RIVER NETWORKS AS ECOLOGICAL CORRIDORS FOR SPECIES POPULATIONS AND WATER-BORNE DISEASE
1. RIVER NETWORKS
AS ECOLOGICAL CORRIDORS
FOR SPECIES
POPULATIONS AND WATER-BORNE
DISEASE
Andrea Rinaldo
!
!
Laboratory of Ecohydrology ENAC/IIE/ECHO Ecole Polytechnique Fédérale Lausanne (EPFL) CH
Dipartimento ICEA Università di Padova
2. 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
3. questions of scientific & societal relevance
(population migrations, loss of biodiversity, hydrologic
controls on the spreading of Cholera, meta-history)
explore two critical characteristics (directional dispersal
& network structure as environmental matrix)
for spreading of organisms, species & water-borne disease
10. TOOLS - about the progress (recently) made on
how to decode the mathematical language
of the geometry of Nature
11. DTM - GRID
(Planar view)
DTM – GRID format
(Perspective – North towards bottom)
12. 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
26. network → oriented
graph made by nodes
& edges
models of reactive transport
COUPLED MODELS
NODAL REACTIONS
TRANSPORT
MODELS BETWEEN
NODES
individuals, species, populations (metacommunities)
27. TOOLS 2 - reactive continuous time random walk
Φ(t)
Ψ ( x, t ) pdf of jump &
waiting time
x
diffusion
f (ρ ) reaction
ρ (x,0) ?
∞
+∞
t
−∞
φ (t ) = ∫ dt ' ∫ dx Ψ ( x, t ' )
28. transport + possibly reactions or interactions
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
30. 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)
!
!
!
!
39. v speed of front [L/T]
Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007
isotropic migration – Fisher’s model
v = 2√aD
Murray, 1988
Peano (exact)
Peano (numerical)
a (logistic growth)
40. Relative frequency (%)
geometric constraints imposed by the network
(topology & geometry) impose strong corrections
to the speed of propagation of migratory fronts
Campos et al., Theor. Pop. Biol., 2006; Bertuzzo et al., WRR, 2007
41. what is a node?
strong hydrologic controls
!
!
!
45. $ ρ'
∂ρ
∂2 ρ
= D 2 + rρ & 1− ) +σ ρ η
∂t
∂x
% K(
η
is
a
δ-‐correlated
gaussian
white
noise
Itô
stochastic
calculus
ML
estimates
for
r,K,
σ
$ ρ'
∂ρ
= rρ & 1− ) +σ ρ η
∂t
% K(
Transitional
probability
densities
are
computed
by
numerical
integration
of
the
related
Fokker-‐Planck
equation.
46. Demographic
stochasticity
$ ρ'
∂ρ
∂2 ρ
= D 2 + rρ & 1− ) +σ ρ η
∂t
∂x
% K(
η
is
a
δ-‐correlated
gaussian
white
noise
Itô
stochastic
calculus
ML
estimates
for
r,K,
σ
$ ρ'
∂ρ
= rρ & 1− ) +σ ρ η
∂t
% K(
Transitional
probability
densities
are
computed
by
numerical
integration
of
the
related
Fokker-‐Planck
equation.
48. 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
49. Zebra Mussel
Dreissena polymorpha
larval stages
transported along the
fluvial network
1988
1989
1990
1991
data: Nonindigenous Aquatic
species program USGS 1992
1993
1994
1995
50. Mari et al., in review, 2007
local age-growth model
(4 stages)
!
larval production
!
larval transport
(network)
Zebra Mussel
54. neutral metacommunity model
metacommunity model
every link is a community of
organisms & internal implicit
spatial dynamics
Explicit spatial dynamics
among different communities
the neutral assumption
all 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
55. 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
Pij = (1 − v)
K ij H j
N
∑K
ik
Hk
k =1
H j :habitat capacity link i
K ij :dispersal kernel
run up to steady state
66. patterns -- weak or strong impliations of neutrality?
comparison between geographic ranges of individual
species: a) data b) results from the neutral
metacommunity model (after matching procedure)
67. Bertuzzo et al., submitted, 2008
equiprobability map – ratio between the number of
common species and the number of species in
the central DTA
69. 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
!
76. Codeco, JID, 2001; Pascual et al, PLOS, 2002; Chao et al, PNAS, 2011
continuous SIR model
αI
µI
infected I
γI
persons
p
I
W
β
B
S
K+B
vibrios
vibrios/m3 B
recovered R
µB B
µR
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)
susceptibles S
µS
µH
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)
78. the class of SIB models
Capasso et al, 1979; Codeco, JID, 2001
79. person
!
SIR model for the temporal &
spatial evolution of water-transmitted
disease revisited → network
susceptibles
I(t)
infected
!
S(t)
a few assumptions
!
!
0
50
100
150
200
total population of humans is
t [days]
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
90. Zipf’s distribution of population &
secondary peaks of infection
1400
1200
I(t)
infected
1000
800
600
400
200
0
0
50
100
150
200
time [days]
250
t
300
350
102. effects of rates of loss of acquired immunity (1-5 years)
Weekly Cases [103]
40
30
20
10
0
Jan 11
Jul 11
Jan 12
Jul 12
Jan 13
Jul 13
Jan 14
Rinaldo et al., PNAS, in press
105. recorded cholera
cases in Haiti
(2010-2013)
(normalized)
normalized maximum
eigenvector
Gatto et al, PNAS, 2012; Gatto et al, Am Nat, 2014
106. river networks & biodiversity
!
tradeoff versus neutral models of
the ecology of riparian vegetation
Muneepeerakul et al., JTB, 2007 --> Mari et al., Ecol Lett., 2014
108. Muneepeerakul, Weitz, Levin, Rinaldo, Rodriguez-Iturbe, JTB, 2007
links are essentially patches within a landscape
cointaining sites that are occupied by individual plants
109. 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
110. power laws matter alot - hotspots & geomorphology
!
indeed a frontier of ecological research
Muneepeerakul et al., WRR, 2007
112. →
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
!
113. 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
!
114. 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