“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)