1. "Food Chains with a Scavenger" Penn State Behrend Summer 2005 REU --- NSF/ DMS #0236637 Ben Nolting, Joe Paullet, Joe Previte Tlingit Raven, an important scavenger in arctic ecosystems

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 Experience doing research

3. Scavengers: Animals that subsist primarily on carrion (the bodies of deceased animals) Ravens Beetles

4. Crabs Hyenas, Wolves, and Foxes Vultures Earwigs

5. e.g., x= hare; y =lynx (fox) Introduce scavenger on a simple Lotka-Volterra Food Chain

6. 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

7. Analysis of 2-species model Solutions follow a ln y – b y + c lnx – dx=C

8. More general systems of this type look like: 1. Quadratic (only get terms like x i x j ) 2. Studied to death! But still some open problems (another talk)

9. Volterra Proved: If there is an interior fixed point with x-coord x * : Similar with others coordinates (we’ll use this later)

10. Simple Scavenger Model lynx hare beetle

11. Among other things, a scavenger species z should benefit whenever a predator kills its prey (scavenger eats dead body) xyz is proportional to the number of interactions between scavengers and carrion. The Simple Scavenger Model

12. Note: To simplify the analysis of these systems, it is often convenient to rescale parameters. The number of parameters that you can eliminate depends on the structure of the system.

13. Results for the simple scavenger system Three cases: Fixed point in 2d system: (c,1)

14. Dynamics trapped on cylinders

15. Scavenger dies e>cf+gc+h

16. Scavenger stays bounded e = cf+gc+h

17. Scavenger blows up e<gc+fc+h

18. Case 1: z2 = z1 Main Idea: (return map in z) of PROOF

19. Case 2: z2> z1 => z3>z2 z3<z2 no good! z_i monotone increasing

20. So… z1 z i increasing z1 > z2 => z i decreasing z1 = z2 => z i constant (periodic) Monotone Sequence Theorem: z i either converges or goes to +∞

21. Let (x0,y0,z0) be given having period T in the plane Why?

23. Not Biologists Not Pleased!! I’m NOT pleased Scavenger dies or blows up except on a set of measure zero!

24. We want stable behavior, logistic So let’s make the growth of x logistic: Know (x,y) -> (c, 1-bc) use this to see e<f(1-bc)c+gc+h(1-bc) implies z is unbounded e>f(1-bc)c+gc+h(1-bc) implies z goes extinct e=f(1-bc)c+gc+h(1-bc) implies z to a non-zero limit Still No good!

25. Let’s go back to LV w/o logistic, But put a quadratic death term on the scavenger.

26. Rutter’s slide Average death rate proportional to z, so Adding a quadratic death term makes perfect sense and is not overkill (but needed here!)

27. Globally stable limit cycles on every cylinder! No blow ups or extinctions.

28. Keys to proof: 1)Orbits are confined to cylinders 2)For a particular cylinder, the z nullcline intersects the cylinder at a high point z*. 3)z* is an upper bound for trajectories starting below z*. 4)Every trajectory starting above z* must eventually venture below z*. 5)Very close to xy plane, return map is increasing. 6)Monotone sequence bounded above-> limit. 7)Time averages show you can’t have two limit cycles on the same cylinder.

29. Other possibilities for further research 3 species models w/ scavenger Scavengers affect other species (crowding) Scavenger Ring models More quadratic death terms Etc. etc. etc. Ben Nolting (Alaska)

30. Ring Model