How does variability in cells aging and growth rates influence the malthus parameter?

Abstract: The aim of this study is to compare the growth speed of different cell populations measured by their Malthus parameter. We focus on both the age-structured and size-structured equations. A first population (of reference) is composed of cells all aging or growing at the same rate $\bar v$. A second population (with variability) is composed of cells each aging or growing at a rate $v$ drawn according to a non-degenerated distribution $\rho$ with mean $\bar v$. In a first part, analytical answers -- based on the study of an eigenproblem -- are provided for the age-structured model. In a second part, numerical answers -- based on stochastic simulations -- are derived for the size-structured model. It appears numerically that the population with variability proliferates more slowly than the population of reference (for experimentally plausible division rates). The decrease in the Malthus parameter we measure, around 2% for distributions $\rho$ with realistic coefficients of variations around 15-20\%, is determinant since it controls the {\it exponential} growth of the whole population.