Harvesting an Age-Distributed Population
Part 3: Optimal harvesting
We have now considered two
sustainable harvesting policies: harvesting the same fraction from every
age group, and harvesting only from the youngest age group. The latter probably
produces a much smaller harvest in pounds of meat, but possibly a more valuable
harvest for the sheep farmer. We now ask: What sustainable harvesting policy
would produce the largest possible harvest in terms of numbers of animals?
Another way to ask the question: What combination of harvesting fractions
h1, h2, ..., h12 results in the largest
fraction of the total population being harvested -- while still maintaining
sustainability?
On the face of it, this
is a very difficult question. However, there is a theorem from linear programming
theory that simplifies the question considerably:
Optimal Sustainable Yield
| If a sustainable harvesting policy is optimal, it harvests only from
one or two age classes. If two age classes are harvested, then the older
class is completely harvested. |
| [C. Rorres, ``Optimal Sustainable Yield of a Renewable Resource,'' Biometrics, Vol. 32, 1976, pages 945-948.] |
- For the New Zealand sheep
population, it can be shown (by techniques beyond the level of this course)
that the optimal yield is achieved when h1 = 0.522, h9
= 1, and all the other h's are zero. Find the sustainable age distribution
for this policy, both before and after growth.
- What fraction of the total
population is harvested each year? How does this compare with uniform harvesting
and with lambs-only harvesting?
modules at math.duke.edu
