Here we examine the limiting growth characteristics of a multistage age- structured population of the type considered by {\it H. Inaba} [Math. Popul. Stud. 1, 49 - 77 (1988)]. We prove a theorem on weak ergodicity of nonautonomous populations. This theorem generalizes Inaba’s results concerning weak ergodicity [Math. Biosci. 96, No. 2, 195 - 219 (1989;

Zbl 0698.92020)] and strong ergodicity. The method of proof is based on ideas similar to Inaba. However, we restrict our investigation to a space of real-valued bounded functions with the supremum norm. This allows us to omit the lattice-theoretic apparatus used by Inaba in his 1989 - paper.
The plan of the paper is as follows. In Section 2 we precisely formulate the problem, while Section 3 introduces the property of asymptotic similarity. The notion of asymptotic similarity describes the same features of the multiplicative process was weak ergodicity, but in our case it is more convenient. Section 4 presents our main result on the growth properties of the system, and this result is proved in Section 5. Finally, Section 6 considers both autonomous and nonautonomous growth properties using that main result.