Crump-Mode-Jagers branching process: a numerical approach

Crump-Mode-Jagers branching process: a numerical
approach
Plamen Trayanov, plament@fmi.uni-soa.bg
Department of Probability, Operations Research and Statistics, Faculty of
Mathematics and Informatics, Soa University "St. Kliment Ohridski", 5, J.
Bourchier Blvd, 1164 Soa, Bulgaria
Keywords:
General Branching Process, Leslie matrix projection, demographics,
numerical method, renewal equation
AMS:
60J80
Abstract
The theory of Crump-Mode-Jagers branching processes presents the expected future
population as a solution of a renewal equation (see Jagers [1]).
As explained by
the Renewal Theory, the theoretical solution of this equation is a convolution of
two functions, one of which is called renewal function (see Mitov and Omey [2]).
However in practice it is very time consuming to calculate the renewal function as a
sum of convolutions with increasing order. This paper presents a General Branching
Process model (GBP) relevant for the special case of human population and describes
a numerical method for solving the corresponding renewal equation. The presented
numerical method involves only simple matrix multiplications which results in a very
fast calculation speed. Finally it is shown that the Leslie matrix projection, widely
used in demographics, is actually a special case of the presented numerical solution
and thus shows that this standard demographic method is actually related to the
theory of Crump-Mode-Jagers branching process.
Acknowledgements:
The research was supported by the National Fund for Scien-
tic Research at the Ministry of Education and Science of Bulgaria, grant No DFNI
I02/17.
References
[1] Jagers, P. (1975).
Branching Processes with Biological Applications.
John Wiley
& Sons Ltd.
[2] Mitov, K., Omey, E. (2013).
Renewal Processes.
Springer.
III Workshop on Branching Processes and their Applications
April 7-10, 2015
Badajoz (Spain)