Choosing Fagus sylvatica L. matrix model dimension by sensitivity analysis of the population growth rate with respect to the width of the diameter classes

In this study we use a projection matrix model of order n of an uneven-aged beech stand to analyse how the width of the diameter (physical tree diameter) classes d influence the population growth rate λ0, and to obtain the sensitivity and the elasticity of λ0 with respect to d. Numerical computations using systematic variations in d and n (where n represents the number of diameter classes), three levels of diameter growth, three levels of stand basal area, G = 20, 22 and 24 m2/ha, and three levels of recruitment, R = 200, 520 and 840 stems/ha, were conducted. The lowest values for λ0, sensitivities (∂λ0/∂d) and elasticities (∂λ0/λ0)/(∂d/d), were obtained at the lowest width of the diameter classes, d = 5 cm. The results have also shown that: (a) the estimation of λ0 was not significantly affected by small or even moderate changes in n and/or d; (b) (∂λ0/∂d) and (∂λ0/λ0)/(∂d/d) increased with increasing d; (c) within each quality level (diameter growth level), λ0 increased with increasing R and decreased with increasing G, (∂λ0/∂d) decreased with increasing R and increased with increasing G, and (∂λ0/λ0)/(∂d/d) decreased with increasing R and increased with increasing G. We suggest that narrow size categories should be used in this model to obtain conservative estimates of λ0 and sustainable harvest rates, with minimum sensitivities and elasticities, although the results were not substantially affected by small or even moderate changes in the width and/or number of the diameter classes.