Phenotypic Equilibrium as Probabilistic Convergence in Multi-phenotype   Cell Population Dynamics
  We consider the cell population dynamics with $n$ different phenotypes. Both the Markovian branching process model (stochastic model) and the ordinary differential equation (ODE) system model (deterministic model) are presented, and exploited to investigate the dynamics of the phenotypic proportions. We will prove that in both models, these proportions will tend to constants regardless of initial population states ("phenotypic equilibrium") under weak conditions, which explains the experimental phenomenon in Gupta et al.'s paper. We also prove that Gupta et al.'s explanation is the ODE model under a special assumption. As an application, we will give sufficient and necessary conditions under which the proportion of one phenotype tends to $0$ (die out) or $1$ (dominate). We also extend our results to non-Markovian cases. 
Da-Quan Jiang, Yue Wang, Da Zhou
q-bio.PE