Individually-based Markov processes modeling nonlinear systems in mathematical biology

Abstract

In the present paper the general approach that allows one to construct the individually-based Markov processes describing various systems in mathematical biology (or in other applied sciences) is presented. The Markov processes are of a jump type and the starting point is the related linear equations. They describe at the micro–scale level the behavior of a large number N of interacting entities (particles, agents, cells, individuals, …). The large entity limit (“ N → ∞ ”) is studied and the intermediate level (the meso–scale level) is given in terms of nonlinear kinetic–type equations. Finally the corresponding systems of nonlinear ODEs (or PDEs) at the macroscopic level (in terms of concentrations of the interacting subpopulations) are obtained. 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 well known systems of nonlinear equations in biomathematics. The paper generalises the previous approach resulting in bilinear equations of the Boltzmann–type at the mesoscopic level.