The Effect of Fire in Ponderosa Pine Forests

Transcription

1 Some Examples of Mathematics in Forest Ecology Department of Mathematics and Computer Science Colorado College April 3, 2008

2 Ecology Question The Ecology Question An important issue in forest ecology and management is the role of fire in forest communities.

3 Ecology Question The Ecology Question An important issue in forest ecology and management is the role of fire in forest communities. What factors contribute to the spatial arrangement of trees in a Ponderosa Pine forest?

4 Ecology Question The Ecology Question An important issue in forest ecology and management is the role of fire in forest communities. What factors contribute to the spatial arrangement of trees in a Ponderosa Pine forest? Recruitment (Where do seeds grow?) moisture and light litter depth

5 Ecology Question The Ecology Question An important issue in forest ecology and management is the role of fire in forest communities. What factors contribute to the spatial arrangement of trees in a Ponderosa Pine forest? Recruitment (Where do seeds grow?) moisture and light Fire litter depth death of seedlings removes litter from ground

6 Ecology Question The Ecology Question An important issue in forest ecology and management is the role of fire in forest communities. What factors contribute to the spatial arrangement of trees in a Ponderosa Pine forest? Recruitment (Where do seeds grow?) moisture and light Fire litter depth death of seedlings removes litter from ground By what mechanisms does fire affect the spatial arrangement of trees in a forest?

7 The Model The Model An object-oriented simulation was used to grow trees in a 1-acre square plot for 1000 years. simulation program written in Java simulation of tree growth and fire on 10cm x 10cm grid used growth and reproduction parameters appropriate for Ponderosa Pine

8 The Model The Model An object-oriented simulation was used to grow trees in a 1-acre square plot for 1000 years. simulation program written in Java simulation of tree growth and fire on 10cm x 10cm grid used growth and reproduction parameters appropriate for Ponderosa Pine The Treatments: Initial 5-year fire interval (Control) 10-year fire interval Equal Litter Equal Fire

9 Math Problem #1 How can I create an initial forest with different ages of trees?

10 Math Problem #1 How can I create an initial forest with different ages of trees? Trees grow in height according to: dh dt = r(t)(1 h H ) where h = height, r(t) = variable growth rate, H = maximum height.

11 Math Problem #1 How can I create an initial forest with different ages of trees? Trees grow in height according to: dh dt = r(t)(1 h H ) where h = height, r(t) = variable growth rate, H = maximum height. Suppose r(t) is a constant r.

12 Math Problem #1 How can I create an initial forest with different ages of trees? Trees grow in height according to: dh dt = r(t)(1 h H ) where h = height, r(t) = variable growth rate, H = maximum height. Suppose r(t) is a constant r. We can calculate height from: rt H ln(h) H e H

13 Math Problem #2 How do I choose parameters so that trees grow at the right speed?

14 Math Problem #2 How do I choose parameters so that trees grow at the right speed? Simulate differential equations for one tree in Mathematica. Change parameters until size variables match those in the literature.

15 Math Problem #3 How do I choose a simple fire model on a grid of cells?

16 Math Problem #3 How do I choose a simple fire model on a grid of cells? Direction of fire spread:

17 Math Problem #3 How do I choose a simple fire model on a grid of cells? Direction of fire spread: Probability of fire spread between two cells: fires should travel across the entire grid fires should not burn every cell

18 Math Problem #3 How do I choose a simple fire model on a grid of cells? Percolation theory: The probability of an infinite fire becomes positive when the probability of fire spreading between two cells is greater than 1 2.

19 Math Problem #3 How do I choose a simple fire model on a grid of cells? Percolation theory: The probability of an infinite fire becomes positive when the probability of fire spreading between two cells is greater than 1 2. Choose P =

20 Math Problem #4 How do I measure clustering of trees?

21 Math Problem #4 How do I measure clustering of trees? Quadrat Methods: divide the data into spatial regions at different scales and examine the differences between regions.

22 Math Problem #4 How do I measure clustering of trees? Quadrat Methods: divide the data into spatial regions at different scales and examine the differences between regions. : examine the distributions of various measures of distance within the data points (Diggle 1983): 1. inter-event distances 2. point to event distances 3. nearest neighbor distances

23 The Inter-Event Distance Distribution

24 The Inter-Event Distance Distribution The distribution function for inter-event distances between points distributed randomly on a square is described in Bartlett 1964: f (z) = { π L 4 z 2 L + z 3 L for z L L arcsin( 2L2 2 z 1) z L 2 2 L z 3 L for L 2 < z 2L 2 4 L 2 + 4

25 The Inter-Event Distance Distribution The distribution function for inter-event distances between points distributed randomly on a square is described in Bartlett 1964: f (z) = { π L 4 z 2 L + z 3 L for z L L arcsin( 2L2 2 z 1) z L 2 2 L z 3 L for L 2 < z 2L 2 4 L Use χ 2 analysis to compare this distribution to inter-event distances generated by the simulation

26 T-square Sampling Diggle (1976) recommends the statistic: t N = 1 m m i=1 u i u i v T i where

27 T-square Sampling

28 Comparing Simulations

29 Comparing Simulations The Treatments: Initial, 5-year fire interval (Control), 10-year fire interval Equal Litter, Equal Fire

30 Comparing Simulations The Treatments: Initial, 5-year fire interval (Control), 10-year fire interval Equal Litter, Equal Fire Compare distribution functions (Brieman 1973): χ 2 based on histogram categories Kolmogorov-Smirnov

31 Example A Simulation Comparison Example We will compare: Control vs. Equal Litter Control vs. Equal Fire We will use the distribution function of point to event distances (u). The Question: Which treatment had a greater affect on the distribution of nearest neighbor distances after 500 years?

32 Example A Simulation Comparison Example The χ 2 histogram analysis: D = mn J (ˆp (1) j ˆp (2) j ) 2 m+n j=1 ˆp j Control vs. Equal Litter Control vs. Equal Fire D has a χ 2 distribution with J 1 degrees of freedom.

33 Example A Simulation Comparison Example Comparison of the Distribution Function of u between Simulations Runs Compared D P Control (500 years) Equal Fire (500 years) Control (500 years) Equal Litter (500 years) Equal Fire (500 years) Equal Litter (500 years) 25.4 > 0.99 Initial Control (500 years) 50.7 > 0.99 Initial Equal Fire (500 years) Initial Equal Litter (500 years) < 0.001

34 Example A Simulation Comparison Example Comparison of the Distribution Function of u between Simulations Runs Compared D P Control (500 years) Equal Fire (500 years) Control (500 years) Equal Litter (500 years) Equal Fire (500 years) Equal Litter (500 years) 25.4 > 0.99 Initial Control (500 years) 50.7 > 0.99 Initial Equal Fire (500 years) Initial Equal Litter (500 years) < Equal Litter treatment had more of an effect than Equal Fire treatment. Equal Litter and Equal Fire were very similar. The Control run did not differ from the initial configuration after 500 years, but both Equal Litter and Equal Fire did.

35 Example A Simulation Comparison Example The Kolmogorov-Smirnov analysis: d = max ˆF 1 (x) ˆF 2 (x) d has a known distribution that rejects homogeneity if d is large.

36 Example A Simulation Comparison Example Comparison of the Distribution Function of u between Simulations Runs Compared d P Control (500 years) Equal Fire (500 years) 0.56 < 0.01 Control (500 years) Equal Litter (500 years) 0.70 < 0.01 Equal Fire (500 years) Equal Litter (500 years) 0.37 < 0.01 Initial Control (500 years) 0.39 < 0.01 Initial Equal Fire (500 years) 0.67 < 0.01 Initial Equal Litter (500 years) 0.85 < 0.01

37 Conclusion Further Questions: How does topography affect spatial dynamics of a forest? Will changing the size of the grid on which fire is simulated qualitatively affect observed spatial petterns? Are there better statistical methods to measure more detailed spatial patterns such as the size of clusters?

38 Conclusion Further Questions: How does topography affect spatial dynamics of a forest? Will changing the size of the grid on which fire is simulated qualitatively affect observed spatial petterns? Are there better statistical methods to measure more detailed spatial patterns such as the size of clusters? One biologically motivated problem can involve many areas of mathematics: Calculus, Differential Equations, Statistics, Computer Programming, Probability, Discrete Mathematics

39 Acknowledgements Thank you to: Dr. Steven Janke, my advisor for this project, for countless hours spent helping me pull this together. Dr. Robert Pelayo, Dr. Amelia Taylor, and Dr. Stefan Erickson for invaluable help with figuring out LaTEX and Beamer.

40 Sources P. Diggle, J. Besag, and J.T. Gleaves. Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 (1976), M.S. Bartlett. The spectral analysis of two-dimensional point processes. Biometrika 51 (1964), G. Grimmett. Percolation. Springer-Verlag, New York (1989). P. Diggle. Statistical Analysis of Spatial Point Patterns. Academic Press, New York (1983). L. Breiman. Statistics: With a View Towards Applications. Houghton-Mifflin (1973).