Asymptotic distribution of stage-grouped population models

Matrix models are often used to predict the dynamics of size-structured or age-structured populations. The asymptotic behaviour of such models is defined by their malthusian growth rate λ, and by their stationary distribution w that gives the asymptotic proportion of individuals in each stage. As the coefficients of the transition matrix are estimated from a sample of observations, λ and w can be considered as random variables whose law depends on the distribution of the observations. The goal of this study is to specify the asymptotic law of λ and w when using the maximum likelihood estimators of the coefficients of the transition matrix. We prove that λ and w are asymptotically normal, and the expressions of the asymptotic variance of λ and of the asymptotic covariance matrix of w are given. The convergence speed of λ and w towards their asymptotic law is studied using simulations. The results are applied to a real case study that consists of a Usher model for a tropical rain forest in French Guiana. They permit to assess the number of trees to measure to get a given precision on the estimated asymptotic diameter distribution, which is an important information on tropical forest management.