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
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
-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)
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
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).