Keeping random walks safe from extinction and overpopulation in the presence of life-taking disasters

Abstract

ABSTRACT Recurrence and transience conditions are made explicit in discrete-time Markov chain population models for which random stationary growth alternates with disastrous random life-taking events. These events either have moderate stationary magnitudes or lead to an abrupt population decline. The probability of their occurrence may or may not depend on the population size. These conditions are based on the existence or not of a “weak” carrying capacity, where “weak” means that the carrying capacity can be exceeded, temporarily. In this framework, the population is threatened with extinction, an event whose probability is expressed, as well as the law of the time remaining until this deadline. On the other hand, the population is also threatened by overpopulation, an event whose time to reach a given threshold is expressed, as well as the difference between the population size and the carrying capacity. The theory is that of extreme values for Markov chains and is based on the control of the spectral properties of the northwest truncation of the transition matrix of the original Markov chain with life-taking disasters. The article presents an extension to the case where the process of life-taking disasters is no longer geometric and to the case where the probability of occurrence of a disaster depends on the population size. Both the time to extinction and the time to a given threshold have geometrically decaying distribution tails. The use of the extremal Markov chain in the calculation of the time to overpopulation is innovative.