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.