2. Mortality over Time Population Density Declines through Mortality

3. Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

4. Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

5. Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

6. Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

7. Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) General pattern 1.Unimpeded growth 2.Mortality begins 3.Similar trajectories exhibited once thinning starts 4.At some point thinning slows 1 1 1 1 2 2 2 3 4

8. Self Thinning in Thirty Species Similar slope to thinning line across a range of species

9. Attempts to Explain the Thinning Line

10. An Intuitive Argument Two stands of trees starting at different densities

11. An Intuitive Argument Two stands of trees starting at different densities Thinning occurs as trees increase in size.

12. An Intuitive Argument Two stands of trees starting at different densities Thinning occurs as trees increase in size. Trees cannot grow larger unless enough space is made available through mortality.

13. Yoda et al. (1963) propose the “-3/2 Thinning Law” k ≈ -3/2

14. “-3/2 Thinning” k ≈ -3/2 Allometric relationships: those that scale with body mass They posit an underlying allometric relationship

15. “-3/2 Thinning” k ≈ -3/2 They posit an underlying allometric relationship w = average individual biomass C = constant N = population density -k = slope of thinning line

16. “-3/2 Thinning” k ≈ -3/2 They posit an underlying allometric relationship Why 3/2?

17. An Intuitive Argument

21. Biomass Density  Volume–> m 3  Area  m2  m2

22. An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

23. An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

24. An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

25. An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

26. k ≈ -3/2k ≈ -4/3 Revisiting the “-3/2 Thinning Law” X

27. k ≈ -3/2k ≈ -4/3 A Revised View of the Allometric Relationship Same as the scaling relationship of body mass to maximum density in animals!

28. A General Interpretation of the Thinning Relationship

29. Lemna Sequoia A General Interpretation of the Thinning Relationship

30. Permitted combinations Prohibited combinations

31. Self Thinning Revisited Log mean plant weight (w ) Log density (N) General pattern 1.Unimpeded growth 2.Mortality begins 3.Similar trajectories exhibited once thinning starts 4.At some point thinning slows 4 ?

32. Self Thinning Revisited Log mean plant weight (w ) Log density (N) Growth limited by space Growth limited by resources

33. Self Thinning Revisited Log mean plant weight (w ) Log density (N) Growth limited by resources Resource limitation regulating growth leads to the “Law of Constant Yield”

34. Proof of Constant Yield with a slope = -1 Log mean plant weight Log density Slope ≈ -1 log(N)log(N-z) log Y N Y (N-z)

35. Proof of Constant Yield with a slope = -1 Log mean plant weight Log density Slope ≈ -1 log(N)log(N-z) log Y N Y (N-z) Calculation of slope

36. Proof of Constant Yield with a slope = -1 Log mean plant weight Log density log(N)log(N-z) log Y N Y (N-z) Calculation of slope XX

37. Proof of Constant Yield with a slope = -1 Log mean plant weight Log density log(N)log(N-z) log Y N Y (N-z) Calculation of slope = -1

38. Putting it all together Development of size hierarchies Thinning Law of Constant Yield