Extinction probabilities in microbial predation: A birth and death approach

1981 ◽  
Vol 43 (2) ◽  
pp. 165-181 ◽  
Author(s):  
Gregory Stephanopoulos ◽  
A. G. Fredrickson
Keyword(s):  
Extinction Probabilities ◽  
Microbial Predation

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

On determining absorption probabilities for Markov chains in random environments

1981 ◽  
Vol 13 (2) ◽  
pp. 369-387 ◽  
Author(s):  
Richard D. Bourgin ◽  
Robert Cogburn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Efficient Method ◽  
General Framework ◽  
Random Environments ◽  
Extinction Probabilities ◽  
Environmental Process ◽  
Branching Chains ◽  
Absorption Probabilities

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

1976 ◽  
Vol 8 (2) ◽  
pp. 278-295 ◽  
Author(s):  
Michael Sze
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Continuous Process ◽  
Regularly Varying Functions ◽  
Original Process ◽  
Additional Information ◽  
Embedding Technique ◽  
Extinction Probabilities ◽  
Watson Process ◽  
The Given

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

scholarly journals Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (03) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

Inhibition of inter-microbial predation by chlorinated hydrocarbons

Nature ◽  
1974 ◽  
Vol 249 (5458) ◽  
pp. 673-674 ◽  
Author(s):  
F. WALSH ◽  
R. MITCHELL
Keyword(s):  
Chlorinated Hydrocarbons ◽  
Microbial Predation

Microbial Predation Dynamics

1983 ◽  
pp. 253-264 ◽  
Author(s):  
M. J. BAZIN ◽  
C. CURDS ◽  
A. DAUPPE ◽  
B. A. OWEN ◽  
P. T. SAUNDERS
Keyword(s):  
Microbial Predation

Forward and backward processes in bisexual models with fixed population sizes

1994 ◽  
Vol 31 (02) ◽  
pp. 309-332 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Overlapping Generations ◽  
Weak Limit ◽  
Fixed Number ◽  
Uhlenbeck Process ◽  
Fisher Model ◽  
Extinction Probabilities ◽  
Ornstein Uhlenbeck Process ◽  
Population Sizes ◽  
Watson Process ◽  
Fixed Population

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random. First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N. Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

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