We consider the cell population dynamics with n different phenotypes. Cells
in one phenotype can produce cells in other phenotypes through conversions or
asymmetric divisions. Both the Markov branching process model and the ordinary
differential equation (ODE) system model are presented, and exploited to
investigate the dynamics of the phenotypic proportions. Gupta et al. observed
that with different initial population states, the proportions of different
phenotypes will always tend to certain constants ("phenotypic equilibrium"). In
the ODE system model, they gave a mathematical explanation through assuming the
phenotypic proportions satisfy the Kolmogorov forward equations of an n-state
Markov chain. We give a sufficient and necessary condition under which this
assumption is valid. We also prove the "phenotypic equilibrium" without such
assumption. In the Markov branching process model, more generally, we show the
stochastic explanation of "phenotypic equilibrium" through improving a limit
theorem in Janson's paper, which may be of theoretical interests. 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-Markov cases.Comment: 14 page
