Dynamics of Infinite Dimensional Lotka–Volterra Operators

The majority of research on quadratic stochastic operators (QSOs) has focused on finite-dimensional sets of probability distributions, known as simplices. However, extending this study to the infinite case presents an intriguing challenge. This paper specifically addresses infinite-dimensional Lotka–Volterra operators. Unlike the finite case, the non-compactness of infinite-dimensional simplices complicates the general analysis, so several subclasses of infinite-dimensional Lotka–Volterra operators are introduced. Additionally, Lyapunov functions are constructed for each subclass, enabling the exploration of the dynamics of these operators. The paper also describes the omega limiting sets of the Lotka–Volterra operators.