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)