1. Spatial Lotka-Volterra Systems Joe Wildenberg Department of Physics, University of Wisconsin Madison, Wisconsin 53706 USA

2. Lotka-Volterra Equation Nonlinear r i are growth rates – set to 1 (Coste et. al) a ij are interactions Widely used – chemistry, biology, economics, etc.

3. Interaction Matrix Rows effect species i Columns show how species i effects others Not necessarily symmetric!

4. Spatial Dependence Structure of equations contains no spatial dependence Why include? Real-world systems have it!

5. Spatial Interaction Matrices

6. Rows are permutations of each other All species are identical Circulant Matrices

7. Case “Z” Ring Mathematically simple ii+1i-2

8. Z’s Eigenvalues

10. Case “Y” Goals Ring Interactions decrease with distance i

11. Bees can only fly so far from their hive Interactions with other bees depends on distance Can be influenced by far hives if their neighbors are affected Buzz

12. Case “Y” Goals Ring Interactions decrease with distance Chaotic Populations above 10 -6 (Ovaskainen and Hanski, 2003) i

15. Lyapunov Functions An “energy” function Always positive Equilibrium point has value of zero Value decreases along all orbits If one exists, no periodicity or chaos is possible

16. Ring Lotka-Volterra Lyapunov Function Requirements (Zeeman, 1997) Circulant interaction matrix (all species identical) Real part of the eigenvalues positive

17. Ring Lotka-Volterra Lyapunov Function (cont.) Eigenvalues: Lyapunov function exists if:

18. Case “Z” revisited c 1 = 1, c 2 = b = 1, c N-1 = a = 1 all others zero ii+1i-2

19. Case “Z” revisited (cont.) Largest LELyapunov Function

20. Case “Y” revisited.

21. Case “Y” revisited (cont.) Largest LELyapunov Function

22. Line Systems Not restricted by Lyapunov function Most likely others Real-world examples exist Many ways to create boundary conditions

23. Boundary Conditions Simply sever ring (remove entries in lower left and upper right of A) Hold ends fixed “Mirror” – strengthen connections on opposite side

24. 21 2 “Mirror” 1 1 1 1 1 … …

25. Mirror Y Similar spatio-temporal patterns More restrictive parameter space

26. Mirror Y (cont.) LineRing

27. Line Eigenvalues

28. Line Eigenvalues (cont.)

30. Future Work Understand eigenvalues of line systems Determine Lyapunov function(s) Apply results to real-world systems

31. Thank You!