Let P = [pij] be the probability probability transition matrix of Markov chain, so that pij is the probability that node i contacts node j. We assume that P is aperiodic, irreducible. It may or may not be symmetric. P is double–stochastic matrix, i.e. the sum of elements in each row or column is equal to 1. By the Perron–Frobenius theorem, P has a stationary distribution π = [πi] π = πP (1) whe...

We analyze a multi-type age dependent model for cell populations subject to unidirectional motion, in both a stochastic and deterministic framework. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We adapt results on the large-time convergence of PDE systems and branching processes to our context, where the Perron-Frobenius or Krei...