Microscopic, mesoscopic and macroscopic descriptions of complex systems

In the present paper the general approach that relates microscopic models with those at the mesoscopic scale and then the macroscopic scale is reviewed. The models at the micro-scale level are defined by a large number of interacting entities (particles, agents, cells, individuals, etc), and is in terms of a Markov jump process and related linear evolution equations. The intermediate models refer to the meso-scale level of description of test entities and are given in terms of bilinear Boltzmann-type equations. Mathematical relationships between these three possible descriptions are presented and explicit error estimates are given. The general framework is applied to propose the microscopic and mesoscopic models that correspond to very well known models in biomathematics: the Verhulst logistic equation and the Lotka–Volterra system of equations. The asymptotic time behaviour for the mesoscopic model corresponding to the Verhulst logistic equation is defined. The mesoscopic model corresponding to the Verhulst equation is modified to a new mesoscopic model of DNA denaturation.