# Markov processes of cubic stochastic matrices: {\it Quadratic stochastic processes}

## TL;DR

This paper studies quadratic stochastic processes modeled by Markov processes of cubic matrices, exploring their construction under different multiplications and applications to biological population dynamics.

## Contribution

It constructs a wide class of quadratic stochastic processes for specific cubic matrix notions and multiplications, including biological applications.

## Key findings

- Constructed QSPs for two types of stochastic cubic matrices.
- Analyzed time-dependent behavior of the constructed processes.
- Provided an example modeling population dynamics with twin births.

## Abstract

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births.