scavenger models and chaos
DESCRIPTION
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- β z) y-preys on x z-scavenges on y, eats x. Biological Example. - PowerPoint PPT PresentationTRANSCRIPT
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