Fractional Dispersion and Reproduction Equations

1. Fractional Dispersion and Reproduction Equations Boris Baeumer University of Otago http://www.maths.otago.ac.nz/?baeumer

3. Invasion of Organisms • Complex problem with lots of random factors • What can be predicted? What are the right questions? • Data is usually noisy. • Problems in modeling include • Environmental factors such as grazing, droughts, etc. can have big impact. • Dispersal often driven by occasional extreme event such as storms at the right time or piggybacking on animals, etc. • Where are the parents?

4. Growth Dispersion Invasion models • Classical Fisher equation • Mostly abandoned, spread too slow…

5. Dispersal kernel Invasion models • Classical Fisher equation • Integro-Difference equations • Semi-discrete, hard to analyse.

6. PDE Model • To simplify things take out growth, and assume kernel is homogeneous (k(x,y)=k(x-y)) (and non-negative) • In order to get to a PDE, the kernel needs to be infinitely divisible.

7. Infinitely divisible distributions • Lots of kernels are infinitely divisible, but not all… • Levy-Khintchine representation: • General dispersion equation (c=0):

8. Levy-Khintchine Representation Drift Jump Intensity Gaussian Spread

9. Special cases • Long term limit (scaling limit) • If k decays like a power-law, k converges to a stable distribution, otherwise k converges to a normal distribution • If k is symmetric,

10. Time randomiser (Subordinator) Classical Dispersion Stable laws and fractional PDE’s

11. Q dispersion tensor r repulsiveness Adapting to a Complex Environment • Life does not happen on the complex plane or unit circle! • Use same subordination to speed up time. a attractiveness

12. Predominant Wind Effects or Currents • Need to randomise flow, not classical diffusion. • Need new Mathematics • Baeumer, Haase & Kovacs, Journal of Evolution Equation, 2009 • Baeumer, Kovacs & Meerschaert, Functional Analysis and Evolution Equations, 2008

13. The classical Model

14. The Fractional Model

15. The fractional Dispersal and Reproduction Equation

16. Hawthorn at Porters Pass, NZ • Obtained data set from John Kean (AgResearch) on Hawthorn at Porters Pass (data collected by Peter Williams and Rohan Buxton). • Location of trees via GPS • Age of trees estimated (chop down a few…)

18. 1930 • First tree in 1906 • 2nd tree in 1925

19. 1935 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

20. 1940 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

21. 1945 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

22. 1950 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

23. 1955 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

24. 1960 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934

25. 1965 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

26. 1970 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

27. 1975 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

28. 1980 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

29. 1985 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

30. 1990 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

31. 1995 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

32. 2000 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

33. 2005 • First tree in 1906 • 2nd tree in 1925 • 3rd tree in 1934 • Bunnies controlled in late 50’s

34. Growth (Williams et al, preprint)

35. What about spread? • Hard to quantify • Infested area • Mean distance to original tree • Furthest distance to original tree • Other statistics (function of time) • Proportion of trees at a distance >r • Tree density map

36. Keep it simple • Assume growth independent of density (may depend on time) • Growth is a then a factor in the solution; i.e., C(t,x)=G(t)c(t,x), where c is the solution with zero growth (or number of trees normalised) • Model proportion of trees at a certain distance to Grandpa tree gives probability of tree at a distance greater than r.

37. Simple subordinated equation • Two parameter model • C(r,t) is the proportion of trees at a distance greater than r from original tree