“an omnivore brings chaos”
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“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
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
Biological Example
Rainbow Trout (predator)
Mayfly nymph (Prey)
crayfish
Predator of mayfly nymph
Scavenger of trout carcasses
Crayfish are scavenger & predator
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 nymphy- trout (preys on x)z-scavenges on y, eats x
Notes: Some constants above are 1 by changing variables
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 >0or (1/b,0) 1-bc <=0
We will consider 1-bc >0
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.
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
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
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!
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)
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
Hopf Bifurcations of the interior fixed point
Malorie Winters (2006) found when the interior fixed point experiences a Hopf Bifurcation
( ) f c2 g b c e g b c f c g e f c b c2 b c e b c2 f b2 c2 g f2 c2 f c2 c ( ) g g b c b c f c g b c e
( )g 2
g f c2 2 e f c2 f2 c3 c e2 g c e b c2 f c3 b g c e g e b c2 2 g b c2 g b2 c3 g f c3 b c e f c2
g 0
Her proof relied on Routh Hurwitz and some basic ODE techniques
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)
Determining which: super or sub?
Lots of analysis: James Greene 2007 REU
involved
Center Manifold Thm
Numerical estimates for specific parameters
yyyyxxxxyyxxxyyyxxxyyyyxxyxyyxxx gfgfgggfffggffa )()(||16
1)(161
Super-Super Hopf Bifurcation
e = 11.1 e = 11.3 e = 11.45
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
Further Decreases in βDecrease β: -more cardiod bifurcation diagrams
-distorted different, but same general shape/behavior
However, when β gets to around 4:
Period Doubling Begins!
Return Mapsβ = 3.5
e = 10.6 e = 10.8
e = 10.6e = 10.8
Return MapsPlotted return maps for different values of β:β = 3.5 β = 3.3
period 1
period 2 (doubles)period 1
period 1
period 2period 4
Return Mapsβ = 3.25 β = 3.235
period 8
period 16
Evolution of Attractor e = 11.4 e = 10 e = 9.5
e = 9 e = 8
More Return Mapsβ = 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
Chaosβ = 3.2
Limit cycle - periods keep doubling -eventually chaos ensues-presence of strange attractor
-chaos is not long periodics -period doubling is mechanism
Further Decrease in βAs β decreases chaotic region gets larger/more complex
- branches collide
β = 3.2 β = 3.1
Periodic WindowsPeriodic windows
- stable attractor turns into stable periodic limit cycle - surrounded by regions of strange attractor
β = 3.1
zoomed
Period 3 Implies ChaosYorke’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
1 2 2 2 2 ..... ·52 ·32 ... 2·7 2·5 2·3 ... 9 7 5 3 23422 nm
Period 3 FoundDo 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
Movie (PG-13)
• Took 4 months to run.
• Strange shots in this movie..
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…
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)
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