Table 2.

Matrices used in constructing the age × stage‐classified projection matrix, where ω denotes the number of age classes and s the number of stages

Symbol	Expression	Size	Description
Matrix describing the full population
$\tilde{\mathbf{A}}$	$\tilde{\mathbf{U}}+\tilde{\mathbf{F}}$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Age × stage population projection matrix
Matrices describing transitions and survival of existing individuals
$\tilde{\mathbf{U}}$	${\mathbf{K}}^{\top }\mathbb{D}\mathbf{K}\mathbb{U}$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Age × stage transition and survival matrix
$\mathbb{U}$	${\sum }_{j=1}^{\mathrm{\omega }}({\mathbf{E}}_{\mathit{jj}}\otimes {\mathbf{U}}_j)$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Block diagonal age × stage transition matrix
$\mathbb{D}$	${\sum }_{i=1}^s({\mathbf{E}}_{\mathit{ii}}\otimes {\mathbf{D}}_i)$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Block diagonal age transition matrix
U j	U j	$s\times s$	Stage transition matrix for age class j
D i	D i	$\mathrm{\omega }\times \mathrm{\omega }$	Age transition matrix for stage i
Matrices describing reproduction
$\tilde{\mathbf{F}}$	${\mathbf{K}}^{\top }\mathbb{H}\mathbf{K}\mathbb{F}$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Age × stage fertility matrix
$\mathbb{F}$	${\sum }_{j=1}^{\mathrm{\omega }}({\mathbf{E}}_{\mathit{jj}}\otimes {\mathbf{F}}_j)$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Block diagonal fertility matrix
$\mathbb{H}$	${\sum }_{i=1}^s({\mathbf{E}}_{\mathit{ii}}\otimes {\mathbf{H}}_i)$	$s\mathrm{\omega }\times s\mathrm{\omega }$	Block diagonal age assignment matrix
F j	F j	$s\times s$	Fertility matrix for age class j
H i	H i	$\mathrm{\omega }\times \mathrm{\omega }$	Age assignment matrix for offspring of stage i

The index i denotes stages, $i=1,\dots ,s$; the index j denotes age classes, $j=1,\dots ,\mathrm{\omega }$.