Separable-profile closure for an age-structured population (${\Lambda }_i={\int }_0^{\infty }{\Gamma }_i(a)f_i(a)da$)

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 ${\Lambda }_i{N}_i$. 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 $\left(\begin{array}{cc}-{\Lambda }_1& 2{\Lambda }_2\\ {\Lambda }_1& -{\Lambda }_2\end{array}\right)$. 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.