Survival probabilities and extinction times for some multitype branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 446-462 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Functions ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Survival Probabilities ◽  
Time To Extinction ◽  
Special Cases ◽  
Multitype Branching Processes ◽  
Mean Time To Extinction ◽  
Mean Time

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

Survival probabilities and extinction times for some multitype branching processes

1974 ◽  
Vol 6 (03) ◽  
pp. 446-462 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Functions ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Survival Probabilities ◽  
Time To Extinction ◽  
Special Cases ◽  
Multitype Branching Processes ◽  
Mean Time To Extinction ◽  
Mean Time

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.

On survival probabilities and extinction times for some branching processes

1971 ◽  
Vol 8 (04) ◽  
pp. 633-654 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Lower Bounds ◽  
Generating Function ◽  
Negative Binomial ◽  
Positive Root ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
Mean Time

Summary We consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn (t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn (0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.

On survival probabilities and extinction times for some branching processes

1971 ◽  
Vol 8 (4) ◽  
pp. 633-654 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Lower Bounds ◽  
Generating Function ◽  
Negative Binomial ◽  
Positive Root ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Single Individual ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
Mean Time

SummaryWe consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn(t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn(0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (2) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (02) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

On the Extinction of the S–I–S stochastic logistic epidemic

1989 ◽  
Vol 26 (04) ◽  
pp. 685-694
Author(s):  
Richard J. Kryscio ◽  
Claude Lefèvre
Keyword(s):  
Chemical Reactions ◽  
Logistic Model ◽  
Diffusion Processes ◽  
Birth And Death Processes ◽  
Time To Extinction ◽  
Stochastic Logistic Model ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time ◽  
Quasi Stationary Distribution

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.

scholarly journals Evaluating an icon of population persistence: the Devil's Hole pupfish

2014 ◽  
Vol 281 (1794) ◽  
pp. 20141648 ◽  
Author(s):  
J. Michael Reed ◽  
Craig A. Stockwell
Keyword(s):  
Count Data ◽  
Small Populations ◽  
Vulnerable Species ◽  
Human Intervention ◽  
Microsatellite Data ◽  
Time To Extinction ◽  
Credible Intervals ◽  
Mean Time To Extinction ◽  
Iconic Status ◽  
Mean Time

The Devil's Hole pupfish Cyprinodon diabolis has iconic status among conservation biologists because it is one of the World's most vulnerable species. Furthermore, C. diabolis is the most widely cited example of a persistent, small, isolated vertebrate population; a chronic exception to the rule that small populations do not persist long in isolation. It is widely asserted that this species has persisted in small numbers (less than 400 adults) for 10 000–20 000 years, but this assertion has never been evaluated. Here, we analyse the time series of count data for this species, and we estimate time to coalescence from microsatellite data to evaluate this hypothesis. We conclude that mean time to extinction is approximately 360–2900 years (median 410–1800), with less than a 2.1% probability of persisting 10 000 years. Median times to coalescence varied from 217 to 2530 years, but all five approximations had wide credible intervals. Our analyses suggest that Devil's Hole pupfish colonized this pool well after the Pleistocene Lakes receded, probably within the last few hundred to few thousand years; this could have occurred through human intervention.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (03) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

A bounded growth population subjected to emigrations due to population pressure

1981 ◽  
Vol 18 (03) ◽  
pp. 571-582 ◽  
Author(s):  
A. C. Trajstman
Keyword(s):  
Size Distribution ◽  
Population Level ◽  
Population Pressure ◽  
Population Increase ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
The Times ◽  
Negative Exponential ◽  
Mean Time

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure. The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.

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