HantaHanta--bearing mice: do they bearing mice: do they diffuse or stay put?diffuse or stay put?

Guillermo AbramsonCentro Atómico Bariloche and CONICET, Bariloche, Argentina.

Consortium of the Americas for Interdisciplinary Science, University of New Mexico, Albuquerque, USA.

The diffusion paradigm

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2

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IIISI

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SSISS

SS

MDMMaxKMMMc

ttxM

MDMMaxKMMMcMb

ttxM

∇++−−=∂

∂

∇+−−−=∂

∂

Epidemics of Hantavirus in P. maniculatusAbramson, Kenkre, Parmenter, Yates (2001-2002)

diffusionnonlinear “reaction”(logistic growth)

(Fisher, 1937)uDuurt

txu 2)1(),(∇+−=

∂∂

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 forestimating 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.

Three categories of wrongfulnessOkubo & Levin, Diffusion and Ecological Problems

17P

11P

12P

13P

14P

15P

16P

27PA

21P

22P

23PA

24PA

25PA

26PA

37PA

31P

32P

33PA

34PA

35PA

36PA

47B

41B

42B

43B

44B

45B

46B

57B

51B

52B

53B

54B

55B

56B

67B

61B

62B

63B

64B

65B

66B

77B

71B

72B

73B

74B

75B

76B

The source of the dataGerardo Suzán & Erika Marcé, UNM

Six months of field work in Panamá (2003)

Zygodontomys brevicaudaHost of Hantavirus Calabazo

70 m

Zygodontomys brevicauda, 846 captures:

411 total mice, 188 captured more than once (2-10 times)

0.490.370.13Total

0.550.470.00M

0.420.130.22F

ASAJRecapture probability:

Recapture and age

J: juvenileSA: sub-adultA: adultF: femaleM: male

One mouse (SA, F) recaptured off-site, 200 m away

0 20 40 60 80 1000

10

20

30

40

50

60

No.

of m

ice

weight (g)

Female Male

Z. brevicauda - Weight histogramWeight is weight at first capture193 females218 males

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

75 females recaptured113 males recaptured193 females total218 males total

Weight is weight at first capture

Z. brevicauda

Prob

. of r

ecap

ture

weight (g)

Female Male

Recapture and weight

Distribution of weight

-20 -10 0 10 20 30 40 50 60 70-10

0102030405060708090

100

dt (d

ays)

dr (m)

(Overlapped points plotted offset)Z. brevicauda - 434 recaptures

The recaptures

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…

0 30 60 90 120-60-50-40-30-20-10

0102030405060 Z. brevicauda captured ~10 times

x(t)

(m)

t (days)

A117008 A117039 A117075 A117104 A117281

Individual mouse walks

0 30 60 90 1200

200

400

600

800

1000

1200

<x2 >

t (days)

008 039 075 104 281 average

Z. brevicauda captured ~10 times

“Mean” square displacement?

-80 -60 -40 -20 0 20 40 60 800

20

40

60

80

100

120

140

160

P(dx

), P(

dy)

dx, dy (m)

dx dy Gaussian fit Lorentzian fit

Z. brevicauda, all sites, 434 recaptures

<dx> = -0.88 mσ2

x = 18.7 m2

<dy> = 1.2 mσ2

y = 20.5 m2

PDF of individual displacementsThe population as an ensemble of walkers

-80 -60 -40 -20 0 20 40 60 80-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35P(

dx)

dx

dt ~ 1day dt ~ 1 month dt ~ 2 months

247 steps170 steps17 steps

Z. brevicauda

PDF of individual displacementsAs three ensembles, at three time scales

An ensemble of displacements

An ensemble of displacements

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

0 90 180 2700

5x104

1x105

2x105

2x105

x2

t

Average Linear fit

slope = 686 m2/dD

1 = 342 m2/d

Z. brevicauda - 1-day steps

Ideal mouse walks

0 730 1460 2190 2920 3650 4380 51100

1x104

2x104

3x104

4x104

5x104

6x104

7x104

Z. brevicauda - 1-month steps

x2

t

Average Linear fit

slope = 11 m2/dD30 = 6.5 m2/d

Ideal mouse walks

0 365 7300

1000

2000

3000

4000

5000

6000Z. brevicauda - 2-months steps

x2

t

Average Linear fit

slope = 1.9 m2/dD60 = 0.95 m2/d

Ideal mouse walks

0 10 20 30 40 50 60 70 800

500

1000

1500

2000

2500

3000Z. brevicauda - Diffusion properties

x2 (t)

t

b(1-e-at) D1

D30

D60

a = 0.136655 db = 2869.114 m2

D0 ~ 392 m2/dLD ~ 30 m/dHR ~ 54 m

Mean square displacement

(just an ansatz)

“microscopic”diffusion

home range

“microscopic”diffusion length

In summary

Mouse “transport” is more complex than diffusion

Different subpopulations with different mechanisms

Limited data sets can be used to derive some statistically sensible parameters

Existence of home rangesExistence of “transient” mice

Possibility of analytical models

Analysis of other systems (New Mexico…)

0 30 60 90 120-60-50-40-30-20-10

0102030405060

y(t)

(m)

t (days)

A117008 A117039 A117075 A117104 A117281

Z. brevicauda captured ~10 times

Zygodontomys brevicauda

846 captures:

411 total mice, 188 captured more than once (2-10 times)

2231773115Total

10579188M

11898137F

TotalASAJ

Captured once:

188168182Total

11397160M

757122F

TotalASAJ

Recaptured: