Abstract:
The purpose of this dissertation is to formulate,
analyze and numerically solve the dynamic harvesting problem
for uneven-aged stands. The problem is to find the
optimal numbers of trees to remove from diameter classes
over a finite time horizon and is formulated as a
discrete-time optimal-control problem with bounded control
variables and free terminal point. A solution algorithm
called the method of steepest descent is described and
demonstrated with a whole-stand/diameter-class simulator
for Wisconsin hardwood stands. Optimal management regimes
that maximize present net worth (PNW) from harvests taken
on a 5-year cutting cycle during a 150-year time horizon
are developed for three stumpage value functions.
Harvest regimes derived with the gradient method
contradicted optimal steady-state management regimes
determined with static analysis. Pontryagin's maximum
principle is used to establish optimality conditons for
the dynamic and static optimization problems. Comparison
of these conditions shows that for a stand with any initial
diameter distribution: (1) the optimal transition
regime does not converge to the steady state that maximizes
land expectation value defined by the Faustmann
equation; (2) the PNW of the optimal transition and
steady-state regime is greater than the PNW of the
statically determined regime; and (3) the optimal steady-state
regime is invariant. These results invalidate
the use of investment-efficient diameter distributions.
The gradient method produces stationary solutions for
harvest control variables in the first period and beyond;
however, because of large discount factors in distant
periods, the algorithm fails to provide stationary solutions
for long-term management within reasonable execution
times. To avoid this problem, a restart procedure that
takes advantage of the stability of the first-period solution
and the sequential nature of the problem is developed.
An economic model for harvesting forest stands is
presented and used to contrast the two major timber
harvesting systems: even-aged and uneven-aged management.
In contrast to even-aged management, the value of uneven-aged
stand harvesting cannot be separated into independent
components for stand value and land value. Thus, conclusions
about the most profitable harvesting system depend
on the joint productivity of the land and existing timber.
This result is demonstrated by developing optimal management regimes for Arizona ponderosa pine (Pinus ponderosa Laws.) stands.
