Harvesting Models
Wayne Marcus Getz
Department of Environmental Science, Policy & Management
University of California at Berkeley, USA, [email protected]
Mathematical theories of harvesting biological resources can be traced back to
Martin Faustmann’s 1849 analysis of the optimal interval for the periodic harvesting of
cultivated forest stands. Faustmann’s work presaged two different pathways that
harvesting theory took a hundred years later in the early 1950s. These were R. J. H.
Beverton and S. J. Holt’s cohort analysis and M. B. Schaefer’s maximum sustainable rent
analysis.
The latter matured largely though the work of C. W. Clark and colleagues in
the early to mid 1970’s into the field of
Mathematical Bioeconomics
.
In the late 1970s
cohort theory was linked, as described below, to
Leslie matrix
modeling theory and
extended to include nonlinear stockrecruitment analysis, first begun by W. E. Ricker,
and Beverton and Holt in the early 1950s. Today, harvesting theory has been greatly
extended beyond these roots to include the uncertainties inherent in the demographic
processes of births and deaths, the vagaries of environmental forces impinging on
populations, and the incompleteness of biological models, as well as risk analysis and the
use of sophisticated social and economic instruments for meeting the competing needs of
different stakeholders concerned with the exploitation of any particular biological
resources.
I.
Harvesting Plantations
A.
Value function
Plantation or evenaged forest stand management theory can be traced back to
Faustmann’s 1849 formulation for calculating the optimal rotation period for clear
cutting a stand of trees for timber or pulp, where the stand is then replanted only to be
clearcut again at the end of the next rotation period.
The analysis begins with the
assumption that a net revenue function exists that over time represents that net profit
P
(
t
)
that will be obtained if the stand is clearcut at time
t
.
It follows immediately after clear
cutting that in the new rotation period
P
(0)=0, and also that although
P
(
t
) may initially be
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View Full Documentnegative for small
t
(if we clearcut too soon it may actually cost us more to take this
action than the value of the harvest once sold and the costs of replanting the stand may
also not be covered) the enterprise will not be worth undertaking unless ultimately
P
(
t
)>0.
A point in time may come, say
t
*, at which
P
*=
P
(
t
*)>
P
(
t
) for
t
≠
t
*, where we
assume that if
t
* exists it is unique in satisfying this condition for all
t
∈
[0,
∞
).
If
t
*
exists, this implies that the value of the resource starts decaying in quality beyond
t
*, e.g.
timbers getting knotty with age or wood loses quality as a source of pulp for making
paper. If
t
* does not exist, this implies
P
(
t
) asymptotically approaches
P
* as
→∞
.
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 Fall '10
 WayneM.Getz
 Demography, Net Present Value, Logistic function

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