Seed bank as a persistent problem in matrix population models: From uncertainty to certain bounds

Errors in the construction of matrix population models may lead to erroneous conclusions from modeling case studies, and neglecting the seed bank as a stage in the stage-structured model is a kind of ‘persistent problem’ common to many models of plant species. The problem arises due to uncertainties in the seed-related parameters that cannot be reliably estimated from empirical data. We investigate the consequences that including a seed bank into the underlying life cycle graph may have on λ1, the dominant eigenvalue of the calibrated matrix in the “no-seeds” model. Vital rates of that model were calibrated on a time series of data of the 'identified individuals’ type gained in our former, 11-year case study on a local population of Androsace albana, a short-lived perennial inhabiting alpine heaths. The calibration has resulted in 10 successive population projection matrices expressed in rational numbers, with a variety of λ1s distant from 1 by various amounts in both directions. To include the seed bank, we propose an integer-valued formalism with the following three unknown S-related parameters: S(t), the number of dormant seeds able to germinate at the current year of observation, t; Sp(t), the number of seeds produced this year, and Sm(t), the total number of seeds lost this year for various reasons. These preserve the rational form of all the other vital rates and holding the calibration equations true in the integer numbers, too. Some observable traits of A. albana plants help establish certain finite ranges of the three S-related parameters, and they enable finding the ensuing ranges of model outcomes by merely enumerating all the feasible triples. In this way, we get answers to pertinent questions about the λ1 of the seed model vs. its “no-seeds” counterpart and give a mathematical explanation of what we have obtained. An important conclusion is that the seed bank can principally not change the population growth into decline, nor vice versa, within the feasible ranges of parameters.