On some modes of population growth leading to R. A. Fisher's logarithmic series distribution
D. Kendall. Biometrika, (1948)The author discusses a ``birth-and-death process'' for which the Kolmogorov differential equations assume the form $$ P_n'(t)=-n(a+b)+cP_n(t)+(n-1)a+cP_n-1(t)\\ +(n+1)bP_n+1(t), $$ where $P_n(t)$ is the probability of a population size $n$. The case $c0$ corresponds to mortality and fertility proportional to the actual population size. The $c$-term accounts for an increase by immigration. The generating function of $P_n(t)$ is obtained and it is shown that for small $c$ one obtains approximations to R. A. Fisher's ``logarithmic series distribution'' which has found several applications in biology..
The author discusses a ``birth-and-death process'' for which the Kolmogorov differential equations assume the form $$ P_n'(t)=-n(a+b)+cP_n(t)+(n-1)a+cP_n-1(t)\\ +(n+1)bP_n+1(t), $$ where $P_n(t)$ is the probability of a population size $n$. The case $c0$ corresponds to mortality and fertility proportional to the actual population size. The $c$-term accounts for an increase by immigration. The generating function of $P_n(t)$ is obtained and it is shown that for small $c$ one obtains approximations to R. A. Fisher's ``logarithmic series distribution'' which has found several applications in biology.
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%0 Journal Article
%1 MR0026282
%A Kendall, David G.
%D 1948
%J Biometrika
%K birth_death_processes population_dynamics
%P 6--15
%T On some modes of population growth leading to R. A. Fisher's logarithmic series distribution
%V 35
@article{MR0026282,
added-at = {2009-08-17T00:34:14.000+0200},
author = {Kendall, David G.},
biburl = {https://www.bibsonomy.org/bibtex/27b9925172acc9b5568a7835726252eb2/peter.ralph},
description = {On some modes of population growth leading to {R}. {A}. {F}isher's logarithmic series distribution},
fjournal = {Biometrika},
interhash = {eb31806143fa4a37a1c88c67127a5172},
intrahash = {7b9925172acc9b5568a7835726252eb2},
issn = {0006-3444},
journal = {Biometrika},
keywords = {birth_death_processes population_dynamics},
mrclass = {60.0X},
mrnumber = {MR0026282 (10,133a)},
mrreviewer = {W. Feller},
note = {The author discusses a ``birth-and-death process'' for which the Kolmogorov differential equations assume the form $$ \multline P_n{}'(t)=-{n(a+b)+c}P_n(t)+{(n-1)a+c}P_{n-1}(t)\\ +(n+1)bP_{n+1}(t), \endmultline $$ where $P_n(t)$ is the probability of a population size $n$. The case $c\equiv 0$ corresponds to mortality and fertility proportional to the actual population size. The $c$-term accounts for an increase by immigration. The generating function of $P_n(t)$ is obtained and it is shown that for small $c$ one obtains approximations to R. A. Fisher's ``logarithmic series distribution'' which has found several applications in biology.},
pages = {6--15},
timestamp = {2009-08-17T00:34:14.000+0200},
title = {On some modes of population growth leading to {R}. {A}. {F}isher's logarithmic series distribution},
volume = 35,
year = 1948
}