Harvesting an Age-Distributed Population

Part 2: Harvesting the youngest class

Sheep are not all equally valuable for harvest -- in fact, in world meat markets, lamb is much more valuable than mutton. Thus, the best economic use of the herd might be to harvest only lambs and keep the mature ewes alive to breed more lambs. Is there a sustainable harvesting policy for this case?

If we harvest only from the youngest group, then h₁ = h, the fraction harvested from that group, and all the other h's are zero. In step 2 of Part 1, we asked you to think about the algebraic condition on h's in order to have 1 as an eigenvalue of (I - H)L. With only the first h different from zero, that condition simplifies to

or

where

The quantity R is called the net reproduction rate of the population -- it is, in fact, the average number of daughters born to a ewe in her expected lifetime.

1. Calculate the net reproduction rate R for the New Zealand sheep population.
2. To have a sustainable harvesting policy with only lambs being harvested, we must have (1 - h)R = 1. Find the number h that satisfies this condition. What fraction of the lambs should be harvested each year?
3. Find the stable population distribution x for this harvesting policy. What fraction of the total population should be lambs at the start of the growth period? at the end of the growth period? What fraction of the total population is harvested after the growth period? How does this compare with uniform harvesting?

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