Hallo!

1. Hallo! • Carol Horvitz • Professor of Biology • University of Miami, Florida, USA • plant population biology, spatial and temporal variation in demography • applications to plant-animal interactions, invasion biology, global change, evolution of life span

3. Institute for Theoretical and Mathematical Ecology University of Miami Coral Gables, FL USA Mathematics Steve Cantrell Chris Cosner Shigui Ruan Biology Don De Angelis Carol Horvitz Matthew Potts Marine Science Jerry Ault Don Olson

4. Dynamics of structured populations • N(t+1) = N(t) * pop growth rate • Pop growth rate depends upon • Survival and reproduction of individuals • Survival, growth and reproduction are not uniform across all individuals • Thus the population is structured

5. Demographic structure age stage size space year habitat Modeling dynamics life table matrix life cycle graph Population dynamics: changes in size and shape of populations

6. Regression Log-linear Age vs. stage?

7. Projection n(t+1) = A n(t)

8. Population projection matrix

9. Population projection matrix

10. Population projection matrix

11. Population projection matrix

12. Population projection matrix

13. Life cycle graph

14. try it • Start with 10 in each stage class • multiply and add • row times column

15. Population projection matrix

16. try it • Start with 10 in each stage class • Start with 72, 17, 6 and 5 in the stage classes

17. Population projection matrix

18. try it • Start with 10 in each stage class • n(2) = 121, 3, 4, 7 • Start with 72, 17, 6 and 5 in the stage classes • n(2) = 67,16, 6, 5 • population growth rate = 0.9564

19. Projection n(1) = A n(0) time n(2) = A n(1) n(3) = A n(2) n(4) = A n(3) n(5) = A n(4) n(6) = A n(5)

20. Projection n(t+1) = A n(t)

21. Projection n(1)= A n(0) n(2)= AAn(0) n(3)= AAAn(0) n(4)= AAAAn(0) n(5)= AAAAAn(0) n(6)=AAAAAAn(0)

22. Projection n(t) = At n(0)

23. Projection n(t+1) = A n(t) Each time step, the population changes size and shape. The matrix pulls the population into different shapes. There are some shapes that are ‘ in tune ’ with the environment. For these, the matrix only acts to change the size of the population. In these cases the matrix acts like a scalar.

24. Projection n(t+1) = A n(t) n(t+1) = n(t)

25. Projection n(t+1) = A n(t) Examples: stable stage reproductive values sensitivity to perturbation time variant density dependent other

26. 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 Projection exercises

27. Analytical entities • Dominant eigenvalue • Dominant right eigenvector (ssd) • Dominant left eigenvector (rv) • Derivative of population growth rate with respect to each element in the matrix • Derivative of the logarithm of population growth rate with respect to the logarithm of each element in the matrix