Plant Propagation Fronts and Wind Dispersal

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Plant Propagation Fronts and Wind Dispersal

• Sally Thompson and Gabriel Katul
• Duke University

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Plant Dispersal

• Long timescales
• How do plants explore a landscape?
• Wind dispersal is a common strategy amongst
  plants
• Seek a simple process model

Acer saccharum
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Outline

• Development of modified Fisher Equation
• Conceptual model for upscaling with
  superstatistics (after Beck 2003)
• Case Study Post Glacial Expansion
• Ecohydrological considerations

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Fisher Equation

• Darcys law of mathematical biology
• Growth and spread

Logistic Growth
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Fisher Equation

• Darcys law of mathematical biology
• Growth and spread

Spread (diffusion)
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Modify Fisher Equation

• Diffusion term assumes spread is Gaussian
• More general form dispersal kernels

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A Kernel for Wind Dispersal

• WALD Kernel (Katul et al 2005)
• Lagrangian stochastic model
• Distribution of seed rain
• Leptokurtic cf Gaussian
• Improved predictions cf competing models

W - Wald kernel Zr - seed release height h
- canopy height U - mean wind speed ? - ratio
of vertical velocity fluctuations to U Vt
- seed terminal velocity ? - eddy length scale
O(1)
Katul et al 2005, AmNAT
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A Kernel for Wind Dispersal

• WALD Kernel (Katul et al 2005)
• Lagrangian stochastic model
• Distribution of seed rain
• Leptokurtic cf Gaussian
• Improved predictions cf competing models

W - Wald kernel Zr - seed release height h
- canopy height U - mean wind speed ? - ratio
of vertical velocity fluctuations to U Vt
- seed terminal velocity ? - eddy length scale
O(1)
Katul et al 2005, AmNAT
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Modified Fisher with WALD

• For constant parameters an analytical solution
  exists
• c asymptotic maximum spread rate (L/T)

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Scaling

• Dimensionality Curse

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Conceptual Model
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Half Hour to Annual Scales - Superstatistics

• U at ½ hour scale
• Weibull Distribution UWeib(b,k)
• Nonlinear propagation of wind statistics into c
  (front speed) and Ueff (effective wind speed).
• Numerics - Monte Carlo simulations spanning a
  range in b, k
• Relate statistics of U to statistics of Ueff

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Case Study

• Post glacial expansion of tree ranges in SE USA -
  Pollen record
• Long period - asymptotic speeds ok
• Can we predict spread rates?
• Tree parameters very simple allometry
• Wind climate for 10,000 years ago use data for
  eastern USA today.
• Aim to do better than an order of magnitude
  estimate

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Case Study

• Input parameters
• r 1.9-8.2 kg/yr
• h 14.6-25 m
• Vt 0.67-1.6 m/s
• Zr 9.5-15 m

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Modified Fisher Equation Reasonable
ACRU Acer rubrum ACSA Acer saccharum ACNE
Acer negundo ACSM Acer saccharinum BELE
Betula lenta FRAM Fraxinus
americana FRPE Fraxinus
pennsylvanica PITA Pinus taeda
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Conclusions - Specific

• Modified Fisher Equation robust for a broad
  parameter range
• No extreme events needed
• Analytical solution allowed sensitivity testing
  of parameters
• Decouple process from a

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Conclusions - General

• Improved representation of plant dispersion
  important eg patterned veg.
• Novel approach for upscaling
• Dimensionality curses are common!
• Apply to space as well as time

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Acknowledgments

• General Sir John Monash Foundation

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Half Hour to Annual Scales - Superstatistics

• U stationary O(0.5 hrs)
• Changing U ? nonlinear impact on c

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Half Hour to Annual Scales - Superstatistics

• U stationary O(0.5 hrs)
• Changing U ? nonlinear impact on c

Best representation of annual wind climate
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Numerical Approaches and Superstatistics

• Determine f,g via nonlinear regression

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Analytical Solution

• Analytical Solution for Front Speed

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A Semi-Analytical Model
Turbulence
Turbulence
Super
Super
Scaling
Scaling
Seed Trajectories
Seed Trajectories
-
-
-
-
Up
Up
statistics
statistics
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A Semi-Analytical Model
Step 1 Determine Weibull Statistics thus Ueff
Turbulence
Turbulence
Super
Super
Scaling
Scaling
Seed Trajectories
Seed Trajectories
-
-
-
-
Up
Up
statistics
statistics
25
A Semi-Analytical Model
Step 1 Determine Weibull Statistics thus Ueff
Turbulence
Turbulence
Super
Super
Scaling
Scaling
Seed Trajectories
Seed Trajectories
-
-
-
-
Up
Up
statistics
statistics
Step 2 Determine tree parameters and thus WALD
kernel
26
A Semi-Analytical Model
Step 1 Determine Weibull Statistics thus Ueff
Turbulence
Turbulence
Super
Super
Scaling
Scaling
Seed Trajectories
Seed Trajectories
-
-
-
-
Up
Up
statistics
statistics
Step 2 Determine tree parameters and thus WALD
kernel
Step 3 Solve for the annual plant spread rate