Logistic growth with random density independent disasters

Abstract

A stochastic differential equation for a discontinuous Markov process is employed to model the magnitude of a population which grows logistically between disasters which are proportional to the current population size (density independent disasters). The expected persistence or extinction time satisfies a singular differential-difference equation. When the number of disasters, in the absence of recovery, between carrying capacity and extinction is two, analytical expressions are found for the mean persistence time. A comparison is made with the previously studied case of decrements of constant magnitude. When the two problems are suitably normalized, the mean survival times are quite different for the two models, especially in a critical range of initial population sizes near extinction. The expected survival time of a colonizing species is discussed quantitatively in terms of the parameters of the model. Insight into the nature of the probability density of the survival time is obtained by means of computer simulations. The densities resemble gamma densities and long tails appear when the disasters are density independent, implying a small change of long term survival. When the number of consecutive disasters which take the population from carrying capacity to extinction is large, a singular decomposition is employed to solve the differential-difference equation for the mean persistence time. The results are discussed in terms of population strategies in hazardous environments.