Preface to Special Issue on Mathematics of Social Systems

This special issue is an outgrowth of a minisyposium titled “Mathematics of Social Systems” held at the 9th AIMS Conference on Dynamical Systems, Di↵erential Equations and Applications, held in Orlando, FL in July 2012. Presenters from that session were invited to submit papers that were reviewed using the usual procedures of the DCDS journals, along with additional authors from the field. Mathematics has already had a significant impact on basic research involving fundamental problems in physical sciences, biological sciences, computer science and engineering. Examples include understanding of the equations of incompressible fluid dynamics, shock wave theory and compressible gas dynamics, ocean modeling, algorithms for image processing and compressive sensing, and biological problems such as models for invasive species, spread of disease, and more recently systems biology for modeling of complex organisms and complex patterns of disease. This impact has yet to come to fruition in a comprehensive way for complex social behavior. While computational models such as agent-based systems and well-known statistical methods are widely used in the social sciences, applied mathematics has not to date had a core impact in the social sciences at the level that it achieves in the physical and life sciences. However in recent years we have seen a growth of work in this direction and ensuing new mathematics problems that must be tackled to understand such problems. Technical approaches include ideas from statistical physics, nonlinear partial di↵erential equations of all types, statistics and inverse problems, and stochastic processes and social network models. The collection of papers presented in this issue provides a backdrop of the current state of the art results in this developing new research area in applied mathematics. The body of work encompasses many of the challenges in understanding these discrete complex systems and their related continuum approximations. The paper [3] by Bedrossian and Rodŕıguez considers a general class of nonlocal partial di↵erential equations with nonlinear di↵usion and nonlocal advection. A special class of such problems, namely the parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis is well known in the literature. However, more general models arise in problems involving social aggregation dynamics with first order attraction and nonlinear di↵usion, typical of anti-crowding dynamics. In a previous paper [4] the authors developed a unifying theory for the Patlak-Keller-Segel global existence theory with the local existence and uniqueness theory for less singular nonlocal models in dimensions d 3. However many biologically relevant examples