Bellman-Harris Process Research Articles

Almost sure limit results for the supercritical Bellman-Harris process

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

Norming constants for the finite mean supercritical Bellman-Harris process

Let {Z(t)} be a supercritical Bellman-Harris process with offspring distribution {pk} and lifetime distributionG. It is shown that the finiteness of the offspring mean guarantees the existence of norming constants {C(t)} such that\(\mathop {\lim }\limits_{t \to \infty } {{Z(t)} \mathord{\left/ {\vphantom {{Z(t)} {C(t)}}} \right. \kern-\nulldelimiterspace} {C(t)}} = W\) a.s. for some nondegenerate random variableW. C(t) is theμ-quantile of the distribution function ofZ(t), whereq<μ<1,q being the extinction probability of the process. As a byproduct of the proof, {Z(t)/C(t)} is shown to be “asymptotic” Markov. The theory of weakly stable sums of i.i.d. is used to get characterizations ofW and {C(t)}.

Convergence of age and type distributions in multitype critical Bellman-Harris processes

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.

Convergence of age and type distributions in multitype critical Bellman-Harris processes

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.

A NEW LIMIT THEOREM FOR THE CRITICAL BELLMAN-HARRIS BRANCHING PROCESS

Let be the number of particles in a Bellman-Harris process at time , the distribution function of the lifetimes of the particles, the generating function of the number of offspring of one particle, and .In the case when , where and is slowly varying as , and , as , it is shown that for a function equal either to 1 or to .Bibliography: 11 titles.

On age-dependent branching processes with immigration

We derive the probability generating function for the general Bellman—Harris age dependent branching process with immigration emphasizing the role of immigration parameters. The solution requires solving a single scalar-valued integral equation, namely the usual Bellman—Harris equation. Our results are applied to three particular examples. We derive the equations for the immigration of particles governed by a Bellman—Harris process into a second Bellman—Harris process. In the second example we study Poisson immigration of particles into a time continuous Markov branching process. In the third, we derive the equations for a one-time immigration into a Bellman—Harris process.

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z ( k )( t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.

An actual Bellman-Harris epidemic process. The infection of septoria in barley or wheat

The possibility of a Bellman-Harris process is considered for the epidemic of Septoria in barley or wheat. The integral equations for the expected mean value and variance are solved assuming identical exponential shapes for the exudation of the pycnospores and the transformation function of the process. The predictions of the mathematical model are compared with earlier experimental results reported by Lutey & Fezer (1960).

Lag time in microbe growth as an age-dependent branching process with two phase types.

Abstract We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.

Lag time in microbe growth as an age-dependent branching process with two phase types

We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.

Compounding Bellman-Harris processes allowing immigration

A Bellman-Harris process is considered in which each batch N i of immigrants arriving at time T i founds a colony attracting a secondary stream of immigrants. The asymptotic behavior of the mean number of particles living at time t is obtained in various cases. Some limit theorems are proved for the total number of particles living at time t suitably normalized.

Conditioning a branching process on non-extinction

The device of conditioning a critical or subcritical branching process on non-extinction by time t, and letting t→∞, has been widely used to obtain non-trivial limit theorems. We examine here the implications of this conditioning on the early path of a Bellman-Harris process.

A note on the subcritical generalized age-dependent branching process

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

A note on the subcritical generalized age-dependent branching process

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

Limit theorems for a branching process with disasters

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Limit theorems for a branching process with disasters

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.

Convergence of the Age Distribution in the One-Dimensional Supercritical Age-Dependent Branching Process

The age distribution for a supercritical Bellman-Harris process is proven to converge in probability to a deterministic distribution under assumptions slightly more than finite first moment. If the usual $j \log j$ condition holds, then the convergence can be strengthened to hold w.p. 1. As a corollary to this result, the population size, properly normalized is shown to converge w.p. 1 to a nondegenerate random variable under the $j \log j$ assumption.

The First Birth Problem for an Age-dependent Branching Process

If $B_n$ denotes the time of the first birth in the $n$th generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant $\gamma$ such that $B_n/n \rightarrow \gamma$ as $n \rightarrow \infty$, almost surely on the event of ultimate survival. This strengthens a result of Hammersley, who proved convergence in probability for the more special Bellman-Harris process. The proof depends on a class of martingales which arise from a `collective marks' argument.

Some convergence rate results for the Bellman-Harris process with immigration

This article presents some rate of convergence results for P( X( t)= j), where { X( t), t⩾0} is a Bellman-Harris branching process with an immigration component.

On the subcritical Bellman-Harris process with immigration

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.