Kinetic Model for Vehicular Traffic with Continuum Velocity and Mean Field Interactions

This paper is concerned with the modeling and mathematical analysis of vehicular traffic phenomena. We adopt a kinetic theory point of view, under which the microscopic state of each vehicle is described by: (i) position, (ii) velocity and also (iii) activity, an additional varible that we use to describe the quality of the driver-vehicle micro-system. We use methods coming from game theory to describe interactions at the microscopic scale, thus constructing new models within the framework of the Kinetic Theory of Active Particles; the resulting models incorporate some of the symmetries that are commonly found in the mathematical models of the kinetic theory of gases. Short-range interactions and mean field interactions are introduced and modeled to depict velocity changes related to passing phenomena. Our main goal is twofold: (i) to use continuum-velocity variables and (ii) to introduce a non-local acceleration term modeling mean field interactions, related to, for example, the presence of tollgates or traffic highlights.J.C. and J.N. are partially supported by Junta de Andalucía Project P12-FQM-954 and MINECO Project RTI2018-098850-B-I00. J.C. is supported by Universidad de Granada (“Plan propio de investigación, programa 9”) through FEDER funds. M.Z. was supported by CNRST (Morocco), project “Modèles Mathématiques appliqués à l’environnement, à l’imagerie médicale et aux biosystèmes”

State-Policy Dynamics in Evolutionary Games

International audienceStandard Evolutionary Game Theory framework is a useful tool to study large interacting systems and to understand the strategic behavior of individuals in such complex systems. Adding an individual state to model local feature of each player in this context, allows one to study a wider range of problems in various application areas as networking, biology, etc. In this paper, we introduce such an extension of evolutionary game framework and particularly, we focus on the dynamical aspects of this system. Precisely, we study the coupled dynamics of the policies and the individual states inside a population of interacting individuals. We first define a general model by coupling replicator dynamics and continuous-time Markov Decision Processes and we then consider a particular case of a two policies and two states evolutionary game. We first obtain a system of combined dynamics and we show that the rest-points of this system are equilibria profiles of our evolutionary game with individual state dynamics. Second, by assuming two different time scales between states and policies dynamics, we can compute explicitly the equilibria. Then, by transforming our evolutionary game with individual states into a standard evolutionary game, we obtain an equilibrium profile which is equivalent , in terms of occupation measures and expected fitness to the previous one. All our results are illustrated with numerical analysis

Kinetic Theory and Swarming Tools to Modeling Complex Systems—Symmetry problems in the Science of Living Systems

This MPDI book comprises a number of selected contributions to a Special Issue devoted to the modeling and simulation of living systems based on developments in kinetic mathematical tools. The focus is on a fascinating research field which cannot be tackled by the approach of the so-called hard sciences—specifically mathematics—without the invention of new methods in view of a new mathematical theory. The contents proposed by eight contributions witness the growing interest of scientists this field. The first contribution is an editorial paper which presents the motivations for studying the mathematics and physics of living systems within the framework an interdisciplinary approach, where mathematics and physics interact with specific fields of the class of systems object of modeling and simulations. The different contributions refer to economy, collective learning, cell motion, vehicular traffic, crowd dynamics, and social swarms. The key problem towards modeling consists in capturing the complexity features of living systems. All articles refer to large systems of interaction living entities and follow, towards modeling, a common rationale which consists firstly in representing the system by a probability distribution over the microscopic state of the said entities, secondly, in deriving a general mathematical structure deemed to provide the conceptual basis for the derivation of models and, finally, in implementing the said structure by models of interactions at the microscopic scale. Therefore, the modeling approach transfers the dynamics at the low scale to collective behaviors. Interactions are modeled by theoretical tools of stochastic game theory. Overall, the interested reader will find, in the contents, a forward look comprising various research perspectives and issues, followed by hints on to tackle these

Evolutionary stochastic games

We extend the notion of Evolutionarily Stable Strategies introduced by Maynard Smith and Price (Nature 246: 15-18, 1973) for models ruled by a single fitness matrix A, to the framework of stochastic games developed by Lloyd Shapley (Proc. Natl. Acad. Sci. USA 39: 1095-1100, 1953) where, at discrete stages in time, players play one of finitely many matrix games, while the transitions from one matrix game to the next follow a jointly controlled Markov chain. We show that this extension from a single-state model to a multistate model can be done on the assumption of having an irreducible transition law. In a similar way, we extend the notion of Replicator Dynamics introduced by Taylor and Jonker (Math. Biosci. 40: 145-156, 1978) to the multistate model. These extensions facilitate the analysis of evolutionary interactions that are richer than the ones that can be handled by the original, single-state, evolutionary game model. Several examples are provided