Table 1.

Age × stage‐classified demographic analysis

	Notation	Equation
Model construction
Population vector	$\tilde{\mathbf{N}}$	(3)
Forbidden age‐stage combinations	U i, F i	(8), (9)
Block‐diagonal matrices	$\mathbb{U}$, $\mathbb{F}$, $\mathbb{D}$, $\mathbb{H}$	(10)
Age × stage‐classified projection matrices	$\tilde{\mathbf{U}}$, $\tilde{\mathbf{F}}$, $\tilde{\mathbf{A}}$	(11), (12), (13)
Life table functions
Survivorship vector[s]	ℓ(x), L	(17), (46)
Survivorship of mixed cohort	ℓ(x | π)	(19)
Mortality rate vector	μ(x)	(22)
Mortality rate of mixed cohort	μ(x | π)	(23)
Intra‐cohort selection
Joint age × stage	$\tilde{\mathbf{m}}(x|{\tilde{\mathbf{n}}}_0)$	(25)
Marginal stage	${\tilde{\mathbf{m}}}_{\mathrm{stage}}(x|{\tilde{\mathbf{n}}}_0)$	(26)
Marginal age	${\tilde{\mathbf{m}}}_{\mathrm{age}}(x|{\tilde{\mathbf{n}}}_0)$	(27)
Fertility
Fertility matrix	F x
Weighted fertility vector	f weighted(x)	(29)
Mixed fertility vector	f mixed(x)	(30)
Weighted and mixed fertility vector	f(x)	(32)
Longevity statistics
Fundamental matrix	$\tilde{\mathbf{N}}$	(34)
Moments and variance of longevity	${\tilde{\mathrm{\eta }}}_i$, $V(\tilde{\mathrm{\eta }})$	(35), (36), (37)
Decomposition of variance in longevity	V within, V between	(43), (45)
Distribution of age and stage at death
Joint age × stage distribution	$\tilde{\mathbf{B}}$	(50)
Marginal age distribution	B age	(51)
Marginal stage distribution	B stage	(52)
Life disparity, years of life lost	${\tilde{\mathrm{\eta }}}^{†}$	(57)
Population dynamics
Population projection	$\tilde{\mathbf{n}}(t+1)=\tilde{\mathbf{A}}\tilde{\mathbf{n}}(t)$	(60)
Stable population structure
Marginal age structure	${\tilde{\mathbf{w}}}_{\mathrm{age}}$	(61)
Marginal stage structure	${\tilde{\mathbf{w}}}_{\mathrm{stage}}$	(62)
Reproductive value	v	(63)
Net reproductive rate	R 0	(67)
Mean lifetime reproduction
Next generation matrix	R 11	(73)
Weighted lifetime reproduction	r weighted	(75)
Mixed lifetime reproduction	r mixed	(76)
Total lifetime reproduction	ρ	(77)
Cohort generation time	Γ	(88)
Sensitivity analysis
Sensitivity of output ξ to parameters θ	d ξ/d θ ⊤	(89)

Results are shown for model construction, life table functions, longevity statistics, and population dynamics, with equation numbers in which the results are presented.