Functional Differential Equations in Sustainable Forest Harvesting

Abstract

This article deals with the dynamics of the volume control methodin forest regulation. The sustainability of harvesting a given constantvolume in a simpli¯ed forest model is studied. Depending on the ini-tial age distribution, the cut can lead to the forest exhaustion,or to an asymptotic steady-state uniform age distribution. A continuousmodel is formulated in terms of various kinds of partial di®erentialequations, delay di®erential equations, and non-linear integral equations.Equilib- rium solutions and their stability properties are determined.Discrete models are also obtained, both by direct reasoning and asapproxi- mations to the continuous case. These are used for simulationand graphical exploration of the system behavior. In addition, contrastingvarious discrete and continuous versions was found useful in clarifyingsome issues, in particular, ambiguity/redundancy problems in therela- tion between integral equations and delay di®erential equationsderived from them. The basic problem of evaluating sustainabilityfor an ini- tial distribution remains unsolved, however. Furtherprogress is linked to the asymptotic properties of a second-orderrecurrence relationship. It is hoped that the interplay between thetheory of functional di®er- ential equations and this concrete andeasily interpretable problem in forest management might prove fruitfulin both ¯elds.