Table 1.

Advantages and disadvantages of different frameworks for stochastic age-structured populations. ‘Stochastic’ indicates that the model resolves probabilities of configurations of the population

Theory	Stochastic	Age-dependentrates	Age-structuredpopulations	Age-chart resolved	Interactions	Budding	Fission
Verhulst Eq.	$\times$	$\times$	$\times$	$\times$	$✓$	$\times$	$\times$
McKendrick Eq.	$\times$	$✓$	$✓$	$\times$	$✓$	$✓$ 1	$\times$
Master Eq.	$✓$	$\times$	$\times$	$\times$	$✓$	$✓$	$✓$
Bellman-Harris	$✓$	$✓$	$\times$	$\times$	$\times$	$\times$	$✓$
Leslie matrices	$\times$	$✓$ 2	$✓$	$\times$	$✓$	$\times$	$\times$
Martingale	$✓$	$✓$	$\times$ 3	$\times$	$✓$	$✓$	$✓$
Kinetic theory	$✓$	$✓$	$✓$	$✓$	$✓$	$✓$	$✓$ 4

‘Age-dependent rates’ indicates whether or not a model takes into account birth, death, or fission rates that depend on an individuals age (time after its birth). ‘Age-structured Populations’ indicates whether or not the theory outputs the age structure of the ensemble population. ‘Age-chart resolved’ indicates whether or not a theory outputs the age distribution of all the individuals in the population. ‘Interactions’ indicates whether or not the approach can incorporate population-dependent dynamics such as that arising from a carrying capacity, or from birth processes involving multiple parents. ‘Budding’ and ‘Fission’ describes the model of birth and indicates whether the parent lives or dies after birth

¹ Birth and death rates in the McKendrick-von Foerster equation can be made explicit functions of the total populations size, which must be self-consistently solved [17, 18]

² Leslie matrices discretize age groups and are an approximate method

³ Martingale methods do not resolve the age structure explicitly, but utilize rigorous machinery

⁴ The kinetic approach for fission is addressed later in this work, but not in [16]