  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.
