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A 4-species Food Chain

A 4-species Food Chain. Joe Previte-- Penn State Erie. Joe Paullet-- Penn State Erie Sonju Harris & John Ranola (REU students). R.E.U.?. Research Experience for Undergraduates Usually a summer 100’s of them in science (ours is in math biology) All expenses paid plus stipend ! Competitive

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A 4-species Food Chain

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  1. A 4-species Food Chain Joe Previte-- Penn State Erie Joe Paullet-- Penn State ErieSonju Harris & John Ranola (REU students)

  2. R.E.U.? • Research Experience for Undergraduates • Usually a summer • 100’s of them in science (ours is in math biology) • All expenses paid plus stipend ! • Competitive • Good for resume (2 students get a pub.!) • Experience doing research

  3. This research made possible by NSF-DMS-#9987594 And NSF-DMS-#0236637

  4. Lotka – Volterra 2- species model • e.g., x= hare; y =lynx (fox)

  5. Lotka – Volterra 2- species model • Want DE to model situation (1920’s A.Lotka & V.Volterra) • dx/dt = ax-bxy dy/dt = -cx+dxya → growth rate for xc → death rate for yb → inhibition of x in presence of yd → benefit to y in presence of x

  6. Analysis of 2-species model • Solutions followa ln y – b y + c lnx – dx=C

  7. Analysis • Pretty good qualitative fit of data • No unbounded orbits!, despite not having a logistic term on x • Predicts cycles, not many cycles seen in nature.

  8. 3-species model • 3 species food chain! • x = worms; y= robins; z= eagles dx/dt = ax-bxy =x(a-by)dy/dt= -cy+dxy-eyz =y(-c+dx-ez)dz/dt= -fz+gyz =z(-f+gy)

  9. Analysis – 2000 REU Penn State Erie • Key: For ag=bf ; all surfaces of form z= Kx^(-f/a) are invariant

  10. Cases ag ≠ bf

  11. Open Question (research opportunity) • When ag > bfwhat is the behavior of y as t →∞?

  12. Critical analysis • ag > bf → unbounded orbits • ag < bf → species z goes extinct • ag = bf → periodicity • Highly unrealistic model!! (vs. 2-species) • Result: A nice pedagogical tool • Adding a top predator causes possible unbounded behavior!!!!

  13. 4-species model dw/dt = aw-bxw =w(a-bx)dx/dt= -cx+dwx-exy =x(-c+dw-ey)dy/dt= -fy+gxy - hyz =y(-f+gx-hz) dz/dt= -iz+jyz =z(-i+jy)

  14. Equilibria • (0,0,0,0) • (c/d,a/b,0,0) • ((cj+ei)/dj,a/b,i/j,(ag-bf)/hb) • J(0,0,0,0): 3 -, 1 + eigenvalues (saddle) • J(c/d,a/b,0,0): 2 pure im; 1 -, 1 ~ ag-bf • J((cj+ei)/dj,a/b,i/j,(ag-bf)/hb) 4 pure im!

  15. Each pair of pure imaginary evals corresponds to a rotation: so we have 2 independent rotations θ and φ φ θ

  16. A torus is S^1 x S^1 (ag>bf)

  17. Quasi-periodicity

  18. In case ag > bf; found invariant surfaces! K = w- (cj+ei)/dj ln(w) +b/d x – a/d ln(x) + be/dg y – ibe/dgj ln(y) + beh/dgj z – e(ag-bf)/dgj ln (z) These are closed surfaces so long as ag >bf: Moral: NO unbounded orbits!!

  19. For ag > bf: this should be verifiable! • Someone give me a 4-species historical population time series!, • RESEARCH PROJECT # 2! • (Calling all biologists!) • Try to fit such data to our “surface”.

  20. ag=bf • 4th species goes extinct! • Limits to 3-species ag=bf case

  21. ag< bf death to y and z—back to 2d

  22. Summary • Model contains quasiperiodicity • As in 2-species, orbits are bounded. • ag vs. bf controls (species 1 & 3 ONLY) • cool dynamical analysis of the model • Trapping regions, invariant sets, stable manifold theorem, linearization, some calculus 1 (and 3).

  23. Grand finale: Even vs odd disparity • Hairston Smith Slobodkin in 1960 (biologists) hypothesize that (HSS-conjecture) • Even level food chains (world is brown) (top- down) • Odd level food chains (world is green) (bottom –up) • Taught in ecology courses.

  24. Project #3 – a toughie • Prove the HSS conjecture in the simplified (non-logistic) food chain model with n-species.

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