Some Examples of Stochastic DifferentialEquations in Forestry Research
byHaiganoush K Preisler
Mathematical StatisticianPacific Southwest Research Station
USDA Forest Service
11 October 2005
Black-Scholes formula
)()()()( tdBtSdttStdS
S(t) is value of a stock at time tB(t) is a Brownian process (random noise)
Stochastic differential equation
1997 Nobel Prize in Economic Sciences
Heat Equation
H x t
t
H x t
x
( , ) ( , )
2
2
H(x,t)= temperature at depth x and time t.
Joseph Fourier (1768-1830)
French Mathematician
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
Risks
• Increase in mortality rates observed in soil organisms at 34 - 40o C.
• Tissue cannot survive temperatures greater than 45 - 65o C.
0 20 40 60 80 100 120Time (hrs)
020
4060
8010
012
014
0T
empe
ratu
re (
Cel
sius
)
Temperature Profiles at 10cm Depth Around Sugar Pine Trees
0 20 40 60 80 100 120Time (hrs)
020
4060
8010
012
014
0T
empe
ratu
re (
Cel
sius
)
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.
Rocky Mountain elk(Cervus elaphus)
Km
Km
0 2 4 6 8 10
0
2
4
6
8
10
12
14 a
Km
Km
0 2 4 6 8 10
0
2
4
6
8
10
12
14 b
Fenced area of Starkey Experimental Station in OregonTracks of four elk
Differential equations
r(t) = { x(t), y(t) } = location of elk at time t.
)( ; )( tvdt
dytv
dt
dxyx
Stochastic differential equations
)( ; )( noisetvdt
dynoisetv
dt
dxyx
Km
Km
4 5 6 7
1.5
2.0
2.5
3.0
3.5
4.0
4.5
References
1. 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, 239-254.
2. Preisler, H.K., Ager, A.A., Johnson, B.K., and Kie, J.G. (2004) Modeling animal movements using stochastic differential equations. Environmetrics 15: 643-657.