Scavenger Models and Chaos

James Greene

Dr. Joseph Previte

Scavenger Model #1

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)

y-preys on x

z-scavenges on y, eats x

Biological Example

Crayfish Rainbow Trout

Mayfly

Bounded Orbits

Can show all trajectories are bounded

Use trapping regions and invariant sets

Trapping Region

- surface where

- all trajectories must go in from them

Invariant Sets

- surface where

- stay on and cannot pass

plots

Outward normal

Coordinate planes

0 fvn

0 fvn

)/1(1

1)/1(

1

0),,(.1

b

b

bk

kyxzyxF

bk

kxzyxG

/1

0),,(.2

2

2

bbck

kyzyxH

/13

3

)(

1

0),,(.3

bbcgbfek

kzzyxW/1

4

4

)/(/

0),,(.4

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

Using linear stability analysis:

(0,0,0) – always saddle; (1/b,0,0) – saddle/stable point; (c,1-bc,0) – stable/unstable;((β+e)/(βb+f),0, (β+e)/(βb+f)) – stable/unstable point

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.

Stable f.p. – all negative real part of eigenvalues

Unstable f.p. – at least 1 positive real part of eigenvalue

Characteristic Polynomial

Characteristic polynomial of Jacobian at the interior fixed point:

P(λ)= λ3 +Bλ2 +C λ+D

Zeros of P(λ) yield eigenvalues

Use Routh-Hurwitz analysis on P(λ) to determine the number of eigenvalues with positive, negative, and zero real part

Tells stability of fixed pointReal parts of eigenvalues

All complicated functions of parameters

Critical Value

Analysis tells us that a Hopf bifurcation occurs when coefficients satisfy:

BC-D=0Coefficients are functions of parameters, so parameters must

satisfy:

Malorie Winters 2006 REU

( ) f c2g b c e g b c f c g e f c b c

2 b c e b c2 f b

2c

2 g f2c

2f c

2c ( ) g g b c b c f c g b c e

( )g 2

g f c2

2 e f c2

f2c

3c e

2g c e b c2 f c3 b g c e g e b c

22 g b c

2 g b2c

3 g f c3b c e f c2

g 0

Hopf Bifurcation

A Hopf bifurcation occurs through a fixed point when the fixed point loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane.

Limit cycles are born-can be stable or unstable

Supercritical Hopf bifurcation - stable limit cycles

Subcritical Hopf bifurcation - unstable limit cycles

Movie

Plan of Attack

Fix a set of parameters except e and β: b = 0.9, c = 0.1, f = 0.1, g = 13

Easy to work with parameters

Fix a value of β, starting large

Plot bifurcation diagrams for system for parameter e

Explore behavior of bifurcation diagrams as we lower β

Large β

β > ~19

No Hopf bifurcation occurs

Interior fixed point remains stable

Proving Sub- Super Hopf

For smaller β Hopf bifurcations occur

Prove sub- and supercritical Hopf bifurcations occur at parameter values

In 2 dimensions, exists value “a” such that its sign determines what kind of Hopf bifurcation occurs:

a > 0 Subcritical Hopf

a < 0 Supercritical Hopf

But we are in 3 dimensionsReduce system to plane using center manifold - so can apply this theorem

yyyyxxxxyyxxxyyyxxxyyyyxxyxyyxxx gfgfgggfffggffa )()(||16

1)(

16

1

Center Manifold

Consider a dynamical system which has been linearized:

x’ = f(x)

Linearized system has eigenspaces:

Es = stable eigenspace Eu = unstable eigenspace Ec = center eigenspace ={set of eigenvectors with ={set of eigenvectors with ={set of eigenvectors with

negative real part} positive real part} 0 real part}

Invariant subspaces

Nonlinear system has corresponding invariant manifolds

At equilibrium point, invariant manifolds are tangent to the corresponding invariant eigenspaces

x’ = Ax

Center Manifold

For parameter values: b=0.9, c=0.1, f=0.1, g=13, β=18.5

Numerically solve BC-D=0 for e: e = 11.25271967, 11.41142668

Change variables

Assume w is an invariant function of u,v over time

Center manifold has expression:

w = h(u,v)=k1u+k2v+k3u2+k4v2+k5uv+k6u3+k7u2v +k8uv2+k9v3 +…

Must satisfy:

Obtain center manifold at e = 11.25271967 :

0** tvtut vhuhw

Proving Sub and Super

Calculate sub- and supercritical Hopf bifurcation from center manifold

a > 0 Subcritical Hopf

a < 0 Supercritical Hopf

e = 11.25271967 a = -0.8767103e = 11.41142668 a = -8.1980159

yyyyxxxxyyxxxyyyxxxyyyyxxyxyyxxx gfgfgggfffggffa )()(||16

1)(

16

1

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 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

β = 3.5

e = 10.6 e = 10.8

e = 10.6e = 10.8

Return Maps

Plotted 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

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

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

Evolution of Attractor

e = 11.4 e = 10 e = 9.5

e = 9 e = 8

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

1 2 2 2 2 ..... ·52 ·32 ... 2·7 2·5 2·3 ... 9 7 5 3 23422

nm

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

Biological Implications

Mathematical result:

Decrease in β System exhibits chaos

β – logistic term in species z

K = carrying capacity of species z

β ~

Decrease in β Increase in K

Biological Result:

Increase in carrying capacity Increase in complexity of dynamics

Intuitive result

K

1

Scavenger Model #2

dx/dt=x(1-bx-y-z) b, c, e, f, g, h, β > 0

dy/dt=y(-c+x)

dz/dt=z(-e+fx+gy+hxy-βz)

Adds cubic hxyz term

Represent same species biologically

- more complex

Model analyzed 1st

Analysis

Everything analogous to other model

- fixed points, Hopf bifurcations, bounded orbits

Fixed parameters except e and let e vary

Obtain period doubling and chaotic regions

Discontinuities in Return Maps

Get jumps in return maps

Trying to stay on limit cycleJumping off - to a new structure

2 stable structures coexisting

jumps

Tracking Other Structure

Start on other structure and try to track its evolution in e

Totally different stable structure surrounding previous stable structureCould not find way to stay on surrounding structure

start on jump

on a different structurefall back to previous limit cycle structure

More Problems

More discontinuities

strange

Acknowledgments

All REU Faculty

Behrend REU 2007

Jesse Stimpson

Other REU Participants

NSF Award 0552148