Ecological Risk Assesment (EcRA)

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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
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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
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Heat Equation
Joseph Fourier (1768-1830) French Mathematician
H(x,t) temperature at depth x and time t.
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• Fourier used heat equation to predict temperature
  of the ground at depth x due to the suns
  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

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Risks

• Increase in mortality rates observed in soil
  organisms at 34 - 40o C.
• Tissue cannot survive temperatures greater than
  
• 45 - 65o C.

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Different curves due to different soil
conditions, some of which (e.g. soil moisture)
are measured.
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Rocky Mountain elk (Cervus elaphus)
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Fenced area of Starkey Experimental Station in
Oregon Tracks of four elk
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Differential equations r(t) x(t), y(t)
location of elk at time t.
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• 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,
  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.