View source for Bellman-Harris process

{{MSC|60J80}}

[[Category:Branching processes]]

A special case of an age-dependent branching process (cf. [[Branching process, age-dependent|Branching process, age-dependent]]). It was first studied by R. Bellman and T.E. Harris [[#References|[1]]]. In the Bellman–Harris process it is assumed that particles live, independently of each other, for random periods of time, and produce a random number of new particles at the end of their life time. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154701.png" /> is the distribution function of the life times of the individual particles, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154702.png" /> is the generating function of the number of direct descendants of one particle, and if at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154703.png" /> the age of the particle was zero, then the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154704.png" /> of the number of particles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154705.png" /> satisfies the non-linear integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154706.png" /></td> </tr></table>

If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015470/b0154707.png" /></td> </tr></table>

the Bellman–Harris process is a Markov branching process with continuous time.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Bellman,   T.E. Harris,   "On the theory of age-dependent stochastic branching processes"  ''Proc. Nat. Acad. Sci. USA'' , '''34'''  (1948)  pp. 601–604</TD></TR></table>