DYNAMICS OF A CUBIC STOCHASTIC OPERATOR

Abstract

It is well-known that Lotka-Volterra operators play important role in studying predator-prey
models. In the present paper, we consider discrete-time dynamical systems generated by cubic operators
which we are called separable cubic stochastic operators. That is, we consider cubic stochastic operators
defined as a product of three linear operators defined on a simplex. It is shown that for a separable cubic
stochastic operator defined on a two-dimensional simplex, the vertices of the simplex are fixed points. It is
also proven that the orbit of a separable cubic stochastic operator converges for any starting point taken
from the simplex.