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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
(1 - h)(a1 + a2b1 + a3b1b2 + ... + a12b1b2b3...b11) = 1
or
(1 - h)R = 1,
where
R = a1 + a2b1 + a3b1b2 + ... + a12b1b2b3...b11.
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.
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