Modeling Forest Canopies for Herbivory

1
Modeling Forest Canopies for Herbivory Leon
Kaganovskiy and Margaret Lowman New College of
Florida, Sarasota, FL

• Daily Percentage Eating Model
• d - percentage of leaf eaten daily
• Start with green leaves - 0 eaten.
• Each leaf has user-defined probability to be
  eaten (p_eat)
• If any of 26 neighbors are eaten,
• with probability p_spread this cell will get
  "infected"
• At each step (day) increase amount eaten by d
• Percentage eaten increases continually
• Once reaches max-alive threshold - leaf dead.
• Allow re-growth with probability p_regrow
• Measure averaged percentage of leaves eaten vs
  time.
• N - number of points, p_eat - probability of a
  leaf being attacked by bugs.
• p_spread - probability of spread of bugs from
  leaf to leaf
• p_regrow - probability of re-grow day d -
  daily percentage eaten.

• Overview
• Model growth and dynamics of tree leaves attacked
  by insects using tree data structures.
• 3D percolation/cellular automata ideas.
• Similar to fire and infectious decease spread.
• Build generic stochastic trees.
• Volume of leaves broken into computational cells.
  Uniform spacing, 1-1 map to leaves
• Dynamics of the leaves behavior including
  herbivory, aging, light etc evaluated on the
  computational cells and mapped to tree leaves.
• In addition, developed Matlab program to find
  areas of healthy and eaten leaves using Greens
  theorem.

Typical 5 state model output, steps 1, 10, 20
80
Recursive Tree Creation

• Each step 2, 3 or 4 branches are created at
  random
• Length and radius is predefined fraction of
  parents length and radius
• For 3 branches, position of the end of each
  secondary (child) branch related to the parent
  branch end is

Angles of branches are chosen at random in the
ranges
Small probabilities of bug "attack" and spread of
bugs between computational cells. No states,
plot only average percentage of leaves
eaten. Compare to data collected by student C.
Lawton. Statistics done by S. Lowitt. Growth
tendency compares well with the experiments below

• Starting with level 5, allow branch to end with
  leaves (user-defined probability)
• Employ user-defined radius of tree to stop
  branching procedure

Long term behavior of evergreen species. High
probability of insect outbreaks and state
changes. Oscillations and steady state for each
state.
5-state model results No re-growth in this case.
Percentages of leaves eaten similar to
results of M. Lowman
Include small re-growth probability (10). Leads
to much smaller fraction of the dead leaves.
Greens Theorem Application to Areas of
Leaves Greens formula for area of 2D object
bounded by straight lines

• Typical Tree
• Color-coding of 5 states of leaf
• Young leaf - light-green
• Mature leaf green
• Slightly eaten by bugs light-red
• Large percentage eaten - red
• Dead leaf - black

N - number of points, GtoE - probability of a
leaf being attacked by bugs LGtoG - probability
of going from light-green to green LEtoE -
probability of going from light-eaten (light-red)
to eaten (red) EtoD - probability of going from
eaten (red) to dead AgeD - probability of a leaf
dying due to age

• Five States Model
• Leaves volume broken into 30 M cells
• Same size cells, real time conversion between
  leaves and computational cells
• 5 states transfer from state to state with user
  defined probabilities
• May allow re-growth of new leaves
• Percolation/cellular automata ideas
• If any of the 26 neighbors (3D) of a given leaf
  is attacked by bugs, allow possibility of
  spreading (with user defined probability).
• 2D analog only 8 neighbors, figure

Effects of Age, Height, Light Add variation of
leaves eating due to age, light, and height. As
they increase, the chance of a leaf to be eaten
reduces. Divide leaf volume height into 3 zones.
Set factor in each zone to reduce the
user-defined probability of a leaf being eaten.
Fit piecewise linear function to give the height
factor. Same for the light/age. Each
probability is reduced by all these
factors. Including height, light and age factors
reduces fraction of dead leaves and slightly
increases the proportion of green ones compared
to the case with no re-growth.
Acknowledgements We wish to thank the New
College of Florida Foundation, and the Division
of Natural Sciences for support of this project.
Conclusions Percolation/cellular automata
approach can produce realistic description of
tree canopy being attacked by insects. Good
correlation to experiments is obtained. We will
investigate the possibility of extending
percolation models to entire forest being
attacked by insects.
References M. D. Lowman. 1985, "Temporal and
Spatial Variability in Insect Grazing of the
Canopies of Five Australian Rainforests Tree
Species. Australian Journal of Ecology 10,
7-24 M. D. Lowman. 1987, "The Biomass of
New-England Peppermint (Eucalyptus nova
anglica) in Relation to Insect Damage Associated
with Rural Dieback. Australian Journal of
Ecology 12, 361-371 S. Otto and T. Day. 2007,
Math Modeling in Ecology and Evolution P.
Harrison. 2001, Computational Methods in
Physics, Chemistry and Biology
New College of Florida, Sarasota FL Division of
Natural Sciences Margaret Lowman
www.canopymeg.com Leon Kaganovskiy
www.faculty.ncf.edu/lkaganovskiy/