The description of emerging collective behaviors and self--organization in a group of interacting individuals has gained increasing interest from various research communities in biology, engineering, physics, as well as sociology and economics. In the biological context, the emergent behavior of bird flocks, fish schools or bacteria aggregations, among others, is a major research topic in population dynamics and behavioral ecology. Likewise, emergent economic behaviors, such as distribution of wealth in a modern society and price formation dynamics, or challenging social phenomena such as the formation of choices and opinions, and modern human warfare, are also problems in which emergence of collective behaviors and universal equilibria has been shown. These behaviors widely occur in nature and are part of our daily lives. It is quite surprising and astonishing to learn that they can be fruitfully described and understood by means of suitable mathematical tools. Mathematical modelling using partial differential equations, which touches core areas of physics and engineering, is in fact playing an increasing role in emerging fields such as social, economic and life sciences. Mathematical efforts are gradually gaining strength in this multidisciplinary area. Typically the underlying equations are highly nonlinear, in many cases they are also vectorial systems, and they represent a challenge even for the most modern and sophisticated mathematical-analytical and mathematical-numerical techniques. Among others, multidimensional computations of complex multi-scale phenomena are now within reach. Sophisticated nonlinear analysis deepens our understanding of increasingly complex models. Computational results feed back into the modelling process, and give insight of detailed mechanisms which often cannot be studied by real life experiments. The novelty here is that important phenomena in seemingly different areas such as sociology, economy and rarefied gas dynamics can be described by closely related mathematical models. In this book we present selected research topics which can be regarded as new and challenging frontiers of applied mathematics. These topics are chosen to enlighten the common methodological background underlining the main idea of this book: to identify similar modelling approaches, similar analytical and numerical techniques for systems made out of a large number of "individuals" which show a "collective behavior" and how to obtain from them "average" information. The expertise obtained in dealing with physical situations is considered a basis to model and simulate problems for applications in socio-economic and life sciences, as a newly emerging research field. In most of the selected contributions, the leading idea is that collective behaviors of a group composed by a sufficiently large number of individuals (agents) could be hopefully described using the laws of statistical mechanics as it happens in a physical system composed of many interacting particles. This opens a bridge between statistical physics and socio-economic and life sciences. In particular, powerful methods borrowed from kinetic theory of rarefied gases can been fruitfully employed to construct kinetic equations which describe the emergency of universal structures through their equilibria. The book is subdivided in three parts, and each part is composed by several chapters. Part I deals with the kinetic modeling of microscopic models of simple market economies and financial markets. Some of This kinetic modeling is clearly rather {\em ad hoc}, but if one is willing to accept the analogies between trading agents and colliding particles, then various well established methods from statistical physics are ready for application to the field of economy. Most notably, the numerous tools originally devised for the study of the energy distribution in a rarefied gas can now be used to analyze wealth distributions. Similarly in finance several techniques borrowed from kinetic theory can be used to study the price formation and the presence of power laws. Likewise, microscopic models of both social and political phenomena describing collective behaviors and self-organization in a society can be analyzed within these methods. These topics are the subject of part II. Among others, the modeling of opinion formation and vote intention dynamics has attracted the interest of an increasing number of researchers in the recent years. The last part of the book deals with collective self-driven motion of self-propelled particles such as flocking of birds, schooling of fishes, swarming of bacteria, traffic and crowds movements and general population dynamics. These coherent and synchronized structures are apparently produced without the active role of a leader in the grouping, and can be described within the same concepts of applied mathematics. The idea of publishing a book to highlight these new emerging applications of mathematics came to us at the conclusion of a short series of lectures we organized in Vigevano in november 2008, with the essential support of the Center for Interuniversity Research in Economics for Land of the Universities of Milano-Bicocca, Pavia and Ferrara (CRIET), the Advanced Applied Mathematical and Statistical Sciences Center of the University of Milan (ADAMSS), the Center for Modelling Computing $\&$ Statistics of the University of Ferrara (CMCS). Vigevano is an interesting city on the south of Milan area, known to have one of the most beautiful central squares in Italy, surrounded by magnificent arcades. The main objective of these lectures was to enlighten in an almost elementary way to a large audience the essentials of the mathematical modeling of socio-economic problems, like wealth distributions, opinion formations, market dynamics as well as related topics, including pedestrian traffic, swarming and others collective phenomena. We warmly hope this book could be of great interest to applied mathematicians, physicists, biologists, economists involved in the modelling of complex socio-economic systems, and in aggregation and collective phenomena in general.
Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences
NALDI, Giovanni;PARESCHI, Lorenzo;TOSCANI, Giuseppe
2010
Abstract
The description of emerging collective behaviors and self--organization in a group of interacting individuals has gained increasing interest from various research communities in biology, engineering, physics, as well as sociology and economics. In the biological context, the emergent behavior of bird flocks, fish schools or bacteria aggregations, among others, is a major research topic in population dynamics and behavioral ecology. Likewise, emergent economic behaviors, such as distribution of wealth in a modern society and price formation dynamics, or challenging social phenomena such as the formation of choices and opinions, and modern human warfare, are also problems in which emergence of collective behaviors and universal equilibria has been shown. These behaviors widely occur in nature and are part of our daily lives. It is quite surprising and astonishing to learn that they can be fruitfully described and understood by means of suitable mathematical tools. Mathematical modelling using partial differential equations, which touches core areas of physics and engineering, is in fact playing an increasing role in emerging fields such as social, economic and life sciences. Mathematical efforts are gradually gaining strength in this multidisciplinary area. Typically the underlying equations are highly nonlinear, in many cases they are also vectorial systems, and they represent a challenge even for the most modern and sophisticated mathematical-analytical and mathematical-numerical techniques. Among others, multidimensional computations of complex multi-scale phenomena are now within reach. Sophisticated nonlinear analysis deepens our understanding of increasingly complex models. Computational results feed back into the modelling process, and give insight of detailed mechanisms which often cannot be studied by real life experiments. The novelty here is that important phenomena in seemingly different areas such as sociology, economy and rarefied gas dynamics can be described by closely related mathematical models. In this book we present selected research topics which can be regarded as new and challenging frontiers of applied mathematics. These topics are chosen to enlighten the common methodological background underlining the main idea of this book: to identify similar modelling approaches, similar analytical and numerical techniques for systems made out of a large number of "individuals" which show a "collective behavior" and how to obtain from them "average" information. The expertise obtained in dealing with physical situations is considered a basis to model and simulate problems for applications in socio-economic and life sciences, as a newly emerging research field. In most of the selected contributions, the leading idea is that collective behaviors of a group composed by a sufficiently large number of individuals (agents) could be hopefully described using the laws of statistical mechanics as it happens in a physical system composed of many interacting particles. This opens a bridge between statistical physics and socio-economic and life sciences. In particular, powerful methods borrowed from kinetic theory of rarefied gases can been fruitfully employed to construct kinetic equations which describe the emergency of universal structures through their equilibria. The book is subdivided in three parts, and each part is composed by several chapters. Part I deals with the kinetic modeling of microscopic models of simple market economies and financial markets. Some of This kinetic modeling is clearly rather {\em ad hoc}, but if one is willing to accept the analogies between trading agents and colliding particles, then various well established methods from statistical physics are ready for application to the field of economy. Most notably, the numerous tools originally devised for the study of the energy distribution in a rarefied gas can now be used to analyze wealth distributions. Similarly in finance several techniques borrowed from kinetic theory can be used to study the price formation and the presence of power laws. Likewise, microscopic models of both social and political phenomena describing collective behaviors and self-organization in a society can be analyzed within these methods. These topics are the subject of part II. Among others, the modeling of opinion formation and vote intention dynamics has attracted the interest of an increasing number of researchers in the recent years. The last part of the book deals with collective self-driven motion of self-propelled particles such as flocking of birds, schooling of fishes, swarming of bacteria, traffic and crowds movements and general population dynamics. These coherent and synchronized structures are apparently produced without the active role of a leader in the grouping, and can be described within the same concepts of applied mathematics. The idea of publishing a book to highlight these new emerging applications of mathematics came to us at the conclusion of a short series of lectures we organized in Vigevano in november 2008, with the essential support of the Center for Interuniversity Research in Economics for Land of the Universities of Milano-Bicocca, Pavia and Ferrara (CRIET), the Advanced Applied Mathematical and Statistical Sciences Center of the University of Milan (ADAMSS), the Center for Modelling Computing $\&$ Statistics of the University of Ferrara (CMCS). Vigevano is an interesting city on the south of Milan area, known to have one of the most beautiful central squares in Italy, surrounded by magnificent arcades. The main objective of these lectures was to enlighten in an almost elementary way to a large audience the essentials of the mathematical modeling of socio-economic problems, like wealth distributions, opinion formations, market dynamics as well as related topics, including pedestrian traffic, swarming and others collective phenomena. We warmly hope this book could be of great interest to applied mathematicians, physicists, biologists, economists involved in the modelling of complex socio-economic systems, and in aggregation and collective phenomena in general.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
