A demographic study of the exponential distribution applied to uneven-aged forests

A demographic approach based on a size-structured version of the McKendrick-Von Foerster equation is used to demonstrate a theoretical link between the population size distribution and the underlying vital rates (recruitment, mortality and diameter growth) for the population of individuals whose diameter distribution is negative exponential. This model supports the conclusion that the negative exponential distribution as applied to balanced natural uneven-aged stands is a stable equilibrium model under appropriate assumptions. These assumptions include constant recruitment of stems into the smallest class that balances total mortality, and a simple relation between per capita mortality and diameter growth. A simple maximum likelihood-based solution to parameter estimation in the inverse problem is developed allowing the estimation of recruitment and mortality given reasonable sample of diameters, along with an estimate of population size and diameter growth rate. Two sets of stand dynamics equations are developed that are based on (i) the form of the underlying negative exponential distribution, and (ii) more generally, from the derivation of the McKendrick-Von Foerster equation. Applications of the methods and their assumptions are discussed with regard to both managed and old growth uneven-aged stands. Stands or forests that are close to negative exponential structure and are judged to be reasonably close to steady state will have vital rates that support this model. In contrast, the negative exponential is likely more important and applicable as a pragmatic target distribution when used in managed forests.