Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T17:14:55.885Z Has data issue: false hasContentIssue false

On the theory of birth, death and diffusion processes

Published online by Cambridge University Press:  14 July 2016

A. W. Davis*
Affiliation:
Australian National University

Extract

Several authors have recently discussed the asymptotic properties of stochastic populations which diffuse randomly throughout a given region. Sevast'yanov ([8], [9]) has investigated the extinction probability of a Markovian population in a compact region with an absorbing boundary, his analysis being in terms of “generation times”. Adke and Moyal have considered the spatial dispersion of a population which multiplies according to a simple time-dependent birth-and-death process and undergoes Gaussian diffusion on the real line ([2] and [3]) or on a finite interval with reflecting boundaries [1]. A serious limitation in Adke and Moyal's asymptotic results is that they are conditional upon a finite number of survivors. Moyal [7] has also obtained some basic formulae for a Markovian population diffusing through a general space.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adke, S. R. (1964) A stochastic population diffusing on a finite interval. J. Ind. Statist. Assoc. 2, 3240.Google Scholar
[2] Adke, S. R. The generalized birth and death process and Gaussian diffusion. To appear in J. Math. Anal. Appl. Google Scholar
[3] Adke, S. R. and Moyal, J. E. (1963) A birth, death and diffusion process. J. Math. Anal Appl. 7, 209224.CrossRefGoogle Scholar
[4] Harris, R. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[5] Levinson, N. (1960) Limiting theorems for age-dependent branching processes. Illinois J. Math. 4, 100118.Google Scholar
[6] Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 131.Google Scholar
[7] Moyal, J. E. (1964) Multiplicative population processes. J. Appl. Prob. 1, 267283.CrossRefGoogle Scholar
[8] Sevast'Yanov, B. A. (1958) Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries. Theor. Prob. Appl. 3, 111126.Google Scholar
[9] Sevast'Yanov, B. A. (1961) Extinction conditions for branching stochastic processes with diffusion. Theor. Prob. Appl. 6, 253263.Google Scholar