Extinction in Fragmented Habitats Predicted from Stochastic Birth–death Processes with Density Dependence

Abstract

Habitat loss, the reduction of the habitat area available, is known to greatly reduce resident species' expected time to extinction. This process is widely recognized, if not adequately understood or quantified except in very simple models. However, it is not well understood how the time to extinction will change if the remaining habitat is distributed across a set of smaller, isolated patches, instead of being left in one single, continuous tract. The effect of habitat fragmentation on population persistence under demographic stochasticity has not been resolved. Specifically, it is not known whether a single large population will persist longer than an aggregate set of several smaller populations (with the same total size). Analytical studies of birth–death processes typically report the mean time to extinction for a single population as a function of the maximum population size, but omit higher moments. To estimate the overall persistence time, or the probability of extinction as a function of time, for a set of small populations, the entire distribution of extinction times must be known for a single population of each size. Knowing all the moments of the distribution of extinction times is not adequate, unless one can reconstruct the distribution from them. Here I analyse stochastic birth–death processes with linear density dependence in per capita birth and death rates, and obtain analytical expressions and numerical solutions for the distribution of extinction times in both subdivided and continuous populations. This is a single-species model that deals with demographic stochasticity only, and assumes independence of extinction events in different patches. These assumptions are relaxed elsewhere. Habitat fragmentation, even without any loss of overall area, has a great and detrimental effect on the persistence time of populations across all temporal and spatial scales. The effect is similar across spatial scales, but shifted in time—larger populations take longer to go extinct but the extinction risk relative to that of a smaller or more fragmented population is the same across spatial scales for the available habitat.