Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

1999 ◽  
Vol 36 (02) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang

A population-size-dependent branching process {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ &gt; 1. When the environmental sequence {ξ n } is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q &lt; 1) are studied.


1984 ◽  
Vol 16 (1) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.


1984 ◽  
Vol 21 (01) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner

We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m &gt; 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.


1999 ◽  
Vol 36 (1) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.


1999 ◽  
Vol 36 (01) ◽  
pp. 146-154 ◽  
Author(s):  
Han-xing Wang

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Z n } n≥0 is associated with the stationary environment ξ− = {ξ n } n≥0, let B = {ω : Z n (ω) = for some n}, and q(ξ−) = P(B | ξ−, Z 0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) &lt; 1) = 1) are obtained for the model.


1984 ◽  
Vol 16 (01) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner

We consider a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m &lt; l, m = 1, m&gt; l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m &lt; 1 or m = 1 and mn approaches 1 not slower than n –2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n –1, then Zn /n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an , such that Zn /an converges in probability to a non-degenerate limit. If mn approaches m &gt; 1 not slower than n– α, α &gt; 0, and do not grow to ∞ faster than nß , β &lt;1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.


2004 ◽  
Vol 41 (1) ◽  
pp. 176-186 ◽  
Author(s):  
Lu Zhunwei ◽  
Peter Jagers

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α1 ≤α0 < + ∞ and 0 < C1, C2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C1k-α0 and above by C2k-α1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.


1984 ◽  
Vol 21 (1) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner

We consider a branching-process model {Zn}, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn} necessary conditions for convergence in L1 and L2 and sufficient conditions for almost sure convergence and convergence in L2 of Wn = Zn/mn are given.


2016 ◽  
Vol 53 (2) ◽  
pp. 614-621
Author(s):  
K. B. Athreya ◽  
H.-J. Schuh

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.


2004 ◽  
Vol 41 (01) ◽  
pp. 176-186 ◽  
Author(s):  
Lu Zhunwei ◽  
Peter Jagers

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 &lt; α 1 ≤α 0 &lt; + ∞ and 0 &lt; C 1, C 2 &lt; + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.


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