Crump-Mode-Jagers branching process: a numerical approach 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 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)