Preface

This book is devoted to the design of a unified mathematical approach to the modeling and analysis of large systems constituted by several interacting living entities. It is a challenging objective that needs new ideas and mathematical tools based on a deep understanding of the interplay between mathematical and life sciences. The authors do not naively claim that this is fully achieved, but simply that a useful insight and some significant results are obtained toward the said objective. The source of the contents of this book is the research activity developed in the last 20 years, which involved several young and experienced researchers. This story started with a book edited, at the beginning of this century, by N.B. with Mario Pulvirenti [51], where the chapters of the book presented a variety of models of life science systems which were derived by kinetic theory methods and theoretical tools of probability theory. The contents of [51] were motivated by the belief that an important new research frontier of applied mathematics had to be launched. The basic idea was that methods of the mathematical kinetic theory and statistical mechanics ought to be developed toward the modeling of large systems in life science differently from the traditional application to the fluid dynamics of large systems of classical particles. Here, particles are living entities, from genes, cells, up to human beings. These entities are called, within the framework of mathematics, active particles. This term encompasses the idea that these particles have the ability to express special strategies generally addressing to their well-being and hence do not follow laws of classical mechanics as they can think, namely possess both an individual and a collective intelligence [84]. Due to this specific feature, interactions between particles are nonlinearly additive. In fact, the strategy developed by each particle depends on that expressed by the other particles, and in some cases develops a collective intelligence of the whole viewed as a swarm. Moreover, it often happens that all these events occur in a nonlinear manner. An important conceptual contribution to describing interactions within an evolutive mathematical framework is offered by the theory of evolutive games [186, 189]. Once suitable models of the dynamics at the scale of individuals have been derived, methods of the kinetic theory suggest to describe the overall system