On the emergence of random initial conditions in fluid limits. (9th December 2016)

• Record Type: Journal Article
• Authors: Barbour, A. D.; Chigansky, P.; Klebaner, F. C.
• Is Part Of: Journal of applied probability. Volume 53:Number 4(2016)
• Page Start: 1193
• Page End: 1205
• Publication Date: 2016-12-09
• Subjects: Birth‒death process, -- population dynamics with carrying capacity, -- fluid approximation; Primary 60J80, -- Secondary 92D25, -- 60F05; 519.2
• DOI: 10.1017/jpr.2016.74
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: 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 Kurtz (1970), 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 ∞, 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.