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[[Category:Branching processes]]
A ''Bellmann Harris process'' is
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 {{Cite|BH}}. 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 $G(t)$ is the distribution function of the life times of the individual particles, if $h(s)$ is the generating function of the number of direct descendants of one particle, and if at time $t=0$ the age of the particle was zero, then the generating function $F(t,s)={\rm E}s^{\mu(t)}$ of the number of particles $\mu(t)$ satisfies the non-linear integral equation

$$F(t,s) = \int_0^th(F(t-u,s))dG(u) + s(1-G(t)).$$
If

$$G(t)=1-e^{-\lambda t},\quad t\ge 0,$$
the Bellman–Harris process is a Markov branching process with continuous time.

====References====
{| 
|valign="top"|{{Ref|BH}}|| R. Bellman,   T.E. Harris,   "On the theory of age-dependent stochastic branching processes"  ''Proc. Nat. Acad. Sci. USA'' , '''34''' (1948) pp. 601–604 {{MR|0027466}} 
|}