  Since the seminal work of Powell, the relationships between the population
growth rate, the probability distributions of generation time, and the
distribution of cell age have been known for the bacterial population in a
steady state of exponential growth. Here, we generalize these relationships to
include an unsteady (transient) state for both the batch culture and the mother
machine experiment. In particular, we derive a time-dependent Euler-Lotka
equation (relating the generation time distributions to the population growth
rate) and a generalization of the inequality between the mean generation time
and the population doubling time. To do this, we use a model proposed by
Lebowitz and Rubinow, in which each cell is described by its age and generation
time. We show that our results remain valid for a class of more complex models
that use other state variables in addition to cell age and generation time, as
long as the integration of these additional variables reduces the model to
Lebowitz-Rubinow form. As an application of this formalism, we calculate the
fitness landscapes for phenotypic traits (cell age, generation time) in a
population that is not growing exponentially. We clarify that the known fitness
landscape formula for the cell age as a phenotypic trait is an approximation to
the exact time-dependent formula.
