Dynamics of Orcinus orca Population: An Application of Eigentheory

This study explores the dynamics of the Orcinus orca population using eigentheory models. It examines various age groups within the population and predicts their growth over a span of 100 years. Eigenvalues play a crucial role in understanding the evolution of the population dynamics.

• Orcinus orca
• Population dynamics
• Eigentheory
• Model analysis
• Age groups

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1. An Application of Eigentheory The dynamics of the Orcinus orca population

2. Orcinus orca

3. A model Yearling Juvenile Post- Mature Reproductive

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

5. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P + 0.9775 0.9111 0 0 0 1 k k = + 0.0736 0.9534 0 1 k k 91.11% + 0.0452 0.9804 1 k k 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- Reprod uctive Mature 4.52% 98.04% 95.34%

6. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P + 0.9775 0.9111 0 0 0 1 k k = + 0.0736 0.9534 0 1 k k + 0.0452 0.9804 1 k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years?

7. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P + 0.9775 0.9111 0 0 0 1 k k = + 0.0736 0.9534 0 1 k k + 0.0452 0.9804 1 k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years? 1800 1600 1400 1200 1000 800 600 400 200 0 0 10 20 30 40 50 60 70 80 90 100

8. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P + 0.9775 0.9111 0 0 0 1 k k = + 0.0736 0.9534 0 1 k k + 0.0452 0.9804 1 k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years? 1800 1600 1400 1200 1000 800 600 Eigenvalues are 1.0254 0.9804 0.8342 0.0048 400 200 0 0 10 20 30 40 50 60 70 80 90 100

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

10. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P 0.4000 + 0.9775 0.9111 0 0 0 1 k k = + 0.0736 0.9534 0 1 k k 91.11% 40.00% + 0.0452 0.9804 1 k k 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- Reprod uctive Mature 4.52% 98.04% 95.34%

11. + 0 0.0043 0.1132 0.9111 0.0736 0.9534 0 0 0 0 1 k k Y J M P Y J M P + 0.4 0 0 0 1 k k = + 1 k k + 0.0452 0.9804 1 k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years? 250 200 150 100 50 0 0 10 20 30 40 50 60 70 80 90 100

12. + 0 0.0043 0.1132 0.9111 0.0736 0.9534 0 0 0 0 1 k k Y J M P Y J M P + 0.4 0 0 0 1 k k = + 1 k k + 0.0452 0.9804 1 k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years? 250 200 150 100 Eigenvalues are 0.9945 0.9804 0.8681 0.0020 50 0 0 10 20 30 40 50 60 70 80 90 100

13. And in 1000 years 250 200 150 100 50 0 0 100 200 300 400 500 600 700 800 900 1000

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

15. + 0 0.0043 0.1132 0 0 0 1 k k Y J M P Y J M P 0.4000 0.4856 0.9111 0 0 + 0 1 k k = + 91.11% 0.0736 0.9534 0 1 k k 48.56% + 0.0452 0.9804 1 k k 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- Reprod uctive Mature 4.52% 98.04% 95.34%

16. + 0 0 0.0043 0.1132 0.9111 0.0736 0.9534 0 0 0 0 0 0 1 k 1 k + Y J M P P Y J M P P 0.0043 0.1132 0 0 k k Y J M Y J M + 0.4 0 0 0 0 0 1 k 1 k + 0.4856 0.9111 0 k k = = + 1 k 1 k + 0.0736 0.9534 k k + 0.0452 0.9804 0.0452 0.9804 1 k 1 k + k k Suppose we begin with Y1 = 100 J1 = 100 M1 = 100 P1 = 100 What happens in 100 years? 300 250 200 150 100 Eigenvalues are 1.0000 0.9804 0.8621 0.0024 50 0 0 10 20 30 40 50 60 70 80 90 100

17. And in 1000 years 350 300 250 200 150 100 50 0 0 100 200 300 400 500 600 700 800 900 1000

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. W(n+1) = W(n) + 2W(n-1) + 3W(n-2) for n= 3,4, We could represent this in matrix form as: ( 1) 0 0 3 1 0 2 0 1 1 ( ( 2) 1) W n W n W n W n W n W n = ( ) + ( 1) ( )

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

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

23. The eigenvalues of that matrix are: 2.3744 -0.6872 + 0.8895i -0.6872 - 0.8895i This means that W(n) = O(2.3744n) 10300 Semilog plot of W(n) starting with W(1) = W(2) = W(3) = 1 10250 10200 10150 10100 1050 100 0 100 200 300 400 500 600 700 800 900 1000