Crump-Mode-Jagers branching process: a numerical approach

Crump-Mode-Jagers branching process: a numerical
Plamen Trayanov, [email protected]
Department of Probability, Operations Research and Statistics, Faculty of
Mathematics and Informatics, Soa University "St. Kliment Ohridski", 5, J.
Bourchier Blvd, 1164 Soa, Bulgaria
General Branching Process, Leslie matrix projection, demographics,
numerical method, renewal equation
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.
The research was supported by the National Fund for Scien-
tic Research at the Ministry of Education and Science of Bulgaria, grant No DFNI
[1] Jagers, P. (1975).
Branching Processes with Biological Applications.
John Wiley
& Sons Ltd.
[2] Mitov, K., Omey, E. (2013).
Renewal Processes.
III Workshop on Branching Processes and their Applications
April 7-10, 2015
Badajoz (Spain)