1. “An Omnivore Brings Chaos” Penn State Behrend Summer 2006/7 REUs --- NSF/ DMS #0552148 Malorie Winters, James Greene, Joe Previte Thanks to: Drs. Paullet, Rutter, Silver, & Stevens, and REU 2007

2. R.E.U.? Research Experience for Undergraduates Usually in summer 100’s of them in science (ours is in math biology) All expenses paid plus stipend !! Competitive (GPA important) Good for resume Experience doing research

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

5. z=0; standard Lotka-Volterra dx/dt = x(1 – bx – y) dy/dt = y(-c + x) Everything spirals in to (c, 1 – bc) 1-bc >0 or (1/b,0) 1-bc <=0 We will consider 1-bc >0

6. 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 – important biologically!) Note: No logistic term on y, and z needs one.

7. All positive orbits are bounded Really a glorified calculus 3 proof with a little bit of real analysis For surfaces of the form: x^{1/b} y = K, trajectories are ‘coming in’ for y > 1 Maple pictures

8. OK fine, trajectories are sucked into this region, but can we be more specific? Analyze stable fixed points stable = attracts all close points (Picture in 2D) Stable periodic orbits. Care about stable structures biologically

9. Fixed Point Analysis 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 (why?!) Stability Analysis: Involves linearizing system and analyzing eigenvalues of a matrix (see Dr. Paullet), or take a modeling (math) class!

10. 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. Use slick technique called Routh-Hurwitz to analyze the relevant eigenvalues (Malorie Winters 2006)

11. Hopf Bifurcations A Hopf bifurcation is a particular way in which a fixed point can gain or lose stability. Limit cycles are born (or die) -can be stable or unstable MOVIE

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

13. Two types of Hopf Bifurcations Super critical: stable fixed point gives rise to a stable periodic (or stable periodic becomes a stable fixed point) Sub critical: unstable fixed point gives rise to a unstable periodic (or unstable periodic becomes unstable fixed point)

14. Determining which: super or sub? Lots of analysis: James Greene 2007 REU involved Center Manifold Thm Numerical estimates for specific parameters

15. Super-Super Hopf Bifurcation e = 11.1 e = 11.3 e = 11.45

16. Cardioid Decrease β further: β = 15 Hopf bifurcations at: e = 10.72532712, 11.57454385 e = 10.6 e = 10.8 e = 11.5 e = 11.65 2 stable structures coexisting

17. Further Decreases in β Decrease β: -more cardiod bifurcation diagrams -distorted different, but same general shape/behavior However, when β gets to around 4: Period Doubling Begins!

18. Return Maps β = 3.5 e = 10.6 e = 10.8 e = 10.6 e = 10.8

19. Return Maps Plotted return maps for different values of β: β = 3.5 β = 3.3 period 1 period 2 (doubles)period 1 period 2 period 4

20. Return Maps β = 3.25 β = 3.235 period 8 period 16

21. Evolution of Attractor e = 11.4 e = 10 e = 9.5 e = 9 e = 8

22. More Return Maps β = 3.23 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

23. 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

24. Further Decrease in β As β decreases chaotic region gets larger/more complex - branches collide β = 3.2 β = 3.1

25. Periodic Windows Periodic windows - stable attractor turns into stable periodic limit cycle - surrounded by regions of strange attractor β = 3.1 zoomed

26. Period 3 Implies Chaos Yorke’s and Li’s Theorem - application of it - find periodic window with period 3 - cycle of every other period - chaotic cycles Sarkovskii's theorem - more general - return map has periodic window of period m and - then has cycle of period n

27. Period 3 Found Do not see period 3 window until 2 branches collide β < ~ 3.1 Do appear β = 2.8 Yorke 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

28. Movie (PG-13) Took 4 months to run. Strange shots in this movie..

29. 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…

30. Further research Biological version of this paper Can one trigger chaos with other params in this model Can we get chaos in an even more simplified model Etc. etc. etc. (lots more possible couplings)