Many species of seal have suffered from commercial hunting. They have been killed for their skin, blubber, and meat. The fur trade, in particular, reduced some seal populations to the point of extinction. Today, the greatest threats to seal populations are decline of fish stocks due to over fishing, pollution, disturbance of habitat, entanglement in marine debris, and culling by fishery owners. Some seals have been declared endangered species; other species are carefully managed. Table 4.7 gives the birth and survival rates for the northern fur seal, divided into 2-year age classes. [The data are based on A. E. York and J. R. Hartley, €œPup Production Following Harvest of Female Northern Fur Seals,€ Canadian Journal of Fisheries and Aquatic Science, 38 (1981), pp. 84€“90.] 

(a) Construct the Leslie matrix L for these data and compute the positive eigenvalue and a corresponding positive eigenvector.
(b) In the long run, what percentage of seals will be in each age class and what will the growth rate be? Show that r = 1 if and only if λ₁ = 1. (This represents zero population growth.)

Let Show that l is an eigenvalue of L if and only if g(λ) = 1.]

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Table 4.7 Age (years) Birth Rate Survival Rate 0-2 0.00 0.91 2–4 0.02 0.88 4–6 0.70 0.85 6-8 1.53 0.80 8–10 1.67 0.74 10–12 1.65 0.67 12–14 1.56 0.59 14–16 1.45 0.49 16–18 1.22 0.38 18–20 0.91 0.27 20–22 0.70 0.17 22–24 0.22 0.15 24–26 0.00 0.00