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HarvestingTheoryNotes - Harvesting Models Wayne Marcus Getz...

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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 stock-recruitment 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 even-aged 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 clear-cut 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 clear-cut 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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negative for small t (if we clear-cut 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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HarvestingTheoryNotes - Harvesting Models Wayne Marcus Getz...

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