Population models at stochastic times (original) (raw)

The extinction time of a birth-death diffusion model with general rate catastrophe process

The distribution of the extinction time for a birth -death diffusion model with catastrophes is determined for the case when the catastrophes occurring at a general rate a (x) and having magnitudes with arbitrary distribution function Ht ") (independent of x j. The asymptotic behavior of the expected time to extinction for large initial population size for the such a case is also considered.

Time-resolved extinction rates of stochastic populations

Physical Review E, 2010

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasi-stationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover -- renewal and removal -- is much slower than all other processes. In this case there is a time scale separation in the system which enables one to introduce a short-time quasi-stationary extinction rate W_1 and a long-time quasi-stationary extinction rate W_2, and develop a time-dependent theory of the transition between the two rates. It is shown that W_1 and W_2 coincide with the extinction rates when the population turnover is absent, and present but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.

Extinction Rate of a Population under both Demographic and Environmental Stochasticity

Theoretical Population Biology, 1998

We examined the asymptotic rate of population extinction ; when the population experiences density-dependent population regulation, demographic stochasticity, and environmental stochasticity. We assume discrete-generation population dynamics, in which some parameters fluctuate between years. The fluctuation of parameters can be of any magnitude, including both fluctuation traditionally treated as diffusion processes and fluctuation from catastrophes within a single scheme. We develop a new approximate method of calculating the asymptotic rate of population extinction per year, ;= 0 exp(&x) u(x) dx, where u(x) is the stationary distribution of adult population size from the continuous-population model including environmental stochasticity and population-regulation but neglecting demographic stochasticity. The method can be regarded as a perturbation expansion of the transition operator for population size. For several sets of population growth functions and probability distributions of environmental fluctuation, the stationary distributions can be calculated explicitly. Using these, we compare the predictions of this approximate method with that using a full transition operator and with the results of a direct Monte Carlo simulation. The approximate formula is accurate when the intrinsic rate of population increase is relatively large, though the magnitude of environmental fluctuation is also large. This approximation is complementary to the diffusion approximation.

On time scales and quasi-stationary distributions for multitype birth-and-death processes

arXiv (Cornell University), 2017

We consider a class of birth-and-death processes describing a population made of d sub-populations of different types which interact with one another. The state space is Z d + (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter K which can be thought as the order of magnitude of the total size of the population at time 0. For any fixed finite time span, it is well-known that such processes, when renormalized by K, are close, in the limit K → +∞, to the solutions of a certain differential equation in R d + whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for K large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time t and the qsd. This bound is exponentially small in t, for t log K. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than log K and much smaller than the mean time to extinction, which is exponentially large as a function of K. Let us stress that we are interested in what happens for finite K. We obtain results much beyond what large deviation techniques could provide.

A note on linear equations modeling birth-and-death processes

Mathematical and Computer Modelling, 1992

Cauchy's method of characteristics is applied to derive a comprehensive solution for a class of differential, partial differential and difference-differential equations encountered in the study of branching processes. The results are then used to address an unsolved Markov's generalized birth process. Dover, NJ, (1989).

Mathematical Modeling of Extinction of Inhomogeneous Populations

Bulletin of Mathematical Biology, 2016

Mathematical models of population extinction have a variety of applications in such areas as ecology, paleontology and conservation biology. Here we propose and investigate two types of sub-exponential models of population extinction. Unlike the more traditional exponential models, the life duration of sub-exponential models is finite. In the first model, the population is assumed to be composed clones that are independent from each other. In the second model, we assume that the size of the population as a whole decreases according to the sub-exponential equation. We then investigate the "unobserved heterogeneity", i.e. the underlying inhomogeneous population model, and calculate the distribution of frequencies of clones for both models. We show that the dynamics of frequencies in the first model is governed by the principle of minimum of Tsallis information loss. In the second model, the notion of "internal population time" is proposed; with respect to the internal time, the dynamics of frequencies is governed by the principle of minimum of Shannon information loss. The results of this analysis show that the principle of minimum of information loss is the underlying law for the evolution of a broad class of models of population extinction. Finally, we propose a possible application of this modeling framework to mechanisms underlying time perception.

Birth--death processes with piecewise constant rates

Statistics & Probability Letters, 1992

This paper derives the probability distribution and extinction probabilities for a birth-death process with piecewise constant rates. Such processes are used to model various biological phenomena.

THE POPULATION MOMENTS OF A BIRTH-DEATH DIFFUSION PROCESS WITH

The development of mathematical models for population growth is of great importance in many fields such as ecology, demography and genetics. In this paper. we consider a birth-death diffusion model with immigration for the growth of populations allowing catastrophes. The population moments are derived for this class of diffusion models with immigration interrupted by catastrophes with sizes having a Beta distribution function and constant growth catastrophe rate. An explicit solution to the' moments of such process had been derived. These results can be used in statistical inference problems.