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The paper solves the problem of modeling an outbreak of the species, which begins with the achievement of a specific nontrivial state of a dynamic system. Population process is divided into a sequence of substantially different stages. The model simulates the effect of spontaneous transition from weak fluctuations through accelerated reproductive activity to exceeded ecological capacity of the environment followed by the phase of rapid spontaneous decay. To describe the transition between final stages of the outbreak, the method of controlled functional realization of metamorphoses of trajectory behavior are used. Differential equations are combined with predicative constructions in software environment. The original computational unit is based on discrete-continuous time and allows us to apply the scenario approach to the consideration of environmental situations. The significance of the results is justified by the analysis of the data about outbreaks of small insect pests that eat on a limited number of suitable plants and, under certain conditions, can get out of control of parasitic wasps.

The Moran model with recombination is considered, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ sites (or loci), a finite number of letters (or alleles) at every site, and we do not make any scaling assumptions. In particular, we do not assume a diffusion limit. We consider the following marginal ancestral recombination process. Let $S = \{1,...,n\}$ and $\mathcal A=\{A_1, ..., A_m\}$ be a partition of $S$. We concentrate on the joint probability of the letters at the sites in $A_1$ in individual $1$, $...$, and at the sites in $A_m$ in individual $m$, where the individuals are sampled from the current population without replacement. Following the ancestry of these sites backwards in time yields a process on the set of partitions of $S$, which, in the diffusion limit, turns into a marginalised version of the $n$-locus ancestral recombination graph. With the help of an inclusion-exclusion principle, we show that the type distribution corresponding to a given partition may be represented in a systematic way, with the help of so-called recombinators and sampling functions. The same is true of correlation functions (known as linkage disequilibria in genetics) of all orders. We prove that the partitioning process (backward in time) is dual to the Moran population process (forward in time), where the sampling function plays the role of the duality function. This sheds new light on the work of Bobrowski, Wojdyla, and Kimmel (2010). The result also leads to a closed system of ordinary differential equations for the expectations of the sampling functions, which can be translated into expected type distributions and expected linkage disequilibria., 29 pages, 6 figures

We study Markovian models for population processes in continuous time, addressing questions concerning the behaviour of ensembles of individuals (equilibrium, quasi-equilibrium and time-dependent behaviour) and, in particular, what can be deduced from models for individual behaviour.

In this paper we present various approaches for the transient analysis of a Markovian population process with total catastrophes. We discuss the pros and the cons of these methodologies and point out how they lead to different tractable extensions. As an illustrating example, we consider the nonhomogeneous Poisson process with total catastrophes. The extension of the probabilistic methodologies for analyzing models with binomial catastrophes is also discussed.

The QuaTI System, sponsored by AMIB and DIXTAL, is developing an important database for intensive care in Brazil. The QuaTI System offers significant information for the participating ICUs for their continuous improvement, and gives data, although partial, of demographic characteristics of patients seen at participant ICUs. In our country, there is a steady decline in natality and mortality in the past 30 years, showing an irreversible and accelerated aging population process. Life expectancy at birth is 68 years and expectation at 60 years is more 18 years on average. In 1970, 5.07% of the population was older than 60; in 2000, 8.56%; and in 2002, 9.3%. In the South and Southeast, 10.2% of population are more than 60 years old.

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.

An individual is sampled randomly from a supercritical general branching population and the pedigree process, which centers around this ‘ego’-individual, is studied. The process describes not only lineage backwards and forwards, but also the lives of all individuals involved. Under mild conditions and in several senses, the process is shown to stabilize, as time passes. The limit is a doubly infinite population process, which generalizes the stable age distribution of branching processes and demography. It displays a nice independence structure, and can easily be constructed from the original branching law. The results are applied to certain kin-number problems, the process of ego's ancestors' births, and to the FLM-curve of cell kinetics.

For over 80 years succession theory has played a central role in plant ecology, providing both a predictive tool and organizational scheme. Drawing on this long history of research and observation, Margalef (1968), Odum (1969), Whittaker (1975) and others have identified general trends in community development. Their synthetic treatments have helped focus the efforts of subsequent workers examining the empirical and experimental basis of succession theory (e.g. Connell & Slatyer 1977, Drury & Nisbet 1973, Egler 1975, Horn 1974, Pickett 1976, van Hulst 1978). This more recent work suggests that the classical succession paradigm is seriously flawed and that many long held concepts need to be reexamined.

The problem of the existence of a stationary distribution and the convergence towards it in a certain semistochastic model for the growth of a population is considered. It is assumed that the population grows according to a deterministic equation, but at certain times there are catastrophes, which lead to a decrease in the population level. The hazard function for the occurrence of catastrophes is a function of the population level only. The size of these jumps have a distribution that depends on the population size immediately before the catastrophe. A constructive method for finding the stationary distribution is given.