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This is the presentation given by Andrea Rinaldo in Trento for the opening day of the 2014 Doctoral School.
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Andrea Rinaldo
!!
Laboratory of Ecohydrology ENAC/IIE/ECHO Ecole Polytechnique Fédérale Lausanne (EPFL) CH Dipartimento ICEA Università di Padova
RIVER NETWORKS
AS ECOLOGICAL CORRIDORS
FOR SPECIES
POPULATIONS AND WATER-BORNE DISEASE
PLAN
!tools: reactive transport on networks
nodes (reactions) + branches (transport)
metacommunity & individual-based models
!modeling migration fronts &
human range expansions
!spreading of water-borne disease
hydrologic controls on cholera epidemics
!invasion of vegetation or
freshwater fish species
along fluvial corridors
!hydrochory & biodiversity
explore two critical characteristics (directional dispersal & network structure as environmental matrix)
for spreading of organisms, species & water-borne disease
questions of scientific & societal relevance
(population migrations, loss of biodiversity, hydrologic
controls on the spreading of Cholera, meta-history)
Muneepeerakul et al., JTB, 2007
Rodriguet-Iturbe et al., PNAS, 2012
Carrara et al., PNAS, 2012
Carrara et al., PNAS, 2012
Carrara et al., Am. Nat., 2014
TOOLS - about the progress (recently) made on
how to decode the mathematical language
of the geometry of Nature
DTM - GRID (Planar view)
DTM – GRID format (Perspective – North towards bottom)
remarkable capabilities
to remotely acquire
& objectively
manipulate
accurate descriptions
of natural landforms
over several orders
of magnitude
if I remove the
scale bar …consilience…
Rodriguez-Iturbe & Rinaldo, Fractal River Basins: Chance and Self-Organization, Cambridge Univ, Press, 2007
TOOLS
from O(1) m scales…
the MMRS
random-walk
drainage basin network
(Leopold & Langbein, 1962)
& the resistible
ascent of the
random paradigm
!
Eden growth & self-avoiding random walks !
Rigon et al., WRR, 1998
Huber, J Stat Phys, 1991; Takayasu et al., 1991
Scheidegger’s construction
is exactly solved for
key geometric & topologic features
Rodriguez-Iturbe et al., WRR, 1992 a,b; Rinaldo et al. WRR, 1992
optimal channel networks
Rigon et al., WRR, 1997
Rinaldo et al., PNAS, in press
Peano – exact results & subtleties
(multifractality
binomial multiplicative process & width functions)
Marani et al., WRR, 1991; Colaiori et al., PRE, 2003
TOOLS 1 - comb-like structures, diffusion processes & CTRW framework in terms of density
of particles ρ(x,t)
l
A B
from traditional unbiased random-walks to general cases
!heterogeneous distributions
of spacing, Δx & length of the comb leg, l
AB A B
delay ~ reactions, lifetime distributions
tools - 2
models of reactive transportnetwork → oriented
graph made by nodes & edges
TRANSPORT MODELS BETWEEN
NODES
NODAL REACTIONS
COUPLED MODELS
individuals, species, populations (metacommunities)
TOOLS 2 - reactive continuous time random walk
x
pdf of jump &
waiting time
),( txΨ
)0,(xρ
Φ(t)
reaction)(ρf
∫ ∫∞ +∞
∞−Ψ=
ttxdxdtt )',(')(φ
diffusion
?
a master equation – if we consider many realizations
of independent processes (large number of noninteracting propagules) ρ(i,t) is proportional
to the number of propagules in i at time t
transport + possibly reactions or interactions
hydrochory
!!
human-range expansion, population migration
quantitative model of US colonization 19th century
& transport on fractal networks
Campos et al., Theor. Pop. Biol., 2006 !!
the idea that landscape heterogeneities & need for
water forced settling about fluvial courses
!!
Ammerman & Cavalli Sforza, The Neolithic transition and the Genetics of population in Europe, Princeton Univ. Press 1984
!!!
exact reaction-diffusion model (logistic with rate parameter a for population growth)
!
a little background on Fisher’s fronts
phase plane → the sign of the eigenvalues of an
appropriate Jacobian matrix
determines the nature of the equilibria
!(e.g. Murray, 1993)
a few further mathematical details
the network slows the front! you waste time trapped in the pockets
the Hamilton-Jacobi formalism
Peano’s networkinitial cond
r=1
logistic growth at every node )1()( ρρρ −= af
f
ρ0 1
a
reaction
transport at every timestep each particle moves towards a nearest neighbour
w.p. p= 1 / # nn
SIMULAZIONI
P+=0.5 a=0.5
isotropic migration – Fisher’s model
v = 2√aD Murray, 1988
Peano (exact)
Peano (numerical)
a (logistic growth)
v sp
eed
of fro
nt [
L/T]
Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007
Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007
geometric constraints imposed by the network
(topology & geometry) impose strong corrections
to the speed of propagation of migratory fronts
Rel
ativ
e fr
eque
ncy
(%)
what is a node? !!
strong hydrologic controls
!
Giometto et al., PNAS, 2013
What about variability?
Fisher-‐Kolmogorov Equation
∂ρ
∂t=D∂
2ρ
∂x2+ rρ 1−
ρ
K$
%&'
()
∂ρ
∂t= rρ 1−
ρ
K$
%&'
()+σ ρ η
∂ρ
∂t=D∂
2ρ
∂x2+ rρ 1−
ρ
K$
%&'
()+σ ρ η
ML estimates for r,K, σ
η is a δ-‐correlated gaussian white noise Itô stochastic calculus
Transitional probability densities are computed by numerical integration of the related Fokker-‐Planck equation.
∂ρ
∂t= rρ 1−
ρ
K$
%&'
()+σ ρ η
∂ρ
∂t=D∂
2ρ
∂x2+ rρ 1−
ρ
K$
%&'
()+σ ρ η
Demographic stochasticity
ML estimates for r,K, σ
η is a δ-‐correlated gaussian white noise Itô stochastic calculus
Transitional probability densities are computed by numerical integration of the related Fokker-‐Planck equation.
Front variability
Giometto et al., PNAS, 2013
Take-‐home message
• Fisher-‐Kolmogorov equation correctly predicts the mean features of dispersal !
• The observed variability is explained by demographic stochasticity
Link between scales
Giometto et al., PNAS, 2013
Zebra MusselDreissena polymorpha
1989
1990
1991
1992
1994
1995
1988
1993
data: Nonindigenous Aquatic species program USGS
larval stages transported along the
fluvial network
Zebra MusselMari et al., in review, 2007
local age-growth model (4 stages) !larval production
!larval transport
(network)
Zebra Mussel
Mari et al., WRR, 2011; Mari et al., Ecol. Lett., 2014
river biogeography
!spatial distribution of
biodiversity within a biota
!riparian vegetation
fluvial fauna
freshwater fish
neutral metacommunity model
metacommunity modelevery link is a community of organisms & internal implicit spatial dynamics
Explicit spatial dynamics among different communities
the neutral assumptionall species are equivalent (equal fertility, mortality, dispersion Kernel)
the probability with which a propagule colonizes a site depends only on its relative abundance
patterns of biodiversity emerge because of ecological drift
Hubbel, 2001
neutral metacommunity model
the model
at each timestep an organism is randonly chosen & killed w.p. ν it is substituted by a species non existing (prob of speciation/immigration)
w.p. 1-ν the site is colonized by an organism present in the system
∑=
−= N
kkik
jijij
HK
HKvP
1
)1(
:habitat capacity link ijH
ijK :dispersal kernel
run up to steady state
river biogeography
abundance
# of
spe
cies
20 21 22 24 26 28 21223 25 27 29 210 211
global properties
γ-diversity: total # of species
patterns of abundance
preston plot
river biogeography
α-diversitynumber of
species at local scale
LOCAL PROPERTIES
river biogeography
β-diversity
Jaccard similarity index
a)
b)
abba
abab S
SxJ
−+=
αα)( abS # common species
x distance measured along the network
57.0456
4)( =−+
=xJab
river biogeography
geographic range
area occupied
by a species
ranked species
geog
raph
ic ra
nge
USGS, hydrologic data, NatureServe, Bill Fagan’s ecological data
Mississippi-Missouri freshwater fauna
presence(absence) of 429 species freshwater fish in 421 subbasins
database
α-diversity, β-diversity, γ-diversity, geographic range
fonti
Mississippi-Missouri freshwater fish
α-diversity
runoff
strong correlation
habitat capacity ~ runoff
Muneeperakul, Bertuzzo, Fagan, Rinaldo, Rodriguez-Iturbe, Nature, 2008
distance to outlet
Muneeperakul et al., Nature, May 8 2008
constant
habitat
capacity
per DTA
hydrologic controls
weak interspecific interactions & weak/strong
formulations of the neutral model
model
comparison between geographic ranges of individual species: a) data b) results from the neutral
metacommunity model (after matching procedure)
patterns -- weak or strong impliations of neutrality?
equiprobability map – ratio between the number of common species and the number of species in
the central DTA
Bertuzzo et al., submitted, 2008
environmental resistance R50 for data & the model
is topology reflected in the spatial organization of the species?
!species range & maximum drainage
area – the max area experienced
by a species is that in blue color, range
is cross-hatched red
!containment effect favors colonization
!
Corridors for pathogens of waterborne disease !!
Of cholera epidemics & hydrology
Haiti (2010-2011)
Piarroux et al., Emerging Infectious Diseases, 2011
no elementary correlation between population and cholera cases
Mari et al., J Roy Soc Interface, 2011
continuous SIR model
susceptibles S
infected I
vibrios/m3 B
recovered R
persons
vibriosSBK
B+
βIγ
BBµ
IWp
Iµ Iα
Rµ Sµ Hµ
H: total human population at disease free equilibrium
µ: natality and mortality rate (day-1) β: rate of exposure to contaminated water
(day-1) K: concentration of V. cholerae in water
that yields 50% chance of catching cholera (cells/m3)
α: mortality rate due to cholera(day-1) γ : rate at which people recover from
cholera (day-1) µB:death rate of V. cholerae in the aquatic
environment (day-1) p : infected rate of production of V.
cholerae (cells day-1 person-1) W: volume of water reservoir (m3)
Codeco, JID, 2001; Pascual et al, PLOS, 2002; Chao et al, PNAS, 2011
Chao et al., PNAS, 2011
Capasso et al, 1979; Codeco, JID, 2001
the class of SIB models
0 50 100 150 200t [days]
pers
on
susceptiblesinfected
I(t) S(t)
!SIR model for the temporal &
spatial evolution of water-transmitted disease revisited → network
!a few assumptions
!!
total population of humans is unaffected by the disease
!diffusion of infective humans is small
w.r. to that of bacteria thus set to zero !
density-dependent reaction terms depend on local susceptibles
!!Capasso et al, 1979; Codeco, JID, 2001; Pascual et al, PLOS, 2002; Hartley et al, PNAS, 2006
0 50 100 150 200t [days]
pers
on
susceptiblesinfected
nodes are human communities with population H in which the disease can diffuse & grow
Hydrologic Networks Human-Mobility Network
i
j
i
jRij
Qij
Pij Qij
Mari et al., J Roy Soc Interface, 2011
0 50 100 150 200 250 300 3500
100
200
300
400
500
600
time [days]
infe
cted
uniform population
I(t)
t
Zipf distribution of population size & self-organization
0 50 100 150 200 250 300 3500
200
400
600
800
1000
1200
1400
time [days]
infe
cted
Zipf’s distribution of population & secondary peaks of infection
I(t)
t
spatio-temporal dynamics
initial conditions
the higher the transport rate, the better the system is approximated by a well-mixed reactor (spatially implicit scheme)
refelecting boundary condition at all the leaves
and at the outlet
Bertuzzo et al., J Roy Soc Interface, 2010
0
5
10
15
20
25
Wee
kly
Cas
es [1
03 ]
0
10
20R
ainf
all [
mm
/day
]
prediction
calibration
Nov 10
Jan 11 Mar 11 May 11 Jul 11
Sep 11
100
300
500
prediction
Cum
ulat
ive
Cas
es [1
03 ]
Nov 10 Sep 11
Bertuzzo et al. GRL 2011
Bertuzzo et et al., GRL, 2011
−60% −40% −20% 0 20% 40% 60%
α
∝
γ
β
ρD
φ
∝B
lσ
mθ
Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 110
5
10
15
20
25W
eekl
y C
ases
[103 ]
calibration hindcast
0
10
20R
ainf
all [
mm
/day
]
Rinaldo et al., PNAS, in press
0
5 Nord−Ouest
0
5 Nord
0
5 Nord−Est
0
5
10Artibonite
0
5 Centre
0
5 Grande Anse
0
5 Nippes
0
5
10
15 Ouest
0
5 Sud
0
5 Sud−Est
15
Wee
kly
Cas
es [1
03 ]
Jan 11 May 11 Sep 11 Jan 11 May 11 Sep 11
0
5
10
15
20
25
Wee
kly
Cas
es [1
03 ]
Nov 10 Jan 11 Mar 11 May 11 Jul 11 Sep 11
Wee
kly
Cas
es [1
03 ]
0
10
20
30
40
Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 13 Jan 14
effects of rates of loss of acquired immunity (1-5 years)
Rinaldo et al., PNAS, in press
Jan 11 Jul 11 Jan 12 Jul 12 Jan 13 Jul 130
10
20
30
Wee
kly
Cas
es [1
03 ]0
5
10
15
20
Rai
nfal
l [m
m/d
ay]
Jan 14
Rinaldo et al., PNAS, in press
recorded cholera cases in Haiti (2010-2013) (normalized)
normalized maximum eigenvector
Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014
river networks & biodiversity
!tradeoff versus neutral models of the ecology of riparian vegetation
Muneepeerakul et al., JTB, 2007 --> Mari et al., Ecol Lett., 2014
Mari et al., Ecol. Lett., 2014
Muneepeerakul, Weitz, Levin, Rinaldo, Rodriguez-Iturbe, JTB, 2007
links are essentially patches within a landscape cointaining sites that are occupied by individual plants
the containment effect: the network structure
significantly hinders the dispersal of propagules
across subbasins – less sharing of species
!fragmentation increases species richness (both neutral
& trade-off communities) (diameters ~ species’ link-scale abundance
power laws matter alot - hotspots & geomorphology
!indeed a frontier of ecological research
Muneepeerakul et al., WRR, 2007
remote sensing & (much) hydrologic research
CONCLUSIONS
! rivers as ecological corridors →
containment effects (hydrochory
migrations & spreading of epidemics) !
network structure
provides strong controls & susceptibility
!e.g. secondary peaks of ‘infections’ or
biodiversity hotspots ~ geometric
constraints rather than dynamics
!river networks are possibly
templates of biodiversity → impacts of climate change scenarios on local
and regional biodiversity
!!
CONCLUSIONS -- 2 !
ecohydrological footprints from rivers as ecological corridors
& human mobility
for the spreading of epidemic cholera !
network structure(s) provides controls & susceptibility
!from secondary peaks of infections to
rainfall prediction ~ it’s all in the water
!rainfall drivers –
seasonality, endemicity & impacts of climate change scenarios,
water management, sanitation !!
collaborations
IGNACIO RODRIGUEZ-ITURBE MARINO GATTO AMOS MARITAN
RICCARDO RIGON
the ECHO/IIE/ENAC/EPFL Laboratory ENRICO BERTUZZO, LORENZO MARI, SAMIR SUWEIS
LORENZO RIGHETTO, FRANCESCO CARRARA SERENA CEOLA, ANDREA GIOMETTO
PIERRE QUELOZ, CARA TOBIN, BETTINA SCHAEFLI