The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology

1969 ◽  
Vol 6 (2) ◽  
pp. 249-260 ◽  
Author(s):  
P. Jagers
Keyword(s):  
Branching Process ◽  
The Other ◽  
Mean Square ◽  
Random Functions ◽  
Age Dependent ◽  
Random Limit

Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process.

The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology

1969 ◽  
Vol 6 (02) ◽  
pp. 249-260 ◽  
Author(s):  
P. Jagers
Keyword(s):  
Branching Process ◽  
The Other ◽  
Mean Square ◽  
Random Functions ◽  
Age Dependent ◽  
Random Limit

Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (4) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

SummaryIn a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (04) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

Summary In a standard age-dependent branching process, let Rn (t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn (t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn (t) is asymptotically linear in t. Further, it is found that, for large n, Rn (t) has the shape of a normal density function (of t). Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn (t)}. For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (02) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk (t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk (t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (3) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (2) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk(t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk(t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (04) ◽  
pp. 678-686
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

Mean-square and almost-sure convergence of supercritical age-dependent branching processes

1974 ◽  
Vol 11 (4) ◽  
pp. 678-686 ◽  
Author(s):  
Edgar Z. Ganuza ◽  
S. D. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Age Distribution ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Mean Square Convergence ◽  
Watson Process ◽  
Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

An age dependent branching process with variable lifetime distribution

1972 ◽  
Vol 4 (03) ◽  
pp. 453-474 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Integral Equations ◽  
Branching Process ◽  
Probability Generating Function ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Distribution Functions ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
Number Of Particles

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {G i (t)}, together with a probability generating function, h(s) = Σ k ∞= 0pks k . An ith generation particle lives a random length of time, determined by G i (t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution G i + 1 (t). Let Z i (t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Z i (t) and it is shown that for some constants N i , γ, a, Z i (t)/(N i t γ-1 e αt ) converges in mean square to a proper random variable.

The asymptotic frequencies of the types in a multitype age-dependent branching process allowing immigration

1973 ◽  
Vol 10 (03) ◽  
pp. 652-658
Author(s):  
J. Radcliffe
Keyword(s):  
Renewal Process ◽  
Branching Process ◽  
Almost Sure Convergence ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean

The mean square and almost sure convergence of W(t) = e–αt Z(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

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