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An Application of Eigentheory

An Application of Eigentheory. The dynamics of the Orcinus orca population. Orcinus orca . Yearling. Juvenile. A model. Post-Reproductive. Mature. 91.11%. 97.75%. Yearling. Juvenile. .43%. A model. 7.36%. 11.32%. Post-Reproductive. Mature. 4.52%. 95.34%. 98.04%. 91.11%.

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An Application of Eigentheory

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  1. An Application of Eigentheory The dynamics of the Orcinusorca population

  2. Orcinus orca

  3. Yearling Juvenile A model Post-Reproductive Mature

  4. 91.11% 97.75% Yearling Juvenile .43% A model 7.36% 11.32% Post-Reproductive Mature 4.52% 95.34% 98.04%

  5. 91.11% 97.75% Yearling Juvenile Where : Yk = yearlings in year k Jk= juveniles in year k Mk= matures in year k Pk= post-matures in year k .43% 7.36% 11.32% Post-Reproductive Mature 4.52% 98.04% 95.34%

  6. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years?

  7. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years?

  8. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years? Eigenvalues are 1.0254 0.9804 0.8342 0.0048

  9. 91.11% 40.00% 97.75% Yearling Juvenile .43% A newmodel 7.36% 11.32% Post-Reproductive Mature 4.52% 95.34% 98.04%

  10. 0.4000 91.11% 40.00% 97.75% Yearling Juvenile Where : Yk = yearlings in year k Jk= juveniles in year k Mk= matures in year k Pk= post-matures in year k .43% 7.36% 11.32% Post-Reproductive Mature 4.52% 98.04% 95.34%

  11. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years?

  12. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years? Eigenvalues are 0.9945 0.9804 0.8681 0.0020

  13. And in 1000 years…

  14. 48.56% 91.11% 40.00% 97.75% Yearling Juvenile .43% A third model 7.36% 11.32% Post-Reproductive Mature 4.52% 95.34% 98.04%

  15. 0.4000 91.11% 48.56% 40.00% 97.75% Yearling Juvenile Where : Yk = yearlings in year k Jk= juveniles in year k Mk= matures in year k Pk= post-matures in year k .43% 7.36% 11.32% Post-Reproductive Mature 4.52% 98.04% 95.34%

  16. Suppose we begin with Y1 = 100 J1= 100 M1= 100 P1= 100 What happens in 100 years? Eigenvalues are 1.0000 0.9804 0.8621 0.0024

  17. And in 1000 years…

  18. An Application of Eigentheory Growth Rates of Computational Work

  19. We have a problem whose computational work depends upon some size parameter n W(n+1) = W(n) + 2·W(n-1) + 3·W(n-2) for n = 3,4, …

  20. We could represent this in matrix form as: W(n+1) = W(n) + 2·W(n-1) + 3·W(n-2) for n = 3,4, …

  21. The eigenvalues of that matrix are: 2.3744 -0.6872 + 0.8895i -0.6872 - 0.8895i

  22. Complex numbers The eigenvalues of that matrix are: 2.3744 -0.6872 + 0.8895i -0.6872 - 0.8895i

  23. This means that W(n) = O(2.3744n) Semilog plot of W(n) starting with W(1) = W(2) = W(3) = 1 The eigenvalues of that matrix are: 2.3744 -0.6872 + 0.8895i -0.6872 - 0.8895i

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