Scaling Features of Two Special Markov Chains Involving Total Disasters

Abstract

Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto-Zipf distributions whose divisibility and self-decomposability properties are shown relevant. The study includes: existence and shape of the invariant measure, time-reversal, return time to the origin, contact probability at the origin, extinction probability, height and length of the excursions, a renewal approach to the fraction of time spent in the catastrophic state, scale function, first time to collapse and first-passage times, divisibility properties.

A generating function approach to Markov chains undergoing binomial catastrophes

“…Some Markov catastrophe models involving total disasters are described in [21,33]. (c) If β ∼ δ 1 a move up results in the addition of only one individual, which is the simplest deterministic drift upwards.…”

“…(a) With r (x 0 ) the x 0 − entry of r, ρ −n P (τ x 0 ,0 > n) → r (x 0 ) , as n → ∞, (21) showing that P (τ x 0 ,0 > n) has geometric tails with rate ρ (extinction is fast). (b) With l (y) the y−entry of l, for all x 0 > 0,…”

A generating function approach to Markov chains undergoing binomial catastrophes

“…Consecutive excursions are the iid pieces of this random walk on the non-negative integers. Some Markov catastrophe models involving total disasters are described in [33], [21].…”

“…Density-dependence is shown here to have a significant effect on population persistence (mean time to extinction), with a decreasing mean persistence time at large initial population sizes and causing a relative increase at intermediate sizes. A density-dependent model involving total disasters was designed in [21]. Discrete-time random population dynamics with catastrophes balanced by random growth has a long history in the literature, starting with [25].…”

On Random Population Growth Punctuated by Geometric Catastrophic Events

“…Many Markov chains models have been designed in an attempt to explain the transient and large-time behavior of the size of such populations. Some results concern the evaluation of the risk of extinction and the distribution of the population size in the case of total disasters where all individuals in the population are removed simultaneously [1][2] . Other works deal with the basic immigration process subject to either binomial or geometric catastrophes; that is when the population size is iteratively reduced according either to a binomial or a geometric law [3][4] .…”