1. “An Omnivore Brings Chaos”
Penn State Behrend
Summer 2006/7 REUs --- NSF/ DMS #
James Greene (Penn State), Joe Previte (Penn State Erie)

2. 3 species model Predator (y) Third species Lotka-Volterra
Scavenger of Predator
Predator (y)
Prey (x)
Third species
Lotka-Volterra
Also a predator of x

3. Biological Example Rainbow Trout (predator) crayfish
Predator of mayfly nymph
Scavenger of trout carcasses
Rainbow Trout (predator)
Mayfly nymph (Prey)
Crayfish are scavenger & predator

4. Model dx/dt=x(1-bx-y-z) b, c, e, f, g, β > 0 dy/dt=y(-c+x)
dz/dt=z(-e+fx+gy-βz)
x- mayfly nymph
y- trout (preys on x)
z-scavenges on y, eats x
Notes: Some constants above are 1 by changing variables,
z=0 simpleLotka Volterra

5. Bounding trajectories
Thm For any positive initial conditions, there is a compact region in 3- space where all trajectories are attracted to.
(Moral : Model does not allow species to go to infinity, with no logistic term on y)

6. Fixed Points 5 Fixed Points (0,0,0), (1/b,0,0), (c,1-bc,0),
((β+e)/(βb+f),0, (β+e)/(βb+f)),
(c,(-fc-cβb+e+ β)/(g+ β),(-e+fc+g-gbc)/(g+ β))
only interior fixed point
Want to consider cases only when interior fixed point exists
in positive space.

7. Interior Fixed Point (c,(-fc-cβb+e+ β)/(g+ β),(-e+fc+g-gbc)/(g+ β))
Can be shown that when this is in positive space, all other fixed points are unstable.
Linearization at this fixed point yields eigenvalues that are difficult to analyze analytically.
Used Routh-Hurwitz to analyze the relevant eigenvalues (Malorie Winters REU 2006)

8. Hopf Bifurcations of the interior fixed point
Malorie Winters (REU 2006) found when the interior fixed point experiences a Hopf Bifurcation
Her proof relied on Routh Hurwitz and some basic ODE techniques

9. Super-Super critical Hopf Bifurcation
β =25
vary e

10. Cardioid (sub-super) Decrease β further: β = 15 Hopf bifurcations at:
e = e = 10.8
e = e = 11.65
2 stable structures coexisting

11. Determining type of Hopf: super or sub critical?
Lots of analysis: James Greene 2007 REU
involving
Center Manifold Thm / Hopf Bifurcation Theory
Greene classified the Hopf bifurcation for several specific values (AMS Undergraduate Poster Session)

12. Further Decreases in β Decrease β: -more cardioid bifurcation diagrams
-distorted different, but same general shape/behavior until...
However, when β gets to around 4:
Period Doubling of stable periodic begins!

13. Poincare Return Maps Plotted return maps for different values of β:
β = β = 3.3
period 1
period 2 (doubles)
period 4
period 2
period 1
period 1

14. Return Maps varying e
β = β = 3.235
period 16
period 8

15. More Return Maps Strange Attractor Similar to Lorenz butterfly
β = β = 3.2
As β decreases doubling becomes “fuzzy” region
Classic indicator of CHAOS
Strange Attractor
Similar to Lorenz butterfly
does not appear periodic here

16. Chaos β = 3.2 Limit cycle - periods keep doubling
-eventually chaos ensues-presence of strange attractor
-chaos is not long periodics
-period doubling is mechanism

17. Further Decrease in β
As β decreases chaotic region gets larger/more complex
- branches collide
β = β = 3.1

18. Period 3 Found Do not see period 3 window until 2 branches collide
β < ~ 3.1
Do appear
β = 2.8
Yorke & Li’s Theorem implies periodic orbits of all possible positive integer values
Further decrease in β
- more of the same
- chaotic region gets worse and worse
e = 9

19. Movie Took 4 months to run.
Return maps in e with beta decreasing as movie progresses.
Strange shots in this movie.. multiple chaotic attractors?

20. Wrapup
I think, this is the easiest population model discovered so far with chaos.
The parameters beta and e triggered the chaos
A simple food model brings complicated dynamics.
Tons more to do…