  The growth of a population is often modeled as branching process where each
individual at the end of its life is replaced by a certain number of offspring.
An example of these branching models is the Bellman-Harris process, where the
lifetime of individuals is assumed to be independent and identically
distributed. Here, we are interested in the estimation of the parameters of the
Bellman-Harris model, motivated by the estimation of cell division time.
Lifetimes are distributed according a Gamma distribution and we follow a
population that starts from a small number of individuals by performing
time-resolved measurements of the population size. The exponential growth of
the population size at the beginning offers an easy estimation of the mean of
the lifetime. Going farther and describing lifetime variability is a
challenging task however, due to the complexity of the fluctuations of
non-Markovian branching processes. Using fine and recent results on these
fluctuations, we describe two time-asymptotic regimes and explain how to
estimate the parameters. Then, we both consider simulations and biological data
to validate and discuss our method. The results described here provide a method
to determine single-cell parameters from time-resolved measurements of
populations without the need to track each individual or to know the details of
the initial condition.
