1. Projection exercises Stable age distribution and population growth rate Reproductive value of different ages Not all matrices yield a stable age distribution concentration of reproduction in the last age oscillations

2. Eigenvalues, eigenvectors and the projection equation

3. « Eigen » A natural direction of the dynamics of the matrix Such that if the system starts on this direction, it will stay in the same direction

4. Solution of the projection equation A matrix has many eigenvalues (speeds) A matrix has many eigenvectors (directions)

5. Right eigenvectors These are columns Matrix * column = a new column of the same shape

6. Left eigenvectors These are transpose of rows Row * Matrix = a new row of the same shape

7. Population projection matrix Carreta carreta (Crowder et al. 1994)

8. Try it (we will use mat2valvecs) Multiply the matrix by the column Multiply the row by the matrix

9. Solution of the projection equation A matrix has many eigenvalues (speeds) A matrix has many eigenvectors (directions) The number of individuals in a each stage at some time in the future will depend upon ALL these Eigenvalues are raised to the « t th » power (see 4.49, Caswell 2001, p. 75)

10. To find out what will happen We need to know how fast the system is moving in each direction In other words, the relative impact of lambdas after time (ie after raising them to the « t th » power

11. Raising lambdas to powers Fractional or not? Positive or negative? Absolute value = 1? Real or complex If complex, the absolute magnitude?

12. Raising lambdas to powers Exponential increase Exponential decrease Oscillations Spiraling through the complex plane

13. Directions are weighted The direction associated with a lambda that increases (or decreases) exponentially when raised to successively higher powers will dominate over other directions

14. Some analytical entities Dominant eigenvalue Dominant right eigenvector (ssd) Dominant left eigenvector (rv)

15. Sensitivity: What ’s important to population growth? A bad question! Good questions are more specific Prospective vs. retrospective questions A parameter which does not vary can not contribute to variation in population growth no matter how high its sensitivity is!