Apparent differences in stochastic growth rates based on timing of census: a cautionary note

Matrix population models are increasingly used in both theoretical and applied analysis of population dynamics ‘Matrix Population Models—Construction, Analysis, and Interpretation (2001) 722’. The projected asymptotic growth rate from deterministic models, based on a single matrix, is invariant as to when in the annual cycle the population is censused (typically, either immediately prior to or after breeding). Thus, we expect that given a set of n matrices {A1, A2,…, An}, where each matrix Ai corresponds to a given environmental state i, that the long-term expected stochastic growth rate calculated by the product of matrices selected at random from this set should not depend on whether the matrices in the set are configured based on either a pre- or post-breeding census. However, differences in stochastic growth rate as a function of the timing of the census can arise under conditions where there is significant covariance among the individual matrix elements. Using a seasonal (periodic) matrix modeling approach, we show that such differences are an artifact of how stochasticity is entered into the matrices, particularly when the matrices are structured based on a post-breeding census model. In such cases, the annual projection matrix for a given year is in fact a product of component seasonal matrices from two successive years. Failing to account for this when there is covariance among seasons will lead to a disparity in estimated stochastic growth rate when growth is calculated based on the product of random matrices, as is frequently done. We show that a seasonal matrix approach, where the seasonal matrices are explicitly subscripted for the appropriate year, eliminates the problem, and is a generally robust approach to stochastic modeling.