Calamagrostis model revisited: Matrix calibration as a constraint maximization problem

A matrix model for age-stage-structured population dynamics of Calamagrostis canescens, a perennial grass species colonizing forest clear-cut areas, was calibrated before from a type of data called ‘identified individuals with unknown parents’ ( Logofet, 2008) in order to estimate λ1, the dominant eigenvalue of the projection matrix. A number of methods were applied to tackle the ‘reproductive uncertainty’ in data, and the output variety contained λ1 both greater and less than 1 (Logofet, 2008), leaving the estimation uncertain. An ‘adaptation conjecture’ was then proposed that reduced the calibration to a nonlinear constraint maximization problem and provided for a satisfactory outcome.Two reasons have now caused revisiting. First, the maximization technique has been theoretically comprehended. In particular, an existence-uniqueness theorem has been proved that requires the maximizing solution to be reached at a vertex of the polyhedral of constrains. To facilitate searching for the solution in practice, I use the notion of potential-growth indicator and prove R0 and R1, the known indicators, to be certain linear functions of the uncertain fertility rates in a general class of projection matrix patterns. To solve a conjugate linear maximization problem under the same constraints as for λ1 is both theoretically and technically simpler, and this causes a practical benefit from the indication along with calculation.Second, the former uniform (non-specific) estimate of the upper bounds for the status-specific fertility rates has now conceded to the age-stage-specific estimates inferred from a special case study. These more sophisticated constraints produce respectively more accurate calibration, hence a more reliable estimation of λ1 as the growth potential inherent in the population in a certain environment at a given time.

The woodreed (Calamagrostis canescens) life cycle graph as a sample of polyvariant ontogeny. The stages are: v, virginal; g, generative; ss, subsenile; s, senile; the shorter initial, plantule and immature, stages are incorporated into the 1-year virginal one. Empty boxes designate the age-stage states not observed in the field study, patterned boxes indicate the states participating in reproduction; solid arrows represent ageing and ontogenetic transitions in one year, dashed arrows correspond to reproduction; Latin letters associated with arrows denote the age-stage-specific rates in a matrix model.Figure optionsDownload as PowerPoint slideHighlights
► Calibration under reproductive uncertainty reduces to a constraint maximization of λ1.
► The problem is proved to have a unique solution at a vertex of constraint polyhedral.
► Potential growth indicators R0 and R1 are proved to be linear in the fertility rates.
► Standard solution to the linear problem is of heuristic value for maximization of λ1.
► Fertility rate upper bounds are age-stage-specific for more reliable estimation of λ1.