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The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the well-known transition probability is an ill-conditioned problem. To overcome this difficulty we will rewrite the transition probability in terms of a Gaussian hypergeometric function and subsequently obtain a three-term recurrence relation for its accurate evaluation. We will also study the properties of the hypergeometric function as a solution to the three-term recurrence relation. We will then provide formulas for the gradient and Hessian of the log-likelihood function and conclude the article by applying our methods for numerically computing maximum likelihood estimates in both simulated and real dataset. [ABSTRACT FROM AUTHOR]

We consider a discrete-time population dynamics with age-dependent structure. At every time step, one of the alive individuals from the population is chosen randomly and removed with probability $$q_k$$ depending on its age, whereas a new individual of age 1 is born with probability r. The model can also describe a single queue in which the service order is random while the service efficiency depends on a customer's 'age' in the queue. We propose a mean field approximation to investigate the long-time asymptotic behavior of the mean population size. The age dependence is shown to lead to anomalous power-law growth of the population at the critical regime. The scaling exponent is determined by the asymptotic behavior of the probabilities $$q_k$$ at large k. The mean field approximation is validated by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure. [ABSTRACT FROM AUTHOR]

Markov chains provide excellent statistical models for studying many natural phenomena that evolve with time. One particular class of continuous-time Markov chain, called birth–death processes, can be used for modelling population dynamics in fields such as ecology and microbiology. The challenge for the practitioner when fitting these models is to take measurements of a population size over time in order to estimate the model parameters, such as per capita birth and death rates. In many biological contexts, it is impractical to follow the fate of each individual in a population continuously in time, so the researcher is often limited to a fixed number of measurements of population size over the duration of the study. We show that, for a simple birth–death process, with positive Malthusian growth rate, subject to common practical constraints, there is an optimal schedule for measuring the population size that minimises the expected confidence region of the parameter estimates. Throughout our exposition of the optimal experimental design, we compare it to a simpler equidistant design, where the population is sampled at regular intervals. This is an experimental design worthy of comparison since it can represent a much simpler design to implement in practice. In order to find optimal experimental designs for our population model, we make use of a combination of useful statistical machinery. Firstly, we use a Gaussian diffusion approximation of the underlying discrete-state Markov process, which allows us to obtain analytical expressions for Fisher’s information matrix (FIM), which is crucial to optimising the experimental design. We also make use of the cross-entropy method of stochastic optimisation for the purpose of maximising the determinant of FIM to obtain the optimal experimental designs. Our results show that the optimal schedule devised by others for a simple model of population growth without death can be extended, for large populations, to the two-parameter model that incorporates both birth and death. For the simple birth–death process, we find that the likelihood surface is also problematic and poses serious problems for point estimation and easily defining confidence regions. A Bayesian approach to inference is proposed as a way in which these problems could be circumvented. [ABSTRACT FROM AUTHOR]