Preface

Nonlinear mappings appear throughout mathematics, and their range of applications is immense, including the theory of differential equations, the theory of probability, the theory of dynamical systems, mathematical biology, and statistical physics. Most of the simplest nonlinear operators are quadratic. Even in a one-dimensional setting, the behavior of such operators reveals their complicated structure. If one considers multidimensional analogues of quadratic operators, then the situation becomes more complicated, i.e., the investigation of the dynamical behavior of such operators is very difficult. The history of quadratic stochastic operators and their dynamics can be traced back to Bernstein’s work [18]. The continuous time dynamics of this type of operator was considered by Lotka [134] and Volterra [252]. Quadratic stochastic operators are an important source of analysis in the study of dynamical properties and for modeling in various fields such as mathematical economics, evolutionary biology, population and disease dynamics, and the dynamics of economic and social systems. Unfortunately, up to now, there have been no books devoted to the dynamics of quadratic stochastic operators. This omission in the literature gave us the motivation to write a systematic book about such operators. The general objectives of this book are: (i) to give the first systematic presentation of both analytical and probabilistic techniques used in the study of the dynamics of quadratic stochastic operators and corresponding processes; (ii) to establish a connection between the dynamics of quadratic stochastic operators with the theory of Markov processes; and (iii) to give a systematic introduction to noncommutative or quantum analogues of quadratic stochastic operators and processes. The book addresses the most fundamental questions in the theory of quadratic stochastic operators: dynamics, constructions, regularity, and the connection with stochastic processes. This connection means that the dynamics of such operators can be treated as certain Markov or quadratic processes. This interpretation allows us to use the methods of stochastic processes for a better understanding of the limiting behavior of the dynamics of quadratic operators.