"Food Chains with a Scavenger"

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) Beetles Ravens

4. Earwigs Hyenas, Wolves, and Foxes Crabs Vultures

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

6. Lotka – Volterra 2- species model • Want DE to model situation (1920’s A.Lotka & V.Volterra) • dx/dt = ax-bxy dy/dt = -cx+dxya → growth rate for xc → death rate for yb → inhibition of x in presence of yd → benefit to y in presence of x

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

8. More general systems of this type look like: 1. Quadratic (only get terms like xixj) 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 beetle hare

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 Fixed point in 2d system: (c,1) Three cases:

14. Dynamics trapped on cylinders

15. Scavenger dies e>cf+gc+h

16. Scavenger stays boundede = cf+gc+h

17. Scavenger blows upe<gc+fc+h

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

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

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

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

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

24. Still No good! We want stable behavior, 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

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: Orbits are confined to cylinders For a particular cylinder, the z nullcline intersects the cylinder at a high point z*. z* is an upper bound for trajectories starting below z*. Every trajectory starting above z* must eventually venture below z*. Very close to xy plane, return map is increasing. Monotone sequence bounded above-> limit. 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