The structured demography of open populations in fluctuating environments

Abstract

At the spatial scale relevant to many field studies and management policies, populations may experience more external recruitment than internal recruitment. These sources of recruitment, as well as local demography, are often subject to stochastic fluctuations in environmental conditions. Here, we introduce a class of stochastic models accounting for these complexities and provide analytic methods for understanding their long-term behavior. The population state n(x) of these stochastic models is a function or vector keeping track of densities of individuals with continuous (e.g. size) or discrete (e.g. age) traits x taking values in a compact metric space. This state variable is updated by a stochastic affine equation nt+1 = At+1nt + bt+1 where At+1 is a time varying operator (e.g. an integral operator or a matrix) that updates the local demography, and bt+1 is a time varying function or vector representing external recruitment. When the realized per-capita growth rate of the local demography is negative, we show that all initial conditions converge to the same time-varying trajectory. Furthermore, when A1, A2... and b1, b2, ... are stationary sequences, this limiting behavior is determined by a unique stationary distribution. When the stationary sequences are periodic, uncorrelated, or a mixture of these two types of stationarity, we derive explicit formulas for the mean, within-year covariance, and auto-covariance of the stationary distribution. Sensitivity formulas for these statistical features are also given. The analtyic methods are illustrated with applications to discrete size-structured models of space-limited coral populations, and continuously size-structured models of giant clam populations.