Quasi-stationary distributions for reducible absorbing Markov chains in discrete time

We consider discrete-time Markov chains with one coffin state and a finite set $S$ of transient states, and are interested in the limiting behaviour of such a chain as time $n \to \infty,$ conditional on survival up to $n$. It is known that, when $S$ is irreducible, the limiting conditional distribution of the chain equals the (unique) quasi-stationary distribution of the chain, while the latter is the (unique) $\rho$-invariant distribution for the one-step transition probability matrix of the (sub)Markov chain on $S,$ $\rho$ being the Perron-Frobenius eigenvalue of this matrix. Addressing similar issues in a setting in which $S$ may be reducible, we identify all quasi-stationary distributions and obtain a necessary and sufficient condition for one of them to be the unique $\rho$-invariant distribution. We also reveal conditions under which the limiting conditional distribution equals the $\rho$-invariant distribution if it is unique. We conclude with some examples

Limiting conditional distributions for birth-death processes

In a recent paper one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations

Total variation approximation for quasi-equilibrium distributions

Quasi-stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.Comment: 16 page

Interaction between habitat quality and an Allee-like effect in metapopulations

We construct a stochastic patch occupancy metapopulation model that incorporates variation in habitat quality and an Allee-like effect. Using some basic results from stochastic ordering, we investigate the effect of habitat degradation on the persistence of the metapopulation. In particular, we show that for a metapopulation with Allee-like effect habitat degradation can cause a dramatic decrease in the level of persistence while in the absence of an Allee-like effect this decrease is more gradual

A central limit theorem for a discrete-time SIS model with individual variation

A discrete-time SIS model is presented that allows individuals in the population to vary in terms of their susceptibility to infection and their rate of recovery. This model is a generalisation of the metapopulation model presented in McVinish and Pollett (2010). The main result of the paper is a central limit theorem showing that fluctuations in the proportion of infected individuals around the limiting proportion converges to a Gaussian random variable when appropriately rescaled. In contrast to the case where there is no variation amongst individuals, the limiting Gaussian distribution has a nonzero mean

Quasi-stationarity in populations that are subject to large-scale mortality or emigration

We shall examine a model, first studied by Brockwell et al. [Adv Appl Probab 14 (1982) 709.], which can be used to describe the longterm behaviour of populations that are subject to catastrophic mortality or emigration events. Populations can suffer dramatic declines when disease, such as an introduced virus, affects the population, or when food shortages occur, due to overgrazing or fluctuations in rainfall. However, perhaps surprisingly, such populations can survive for long periods and, although they may eventually become extinct, they can exhibit an apparently stationary regime. It is useful to be able to model this behaviour. This is particularly true of the ecological examples that motivated the present study, since, in order to properly manage these populations, it is necessary to be able to predict persistence times and to estimate the conditional probability distribution of population size. We shall see that although our model predicts eventual extinction, the time till extinction can be long and the stationary exhibited by these populations over any reasonable time scale can be explained using a quasistationary distribution. (C) 2001 Elsevier Science Ltd. All rights reserved

Modelling quasi-stationary behaviour in metapopulations

We consider a Markovian model proposed by Gyllenberg and Silvestrov for studying the behaviour of a metapopulation: a population that occupies several geographically separated habitat patches. Although the individual patches may become empty through extinction of local populations, they can be recolonized through migration from other patches. There is considerable empirical evidence tin the work of Gilpin and Hanski, for example) which suggests that a balance between migration and extinction is reached which enables these populations to persist for long periods. The Markovian model predicts extinction in a finite time. Thus, there has been considerable interest in developing methods which account for the persistence of these populations and which provide an effective means of studying their long-term behaviour before extinction occurs. We shall compare and contrast the methods of Gyllenberg and Silvestrov (pseudo-stationary distributions) and those of Day and Possingham, which are based on the classical notion of a quasi-stationary distribution. We present here a convincing rationale for the latter, using limits of conditional probabilities. (C) 1999 IMACS/Elsevier Science B.V. All rights reserved

Development and testing of a genetic marker-based pedigree reconstruction system 'PR-genie' incorporating size-class data

For wildlife populations, it is often difficult to determine biological parameters that indicate breeding patterns and population mixing, but knowledge of these parameters is essential for effective management. A pedigree encodes the relationship between individuals and can provide insight into the dynamics of a population over its recent history. Here, we present a method for the reconstruction of pedigrees for wild populations of animals that live long enough to breed multiple times over their lifetime and that have complex or unknown generational structures. Reconstruction was based on microsatellite genotype data along with ancillary biological information: sex and observed body size class as an indicator of relative age of individuals within the population. Using body size-class data to infer relative age has not been considered previously in wildlife genealogy and provides a marked improvement in accuracy of pedigree reconstruction. Body size-class data are particularly useful for wild populations because it is much easier to collect noninvasively than absolute age data. This new pedigree reconstruction system, PR-genie, performs reconstruction using maximum likelihood with optimization driven by the cross-entropy method. We demonstrated pedigree reconstruction performance on simulated populations (comparing reconstructed pedigrees to known true pedigrees) over a wide range of population parameters and under assortative and intergenerational mating schema. Reconstruction accuracy increased with the presence of size-class data and as the amount and quality of genetic data increased. We provide recommendations as to the amount and quality of data necessary to provide insight into detailed familial relationships in a wildlife population using this pedigree reconstruction technique.Robert C. Cope, Janet M. Lanyon, Jennifer M. Seddon, and Philip K. Pollet