Tree size distribution across forest communities

1. Tree size distribution across forest communities Fangliang He Department of Renewable Resources University of Alberta

2. Trade-offs in ecology • Tradeoffs in mechanisms: shade tolerant vs intolerant, specialist vs generalist, rvsk species, colonization vs competition • Tradeoffs in structure: rare vs abundant, small vs large, low vs tall

3. No of stems Tree size Tree size distribution: number versus size With few exceptions forest communities have many small trees but a few large trees. The reverse relationship between the number of trees and size may result from many mechanisms: self-thinning (competition-density effect), growth-mortality tradeoff, allometric scaling, statistical processes, self-organized criticality, etc.

4. Mean weight/genet (g) Genet density (#stems/m2) Kay & Harper (JE 1974) -3/2 thinning rule

5. Enquist, Brown & West (Nature 1998) Enquist & Niklas (Nature 2001) -4/3 thinning rule (allometric scaling of energy in a saturated ecosystem)

6. Self-organized criticality or statistical properties? Self-organized critical phenomena is exhibited by driven systems which reach a critical state by their intrinsic dynamics, independently of the value of any control parameter. The archetype of a self-organized critical system is a sand pile. Sand is slowly dropped onto a surface, forming a pile. As the pile grows, avalanches occur which carry sand from the top to the bottom of the pile. At least in model systems, the slope of the pile becomes independent of the rate at which the system is driven by dropping sand. This is the (self-organized) critical slope.

7. Self-organized criticality or statistical properties? Reed & Hughes (JTB 2002) Chu & Adami (PNAS 1999)

8. Growth-mortality dynamics A forward Kolmogorov equation: where f(x) is the number of stems in size class x, g(x) is size dependent mean growth rate, m(x) is mortality rate. (The variance term can be ignored.) Probability density function of size distribution: Kohyama et al. J. Ecol. 91:797-806 (2003) Coomes et al. Ecol. Lett. 6:980-989 (2003)

9. Growth-mortality dynamics Exponential distribution Weibull distribution Pareto distribution (c-b=1) Muller-Landau et al. Ecol. Lett. 9:589-602 (2005)

10. log(no of stems) log(no of stems) log(no of stems) log(DBH) DBH log(DBH) Weibull Exponential Pareto

11. The derivation of tree size distribution based on the forward Kolmogorov equation requires an equilibrium state of the growth and mortality dynamics and total population size to be stable. Next we take a different approach to show that these conditions are not necessary. The new approach considers deterministic growth but stochastic colonization/mortality.

12. Exponential growth with a random harvest process The power-law model This model can arise from exponential growth with a random harvest (killing) process: Harvest process Exponential growth

13. Harvest process Exponential growth

14. where Weibull distribution: the most widely used tree size model Harvest process Power growth

15. Weibull distribution: interpretation • An even-aged community where all trees have the same growth rate, but different (exponential) harvesting time. Equivalent to observing a single tree at different time. • An uneven-aged community where trees have the same growth rate, but the mortality and colonization is a stochastic process following an exponential distribution.

16. Enquist & Niklas (Nature 2001) The Gentry data (226 sites)

17. How well the Weibull distribution works?

18. Test the assumption of the power law growth rate? Power growth Weibull

19. Barro Colorado Island

20. a  = 0.6856 The Panama data

21. Average age of the trees in Panama a Empirical estimate of the age of BCI trees: 30-50 yrs

22. Summary • There is no universal law governing tree size distribution. Tree size distribution is more realistically modeled by mortality-growth dynamics. • Weibull distribution of tree size arises from deterministic growth (with trees having the same growth rate) but stochastic mortality and colonization following an exponential distribution. • Enquist et al. metabolic-based power-law model is a special case of the mortality-growth dynamics.