Download Diffusion and home ranges in mice movement

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
Oh, my God, Kenkre has told everything!
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
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!)
dM S
MSM
= bM − c M S −
− aM S M I ,
dt
K
dM I
MIM
= −c M I −
+ aM S M I ,
dt
K
MS (t) : Susceptible mice
MI (t) : Infected mice
M(t)= MS (t)+MI (t): Total
mouse population
carrying capacity
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
BIFURCATION
b
Kc =
a (b − c )
The carrying capacity controls a bifurcation in the equilibrium
value of the infected population.
The susceptible population is always positive.
The same model, with vertical transmission
MS and MI
dM S
M M
= b S M − cM S − S − aM S M I ,
dt
K
M M
dM I
= b I M − cM I − I + aM S M I ,
dt
K
14
a = 0.1
12
c = 0.1
bS = 1.0
10
8
bI = 0.0
6
4
2
0
0
2
4
6
8
10
12
14
K Kc
16
18
20
The same model, with vertical transmission
MS and MI
dM S
M M
= b S M − cM S − S − aM S M I ,
dt
K
M M
dM I
= b I M − cM I − I + aM S M I ,
dt
K
14
a = 0.1
12
c = 0.1
bS = 0.99
10
8
bI = 0.01
6
4
2
0
0
Kc = 0
2
4
6
8
10
K
12
14
16
18
20
The same model, with vertical transmission
MS and MI
dM S
M M
= b S M − cM S − S − aM S M I ,
dt
K
M M
dM I
= b I M − cM I − I + aM S M I ,
dt
K
14
a = 0.1
12
c = 0.1
bS = 0.9
10
8
bI = 0.1
6
4
2
0
0
Kc = 0
2
4
6
8
10
K
12
14
16
18
20
The same model, with vertical transmission
MS and MI
dM S
M M
= b S M − cM S − S − aM S M I ,
dt
K
M M
dM I
= b I M − cM I − I + aM S M I ,
dt
K
14
a = 0.1
12
c = 0.1
bS = 0.8
10
8
bI = 0.2
6
4
2
0
0
Kc = 0
2
4
6
8
10
K
12
14
16
18
20
The same model, with vertical transmission
MS and MI
dM S
M M
= b S M − cM S − S − aM S M I ,
dt
K
M M
dM I
= b I M − cM I − I + aM S M I ,
dt
K
14
a = 0.1
12
c = 0.1
bS = 0.5
10
8
bI = 0.5
6
4
2
0
0
Kc = 0
2
4
6
8
10
K
12
14
16
18
20
Temporal behavior
K=K(t)
A “realistic” time dependent carrying capacity induces the
occurrence of extinctions and outbreaks as controlled by the
environment.
Temporal behavior of real mice
critical
population
Nc=Kc(b-c)
Real populations of susceptible and infected deer mice at
Zuni site, NM. Nc=2 is the “critical population” derived from
approximate fits.
THE DIFFUSION PARADIGM
∂u ( x, t )
= r u (1 − u ) + D ∇ 2u
∂t
(Fisher, 1937)
diffusion
nonlinear “reaction”
(logistic growth)
Epidemics of Hantavirus in
P. maniculatus
Abramson, Kenkre, Parmenter, Yates (2001-2002)
∂M S ( x, t )
M M
= b M − c M S − S − a M S M I + DS ∇ 2 M S ,
∂t
K ( x)
∂M I ( x, t )
M M
= − c M I − I + a M S M I + DI ∇ 2 M I ,
∂t
K ( x)
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.
THE SOURCE OF THE DATA
Gerardo Suzán & Erika Marcé, UNM
Six months of field work in Panamá (2003)
17
P
27
PA
37
PA
47
B
57
B
67
B
77
B
16
P
26
PA
36
PA
46
B
56
B
66
B
76
B
15
P
25
PA
35
PA
45
B
55
B
65
B
75
B
14
P
24
PA
34
PA
44
B
54
B
64
B
74
B
13
P
23
PA
33
PA
43
B
53
B
63
B
73
B
12
P
22
P
32
P
42
B
52
B
62
B
72
B
11
P
21
P
31
P
41
B
51
B
61
B
71
B
Zygodontomys brevicauda
Host of Hantavirus Calabazo
60 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-)
N
100
50
0
-50
-100
Peromyscus maniculatus
-100
-50
0
200 m
50
100
Host of Hantavirus Sin Nombre
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
SA
A
Z. Brevicauda 0.13
0.37*
0.49
P. maniculatus 0.32
0.48
0.58
J: juvenile
SA: sub-adult
A: adult
*One mouse (SA, F) recaptured off-site, 200 m away
Different types of movement
Adult mice Ö diffusion within a home range
Sub-adult mice Ö run away to establish
a home range
Juvenile mice Ö excursions from nest
Males and females…
The recaptures
60
Z. brevicauda
40
200
0
-20
P. maniculatus
all sites, all mice
100
-40
-60
0
20
40
60
80
Δt (days)
100
120
140
Δx (m)
Δx (m)
20
0
-100
-200
0
30 60 90 120 150 180 210 240 270 300 330
time (days)
x(t) (m)
MOUSE WALKS
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
Z. brevicauda captured ~10 times
926
30
799
835
801
0
P.m. tag 3460
897
898
899
953
927
A117008
A117039
A117075
A117104
A117281
0
30
60
t (days)
90
925
-30
744
745
954
771
-60
834
120
-40
0
40
80
Julian date 2450xxx
(Sept. 97 to May 98)
An ensemble of displacements
An ensemble of displacements
An ensemble of displacements
…representing
the walk of an
“ideal mouse”
PDF of individual displacements
As three ensembles, at three time scales:
0.35
Z. brevicauda
0.30
247 steps
170 steps
17 steps
0.25
dt ~ 1day
dt ~ 1 month
dt ~ 2 months
0.15
0.10
0.05
0.016
q(x)
p(x) (renormalized)
Gaussian fit
0.03
0.02
0.00
-0.05
0.012
-80
-60
-40
-20
0
dx
20
40
60
80
p(x)
P(dx)
0.20
1 day intervals
0.01
P. maniculatus
0.00
-100 -50
0
50 100
x (m)
0.008
0.004
0.000
-150
-100
-50
0
x (m)
50
100
150
Mean square displacement
3000
2
<Δx >, <Δy > (m )
600
2500
2
<Δx >, <Δy > (m )
0
2000
2
200
2
<Δx >
2
<Δy >
2
2
2
400
0
30
60
90
1500
1000
East-West direction
North-South direction
500
t (days)
0
Z. brevicauda (Panama)
0
30
60
90
120
150
180
t (days)
P. maniculatus (New Mexico)
Confinements to diffusive motion
• Home ranges
• Capture grid
L
• Combination of bothG
G
L
A harmonic model for home ranges
U2
U1
P1(x)
-xc/2
U3
P3(x)
P2(x)
L/2
L/2
L/2
xc1
xc2
xc3
-G/2
G/2
xc/2
∂P( x, t ) ∂ ⎡ dU ( x )
⎤
2
=
+
∇
P
(
x
,
t
)
D
P ( x, t )
⎢
⎥
∂t
∂x ⎣ dx
⎦
PDF of an animal
Time dependent MSD
11
L=G
0
0
L=∞
3000
2500
2
saturation
2000
2
200
2
<Δx >
2
<Δy >
2
2
2
22>/(G22/12)
<x
<x >/(G /12)
400
<Δx >, <Δy > (m )
L>G
2
<Δx >, <Δy > (m )
600
30
60
90
1500
1000
L<G
500
t (days)
initially diffusive ~t0
Z. brevicauda (Panama)
00
00
0
30
East-West
direction
box
potential,
box
potential,
North-South direction
60
P. maniculatus
2
2
Dt/G
Dt/G
concentric
concentricwith
with
the
90 window
120
150
180
the
window
t (days)
0.3
(New0.3
Mexico)
Saturation of the MSD
1.0
2
0.8
2
2
<<x >> /(G /6)
L /6
0.6
harmonic numerical
harmonic analytical
box numerical
box analytical
asymptotics
0.4
from
measurements 0.2
0.0
0.0
0.5
1.0
1.5
L/G
resulting value
2.0
2.5
Application of
the use of the
saturation
curves to
calculate the
home range
size of P.
maniculatus
(NM average)
Periodic arrangement of home ranges
…
…
2.0
a
a/G
1.5
1.0
2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5
0.0
0.0
2
Δx /(G /6)
0.5
1.0
1.5
L/G
2.0
2.5
Periodic arrangement of home ranges
1.0
Measurement 1 (G1 = 1)
Measurement 2 (G2 = 0.5)
0.8
Measurement 3 (G3 = 0.75)
Δx12 = f (L / G1 , a / G1 )
0.6
a
Δx22 = f (L / G2 , a / G2 )
intersection
Δx32 = f (L / G3 , a / G3 )
0.4
0.2
0.0
0.0
0.2
0.4
0.6
L
0.8
1.0
SUMMARY
¾Simple model of infection in the mouse population
¾Important effects controlled by the environment
¾Extinction and spatial segregation of the infected population
¾Propagation of infection fronts
¾Delay of the infection with respect to the suceptibles
¾Mouse “transport” is more complex than diffusion
¾Different subpopulations with different mechanisms
•Existence of home ranges
•Existence of “transient” mice
¾Limited data sets can be used to derive some statistically
sensible parameters: D, L, a
¾Possibility of analytical models
Thank you!
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:
⎛
M ⎞
∂M ( x, t )
⎟⎟ + D∇ 2 M
= (b − c) M ⎜⎜1 −
∂t
⎝ (b − c) K ⎠
(Fisher, 1937)
There exist solutions of this equations in the form
of a front wave traveling at a constant speed.
Traveling
waves of
the
complete
system
Allowed speeds:
vS ≥ 2 D(b − c)
vI ≥ 2 D[− b + aK (b − c)]
Depends on K and a
Two regimes of propagation:
v I < vS if K < K 0
v I = vS if K > K 0
2b − c
K0 =
a (b − c )
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).