Law of Large Numbers for DNA Distribution in an Interacting Cell Population

Abstract

A continuous measure-valued model is obtained as the limit of discrete finite-dimensional Markov processes describing the evolution of a population of interacting cells (classified by their DNA content), as the initial number of individuals diverges and the DNA production unit tends independently to zero. The limit is identified by a nonlinear evolution equation, which is shown to have a unique solution by a contraction argument in a suitable metric space. In the critical case the solution of the limit equation can be viewed as the family of the one-time probability distributions of a nonlinear Markov process.