= Separable-profile closure for an age-structured population
{title2=$\Lambda_i=\int_0^\infty\Gamma_i(a)f_i(a)\,da$}
For $n_i(a,t)=N_i(t)f_i(a)$ with $\int f_i=1$, integrating the <age-structured population equation> replaces its exit integral by $\Lambda_iN_i$. For one-to-one transition from stage one to stage two and two-to-one reproduction from stage two to stage one, the population <matrix> is $\left(\begin{smallmatrix}-\Lambda_1&2\Lambda_2\\\Lambda_1&-\Lambda_2\end{smallmatrix}\right)$. This follows by integrating the age <derivative> and substituting the incoming <boundary conditions>. Exact time-independent positive profiles also force a constant stage ratio and common exponential growth through those <boundary conditions>. Thus the constant-coefficient closure is not an exact transient description for arbitrary age-dependent <hazard functions> and arbitrary <initial conditions>; constant <hazard functions> are an important exception for the integrated populations.
Back to article page
