Diffusion and home ranges in mice movement Guillermo Abramson Statistical Physics Group, Centro Atómico Bariloche and C

1. Diffusion and home ranges in mice movement Guillermo Abramson Statistical Physics Group, Centro Atómico Bariloche and CONICET Bariloche, Argentina. with L. Giuggioli and V.M. Kenkre

2. Oh, my God, Kenkre has told everything!

3. OUTLINE • The basic model • Implications of the bifurcation • Lack of vertical transmission • Temporal behavior • Traveling waves • The diffusion paradigm • Analysis of actual mice transport • Model of mice transport

4. Population dynamics + Contagion + (Mice movement) Mathematical model THREE FIELD OBSERVATIONS AND A SIMPLE MODEL • Strong influence by environmental conditions. • Sporadical dissapearance of the infection from a population. • Spatial segregation of infected populations (refugia). Single control parameter in the model simulate environmental effects. The other two appear as consequences of a bifurcation of the solutions.

5. carrying capacity BASIC MODEL (no mice movement yet!) MS (t) : Susceptible mice MI (t) : Infected mice M(t)= MS (t)+MI (t): Total mouse population Rationale behind each term Births: bMonly of susceptibles,all mice contribute to it Deaths:-cMS,I infection does not affect death rate Competition:-MS,I M/K population limited by environmental parameter Contagion:  aMS MI  simple contact between pairs

6. BIFURCATION The carrying capacity controls a bifurcation in the equilibrium value of the infected population. The susceptible population is always positive.

7. The same model, with vertical transmission

8. The same model, with vertical transmission

9. The same model, with vertical transmission

10. The same model, with vertical transmission

11. The same model, with vertical transmission

12. Temporal behavior K=K(t) A “realistic” time dependent carrying capacity induces the occurrence of extinctions and outbreaks as controlled by the environment.

13. critical population Nc=Kc(b-c) Temporal behavior of real mice Real populations of susceptible and infected deer mice at Zuni site, NM. Nc=2 is the “critical population” derived from approximate fits.

14. THE DIFFUSION PARADIGM (Fisher, 1937) diffusion nonlinear “reaction” (logistic growth) Epidemics of Hantavirus inP. maniculatus Abramson, Kenkre, Parmenter, Yates (2001-2002)

15. Three categories of wrongfulness Okubo & Levin, Diffusion and Ecological Problems Wrong but useful: the simplest diffusion models cannot possibly be exactly right for any organism in the real world (because of behavior, environment, etc). But they provide a standardized framework for estimating one of ecology most neglected parameters: the diffusion coefficient. Not necessarily so wrong: diffusion models are approximations of much more complicated mechanisms, the net displacements being often described by Gaussians. Woefully wrong: for animals interacting socially, or navigating according to some external cue, or moving towards a particular place.

16. 17 P 47 B 57 B 67 B 77 B 27 PA 37 PA 26 PA 36 PA 16 P 46 B 56 B 66 B 76 B 25 PA 35 PA 15 P 45 B 55 B 65 B 75 B Puas 24 PA 34 PA 14 P 44 B 54 B 64 B 74 B 23 PA 33 PA 13 P 43 B 53 B 63 B 73 B 22 P 32 P 12 P 42 B 52 B 62 B 72 B 21 P 31 P 11 P 41 B 51 B 61 B 71 B 60 m THE SOURCE OF THE DATA Gerardo Suzán & Erika Marcé, UNM Six months of field work in Panamá (2003) Zygodontomys brevicauda Host of Hantavirus Calabazo

17. 200 m THE SOURCE OF THE DATA Terry Yates, Bob Parmenter, Jerry Dragoo and many others, UNM Ten years of field work in New Mexico (1994-) Peromyscus maniculatus Host of Hantavirus Sin Nombre

18. Recapture and age Zygodontomys brevicauda, 846 captures: 411 total mice, 188 captured more than once (2-10 times) P. maniculatus: 3826 captures: 1589 total mice, 849 captured more than once (2-20 times) Recapture probability: J: juvenile SA: sub-adult A: adult *One mouse (SA, F) recaptured off-site, 200 m away

19. Different types of movement Adult mice  diffusion within a home range Sub-adult mice  run away to establish a home range Juvenile mice  excursions from nest Males and females…

20. The recaptures Z. brevicauda

21. MOUSE WALKS Julian date 2450xxx (Sept. 97 to May 98)

22. An ensemble of displacements

23. An ensemble of displacements

24. An ensemble of displacements …representing the walk of an “ideal mouse”

25. P. maniculatus 1 day intervals PDF of individual displacements As three ensembles, at three time scales:

26. Mean square displacement Z. brevicauda (Panama) P. maniculatus (New Mexico)

27. Confinements to diffusive motion • Home ranges • Capture grid • Combination of both

28. PDF of an animal A harmonic model for home ranges MSD FPE

29. L = G L > G L =  L < G initially diffusive ~t Z. brevicauda (Panama) P. maniculatus (New Mexico) Time dependent MSD saturation box potential, concentric with the window

30. Saturation of the MSD Application of the use of the saturation curves to calculate the home range size of P. maniculatus (NM average) from measurements resulting value

31. … … a Periodic arrangement of home ranges

32. intersection Periodic arrangement of home ranges

33. SUMMARY • Simple model of infection in the mouse population • Important effects controlled by the environment • Extinction and spatial segregation of the infected population • Propagation of infection fronts • Delay of the infection with respect to the suceptibles • Mouse “transport” is more complex than diffusion • Different subpopulations with different mechanisms • Existence of home ranges • Existence of “transient” mice • Limited data sets can be used to derive some statistically sensible parameters: D, L, a • Possibility of analytical models

34. Thank you!

35. TRAVELING WAVES How does infection spread from the refugia? The sum of the equations for MS and MI is Fisher’s equation for the total population: (Fisher, 1937) There exist solutions of this equations in the form of a front wave traveling at a constant speed.

36. Traveling waves of the complete system Allowed speeds: Depends on K and a

37. Two regimes of propagation: The delay D is also controlled by the carrying capacity

38. 1. Spatio-temporal patterns in the Hantavirus infection, by G. Abramson and V. M. Kenkre, Phys. Rev. E 66, 011912 (2002). 2. Simulations in the mathematical modeling of the spread of the Hantavirus, by M. A. Aguirre, G. Abramson, A. R. Bishop and V. M. Kenkre, Phys. Rev. E 66, 041908 (2002). 3. Traveling waves of infection in the Hantavirus epidemics, by G. Abramson, V. M. Kenkre, T. Yates and B. Parmenter, Bulletin of Mathematical Biology 65, 519 (2003). 4. The criticality of the Hantavirus infected phase at Zuni, G. Abramson (preprint, 2004). 5. The effect of biodiversity on the Hantavirus epizootic, I. D. Peixoto and G. Abramson (preprint, 2004). 6. Diffusion and home range parameters from rodent population measurements in Panama, L. Giuggioli, G. Abramson, V.M. Kenkre, G. Suzán, E. Marcé and T. L. Yates, Bull. of Math. Biol (accepted, 2005). 7. Diffusion and home range parameters for rodents II. Peromyscus maniculatus in New Mexico, G. Abramson, L. Giuggioli, V.M. Kenkre, J.W. Dragoo, R.R. Parmenter, C.A. Parmenter and T.L. Yates (preprint, 2005). 8. Theory of home range estimation from mark-recapture measurements of animal populations, L. Giuggioli, G. Abramson and V.M. Kenkre (preprint, 2005).