Age-structured population model

An age-structured population model tracks a density $n(a,t)$ transported toward greater age, reduced by age-dependent mortality, and replenished at age zero by births.

The continuous-age transport version is the Von Foerster equation. Discrete age-class models are other members of this broader modeling class.

The Von Foerster equation transports a population density through age while mortality removes individuals. A birth boundary condition at age zero closes the model; for a prescribed fertility rate it has the form $n(0,t)=\int b(a,t)n(a,t)da$.

A seed-bank population recurrence includes delayed recruitment from seeds that survive more than one season. For $a,b\ge 0$, recruitment function $N$ with $N(0)=0$ and $N^{\prime }(0)=1$ gives extinction multipliers solving ${\lambda }^2-a\lambda -b=0$. If $N$ is strictly concave, bounded and increasing, a positive fixed point exists uniquely exactly when $a+b>1$; it solves $M=N((a+b)M)$. The dominant extinction multiplier then exceeds one.

For birth rate $b(a)$, death rate $d(a)$ and exponential population growth rate $r$, the Euler-Lotka equation is

For serial stages with survival functions $S_i(a)=\exp [-{\int }_0^a{\Gamma }_i(s)ds]$, set $H_i(r)={\int }_0^{\infty }{\Gamma }_i(a)S_i(a)e^{-ra}da$. If completion of stage two produces $m$ new stage-one individuals, a common exponential mode requires $m{H}_1(r)H_2(r)=1$. Indeed its age profiles are ${\varphi }_i(a)={\varphi }_i(0)e^{-ra}{S}_i(a)$, while the incoming fluxes give ${\varphi }_2(0)=H_1{\varphi }_1(0)$ and ${\varphi }_1(0)=m{H}_2{\varphi }_2(0)$. Cancelling nonzero amplitudes proves the equation. Under ordinary positive residence-time assumptions a positive root exists exactly when $m{H}_1(0)H_2(0)>1$. The normalized profiles are $e^{-ra}{S}_i(a)/\int e^{-rs}{S}_i(s)ds$.

For annual survival probabilities $1-{\mu }_i$, age-specific offspring numbers $b_a$, and a population multiplier $\gamma$, the discrete Euler-Lotka equation isIts value at $\gamma =1$ is the expected lifetime offspring count of a newborn under the fixed vital rates. A value above one implies a growing mode $\gamma >1$, while a value below one implies decline.