“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

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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


crayfish

Scavenger of trout carcasses

Predator of mayfly nymph

Biological Example

Rainbow Trout (predator)

Mayfly nymph (Prey)

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 nymph

    y- 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 >0

    or (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

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


Super-Super Hopf Bifurcation

e = 11.1 e = 11.3 e = 11.45


Cardioid

2 stable structures coexisting

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


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!


Return Maps

e = 10.8

e = 10.6

β = 3.5

e = 10.6 e = 10.8


Return Maps

Plotted return maps for different values of β:

β =3.5 β =3.3

period 1

period 2 (doubles)

period 4

period 2

period 1

period 1


Return Maps

β = 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 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


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


Further Decrease in β

As β decreases chaotic region gets larger/more complex

- branches collide

β = 3.2 β = 3.1


Periodic Windows

Periodic windows

- stable attractor turns into stable periodic limit cycle

- surrounded by regions of strange attractor

β = 3.1

zoomed


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


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


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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