Historical behavior for a class of Lotka–Volterra systems

Abstract

It is well-known that discrete-time Kolmogorov systems play important role in studying predator-prey models with interactions. A topic of this paper deals with stochastic discrete-time Kolmogorov systems which are called Lotka-Volterra operators. In the present paper, we provide a new class of LV operators, and establish that each of operator from this class has historical behavior, that is, their time averages diverge.

Discrete time model of sexual systems

“…However, Zakharevich's findings, as published in [28] demonstrates that the limit of (3) does not exist for the mapping of described by The findings, suggest that Ulam's conjecture is generally false. Moreover, [29] , [30] , [31] present various extensions of the mapping (4) .…”