On Kinetic Equations Modeling Evolution of Systems in Mathematical Biology

TL;DR

This paper develops a rigorous formalism for modeling the kinetic evolution of biological systems using dual BBGKY hierarchy and Vlasov-type equations, providing a mathematical foundation for understanding interacting entities in biology.

Contribution

It introduces a formalism based on the dual BBGKY hierarchy and derives the mean field limit leading to Vlasov-type kinetic equations for biological systems.

Findings

Established the dual Vlasov hierarchy for limit marginal observables.

Connected the dual Vlasov hierarchy with Vlasov-type kinetic equations.

Provided a rigorous mathematical framework for biological kinetic modeling.

Abstract

We develop a rigorous formalism for the description of the kinetic evolution of interacting entities modeling systems in mathematical biology within the framework of the evolution of marginal observables. For this purpose we construct the mean field asymptotic behavior of a solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables of the dynamical systems based on the Markov jump processes, exhibiting the intrinsic properties of the living entities. The constructed scaling limit is governed by the set of recurrence evolution equations, namely by the dual Vlasov-type hierarchy. Moreover, the relationships of the dual Vlasov hierarchy for the limit marginal observables with the Vlasov-type kinetic equation is established.