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

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., 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

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


Take-‐home message

• Fisher-‐Kolmogorov equation correctly predicts the mean features of dispersal !

• The observed variability is explained by demographic stochasticity

Link between scales


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

]


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)


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


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

RIVER NETWORKS AS ECOLOGICAL CORRIDORS FOR SPECIES POPULATIONS AND WATER-BORNE DISEASE

Education

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