1. Hopf Bifurcations on a Scavenger/Predator/Prey System
Malorie Winters
Dr. Joseph Previte

2. Lotka-Volterra dx/dt = x(1 – bx – y) dy/dt = y(-c + x)
Everything spirals in to (c, 1 – bc)
Introduce scavenger onto this system
A little background: Basic Lotka-Volterra equations with logistic term on x (bx-term). To simplify, there was a change in variables done 3 times. In this system, all trajectories spiral in toward fixed point (c, 1-bc), when it is positive.

3. The Model dx/dt = x(1 – bx – y – z) dy/dt = y(-c + x)
dz/dt = z(-e + fx + gy + hxy – βz)
Lotka-Volterra with logistic on x
Scavenger z with linear and quadratic death
Explain parameters (especially in dz/dt): e = linear death, beta = quadratic death, f & g = z benefits from dead x & y, h = z benefits from leftover x killed by y
Logistic because Britney did model without b and had unbounded growth
resource limitations

4. Owl
Crow
Rabbit

5. Fixed points & Stability
1. (0, 0, 0) – saddle point
2. (1/b, 0, 0) – saddle/stable point
stable/unstable point
stable/unstable point
- when stable, scavenger becomes extinct
Linearized the system and used the eigenvalues of Jacobian to determine the stability of the points
- talk about what species become extinct & when

6. Interior Fixed Point Linearization of system at fixed point
Eigenvalues of Jacobian are messy
Characteristic polynomial of Jacobian
Routh Hurwitz Analysis
Linearization – Did the same as other fixed points, linearized, Jacobian, etc.
Char poly – A,B,C,D polynomials of parameters

7. Routh Hurwitz Method Categorizes real parts of roots of a polynomial
Array

8. To finish array Count number of sign changes in first column
Gives number of roots with positive real part

9. Example
Row 1
Row 2
2 sign changes means 2 roots with pos. real part

10. Example RH array predicted 2 roots with positive real part.
Actual solutions:

11. Results Routh Hurwitz analysis on characteristic polynomial
For our system A = 1
CB – D > 0, stable
no sign changes, all eigenvalues neg. real part
CB – D < 0, unstable
two sign changes, two eigenvalues pos. real part
CB – D = 0, special case
B, C, & D are polynomials made up of parameters
- when unstable, someone becomes extinct
Will return to this special case in a few minutes

12. CB – D

13. Hopf Bifurcations Movie Hopf Bifurcation Theorem
Suppose you have a family of systems of ODEs parameterized by s and is a fixed point, then a Hopf bifurcation occurs when a pair of complex conjugate eigenvalues of the Jacobian cross the imaginary axis non-tangentially and not at zero.
Movie
Hopf Bifurcation Theorem
Guarantees a limit cycle

14. Idea If there is a path in parameter space and at with
then the interior fixed point has a Hopf
bifurcation at .

15. Example
Let
Then at and
and

16. Results
Bifurcation diagram
(Explain diagram)

17. Region 1: Before Bifurcation
Trajectories begin red
and travel in toward purple
(stable node).
All species survive.
Stable Node

18. Region 2: Stable limit cycle
Trajectories begin red and travel toward purple (stable limit cycle).
Oscillatory behavior in all species.
Stable Limit Cycle
Unstable Node

19. Region 3: Multiple limit cycles
Unstable Limit Cycle
It depends on the initial conditions whether oscillations occur or not.
Draw on board – explain limit cycles much better
Stable Limit Cycle
Stable Center

20. Region 4: After Limit Cycles
Trajectories begin red
and travel in toward blue
(stable node).
All species survive.
Stable Node

21. Biology
Eventual behavior dependent upon parameters and initial conditions for some of these systems
Biologically viable and interesting?
Does it exist in nature?

22. Notes/Questions
Bifurcations possible by varying all parameters except b
Do unbounded orbits exist?

23. Acknowledgements Thanks to Dr. Joseph Previte Behrend REU 2006
NSF Award