The structured demography of open populations in fluctuating environments

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, provide analytic methods for understanding their long‐term behaviour and illustrate the application of methods to two marine populations. 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 n t +1  =  A t +1 n t  +  b t +1 where is A t +1 is a time varying operator (e.g. an integral operator or a matrix) that updates the local demography and b t +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 A 1 , A 2 ,… and b 1 , b 2 ,… are stationary sequences, this limiting behaviour 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 autocovariance of the stationary distribution. Sensitivity formulas for these statistical features are also given. The analytic methods are illustrated with applications to discrete size‐structured models of space‐limited coral populations and continuously size‐structured models of giant clam populations.