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Population Model with Competition

This overview presents a mathematical model of population dynamics under competition. It details the identification and analysis of stationary points, including their stability determined by eigenvalues of the associated matrix. Key observations include conditions for stationary points depending on parameters and the implications of calculated eigenvalues on stability, leading to practical applications in understanding competitive species interactions. Two distinct phase portraits illustrate the behavior of stationary points and their stability characteristics.

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Population Model with Competition

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  1. Population Model with Competition Doreen De Leon Math 191T, September 23, 2005

  2. Overview • The model • Analysis of stationary points • Example

  3. A Population Model of Competition

  4. Stationary Points

  5. Other Stationary Points

  6. Observations • The x-coordinate of the first stationary point is positive only if bm > 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).

  7. Type of Stationary Points • We need to find the eigenvalues of the above matrix • Positive (negative) eigenvalues imply unstable (stable) stationary point

  8. The Eigenvalues • We determine the eigenvalues of the matrix L to be:

  9. Implications of Eigenvalues

  10. Example Phase Portrait

  11. Stationary Points

  12. Another Phase Portrait

  13. Stationary Points

  14. 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)

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