How does variability in cell aging and growth rates influence the Malthus parameter?

Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 [15]. A cell in a controlled culture grows at a constant rate $v>0$, but this rate can differ from one individual to another. The biological question we address here states as follows. How does individual variability in the growth rate influence the growth speed of the population? The growth speed of the population is measured by the Malthus parameter we define thereafter, also called in the literature fitness. Even if the variability in the growth rate among cells is small, with a distribution of coefficient of variation around 10%, and even if its influence on the Malthus parameter would be still smaller, such an influence may become determinant since it characterises the exponential growth speed of the population.