instead of .

Let be a family of functions defined on , which satisfy the following conditions:(i) is a probability measure on for any fixed ;(ii), where for any fixed .

In this case, a qso (1) on measurable space is defined as follows: where for arbitrary measure .

In this paper, we consider a Poisson qso which is a Poisson distribution with a positive real parameter defined on by the equation for any .

Let be a set of all probability measures on and let be a probability measure on for any .

*Definition 1. *A quadratic stochastic operator (5) is called a Poisson qso if, for any , the probability measure is the Poisson distribution with positive real parameters , where .

Assume that is the trajectory of the initial point , where for all , with .

In this paper, we will study limit behavior of trajectories of Poisson qso.

#### 3. Ergodicity and Regularity of qso

Let us consider a qso (5) defined on countable set . Let be the trajectory of the initial point , where for all .

*Definition 2. *A measure is called a fixed point of a qso if .

Let be the set of all fixed points of qso .

*Definition 3. *A qso is called regular if, for any initial point , the limit
exists.

In measure theory, there are various notions of the convergence of measures: weak convergence, strong convergence, and total variation convergence. Below we consider strong convergence.

*Definition 4. *For a measurable space, a sequence is said to converge strongly to a limit if
for every set .

If is a countable set, then a sequence converges strongly to a limit if and only if for every singleton .

In statistical mechanics, the ergodic hypothesis proposes a connection between dynamics and statistics. In the classical theory, the assumption was made that the average time spent in any region of phase space is proportional to the volume of the region in terms of the invariant measure. More generally, such time averages may be replaced by space averages.

For nonlinear dynamical systems, Ulam [11] suggested an analogous measure-theoretic ergodicity with following ergodic hypothesis.

*Definition 5. *A nonlinear operator defined on is called ergodic, if the limit
exists for any .

On the ground of numerical calculations for quadratic stochastic operators defined on with finite , Ulam [11] conjectured that the ergodic theorem holds for any such qso .

In 1977, Zakharevich [12] proved that this conjecture is false in general. He considered the following operator on : and he proved that such operator is nonergodic transformation. Later in [13], the sufficient condition to be nonergodic transformation was established for qso defined on .

In the next section, we will show that Ulam’s conjecture is true for some class of Poisson qso.

#### 4. Ergodicity and Regularity of Poisson qso

Let defined in (5) be a Poisson qso. We consider the following cases.

##### 4.1. Homogenious Poisson qso

We call a Poisson quadratic stochastic operator (5) homogenious, if , for any , that is, . Then for arbitrary measure where , that is, .

Thus for any , that is, , and we have the following statement.

Proposition 6. *A homogenious Poisson qso is a regular transformation.*

##### 4.2. Poisson qso with Two Different Parameters

We consider a Poisson qso such that

For any initial measure let where . It is easy to show that for Poisson distribution Then for any measure , we have

By simple calculations, we have

Thus, by using induction on the sequence , we produce the following recurrent equation: where , Besides, for parameters and , we have the following recurrent equations:

It is obvious that the limit behavior of the recurrent equation (18) is fully determined by limit behavior of recurrent equations (19).

Since , where and , the recurrent equations (19) are rewritten as follows: with , , and .

Solving the following quadratic equation we have single fixed point and denoted it as (see Figure 1). Using simple calculus (see Figure 1), one can show that any trajectory of the qso (20) defined on one-dimensional simplex converges to this fixed point; that is, qso (20) is regular transformation, so that it is ergodic.

(a) When and |

(b) When and |

Thus, for any initial measure , we have Then, passing to limit in (18), for any singleton , we have

Thus, for any initial measure , the strong limit of the sequence exists and is equal to the convex linear combination of two Poisson measures and . It is evident that .

As corollary we have following statement.

Proposition 7. *A Poisson qso with two different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.*

##### 4.3. A Poisson qso with Three Different Parameters

We consider a Poisson qso such that For any initial measure , let where . It is easy to show that, for Poisson distribution with parameter , we have Then, for any measure , we have By simple calculations, we have

Thus, by using induction on sequence , we produce the following recurrent equation: where . Besides, for parameters , and , we have the following recurrent equations:

It is obvious that the limit behavior of the recurrent equation (30) is fully determined by limit behavior of recurrent equations (31).

Since , where , , and , the recurrent equations (31) are rewritten as follows: where .

Starting from arbitrary initial data, we iterate the recurrence equations (31) and observe their behavior after a large number of iterations. The resultant diagram in the space with , , and some fixed are shown in Figure 2. In this diagram, blue color corresponds to the converges of the trajectory.

(a) Diagram when |

(b) Diagram when |

One can prove that for any values of parameters , , and the nonlinear transformation (31) has a single fixed point and, respectively, it is regular transformation.

If these parameters are very small, for instance, , , and , then any trajectory converges to . But, if they are rather large, for instance, , , and , then any trajectory converges to .

As above, from (31) it follows that for any singleton the limit of the sequence exists and equals Thus, the strong limit of the sequence exists and equals convex linear combination of three Poisson measures , , and . It is evident that .

As corollary we have following statement.

Proposition 8. *A Poisson qso with three different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.*

#### 5. Conclusion

In this paper, we present a construction of Poisson quadratic stochastic operators and prove their regularity when the number of different parameters is less than or equal to three. The Poisson quadratic stochastic operators with any finitely many different parameters and countably many different parameters will be considered in another paper.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This research was supported by Ministry of Higher Education Malaysia (MOHE) under Grant FRGS14-116-0357.