Diffusion and home ranges Diffusion and home ranges in micein mice movementmovement

Guillermo AbramsonStatistical Physics Group, Centro Atómico Bariloche and CONICET

Bariloche, Argentina.

with L. Giuggioli and V.M. Kenkre

Oh, my God, Kenkre has told everything!

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

OUTLINE

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).

Population dynamics+

Contagion+

(Mice movement)

Mathematical model

Single control parameter in the model simulate environmental effects.

The other two appear as consequences of a bifurcation of the solutions.

BASIC MODEL (no mice movement yet!)

,

,

ISI

II

ISS

SS

MMaK

MMMcdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

Rationale behind each termBirths: bM → only of susceptibles, all mice contribute to itDeaths: -cMS,I → infection does not affect death rateCompetition: -MS,I M/K → population limited by environmentalparameter Contagion: ± aMS MI → simple contact between pairs

MS (t) : Susceptible mice

MI (t) : Infected mice

M(t)= MS (t)+MI (t): Total mouse population

carrying capacity

The carrying capacity controls a bifurcation in the equilibrium value of the infected population.

The susceptible population is always positive.

)( cbabKc −

=

BIFURCATION

The same model, with vertical transmission

,

,

ISI

III

ISS

SSS

MMaK

MMMcMbdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14M

S and

MI

K

0.00.1

1.01.0

==

==

I

S

bbca

cK

The same model, with vertical transmission

,

,

ISI

III

ISS

SSS

MMaK

MMMcMbdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14M

S and

MI

K

01.099.0

1.01.0

==

==

I

S

bbca

0=cK

The same model, with vertical transmission

,

,

ISI

III

ISS

SSS

MMaK

MMMcMbdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14M

S and

MI

K

1.09.0

1.01.0

==

==

I

S

bbca

0=cK

The same model, with vertical transmission

,

,

ISI

III

ISS

SSS

MMaK

MMMcMbdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14M

S and

MI

K

2.08.0

1.01.0

==

==

I

S

bbca

0=cK

The same model, with vertical transmission

,

,

ISI

III

ISS

SSS

MMaK

MMMcMbdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14M

S and

MI

K

5.05.0

1.01.0

==

==

I

S

bbca

0=cK

Temporal behavior

A “realistic” time dependent carrying capacity induces the occurrence of extinctions and outbreaks as controlled by the environment.

K=K(t)

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.

critical populationNc=Kc(b-c)

THE DIFFUSION PARADIGM

,)(

),(

,)(

),(

2

2

IIISI

II

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

THE SOURCE OF THE DATA

Gerardo Suzán & Erika Marcé, UNM

Six months of field work in Panamá (2003)

Zygodontomys brevicaudaHost of Hantavirus Calabazo

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

60 m

200 m-100 -50 0 50 100

-100

-50

0

50

100 N

Terry Yates, Bob Parmenter, Jerry Dragoo and many others, UNM

Peromyscus maniculatusHost of Hantavirus Sin Nombre

Ten years of field work in New Mexico (1994-)

THE SOURCE OF THE DATA

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)

0.580.480.32P. maniculatus

0.490.37*0.13Z. Brevicauda

ASAJ

Recapture probability:

Recapture and age

J: juvenileSA: sub-adultA: adult

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

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…

The recaptures

0 20 40 60 80 100 120 140-60

-40

-20

0

20

40

60

Δx (m

)

Δt (days)

0 30 60 90 120 150 180 210 240 270 300 330-200

-100

0

100

200

Δx (m

)

time (days)

P. maniculatusall sites, all mice

Z. brevicauda

MOUSE WALKS

-40 0 40 80

-60

-30

0

30

744745

771

799801

834

835 897

898

899

925

926

927

953

954

P.m. tag 3460

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

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

An ensemble of displacements

An ensemble of displacements

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

-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

0.35

P(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:

-150 -100 -50 0 50 100 1500.000

0.004

0.008

0.012

0.016

-100 -50 0 50 1000.00

0.01

0.02

0.03 p(x) (renormalized) Gaussian fit

p(x)

x (m)

q(x)

x (m)

1 day intervalsP. maniculatus

Mean square displacement

0 30 60 900

200

400

600

<Δx2 >,

<Δy

2 > (m

2 )

t (days)

<Δx2> <Δy2>

Z. brevicauda (Panama) 0 30 60 90 120 150 1800

500

1000

1500

2000

2500

3000

<Δx2 >,

<Δy

2 > (m

2 )t (days)

East-West direction North-South direction

P. maniculatus (New Mexico)

Confinements to diffusive motion

• Home ranges

• Combination of both

• Capture grid

L

G

LG

A harmonic model for home ranges

xc3xc2xc1 xc/2-xc/2 G/2-G/2

L/2

U1U2 U3

P3(x)P2(x)P1(x)

L/2L/2

),(),()(),( 2 txPDtxPdx

xdUxt

txP∇+⎥⎦

⎤⎢⎣⎡

∂∂

=∂

∂

PDF of an animal

Time dependent MSD

0 0.30

1

Dt/G2

<x2 >

/(G

2 /12)

0 0.30

1

Dt/G2

<x2 >

/(G

2 /12)

L = ∞

L < G

L > G

L = G

saturation

initially diffusive ~tbox potential, concentric with the window

box potential, concentric with the window

0 30 60 900

200

400

600

<Δx2 >,

<Δy

2 > (m

2 )

t (days)

<Δx2> <Δy2>

Z. brevicauda (Panama) 0 30 60 90 120 150 1800

500

1000

1500

2000

2500

3000

<Δx2 >,

<Δy

2 > (m

2 )t (days)

East-West direction North-South direction

P. maniculatus (New Mexico)

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

harmonic numerical harmonic analytical box numerical box analytical asymptotics

<<x2 >>

/(G

2 /6)

L/G

L2/6

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

Periodic arrangement of home ranges

a

……

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

a/G

L/G

00.10.20.30.40.50.60.70.80.91.0

Δx2/(G2/6)

Periodic arrangement of home ranges

( )( )( )33

23

2222

1121

/,/

/,/

/,/

GaGLfx

GaGLfx

GaGLfx

=Δ

=Δ

=Δ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Measurement 1 (G1 = 1)Measurement 2 (G2 = 0.5)Measurement 3 (G3 = 0.75)

a

L

intersection

SUMMARYSimple 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

Thank you!

TRAVELING WAVES

The sum of the equations for MS and MI is Fisher’s equation for the total population:

There exist solutions of this equations in the form of a front wave traveling at a constant speed.

How does infection spread from the refugia?

MDKcb

MMcbt

txM 2

)(1)(),(

∇+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−=∂

∂

(Fisher, 1937)

[ ])(2

)(2

cbaKbDv

cbDv

I

S

−+−≥

−≥

Allowed speeds:

Depends on K and a

Traveling waves of the complete system

Two regimes of propagation:

)(2

0 cbacbK

−−=

0

0

if if

KKvvKKvv

SI

SI

>=<<

The delay Δ is also controlled by the carrying capacity

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).