REU 2005

1. REU 2005 Predator Prey Population Models

2. REU ’05 • Often know how populations change over time (e.g. birth rates, predation, etc.), as opposed to knowing a ‘population function’ Differential Equations! • Knowing how population evolves over time w/ initial population  population function

3. Example – Hypothetical rabbit colony lives in a field, no predators. Let x(t) be population at time t; Want to write equation for dx/dt Q: What is the biggest factor that affects dx/dt? A: x(t) itself! more bunnies  more baby bunnies

4. 1st Model—exponential, MalthusianSolution: x(t)=x(0)exp(at)

5. Critique • Unbounded growth • Non integer number of rabbits • Unbounded growth even w/ 1 rabbit! Let’s fix the unbounded growth issue dx/dt = ????

6. Logistic Model • dx/dt = ax(1-x/K) K-carrying capacity we can change variables (time) to get dx/dT = x(1-x/K) • Can actually solve this DE

7. Solutions: • Critique: • Still non-integer rabbits • Still get rabbits with x(0)=.02

8. Predator Prey • Today we have 2 species; one predator y(t) (e.g. wolf) and one its prey x(t) (e.g. hare)

9. Actual Data

10. 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

11. Model Simplification 3 • Could get rid of __________constants—how? • dx/dt= ax-bxy = x(a-by) dy/dt=-cx+dxy = y(-c+dx)

12. Called Lotka-Volterra Equations, Lotka & Volterra independently studied this post WW I. • Fixed points: (0,0), (c/d,a/b)

13. Solution vs time

14. The Phase Portrait (x vs. y)

15. No logistic term • Note in 1 species model, all orbits grow unbounded. • In 2-species model, all orbits (most anyway) stay bounded. • What happens in 3 or 4 species?!

16. What are you going to do? • Try to use analysis to argue that this is indeed the phase portrait.

17. Poincare’-Bendixson** • dx/dt= f(x,y) dy/dt= g(x,y); f & g nice If (x(t),y(t)) is bounded then as t increases, (x(t),y(t)) must approach:1. a fixed point2. a periodic orbit3. a cycle of one or more fixed point(s) connected by hetero/homoclincs. ** only works in the plane!

18. OK what now? • 3 species food chain! • x = worms; y= robins; z= eagles dx/dt = ax-bxydy/dt= -cy+dxy-eyzdz/dt= -fz+gyz

19. Analysis – 2000 REU Penn State Erie • For ag=bf ; get invariant surfaces, numerical solutions are periodic.

20. Invariant Surfaces { dx/dt= f(x,y,z) dy/dt= g(x,y,z) dz/dt= h(x,y,z); f,g, and h niceIf S is a smooth surface with normal vector n at (x,y,z) and dot (n,<f,g,h>)=0 at all (x,y,z) thenthe surface S is invariant w.r.t. (1). (1)

21. 3d time plot for ag=bf

22. Linearization of 3d system (informal stable manifold Thm)If at fixed point P, Jac(P) has: 3 + real part evals – unstable 3 - real part evals – stable a mix– a generalized saddle 0 real parts– (pure imaginary) no good

23. Cases ag ≠ bf

24. Trapping Surfaces { dx/dt= f(x,y,z) dy/dt= g(x,y,z) dz/dt= h(x,y,z); f,g, and h niceIf S is a smooth surface with normal vector n at (x,y,z) ‘upward’ dot (n,<f,g,h>)>0 at all (x,y,z) thenall trajectories of (1) pass through S upward. (1)

25. ag<bf

26. ag>bf

27. 3 species Open Question (research opportunity) • When ag > bfwhat is the behavior of y as t →∞?

28. 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!!!!

29. What are you going to do? • Verify many of the statements made using: 1. trapping regions 2. invariant sets 3. linearization

30. 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)

31. 2004 REU analysis • Orbits bounded again as in n=2 • Quasi periodicity • ag<bf gives death to top 2 • ag=bf gives death to top species • ag>bf gives quasi-periodicity

32. In case ag > bf; invariant surface [Volterra] K = x- x0 ln(x) +b/d y – y0 ln(y) + be/dg z – be/dg z0 ln(z) + beh/dgj w – e/j w0 ln (w) These are closed surfaces so long as ag >bf: Moral: NO unbounded orbits!!

33. For ag > bf: this should be verifiable! • Someone give me a 4-species historical population time series!, “RESEARCH PROJECT”! (Try to fit such data to our “surface”.

34. ag=bf • 4th species goes extinct! • Limits to 3-species ag=bf case

35. ag< bf death to y and z—back to 2d

36. 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!

37. Each pair of pure imaginary evals corresponds to a rotation: so we have 2 independent rotations θ and φ φ θ

38. A torus is S^1 x S^1 (ag>bf)

39. Quasi-periodicity

40. 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) (n=2, n=4 orbits bounded) • Odd level food chains (world is green) (bottom –up) (n=1, 3 orbits unbounded) • Taught in ecology courses.

41. Project #1 • Prove the asymptotic behavior of y in the 3-species model. • Obtain 4 species data and fit the data to our model. Project #2

42. Project #3 • Prove that in the n-species cases (n even) that the Jacobian about P always has imaginary eigenvalues. (n=4 is done) • Examine the odd cases where unbounded orbits exist, completely characterize these cases.

43. Project #4 • In ag>bf n=4, it appears that all orbits stay on a torus (the linearization is not enough)(a) amass numerical evidence to verify(b) prove it (!?) This may involve series solutions, trying to find an invariant set for special parameters, or some very sophisticated math (KAM theory)