The extinction time of a birth, death and catastrophe process and of a related diffusion model

1985 ◽  
Vol 17 (01) ◽  
pp. 42-52 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Population Size ◽  
Diffusion Processes ◽  
Death Process ◽  
Arbitrary Distribution ◽  
Initial Population ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction ◽  
Initial Population Size

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

The extinction time of a birth, death and catastrophe process and of a related diffusion model

1985 ◽  
Vol 17 (1) ◽  
pp. 42-52 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Population Size ◽  
Diffusion Processes ◽  
Death Process ◽  
Arbitrary Distribution ◽  
Initial Population ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction ◽  
Initial Population Size

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (02) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (2) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes

1983 ◽  
Vol 20 (2) ◽  
pp. 305-321 ◽  
Author(s):  
J. Grasman ◽  
D. Ludwig
Keyword(s):  
Stochastic Processes ◽  
Diffusion Approximation ◽  
Transition Matrix ◽  
Branching Process ◽  
Substantial Improvement ◽  
Death Process ◽  
Asymptotic Approximations ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

Extinction times for a general birth, death and catastrophe process

2004 ◽  
Vol 41 (4) ◽  
pp. 1211-1218 ◽  
Author(s):  
Ben Cairns ◽  
P. K. Pollett
Keyword(s):  
Population Size ◽  
Rate Coefficients ◽  
Death Process ◽  
Transition Rates ◽  
Linear Rate ◽  
Expected Time ◽  
Current Population ◽  
Time To Extinction ◽  
Limited Class ◽  
Birth Death

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

scholarly journals Experimental demonstration of a two-phase population extinction hazard

2011 ◽  
Vol 8 (63) ◽  
pp. 1472-1479 ◽  
Author(s):  
John M. Drake ◽  
Jeff Shapiro ◽  
Blaine D. Griffen
Keyword(s):  
Population Size ◽  
Time Distribution ◽  
Proportional Hazards ◽  
Initial Population ◽  
Extinction Time ◽  
Habitat Size ◽  
Two Phase ◽  
Population Extinction ◽  
Initial Population Size ◽  
Phase Population

Population extinction is a fundamental biological process with applications to ecology, epidemiology, immunology, conservation biology and genetics. Although a monotonic relationship between initial population size and mean extinction time is predicted by virtually all theoretical models, attempts at empirical demonstration have been equivocal. We suggest that this anomaly is best explained with reference to the transient properties of ensembles of populations. Specifically, we submit that under experimental conditions, many populations escape their initially vulnerable state to reach quasi-stationarity, where effects of initial conditions are erased. Thus, extinction of populations initialized far from quasi-stationarity may be exposed to a two-phase extinction hazard. An empirical prediction of this theory is that the fit Cox proportional hazards regression model for the observed survival time distribution of a group of populations will be shown to violate the proportional hazards assumption early in the experiment, but not at later times. We report results of two experiments with the cladoceran zooplankton Daphnia magna designed to exhibit this phenomenon. In one experiment, habitat size was also varied. Statistical analysis showed that in one of these experiments a transformation occurred so that very early in the experiment there existed a transient phase during which the extinction hazard was primarily owing to the initial population size, and that this was gradually replaced by a more stable quasi-stationary phase. In the second experiment, only habitat size unambiguously displayed an effect. Analysis of data pooled from both experiments suggests that the overall extinction time distribution in this system results from the mixture of extinctions during the initial rapid phase, during which the effects of initial population size can be considerable, and a longer quasi-stationary phase, during which only habitat size has an effect. These are the first results, to our knowledge, of a two-phase population extinction process.

scholarly journals On the times of births in a linear birthprocess

1971 ◽  
Vol 12 (4) ◽  
pp. 473-475 ◽  
Author(s):  
Marcel F. Neuts ◽  
Sidney I. Resnick
Keyword(s):  
Population Size ◽  
Order Statistics ◽  
Random Variables ◽  
Independent Random Variables ◽  
Death Process ◽  
Initial Population ◽  
Birth Process ◽  
Initial Population Size ◽  
The Times

AbstractIf S1, S2,… are the times of successive births in a pure birth-process with linear birthrate and i > 0 individuals initially, then given that there are k > 0 births in (0, t), the random variables eλS1—1, eλS2−1,λS1−1 are distributed (conditionally) like the order statistics of k independent random variables, all uniformly distributed on the interval (0, eλt–1), regardless of the initial population size. A similar property holds for the pure death process.

scholarly journals Extinction time of the logistic process

2021 ◽  
Vol 58 (3) ◽  
pp. 637-676
Author(s):  
Eric Foxall
Keyword(s):  
Population Model ◽  
System Size ◽  
Complete Classification ◽  
Reproductive Rate ◽  
Death Process ◽  
Initial Population ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Stochastic Population Model

AbstractThe logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction and a phase transition, and a lot can be learned about the process by studying its extinction time, $\tau_n$ , as a function of system size n. A number of existing results describe the scaling of $\tau_n$ as $n\to\infty$ for various choices of reproductive rate $r_n$ and initial population $X_n(0)$ as a function of n. We collect and complete this picture, obtaining a complete classification of all sequences $(r_n)$ and $(X_n(0))$ for which there exist rescaling parameters $(s_n)$ and $(t_n)$ such that $(\tau_n-t_n)/s_n$ converges in distribution as $n\to\infty$ , and identifying the limits in each case.

The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes

1983 ◽  
Vol 20 (02) ◽  
pp. 305-321
Author(s):  
J. Grasman ◽  
D. Ludwig
Keyword(s):  
Stochastic Processes ◽  
Diffusion Approximation ◽  
Transition Matrix ◽  
Branching Process ◽  
Substantial Improvement ◽  
Death Process ◽  
Asymptotic Approximations ◽  
Birth And Death Process ◽  
Expected Time ◽  
Time To Extinction

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

Extinction times for a general birth, death and catastrophe process

2004 ◽  
Vol 41 (04) ◽  
pp. 1211-1218 ◽  
Author(s):  
Ben Cairns ◽  
P. K. Pollett
Keyword(s):  
Population Size ◽  
Rate Coefficients ◽  
Death Process ◽  
Transition Rates ◽  
Linear Rate ◽  
Expected Time ◽  
Current Population ◽  
Time To Extinction ◽  
Limited Class ◽  
Birth Death

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

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