Abstract
We consider models of population size dependent branching processes with the feature that they are supercritical when population reaches some threshold K, near critical around that value, and subcritical below it. Although such population die out with probability one, their time to extinction is large. We show that this time is exponential in K. Approximations to the populations size in various domains are given, and a problem of small initial population size is discussed. From a technical point of view, analysis of such processes involves techniques of density dependent models as small random perturbations of dynamical systems, and sizedependent bounds.
Original language  English 

Title of host publication  Lecture Notes in Statistics  Proceedings 
Editors  M V Gonzalez, I M del Puerto, R Martinez, M Molina, M Mota, A Ramos 
Place of Publication  Germany 
Publisher  SpringerVerlag London Ltd. 
Pages  71  78 
Number of pages  8 
Volume  197 
ISBN (Print)  9783642111549 
DOIs  
Publication status  Published  2010 
Event  Workshop on Branching Processes and their Applications 2009  Badajoz, Spain Duration: 20 Apr 2009 → 23 Apr 2009 Conference number: 1st http://www.springer.com/us/book/9783642111549 (Conference Proceedings) 
Conference
Conference  Workshop on Branching Processes and their Applications 2009 

Abbreviated title  WBPA 2009 
Country  Spain 
City  Badajoz 
Period  20/04/09 → 23/04/09 
Internet address 

Cite this
Klebaner, F. C. (2010). Approximations in populationdependent branching processes. In M. V. Gonzalez, I. M. del Puerto, R. Martinez, M. Molina, M. Mota, & A. Ramos (Eds.), Lecture Notes in Statistics  Proceedings (Vol. 197, pp. 71  78). SpringerVerlag London Ltd.. https://doi.org/10.1007/9783642111563_5