Population Model with Competition

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Population Model with Competition

• Doreen De Leon
• Math 191T, September 23, 2005

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Overview

• The model
• Analysis of stationary points
• Example

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A Population Model of Competition
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Stationary Points
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Other Stationary Points
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Observations

• The x-coordinate of the first stationary point is
  positive only if bm gt dk
• The point (0,0) is only semistable
• There is always one stationary point calculated
  with an x-coordinate less than zeros (which means
  it is physically impossible).

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Type of Stationary Points

• We need to find the eigenvalues of the above
  matrix
• Positive (negative) eigenvalues imply unstable
  (stable) stationary point

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The Eigenvalues

• We determine the eigenvalues of the matrix L to
  be

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Implications of Eigenvalues
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Example Phase Portrait
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Stationary Points
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Another Phase Portrait
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Stationary Points
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Conclusions of the Phase Portratis

• For both plots, (0, 2) is a stable node
• In the second plot, we also have a stable focus,
  (3/7, 8/7)