Table 3:

Stage and Age: Stable Population

Quantity	Equation	Notes
Stable Population	A population with unchanging stage-and-age structure and exponential growth rate r	Fitness equals r
Stable Stage-structure	In stable population, each age a has unchanging stage structure. New-borns have stable stage-structure u1. Then can deduce stable stage-structure for every age	If only age matters, There is effectively just one stage, and u1 collapses to a number
Renewal equation	${\sum }_a{e}^{-ra}{\mathbf{F}}_a{\mathbf{L}}_a{\mathbf{u}}_1={\mathbf{u}}_1$	Says that discounted reproduction by stable stage-structure must produce same structure
Renewal matrix	$\mathbf{A}(r)={\sum }_a{e}^{-ra}{\mathbf{F}}_a{\mathbf{L}}_a$	This matrix has dominant eigenvalue 1 and corresponding right eigenvector u1