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Scavenger Models and Chaos

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.

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Scavenger Models and Chaos

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  1. Scavenger Models and Chaos James Greene Dr. Joseph Previte

  2. 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

  3. Biological Example Crayfish Rainbow Trout Mayfly

  4. 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

  5. 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

  6. 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

  7. 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 point Real parts of eigenvalues All complicated functions of parameters

  8. Critical Value Analysis tells us that a Hopf bifurcation occurs when coefficients satisfy: BC-D=0 Coefficients are functions of parameters, so parameters must satisfy: Malorie Winters 2006 REU

  9. 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

  10. 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

  11. Large β β > ~19 No Hopf bifurcation occurs Interior fixed point remains stable

  12. 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 dimensions Reduce system to plane using center manifold - so can apply this theorem

  13. 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

  14. 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 :

  15. Proving Sub and Super Calculate sub- and supercritical Hopf bifurcation from center manifold a > 0 Subcritical Hopf a < 0 Supercritical Hopf e = 11.25271967a = -0.8767103 e = 11.41142668 a = -8.1980159

  16. Super-Super Hopf Bifurcation e = 11.1 e = 11.3 e = 11.45

  17. 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

  18. 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!

  19. Return Maps β = 3.5 e = 10.6 e = 10.8 e = 10.8 e = 10.6

  20. 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

  21. Return Maps β = 3.25 β = 3.235 period 8 period 16

  22. 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

  23. 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

  24. Evolution of Attractor e = 11.4 e = 10 e = 9.5 e = 9 e = 8

  25. Further Decrease in β As β decreases chaotic region gets larger/more complex - branches collide β = 3.2 β = 3.1

  26. Periodic Windows Periodic windows - stable attractor turns into stable periodic limit cycle - surrounded by regions of strange attractor β = 3.1 zoomed

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. Discontinuities in Return Maps Get jumps in return maps Trying to stay on limit cycle Jumping off - to a new structure 2 stable structures coexisting jumps

  33. Tracking Other Structure Start on other structure and try to track its evolution in e Totally different stable structure surrounding previous stable structure Could not find way to stay on surrounding structure start on jump fall back to previous limit cycle structure on a different structure

  34. More Problems More discontinuities strange

  35. Acknowledgments All REU Faculty Behrend REU 2007 Jesse Stimpson Other REU Participants NSF Award 0552148

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