A random walk with catastrophes

Abstract

Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.

A generating function approach to Markov chains undergoing binomial catastrophes

“…This model was considered by [25], [2]. We have c • X n ∼bin(X n , c) hence the name binomial catastrophe.…”

On Random Population Growth Punctuated by Geometric Catastrophic Events

“…The binomial effect is appropriate when, on a catastrophic event, the individuals of the current population each die or survive in an independent and even way, resulting in a drastic depletion of individuals at each step [4][5][6] .…”

Large deviations in a population dynamics with catastrophes

“…Stochastic models with catastrophes are studied since 70's and recieved a great attention of probability community, see [1] for the probably first systematic review about such processes, and see [2] for short historical overview and more references. These models are used in analyzing a growth of a population subject to catastrophes due large-scale death or emigrations of a population.…”

“…In [3] for the population dynamic, ξ(t) defined by (1,2) in the following, with linear growth and uniform catastrophes we proved the local large deviation principle (LLDP): we established a rough logarithmic asymptotic for the probability of the scaling version ξ T (t), t ∈ [0, 1], defined by (3), to be in a small neighborhood of a continuous function. Here, based on the work [3] we established a large deviation principle (LDP) on the state space at the end of the interval of observation of the process: we find the logarithmic asymptotic for the probability P(ξ T (1) > x).…”