A general age-dependent branching process. I

Crump, Kenny S.; Mode, Charles J.

doi:10.1016/0022-247x(68)90005-x

Cited by 120 publications

(52 citation statements)

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“…We need a particular kind of Crump-Mode-Jagers process [9], namely a BellmanHarris process [18], [1]. Let X be a random variable of mean µ which takes nonnegative integer values.…”

Section: Towards Applicationsmentioning

confidence: 99%

Large Deviations for the Weighted Height of an Extended Class of Trees

Broutin

Devroye

2006

Algorithmica

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We use large deviations to prove a general theorem on the asymptotic edge-weighted height H n of a large class of random trees for which H n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees [11], [13], random recursive trees [12] and plane oriented trees [23] for instance. New applications include the heights of some random lopsided trees [19] and of the intersection of random trees.1. Introduction. This paper gives general laws of large numbers for the height of a class of edge-weighted random trees, which includes as special cases random binary search trees [11], random recursive trees, random plane oriented trees [23] and random split trees [16]. However, it also covers random k-ary trees not analyzed until now. The paper extends the earlier theorems of Devroye [11], [12], [15] where the theory of branching processes was used for this purpose. A special kind of branching random walk permitted Biggins and Grey [5] to obtain the asymptotic height of various random trees including random binary search trees and random recursive trees. We propose in this paper a method based on large deviations for sums of independent random variables. The closest approach was the one of Biggins [4] which used multidimensional branching processes. Our method makes intensive use of Cramér's theorem for large deviations [17], [10] and some properties of the rate functions it defines. The height is characterized as the solution of a two-dimensional optimization problem involving Cramér's functions. We apply our method in some cases where these functions can be expressed in a closed form. In particular, we are able to obtain the height for random binary search trees, random recursive trees, random median-of-(2k + 1) trees and random lopsided trees, thus extending the class of trees covered by a single theorem.We first present the main result and its proof, taking for granted some results about large deviations. The proofs for these have been put in the Appendix. We next make the link between trees of random variables and random trees of size n, leaving the most interesting part on applications as a concluding section.

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“…We need a particular kind of Crump-Mode-Jagers process [9], namely a BellmanHarris process [18], [1]. Let X be a random variable of mean µ which takes nonnegative integer values.…”

Section: Towards Applicationsmentioning

confidence: 99%

Large Deviations for the Weighted Height of an Extended Class of Trees

Broutin

Devroye

2006

Algorithmica

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“…For appropriate choices of b and d, some of the models described in Section 1.2 can be recovered. If b(x, a) = b(a) and d(x, a) = d(a), we recover the age-dependent birth and death processes of [22,23,31,48,54]. The choice b(x, a) = b(a) and d(x, a, ZU (x, a)) = d(a) + η Z, 1 gives the rates used by [42].…”

Section: A Microscopic Modelmentioning

confidence: 99%

“…Kendall [54], Crump and Mode [22,23], Jagers [48,49], Doney [31], study birth and death processes in which an individual can give birth at several random times during its life, with rates that may depend on its age. These processes are called age-structured birth and death processes.…”

Section: Stochastic Modelsmentioning

confidence: 99%

Stochastic and deterministic models for age-structured populations with genetically variable traits

Ferrière

Tran

2009

ESAIM: Proc.

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Abstract. Understanding how stochastic and non-linear deterministic processes interact is a major challenge in population dynamics theory. After a short review, we introduce a stochastic individualcentered particle model to describe the evolution in continuous time of a population with (continuous) age and trait structures. The individuals reproduce asexually, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. In a large population limit, the random process converges to the solution of a Gurtin-McCamy type PDE. We show that the random model has a long time behavior that differs from its deterministic limit. However, the results on the limiting PDE and large deviation techniquesà la Freidlin-Wentzell provide estimates of the extinction time and a better understanding of the long time behavior of the stochastic process. This has applications to the theory of adaptive dynamics used in evolutionary biology. We present simulations for two biological problems involving life-history trait evolution when body size is plastic and individual growth is taken into account.Résumé. Nous illustrons ici certaines interactions entre processus probabilistes et déterministes en dynamique des populations. Après une courte revue, nous introduisons un modèle particulaire stochastique individu-centré pour décrire l'évolution en temps continu d'une population structurée en age (continu) et en trait. Les individus se reproduisent de façon asexuée, vieillissent, interagissent et meurent. Le trait est une propriété de l'individu mesurée par un vecteur de dimension d ; le trait est héritable et peut influencer les taux de naissance et de mort ainsi que les interactions entre individus. Par mutation, le trait peut varier entre parent et enfant. Dans une limite de grande population, notre processus aléatoire converge vers la solution d'une EDP du type Gurtin-McCamy. Nous montrons que le comportement en temps long du processus aléatoire diffère de celui de sa limite déterministe. Cependant, les deux types de modèles sont complémentaires : les résultats sur l'EDP limite et des techniques de grandes déviationsà la Freidlin-Wentzell, nous fournissent des estimées du temps d'extinction et une description précise du comportement en temps grand de notre processus aléatoire, avec des applicationsà la théorie des "dynamiques adaptatives". Nous présentons ensuite sur des simulations deux problèmes biologiques liés aux conséquences de plasticité de la taille lorsque la croissance individuelle est prise en compte, ainsi qu'à l'évolution des stratégies d'histoire de vie.

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“…Kendall [30], Crump and Mode [9,10], Jagers [26] and Doney [13] have studied birth and death processes in which a particle can give birth many times during its life, with a rate that depends on its age.…”

Section: Vc Tranmentioning

confidence: 99%

Large population limit and time behaviour of a stochastic particle model describing an age-structured population

Tran

2008

ESAIM: PS

107

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Abstract.We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the supercritical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.Mathematics Subject Classification. 60J80, 60K35, 92D25, 60F10.

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A general age-dependent branching process. I

Cited by 120 publications

References 1 publication

Large Deviations for the Weighted Height of an Extended Class of Trees

Large Deviations for the Weighted Height of an Extended Class of Trees

Stochastic and deterministic models for age-structured populations with genetically variable traits

Large population limit and time behaviour of a stochastic particle model describing an age-structured population

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