Simple stochastic populations in habitats with bounded and varying carrying capacities

Abstract

A population consisting of one single type of individuals where reproduction is seasonal, and by means of asexual binary-splitting with a probability, which depends on the carrying capacity of the habitat, K and the present population is considered. Current models for such binary-splitting populations do not explicitly capture the concepts of early and late extinctions. A new parameter v, called the ‘scaling parameter’ is introduced to scale down the splitting probabilities in the first season, and also in subsequent generations in order to properly observe and record early and late extinctions. The modified model is used to estimate the probabilities of early and late extinctions, and the expected time to extinction in two main cases. The first case is for fixed and large K, where a new and more general upper bound for the expected time to extinction is proposed to be evK. The risk of such populations going extinct is found to be of the order O _ p 1 (2v−1)K _ . The second case considers a scenario where the carrying capacity of the habitat varies in each season between two values L (for low) and H (for high), randomly chosen with equal probabilities to represent either a good or a bad season respectively. Both cases yielded similar results with the probability of early extinction tending to zero as v increases, and the probability of late extinction tending to one as v increases. Keywords: habitat; carrying capacity; branching processes; supercritical; subcritical; extinction time; binary-splitting. v