Table 4:

Stage and Age: Cohorts to Fitness

Quantity	Equation	Notes
Generation matrix	${\mathbf{A}}_0={\sum }_a{\mathbf{F}}_a{\mathbf{L}}_a$	Given a cohort, determines stage-composition of offspring
Dominant eigenvalue of A0	R0	Net Reproductive Rate
Corresponding right eigenvector of A0	c, dimension equals number of stages, sum of components = 1	Stage-structure of stable cohort
Corresponding left eigenvector of A0	d, dimension equals number of stages, normalized so that (d, c) = 1	Generational reproductive value of stages in a cohort
Stable Cohort	Newborn cohort structure c with stage-structure c	A stable cohort’s offspring generations grow at rate R0
Level of Reproduction by Stable Cohort	R0	Net Reproductive Rate
Age-distribution of Stable Cohort Reproduction	${\varphi }_a=({\mathbf{d}}^T{\mathbf{F}}_a{\mathbf{L}}_a\mathbf{c})/R_0$, superscript indicates transposed vector	Fraction of stable cohort’s reproduction that occurs at age a.
Mean Age of Reproduction by a Cohort	$T_c={\sum }_{a}a{\varphi }_a$	Also called Cohort Generation Time
Mean Square Age of Reproduction	${\mu }_2={\sum }_a{a}^2{\varphi }_a$
Age-dispersion of Reproduction	$V_a=({\mu }_2-T_c^2)$
Stage-dispersion of Reproduction	$V_s={\xi }_1-T_c^2$
	${\mathbf{B}}_1=(1/R_0){\sum }_{a}a{\mathbf{F}}_a{\mathbf{L}}_a$
	$\mathbf{Z}=\mathbf{c}\hspace{0.17em}{\mathbf{d}}^T$
	$\mathbf{h}={\{\mathbf{I}-([{\mathbf{A}}_0/R_0]-\mathbf{Z})\}}^{-1}$
	${\xi }_1=({\mathbf{d}}^T{\mathbf{B}}_1\mathbf{h}{\mathbf{B}}_1\mathbf{c})$
Simple stage-age structured approximation	$r_1=(\log R_0)/T_c$	R0 and Tc differ compared to Dublin-Lotka approximation
Better stage-age structured approximation	$r_2=r_1+[V_a+2V_s]{(\log R_0)}^2/2T_c^3$	takes age and stage dispersion of reproduction into account