1. Some Examples of Stochastic Differential
Equations in Forestry Research by
Haiganoush K Preisler
Mathematical Statistician
Pacific Southwest Research Station
USDA Forest Service
11 October 2005

2. Black-Scholes formula
1997 Nobel Prize
in Economic Sciences
S(t) is value of a stock at time t
B(t) is a Brownian process (random noise)
Stochastic differential equation

3. Heat Equation H(x,t)= temperature at depth x and time t.
Joseph Fourier ( )
French Mathematician
H(x,t)= temperature at depth x and time t.

4. Fourier used heat equation to predict temperature of the ground at depth x due to the sun’s heating.
Concluded
At certain depths temperatures are out of step with surface temperature (phase lag).
Six month lag at 2-3 m depth.
(i.e., hotter in winter, cooler in summer).
Good depth for cellars.
Permafros

5. Risks
Increase in mortality rates observed in soil organisms at o C.
Tissue cannot survive temperatures greater than
o C.

6. 20 40 60 80
100
120
Time (hrs)
140
Temperature (Celsius)
Temperature Profiles at 10cm Depth Around
Sugar Pine Trees
Temperature Profiles at 10 cm Depth Around
Sequoia Trees
Different curves
due to different soil conditions, some of which (e.g. soil moisture) are measured.

8. Rocky Mountain elk
(Cervus elaphus)

9. Fenced area of Starkey Experimental Station in Oregon
Tracks of four elk Km 2 4 6 8 10 12 14 a b

10. Differential equations
r(t) = { x(t), y(t) } = location of elk at time t.
Stochastic differential equations

14. References
Preisler, H.K., Haase, A.M. and Sackett, S.S. (2000). Modeling and risk assessment for soil temperatures beneath prescribed forest fires. Environmental and Ecological Statistics 7,
2. Preisler, H.K., Ager, A.A., Johnson, B.K., and Kie, J.G. (2004) Modeling animal movements using stochastic differential equations. Environmetrics 15: