On the emergence of random initial conditions in fluid limits. (English) Zbl 1356.60053

Barbour, A. D.; Chigansky, P.; Klebaner, F. C.

J. Appl. Probab. 53, No. 4, 1193-1205 (2016).

Summary: In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of T. G. Kurtz [J. Appl. Probab. 7, 49–58 (1970; Zbl 0191.47301)], such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to \(\infty\), the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.

MSC:

60F17	Functional limit theorems; invariance principles
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F05	Central limit and other weak theorems
92D25	Population dynamics (general)

Keywords:

birth-death process; population dynamics; fluid limits; carrying capacity

Citations:

Zbl 0191.47301