Transcript A 4-species Food Chain - Pennsylvania State University
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 Good for resume (2 students get a pub.!) Experience doing research
This research made possible by NSF-DMS-#9987594 And NSF-DMS-#0236637
Lotka – Volterra 2- species model e.g., x= hare; y =lynx (fox)
Lotka – Volterra 2- species model Want DE to model situation (1920’s A.Lotka & V.Volterra) dx/dt = ax-bxy dy/dt = -cx+dxy a → growth rate for x c → death rate for y b → inhibition of x in presence of y d → benefit to y in presence of x
Analysis of 2-species model Solutions follow a ln y – b y + c lnx – dx=C
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.
3-species model 3 species food chain!
x = worms; y= robins; z= eagles dx/dt = ax-bxy dy/dt= -cy+dxy-eyz dz/dt= -fz+gyz =x(a-by) =y(-c+dx-ez) =z(-f+gy)
Analysis – 2000 REU Penn State Erie Key: For ag=bf ; all surfaces of form z= Kx^(-f/a) are invariant
Cases ag ≠ bf
Open Question (research opportunity) When ag > bf what is the behavior of y as t → ∞?
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!!!!
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)
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!
Each pair of pure imaginary evals corresponds to a rotation: so we have 2 independent rotations θ and φ θ φ
A torus is S^1 x S^1 (ag>bf)
Quasi-periodicity
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!!
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”.
ag=bf 4 th species goes extinct!
Limits to 3-species ag=bf case
ag< bf death to y and z —back to 2d
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).
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.
Project #3 – a toughie Prove the HSS conjecture in the simplified (non-logistic) food chain model with n species.