A 4-species Food Chain

Joe Previte-- Penn State Erie

Joe Paullet-- Penn State ErieSonju Harris & John Ranola (REU students)

R.E.U.?

Research Experience for UndergraduatesUsually a summer 100’s of them in science (ours is in math

biology)All expenses paid plus stipend !CompetitiveGood 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

(1920’s A.Lotka & V.Volterra)dx/dt = ax-bxy

dy/dt = -cx+dxy

a → growth rate for xc → death rate for yb → inhibition of x in presence of yd → 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 dataNo unbounded orbits!, despite not having

a logistic term on xPredicts 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 orbitsag < bf → species z goes extinctag = bf → periodicity

Highly unrealistic model!! (vs. 2-species)Result: A nice pedagogical toolAdding 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

4th species goes extinct!

Limits to 3-species ag=bf case

ag< bf death to y and z—back to 2d

Summary

Model contains quasiperiodicityAs 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.