Does averaging overestimate or underestimate population growth? It depends

Abstract

When a matrix population model is nonautonomous, i.e., when it represents a set of single-time-step ("annual") PPMs, L(t), t = 0, 1, …, T – 1, each corresponding to a fixed life cycle graph, then each of the annual matrices generates its own set of model results to characterize the population. In particular, λ1(t), the asymptotic growth rate, varies with t and may result in alternating predictions of population viability. A logical way to characterize the population over the total period of observations is to average the given set of T PPMs, and I have proved the correct mode of averaging to be the pattern-geometric average. It means finding a matrix, G, such that its Tth power equals the product of T annual matrices (in the chronological order), while its pattern does correspond to the given life cycle graph. In practical cases however, the mathematical problem of pattern-geometric average has no exact solution for a fundamental mathematical reason. Nevertheless, the approximate solutions have revealed a fair precision of approximation in recent case studies of alpine short-lived perennials (Eritrichium caucasicum and Androsace albana), resulting in quite certain predictions of population viability by means of λ1(G), the dominant eigenvalue of the average matrix.An alternative approach to estimate the viability leades to the concept of stochastic environment (represented by the given PPMs to be chosen at random) and the ensuing stochastic growth rate, λS. In the case studies, the λSs have been estimated by a direct method of Monte Carlo simulations, all the E. caucasicum estimates having unambiguously been less than λ1(G), whereas those for A. albana being certainly greater than λ1(G). There should be a general mathematical reason for this, too, thus a challenge to the theory of nonnegative matrices.