Birth and Death Process

El proceso de nacimiento y muerte es usado frecuentemente en modelos matemáticos de sistemas de colas. En el contexto del proceso de nacimiento y muerte podemos derivar algunos resultados que describen el comportamiento de los sistemas de colas en general. Este proceso es un caso especial de proceso de tiempo continuo de Markov, en donde el estado transitorio solo tiene dos tipos: “nacimiento”, el cual incrementa el estado variable en uno y “muertes”, que disminuye el estado en uno. El nombre del modelo viene de una aplicación común, el uso de estos modelos para representar la medida de la población en la actualidad como transiciones son literalmente nacimientos y muertes. El Proceso de Nacimiento y Muerte tiene muchas aplicaciones en demografía, teoría de colas, ingeniería de alto rendimiento, epidemiología o en biología. Puede ser usado, por ejemplo, en el estudio de la evolución de una bacteria, el número de personas con una enfermedad dentro de una población, el número de clientes en línea en el supermercado, etc.

The work of Aiello and Freedman on a single species growth with stage structure has received much attention in the literature in recent years. Their model predicts a positive steady state as the global attractor and thus suggests that stage structure does not generate the sustained oscillations frequently observed in nature. This work inevitably stirred some controversy. Subsequent works by other authors suggest that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delay can lead to models that produce periodic solutions. We review this work from a fresh biological angle: growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computatio...

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t>0. We assume that X has infinite volume and, for each α>1, we consider the set Θ_α of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θ_α, will never leave Θ_α,...

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R^d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure mu as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of "infinite length" will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in R^d). We prove that, in the low activity-high temperature regime, the generators of the Kawasaki dynamics converge to the generator of a Glauber dynamics. The convergence is on the set of exponential functions, in the L^2(μ)-norm. Furthermore, additionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-dimensional distributions of the processes.

General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.

Background Innate immunity is the ancient defense system of multicellular organisms against microbial infection. The basis of this first line of defense resides in the recognition of unique motifs conserved in microorganisms, and absent in the host. Peptidoglycans, structural components of bacterial cell walls, are recognized by Peptidoglycan Recognition Proteins (PGRPs). PGRPs are present in both vertebrates and invertebrates. Although some evidence for similarities and differences in function and structure between them has been found, their evolutionary history and phylogenetic relationship have remained unclear. Such studies have been severely hampered by the great extent of sequence divergence among vertebrate and invertebrate PGRPs. Here we investigate the birth and death processes of PGRPs to elucidate their origin and diversity. Results We found that (i) four rounds of gene duplication and a single domain duplication have generated the major variety of present vertebrate PGRP...

In some species, histone gene clusters consist of tandem arrays of each type of histone gene, whereas in other species the genes may be clustered but not arranged in tandem. In certain species, however, histone genes are found scattered across several different chromosomes. This study examines the evolution of histone 3 (H3) genes that are not arranged in large clusters of tandem repeats. Although H3 amino acid sequences are highly conserved both within and between species, we found that the nucleotide sequence divergence at synonymous sites is high, indicating that purifying selection is the major force for maintaining H3 amino acid sequence homogeneity over long-term evolution. In cases where synonymous-site divergence was low, recent gene duplication appeared to be a better explanation than gene conversion. These results, and other observations on gene inactivation, organization, and phylogeny, indicated that these H3 genes evolve according to a birth-and-death process under stro...

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.