Domain of Attraction of the Quasi-Stationary Distribution for the Linear Birth and Death Process with Killing

The model of linear birth and death processes with killing has been studied by Karlin and Tavar (1982). This paper is concerned with three problems in connection with quasi-stationary distributions (QSDs) for linear birth-death process  with killing on a semi-infinite lattice of integers. The first problem is to determine the decay parameter  of . We have  where , ,  are the birth, death and killing rates in state , respectively. The second one is to prove the uniqueness of the QSD which is a geometric distribution. It is interesting to find that the unkilled process has a one-parameter family of QSDs while the killed process has precisely one QSD. The last one is to solve the domain of attraction problem, that is, we obtain that any initial distribution is in the domain of attraction of the unique QSD for . Our study is motivated by the population genetics problem.     

Exponential convergence to quasi-stationary distribution and $$$$-process

For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the \(Q\)-process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.     

Parasite population dynamics within a dynamic host population

Summary We propose a stochastic process model for a parasite population living within a host population. The host population is described by an immigration-death process. The parasite population in one host is an immigration-birth-death-emigration process. The death of all parasites at the moment of death of their host is regarded as emigration. We derive explicit expressions for the distributions of the size of the host population, of the parasite load of one host individual and of the parasite population in the total host population and obtain conditions for the existence of limiting distributions if time is tending to infinity. For particular lifetime distributions of hosts including parasite induced mortality and heterogeneous infection risk we finally derive properties of the limiting distributions.Dedicated to Klaus Krickeberg on the occasion of his 60th birthday     

CONSIDERATION OF STOCHASTIC DEMOGRAPHY IN THE DESIGN AND MANAGEMENT OF BIOLOGICAL RESERVES

A generalized birth–and–death process serves as a simple, flexible model for computing the expected persistence time of a small population in a random world. We may reparametrize the model in ways that allow explicit incorporation of density dependence, random differences in events experienced by different individuals, and random environmental variation experienced by all individuals in concert. This model seems to capture the important features of real population dynamics for purposes of computing the mean persistence time, even though the underlying mechanisms presumed in the mathematics of the model are decidedly unrealistic. The lack of isomorphism between birth and death rates, as they feature in the model, and vital rates of real biological populations can lead to extremely misleading results, if the classic formulation, rather than the reparametrization is applied without due circumspection. Using the reparametrized model, we find that environmental variation poses a greater problem for population persistence than does individual variation. In particular, with purely individual variation, the expected persistence time increases approximately with the power of the ceiling on population size; but with purely environmental variation, the expected persistence time increases somewhat less than linearly with the size of the population ceiling. The birth–and–death process model can also be applied to calculating the persistence time of a population on an ensemble of reserves which are linked by natural migration or by deliberate reintroduction programs. Results of this model, for an idealized ensemble, show that multiple independent reserves with a sufficient recolonization rate (natural or otherwise) will confer a longer persistence time than a single reserve with the same total carrying capacity, but in the absence of recolonization, the system of smaller separate reserves confers a shorter persistence time than the single large reserve.     

Ultracontractivity for Markov Semigroups and Quasi-Stationary Distributions

We prove the existence and uniqueness of quasi-stationary distributions for symmetric Markov processes. In particular, we show that if its Markov semigroup is intrinsic ultracontractive, then there exists a unique quasi-stationary distribution. We apply our results to one-dimensional diffusion processes.     

On the link between infinite horizon control and quasi-stationary distributions

We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function becomes infinite, and we characterize the optimal Markov control in this limit. To achieve this, we present a new proof of the dynamic programming principle based upon a pseudo-Markov property for controlled jump processes. We also prove the convergence to a unique quasi-stationary distribution of non-linear branching processes controlled by a Markov control conditioned on non-extinction.     

Semi-Markov processes and α-invariant distributions

A semi-Markov process is easily made Markov by adding some auxiliary random variables. This paper discusses the I-type quasi-stationary distributions of such “extended” processes, and the α-invariant distributions for the corresponding Markov transition probabilities; and we show that there is an intimate relation between the two. The results have relevance in the study of the time to “absorption” or “death” of semi-Markov processes. The particular case of a terminating renewal process is studied as an example.     

Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution—actually an infinite family of quasi-stationary distributions— if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step (killing) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth–death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.      

Conditions for the existence of quasi-stationary distributions for birth–death processes with killing

We consider birth–death processes on the nonnegative integers, where {1,2,…}$\{1,2,\dots \}$ is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 (killing) may occur from any state. Assuming that absorption at 0 is certain we are interested in additional conditions on the transition rates for the existence of a quasi-stationary distribution. Inspired by results of Kolb and Steinsaltz [M. Kolb, D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing, Ann. Probab. 40 (2012) 162–212] we show that a quasi-stationary distribution exists if the decay rate of the process is positive and exceeds at most finitely many killing rates. If the decay rate is positive and smaller than at most finitely many killing rates then a quasi-stationary distribution exists if and only if the process one obtains by setting all killing rates equal to zero is recurrent.     

New methods for determining quasi-stationary distributions for markov chains

We shall be concerned with the problem of determining quasi-stationary distributions for Markovian models directly from their transition rates Q. We shall present simple conditions for a μ-invariant measure m for Q to be μ-invariant for the transition function, so that if m is finite, it can be normalized to produce a quasi-stationary distribution.     

Optimal control of a birth and death epidemic process

We employ a birth and death process to describe the spread of an infectious disease through a closed population. Control of the epidemic can be effected at any instant by varying the birth and death rates to represent quarantine and medical care programs. An optimal strategy is one which minimizes the expected discounted losses and costs resulting from the epidemic process and the control programs over an infinite horizon. We formulate the problem as a continuous-time Markov decision model. Then we present conditions ensuring that optimal quarantine and medical care program levels are nonincreasing functions of the number of infectives in the population. We also analyze the dependence of the optimal strategy on the model parameters. Finally, we present an application of the model to the control of a rumor.     

Slow and fast scales for superprocess limits of age-structured populations

A superprocess limit for an interacting birth-death particle system modeling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence.     

A microscopic interpretation for adaptive dynamics trait substitution sequence models

We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability μ$\mu$ of mutation at each birth event. We introduce a renormalization parameter K$K$ scaling the size of the population, which leads, when K→+∞$K\to +\infty$, to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population (K→+∞$K\to +\infty$) and of small mutations (μ→0$\mu \to 0$), we prove that a timescale separation between the birth and death events and the mutation events occurs and that the interacting particle microscopic process converges for finite dimensional distributions to the biological model of evolution known as the “monomorphic trait substitution sequence” model of adaptive dynamics, which describes the Darwinian evolution in an asexual population as a Markov jump process in the trait space.     

Generating Birth and Death Processes

Associated with an ordered sequence of an even number 2N of positive real numbers is a birth and death process (BDP) on {0, 1, 2,…, N} having these real numbers as its birth and death rates. We generate another birth and death process from this BDP on {0, 1, 2,…, 2N}. This can be further iterated. We illustrate with an example from tan(kz). In BDP, the decay parameter, viz., the largest non-zero eigenvalue is important in the study of convergence to stationarity. In this article, the smallest eigenvalue is found to be useful.     

An approximation scheme for quasi-stationary distributions of killed diffusions

In this paper we study the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is killed at a smooth rate and then regenerated at a random location, distributed according to the weighted empirical occupation measure. We show that the weighted occupation measures almost surely comprise an asymptotic pseudo-trajectory for a certain deterministic measure-valued semiflow, after suitably rescaling the time, and that with probability one they converge to the quasi-stationary distribution of the killed diffusion. These results provide theoretical justification for a scalable quasi-stationary Monte Carlo method for sampling from Bayesian posterior distributions.     

Total variation approximation for quasi-equilibrium distributions,II

Quasi-stationary distributions have been used in biology to describe the steady state behaviour of Markovian population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. However, they have substantial drawbacks; a Markov process may not possess any, or may have several, and their probabilities can be very difficult to determine. Here, we consider conditions under which an apparent stochastic equilibrium distribution can be identified and computed, irrespective of whether a quasi-stationary distribution exists, or is unique; we call it a quasi-equilibrium distribution. The results are applied to multi-dimensional Markov population processes.     

Measure-valued solutions for a hierarchically size-structured population

We present a hierarchically size-structured population model with growth, mortality and reproduction rates which depend on a function of the population density (environment). We present an example to show that if the growth rate is not always a decreasing function of the environment (e.g., a growth which exhibits the Allee effect) the emergence of a singular solution which contains a Dirac delta mass component is possible, even if the vital rates of the individual and the initial data are smooth functions. Therefore, we study the existence of measure-valued solutions. Our approach is based on the vanishing viscosity method.     

On the cycle maximum of birth–death processes and networks of queues

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.     

Penalization for Birth and Death Processes

In this paper we study a transient birth and death Markov process penalized by its sojourn time in 0. Under the new probability measure the original process behaves as a recurrent birth and death Markov process. We also show, in a particular case, that an initially recurrent birth and death process behaves as a transient birth and death process after penalization with the event that it can reach zero in infinite time. We illustrate some of our results with the Bessel random walk example.