Ecosystem Simulations

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Ecosystem Simulations Chaos

• Robb Sinn
• Stetson University
• November 2, 2005

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About Me

• Former High School Teacher
• Mathematics Background
• Dynamical Systems (Chaos)
• Game Theory
• Math-Ed Background
• Develop Math Investigations for Kids

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Ecosystem Idea

• Can we present advanced mathematics to 8th grade
  kids?
• Yes!!
• TI-83 Easy-Iterations
• Focus on Basic Algebra
• Develop Extensions into Advanced Algebra

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Idea for Next Half Hour

• Investigate 4 ecosystem simulations
• Each can be iterated on a TI-83
• By an 8th grader!!
• Along the way, show
• Basic dynamics
• Chaos

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Easy Ecosystem 1

• Let
• D Number of deer at end of year
• d Number of deer at beginning of year
• Then iterating the following equation will
  simulate a basic ecosystem

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Parameters

• We have an ecosystem of the form
• F A limiting food supply value
• r rate of species growth

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Notice

• Negative Initial Population Values?
• Works for all m values less than 1 (in abs.
  val.).
• When m gt 1, what happens?
• Excel Dynamics XLS

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Dynamics Theorems

• If p is fixed point of the 1-D map f(x), then it
  lies on the line y x .
• Find all fixed points by solving f(x) x .
• Take the derivative f(p)
• If f(p) lt 1 , p is an attractor.
• If f(p) gt 1 , p is a repeller.

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Easy Ecosystem 2

• Let
• D Number of deer at end of year
• d Number of deer at beginning of year
• Then iterating the following equation will
  simulate the logistic ecosystem

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Parameters

• k Proportionality Constant
• L Maximum Population (Set 1)
• Math Notes
• Fixed points x 0 x (k 1) / k
• Maximum ( ½ , k / 4 )

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Chaos Theory Bifurcations

• An odd thing happens as k increases
• When k lt 3
• Have fixed point at x (k 1) / k
• It attracts
• When k gt 3
• Have fixed point at x (k 1) / k
• It repels
• There is an attracting, period 2 orbit
• Excel Logistic Example XLS

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Not so Easy Ecosystem 3

• Let
• W Number of whales at end of year
• w Number of whales at beginning of year
• This will simulate an ecosystem for an endangered
  species

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Interesting Behavior

• Try iterating with different initial populations
  values
• p0 10 p goes to zero
• p0 50 p goes to 73
• p0 100 p goes to zero
• Simulates re-introduction problem
• True chaos easy to find XLS

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Predator-Prey Ecosystem 4

• Suppose we have the following two ecosystems
• Rabbits
• Foxes

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Coupled Ecosystems

• We can iterate both at once
• An uncoupled iteration
• Equilibrium values are independent

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Add Partial Predation

• Suppose the foxes are eating rabbits
• The fox food supply increases
• The increase in f is a multiple of the rabbit
  population

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Add Full Predation

• If rabbits are being eaten
• The rabbit population decreases
• The decrease in r is a multiple of the fox
  population

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Matrix Iteration

• For any matrix with
• distinct eigenvalues
• all less than 1 (in abs. val.)
• The matrix converges to the zero matrix under
  iteration