scholarly journals Extinction and explosion of nonlinear Markov branching processes

2007 ◽  
Vol 82 (3) ◽  
pp. 403-428 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Processes ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Expected Time ◽  
Time To Extinction ◽  
Positive Jump ◽  
Unique Invariant Measure ◽  
Necessary And Sufficient ◽  
Forward Equations

AbstractThis paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (4) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

scholarly journals Boundary-value problems in a layer for evolutionary pseudo-differential equations with integral conditions

2018 ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Well Posedness ◽  
Integral Conditions ◽  
Pseudo Differential Equation ◽  
Necessary And Sufficient ◽  
Well Posed

Boundary-value problems for evolutionary pseudo-differential equations with an integral condition are studied. Necessary and sufficient conditions of well-posedness are obtained for these problems in the Schwartz spaces. Existence of a well-posed boundary-value problem is proved for each evolutionary pseudo-differential equation.

Inequalities for branching processes

1966 ◽  
Vol 3 (01) ◽  
pp. 261-267 ◽  
Author(s):  
C. R. Heathcote ◽  
E. Seneta
Keyword(s):  
Generating Function ◽  
Conditional Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Random Variable ◽  
Expected Time ◽  
Time To Extinction ◽  
Explicit Formulae

Summary If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn (s) = Fn– 1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn (s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim n→∞ m −n {1−Fn (s)}, 0 ≦ s ≦ 1 where m = F′ (1) < 1 and F′′ (1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn (s) − Fn (0)} [1 – Fn (0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

Branching processes in generalized autoregressive conditional environments

2016 ◽  
Vol 48 (4) ◽  
pp. 1211-1234 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Ebola Virus ◽  
Negative Binomial ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Offspring Number ◽  
Generalized Poisson ◽  
Novel Approach ◽  
Necessary And Sufficient ◽  
Near Extinction

AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

Stability of a thermoelastic mixture with second sound

2018 ◽  
Vol 24 (6) ◽  
pp. 1692-1706 ◽  
Author(s):  
Margareth S. Alves ◽  
Marcio V. Ferreira ◽  
Jaime E. Muñoz Rivera ◽  
O. Vera Villagrán
Keyword(s):  
Initial Data ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Second Sound ◽  
Dimensional Model ◽  
One Dimensional ◽  
The One ◽  
Necessary And Sufficient

We consider the one-dimensional model of a thermoelastic mixture with second sound. We give a complete characterization of the asymptotic properties of the model in terms of the coefficients of the model. We establish the necessary and sufficient conditions for the model to be exponential or polynomial stable and also the conditions for which there exist initial data for where the energy is conserved.

scholarly journals On expected durations of birth–death processes, with applications to branching processes and SIS epidemics

2016 ◽  
Vol 53 (1) ◽  
pp. 203-215 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton ◽  
Peter Neal
Keyword(s):  
Asymptotic Expression ◽  
Branching Processes ◽  
Random Variable ◽  
Death Process ◽  
Single Individual ◽  
Expected Time ◽  
Time To Extinction ◽  
Number Of Individuals ◽  
Sis Epidemic ◽  
Birth Death

Abstract We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R* is insensitive to the distribution of Q.

scholarly journals Nonoscillation of the solutions of impulsive differential equations of third order

2001 ◽  
Vol 14 (3) ◽  
pp. 309-316 ◽  
Author(s):  
Margarita Boneva Dimitrova
Keyword(s):  
Differential Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Nonoscillatory Solution ◽  
Necessary And Sufficient Conditions ◽  
Third Order ◽  
Impulse Effect ◽  
Necessary And Sufficient

Necessary and sufficient conditions are found for existence of at least one bounded nonoscillatory solution of a class of impulsive differential equations of third order and fixed moments of impulse effect. Some asymptotic properties of the nonoscillating solutions are investigated.

On subordinacy and spectral multiplicity for a class of singular differential operators

1998 ◽  
Vol 128 (3) ◽  
pp. 549-584 ◽  
Author(s):  
D. J. Gilbert
Keyword(s):  
Differential Operators ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Positive Lebesgue Measure ◽  
Spectral Multiplicity ◽  
Limit Circle Case ◽  
Separated Boundary Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Adjoint Operators

The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the formis investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

scholarly journals LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

2020 ◽  
pp. 15-18
Author(s):  
O. Makarchuk ◽  
K. Salnik
Keyword(s):  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Random Variable ◽  
Random Series ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Lévy’S Theorem ◽  
Distribution Matrix ◽  
Necessary And Sufficient

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

scholarly journals The Population Dynamics of File-Sharing Peer-to-Peer Networks

2016 ◽  
Author(s):  
Dang Dinh Trang ◽  
Roland Pereczes ◽  
Sándor Molnár
Keyword(s):  
Population Dynamics ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
File Sharing ◽  
Peer To Peer ◽  
Modeling Framework ◽  
Population Extinction ◽  
P2p Systems ◽  
Necessary And Sufficient ◽  
And Control

Recently  peer-to-peer  file-sharing applications  have  shown  an  extreme  popularity  and the  workload  generated  to  the  Internet  has  been dominated  by  the  traffic  coming  from  these applications.  In  this  paper  we  develop  a  simple  but effective  mathematical  model  to  capture  the  file population  dynamics  of  such  systems.  Our  modeling framework  is  based  on  the  theory  of  branching processes. We describe analytically the behavior of the proposed  model.  The  precise  characterization  of  the necessary  and  sufficient  conditions  of  population extinction  or  explosion  is  given  based  on  the  system parameters.  We  also  present  the  expected  ratio  of active,  passive  and  dead  peers  for  the  long-term regime.  We  validate  and  demonstrate  our  results  in several  simulation  studies.  Based  on  our  results  we propose  a  number  of  engineering  guidelines  to  the design and control of file-sharing P2P systems.

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