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-# And NSF-DMS-#

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

Lotka – Volterra 2- species model (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 Want DE to model situation

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 =x(a-by) dy/dt= -cy+dxy-eyz =y(-c+dx-ez) dz/dt= -fz+gyz=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.