Discrete Time Branching Process

2014 ◽  
pp. 17-46
Author(s):  
Rinaldo B. Schinazi
Keyword(s):  
Discrete Time ◽  
Branching Process

Multitype processes with reproduction-dependent immigration

1998 ◽  
Vol 35 (02) ◽  
pp. 281-292
Author(s):  
Ibrahim Rahimov
Keyword(s):  
Limit Theorem ◽  
Discrete Time ◽  
Branching Process ◽  
Martingale Approach ◽  
Offspring Distribution ◽  
Second Moments ◽  
Branching Process With Immigration

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.

On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

1966 ◽  
Vol 3 (02) ◽  
pp. 403-434 ◽  
Author(s):  
E. Seneta ◽  
D. Vere-Jones
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Recent Work ◽  
Discrete Time ◽  
Branching Process ◽  
Stationary Distributions

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (02) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (4) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

scholarly journals A Branching Process Model of Evolutionary Rescue

2021 ◽  
Author(s):  
Peter Olofsson ◽  
Ricardo B. R. Azevedo
Keyword(s):  
Population Growth ◽  
Mutation Rate ◽  
Discrete Time ◽  
Growth Curve ◽  
Process Model ◽  
Branching Process ◽  
Selective Advantage ◽  
Population Declines ◽  
Beneficial Mutations ◽  
Evolutionary Rescue

Evolutionary rescue is the process whereby a declining population may start growing again, thus avoiding extinction, via an increase in the frequency of beneficial genotypes. These genotypes may either already be present in the population in small numbers, or arise by mutation as the population declines. We present a simple two-type discrete-time branching process model and use it to obtain results such as the probability of rescue, the shape of the population growth curve of a rescued population, and the time until the first rescuing mutation occurs. Comparisons are made to existing results in the literature in cases where both the mutation rate and the selective advantage of the beneficial mutations are small.

A branching-process solution of the polydisperse coagulation equation

1984 ◽  
Vol 16 (01) ◽  
pp. 56-69 ◽  
Author(s):  
John L. Spouge
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Time ◽  
Coagulation Equation ◽  
Multitype Branching Processes ◽  
Irreversible Aggregation ◽  
Image Position ◽  
Supercritical Branching Processes ◽  
Critical Branching Process

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time tc when diverges corresponds to a critical branching process, while post-critical times t&gt; tc correspond to supercritical branching processes.

Moments for a general branching process in a semi-Markovian environment

1980 ◽  
Vol 17 (2) ◽  
pp. 341-349 ◽  
Author(s):  
Craig Whittaker ◽  
Richard M. Feldman
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Markovian Environment ◽  
Iterative Computation ◽  
Second Moments ◽  
Environmental Process

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

1989 ◽  
Vol 26 (3) ◽  
pp. 431-445 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Limit Distribution ◽  
Branching Process ◽  
Linear Growth ◽  
Left Eigenvector ◽  
Size Dependent ◽  
Second Moments ◽  
The Mean ◽  
Critical Process

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

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