Projects per year
Abstract
Density dependent Markov population processes in large populations of size N were shown by Kurtz (1970), (1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, and to exhibit stochastic fluctuations about this deterministic solution on the scale √N that can be approximated by a diffusion process. Here, motivated by an example from evolutionary biology, we are concerned with describing how such a process leaves an absorbing boundary. Initially, one or more of the populations is of size much smaller than N, and the length of time taken until all populations have sizes comparable to N then becomes infinite as N → ∞. Under suitable assumptions, we show that in the early stages of development, up to the time when all populations have sizes at least N1α for 1/3 < α < 1, the process can be accurately approximated in total variation by a Markov branching process. Thereafter, it is well approximated by the deterministic solution starting from the original initial point, but with a random time delay. Analogous behaviour is also established for a Markov process approaching an equilibrium on a boundary, where one or more of the populations become extinct.
Original language  English 

Pages (fromto)  11901211 
Number of pages  22 
Journal  Advances in Applied Probability 
Volume  47 
Issue number  4 
DOIs  
Publication status  Published  2015 
Keywords
 Markov population process
 boundary behaviour
 branching process
Projects
 2 Finished

Measurevalued analysis of stochastic populations
Klebaner, F., Barbour, A., Hamza, K. & Jagers, P.
Australian Research Council (ARC), Monash University, University of Melbourne, Chalmers Tekniska Högskola (Chalmers University of Technology)
1/07/15 → 30/12/18
Project: Research

Stochastic Populations: Theory and Applications
Klebaner, F., Barbour, A. P., Hamza, K. & Jagers, P.
Australian Research Council (ARC), University of Melbourne
3/01/12 → 30/09/15
Project: Research