1. Simulating Spatial Partial Differential Equations with Cellular Automata By Brian Strader Adviser: Dr. Keith Schubert Committee: Dr. George Georgiou Dr. Ernesto Gomez

2. Introduction & Background Topics Covered Partial Differential Equation, Cellular Automata (CA), & Biology Converting Differential Equations to CA CA Theoretical Constraints Convergence Maps & Guidelines

3. Introduction & Background Cellular Automata (CA) CA Model uses simple rules about changes with time. Rules are localized and involve the values of cell neighbors. The set of rules are applied to the cells with the matrix after each time period.

4. Introduction & Background Conway’s Game of Life Survival Rule: 2-3 Neighbors Death by Overpopulation: 4+ Neighbors

5. Introduction & Background Conway’s Game of Life Death by Isolation: 1 or Less Neighbors Birth: 3 Neighbors

6. Introduction & Background Conway’s Game of Life t = 0

7. Introduction & Background Conway’s Game of Life t = 1

8. Introduction & Background Conway’s Game of Life t = 2

9. Introduction & Background Conway’s Game of Life t = 3

10. Introduction & Background Celluar Automata Simulation

11. Introduction & Background Celluar Automata Simulation

12. Introduction & Background Spatial Partial Diff. Equations Changes with respect to time. Part of the equation depends on changes in space.

13. Introduction & Background Vegetation Patterns

14. Introduction & Background CA Advantages Simple Rules - easy to understand Discretized Local Problem View Highly Parallelizable

15. Converting Differential Equations to CA Diff. Equation Form Conditions: for n(u) = u p where p <= 1 for o(u) = u p where p <= 1

16. Converting Differential Equations to CA Diff. Equation Form Conditions: for n(u) = u p where p <= 1 for o(u) = u p where p <= 1

17. Converting Differential Equations to CA Diff. Equation Form Conditions: for n(u) = u p where p <= 1 for o(u) = u p where p <= 1

18. Converting Differential Equations to CA Discretization Techniques

19. Converting Differential Equations to CA Size of h x Large h x Small h x

20. Converting Differential Equations to CA Euler’s Methods Forward Euler’s Method:

21. Converting Differential Equations to CA Size of h t

22. Converting Differential Equations to CA Euler’s Methods Backward Euler’s Method:

23. Converting Differential Equations to CA Euler’s Methods Forward Euler’s Method: Backward Euler’s Method:

24. Converting Differential Equations to CA Euler’s Methods Forward Euler’s Method: 12345 i=1 j j-1 j+1 3.25.77.39.2-7.5 i=2 j j-1 j+1

25. CA Theoretical Constraints General Linear Form

26. CA Theoretical Constraints Convergence and Divergence

27. CA Theoretical Constraints Z-Transform Time Domain Frequency Domain Discrete Form of Laplace Transform and related to the Fourier Transform Transformation makes life easier zeros when f(z)=0 poles when g(z)=0

28. CA Theoretical Constraints Z-Transform

29. CA Theoretical Constraints Z-Transform 1. Perform z-transform 2. Solve for Uj 3. Find poles and zeros for Uj=f(z)/g(z) 4. Set poles and zeros values of z < 1 to converge

30. CA Theoretical Constraints Forward Euler’s Constraints Forward Euler’s Linear Form: Zeros Constraint:

31. CA Theoretical Constraints Forward Euler’s Constraints Forward Euler’s Linear Form: Poles Constraint:

32. CA Theoretical Constraints Backward Euler’s Constraints Backward Euler’s Linear Form: Zeros Constraint:

33. CA Theoretical Constraints Backward Euler’s Constraints Backward Euler’s Linear Form: Poles Constraint:

34. Convergence Maps & Guidelines CA Sim 12345 i=1 j j-1 j+1 1.11.92.82.65.4 i=2 j j-1 j+1 0.10.350.270.40.57 i=n j j-1 j+1 0.110.340.270.40.56 i=n-1 j j-1 j+1... < 10 -10

35. Convergence Maps & Guidelines CA Sim 12345 i=1 j j-1 j+1 1.11.92.82.65.4 i=2 j j-1 j+1 541-5623-897456878 i=n j j-1 j+1 1.2872927-722-256 i=n-1 j j-1 j+1... > 10 10

36. Convergence Maps & Guidelines CA Sim 12345 i=1 j j-1 j+1 1.11.92.82.65.4 i=2 j j-1 j+1 1.12.1345.1 i=4000 j j-1 j+1 12.13.13.95 i=3999 j j-1 j+1...

37. Convergence Maps & Guidelines Forward Convergence Map

38. Convergence Maps & Guidelines Backward Convergence Map

39. Convergence Maps & Guidelines a Parameters

40. Convergence Maps & Guidelines a Parameters a1a1

41. Convergence Maps & Guidelines a Parameters a2a2

42. Convergence Maps & Guidelines Forward Constraints Poles Constraint:

43. Convergence Maps & Guidelines Backward Constraints

44. Convergence Maps & Guidelines Simulation Speed

45. Convergence Maps & Guidelines a 3 Vertical Constraint

46. Convergence Maps & Guidelines a 3 Vertical Constraint Zeros Constraint:

47. Convergence Maps & Guidelines Substituting U j-1 and U j+1 Boundary Zero Values 0.110.340.270.40.56 j j-1 j+1 00

48. Convergence Maps & Guidelines Zeros Boundary Constraint

49. Convergence Maps & Guidelines Zeros Boundary Constraint

50. Convergence Maps & Guidelines Guidelines If ((upperZero and lowerPole intersects) and (intesection < initial point)) then htMax = intersection * safetyBuffer; Else htMax = initial point * safetyBuffer; End ht = userInput( < htMax); hx=lowerPole(ht);

51. Convergence Maps & Guidelines Guidelines Example

52. Conclusion Partial Diff -> CA

53. Conclusion Theoretical Constraints Zeros Constraint: Poles Constraint:

54. Conclusion Guidelines If ((upperZero and lowerPole intersects) and (intesection < initial point)) then htMax = intersection * safetyBuffer; Else htMax = initial point * safetyBuffer; End ht = userInput( < htMax); hx=lowerPole(ht);

55. Conclusion Future Work Proofs of Observations Quadratic General Form: Efficient Parallelization Simulation Error

56. Conclusion References Paul Rochester. Euler's Numerical Method for Solving Differential Equations. November 2009. http://people.bath.ac.uk/prr20/ma10126webpage.html Region of Convergence. Wikipedia. November 2009. http://en.wikipedia.org/wiki/Z- transform Keith Schubert. Cellular automaton for bioverms, October 2008. Jane Curnutt, Ernesto Gomez, and Keith Evan Schubert. Patterned growth in extreme environments. 2007. Cell Image - http://askabiologist.asu.edu/research/buildingblocks/images/cell.jpg Martin Gardner. The fantastic combinations of john conway’s new solitaire game ”life”. Scientific American, (223):120–123, 1970. T.A. Burton, editor. Modeling and Differential Equations in Biology. Pure and Applied Mathematics. Marcel Dekker Inc., 1980. J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi1. Diversity of vegetation patterns and desertification. Physical Review Letters, 87(19), November 2001.

57. Conclusion Acknowledgements and Questions?