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Difference between revisions of "Bellman-Harris process"

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{{MSC|60J80}}
 
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[[Category:Branching processes]]
 
[[Category:Branching processes]]
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A ''Bellmann Harris process'' is
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a special case of an age-dependent branching process (cf.
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[[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
  
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 <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
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$$F(t,s) = \int_0^th(F(t-u,s))dG(u) + s(1-G(t)).$$
 
 
<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
 
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>
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$$G(t)=1-e^{-\lambda t},\quad t\ge 0,$$
 
 
 
the Bellman–Harris process is a Markov branching process with continuous time.
 
the Bellman–Harris process is a Markov branching process with continuous time.
  
 
====References====
 
====References====
{|
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{| |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}} |}
|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}}
 
|}
 

Revision as of 19:31, 28 October 2014

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL] A Bellmann Harris process is a special case of an age-dependent branching process (cf. Branching process, age-dependent). It was first studied by R. Bellman and T.E. Harris [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

How to Cite This Entry:
Bellman-Harris process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bellman-Harris_process&oldid=26348
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article