UNIVERSITÀ DEGLI STUDI DI PADOVA
Sede Amministrativa: Università degli Studi di Padova
Dipartimento di Ingegneria Idraulica, Marittima, Ambientale e Geotecnica - IMAGE
SCUOLA DI DOTTORATO DI RICERCA IN SCIENZE DELL’INGEGNERIA CIVILE
ED AMBIENTALE
INDIRIZZO: AMBIENTE E TERRITORIO
XX° CICLO
RIVER NETWORKS AS ECOLOGICAL
CORRIDORS
Direttore della Scuola: Ch.mo Prof. Andrea Rinaldo
Supervisore: Ch.mo Prof. Andrea Rinaldo
Ch.mo Prof. Ignacio Rodriguez-Iturbe
Dottorando: Enrico Bertuzzo
31 GENNAIO 2008
Abstract
River networks and the transport processes that take place in them provide a
natural integrating framework for the study of hydrologic, biologic and ecologic
processes in river basins. The profound commonalities existing among all types
of river basins and their drainage networks, together with the key role that
these structures play in the above dynamics, encourage the search for general
behaviours. The aim of this work is to put the basis for a general framework for
the analysis of complex system associated with dendritic landscapes. In particular we investigate how the environmental matrix constituted by the ecological
corridors defined by the river network could affect patterns and dynamics of the
system itself. We first analyze invasion, an ecological process that describe the
growth and the spreading of a species in a new territory, finding that the speed
of colonization is strongly affected by the structure of the network and the bias
of the transport. These hydrological controls provide a null model for the comparison with more complex ecologic processes like the spreading of waterborne
diseases. We compare epidemiological data from the real world with the spacetime evolution of infected individuals predicted by a theoretical scheme based
on reactive transport of infective agents through a biased network portraying
actual river pathways. The scheme is remarkably capable of reproducing actual
outbreaks and shows that spatial distribution of different communities and how
they are interconnected trough the river network, could indeed affect epidemic
dynamics. The previous models are then generalized studying river biogeography.
We analyse how the dispersion and growth of several species that compete for the
same resources control river biodiversity. We propose a neutral metacommunity
model that incorporates network structure. The scheme, along with a proper description of the habitat capacity distribution, is able to simultaneously reproduce
several biodiversity patterns of the Missisiippi-Missouri freshwater fishes biota.
Overall the results represent a first step toward the understanding of general
hydrologic controls on complex ecologic systems.
Sommario
Le reti fluviali e i processi di trasporto che avvengono al loro interno, forniscono
un modello integrato per lo studio di processi idrologici, biologici ed ecologici
che si sviluppano nei bacini fluviali. Le profonde somiglianze tra tutte i tipi di
bacini e delle reti di drenaggio, assieme al ruolo che queste strutture giocano
nei processi citati, incoraggia la ricerca di comportamenti universali. Lo scopo
di questa ricerca e porre le basi per un modello generale per l’analisi di sistemi
complessi che si sviluppano su reti fluviali. In particolare si è studiato come
la matrice ambientale costituita dai corridoi ecologici definiti dalle reti fluviali
possa influenzare la dinamica stessa dei processi. Si è analizzato innanzitutto
la colonizzazione, il processo ecologico che descrive la crescita e la diffusione di
una specie in un nuovo territorio, mostrando che la velocità di colonizzazione è
fortemente influenzata dalla struttura della rete e dal bias del trasporto. I controlli idrologici trovati forniscono un modello nullo per il confronto con processi
ecologici più complessi come la diffusione di malattie che hanno come vettore
l’acqua. Si sono comparati dati epidemiologici con l’evoluzione spazio temporale
degli infetti predetta da un modello matematico basato sul trasporto reattivo
degli agenti infettivi attraverso la rete fluviale. Il modello riesce a riprodurre
in modo soddisfacente le epidemie registrate mostrando come la distribuzione
spaziale di diverse comunità e il modo in cui sono interconnesse attraverso la
rete possano influenzare la dinamica dell’epidemia. I modelli proposti sono stati
quindi generalizzati con lo studio della biogeografia fluviale. Si è studiato come
la dispersione e la crescita di molte specie che competono per le stesse risorse
determinano la biodiversità fluviale. E stato proposto un modello neutrale a
metacomunità che incorpora la struttura della rete. Il modello, assieme ad una
appropriata descrizione della capacità portante, è in grado di riprodurre contemporaneamente diverse distribuzioni di biodiversità dei pesci d’acqua dolce
che popolano il bacino del Mississippi-Missouri. Complessivamente i risultati
raggiunti rappresentano un primo passo verso la comprensione dei controlli idrologici in sistemi ecologici complessi
Contents
1 Introduction
9
2 Invasion Models
17
2.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Isotropic Invasion Models and Migration Fronts . . . . . . . . . .
21
2.4
Biased Random Walks on Oriented Graphs . . . . . . . . . . . . .
23
2.5
Reaction Random Walks on Oriented Graphs . . . . . . . . . . .
28
2.6
Computational Results . . . . . . . . . . . . . . . . . . . . . . . .
33
2.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.8
Appendix A: Fisher Equation . . . . . . . . . . . . . . . . . . . .
46
2.9
Appendix B: Biased Telegraph Model . . . . . . . . . . . . . . . .
48
3 Spreading of Diseases Models
59
3.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.3
Theoretical Approach . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.4
The Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.5
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . .
76
3.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4 Neutral Metacommunity Models on River Networks
4.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
85
85
6
CONTENTS
4.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.3
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.4
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.5
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . .
99
4.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
List of Figures
2.1
Examples of networks . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Front speed function of the growth rate . . . . . . . . . . . . . . .
26
2.3
Peano Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.4
Peano Basin: details for the analytical derivation . . . . . . . . .
32
2.5
Comparison between numerical and analytical front speed . . . .
34
2.6
Sketches of the numerical simulation of the colonization process .
36
2.7
Travelling front of the density wave along the backbone . . . . . .
37
2.8
Extraction of the River Tanaro network . . . . . . . . . . . . . . .
38
2.9
Numerical simulations on the River Tanaro network . . . . . . . .
39
2.10 Comparison between Fisher model, telegraph model and reaction
random-walk model on Peano network . . . . . . . . . . . . . . .
42
2.11 First colonization time distributions . . . . . . . . . . . . . . . . .
44
2.12 Front speed function of the growth rate . . . . . . . . . . . . . . .
54
3.1
Map of the KwaZulu-Natal province of South Africa . . . . . . . .
68
3.2
Spatial representation of the health districts of the KwaZulu-Natal
province . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.3
Temporal evolution of the new weekly cholera cases . . . . . . . .
70
3.4
Hydrographic map of KwaZulu-Natal province . . . . . . . . . . .
71
3.5
Maps of cholera cases and population size
. . . . . . . . . . . . .
74
3.6
Average cholera incidence . . . . . . . . . . . . . . . . . . . . . . .
75
3.7
Comparison between data and simulated temporal evolution of the
epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
77
8
LIST OF FIGURES
3.8
Comparison between data and simulated spatial evolution of the
epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.9
Comparison between data and simulated cholera incidence . . . .
80
4.1
Maps of local species richness and average annual runoff production 90
4.2
Local species richness as a function of the topological distance
from the outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.3
Frequency distribution of local species richness . . . . . . . . . . .
92
4.4
Rank-range curve . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.5
Jaccard’s similarity index (JSI) as a function of topological distance between pairs of DTAs . . . . . . . . . . . . . . . . . . . . .
4.6
94
relationship between average annual runoff production and local
species richness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.7
comparison between local species richness of adjacent pairs of DTA 101
4.8
Effects of network structure and climatic gradient on the spatial
profile of local species richness . . . . . . . . . . . . . . . . . . . . 103
4.9
The dispersal kernels and their associated decaying patterns of
Jaccard’s similarity index . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter 1
Introduction
River networks define ecological corridors along which many hydrologic, biologic
and ecologic processes take place. The dynamics of the complex systems developed in a river basin can be affected by the structure of the river network
and by how it interconnects different sites of the drained area. Relevant are the
cases in which the process itself interplays with the development of the drainage
network. The fractal properties of the river network are, in fact, recognized to
be the result of an optimization process that minimizing the energy expenditure
in transporting the rainfall water to the outlet [Rodriguez-Iturbe and Rinaldo,
1997]. Moreover the geomorphological features of the channel network determines the hydrologic response of a basin. At a scale of an organized network, in
fact, the variance of the residence time distribution due to the morphology of the
network structure (geomorphological dispersion) is grater than the one related
to the heterogeneities of the convection field (hydrodynamic dispersion) [Rinaldo
et al. 1991]. Closely related to the hydrologic response at catchment scale are
the transport processes of passive and reactive solutes advected and dispersed
by the hydrological carrier. The residence time of these solutes depends on the
residence time of the water flow and on the particular reaction involved (e.g. decay, sorption-desorption, ion exchange). The role played by the network on the
transport of solutes at basin scale has been investigated both from a theoretical
point of view [Botter et al. 2005; Rinaldo et al. 2005; Rinaldo et al. 2006a] and
9
10
CHAPTER 1. INTRODUCTION
with a significant case study on the leaching of nitrates on the Venice lagoon
[Rinaldo et al. 2006a].
Models of hydrologic response and solute transport on large basin scale are
widespread in the scientific community however models of dispersion processes
that involve biological and ecological interactions on large temporal and spatial
scale are still elusive. Attempts have been recently made studying riparian vegetation in large river systems [Muneepeerakul et al., 2007, 2008b]. The aim of
this work is to put the basis for a general framework for the study of a wide
spectrum of transport phenomena (e.g. invasion process, dispersion of propagules or infective agents, dispersion and distribution of species population) and to
investigate the hydrologic controls on these processes. This is tackled with the
study of basic ecological processes on river networks from a theoretical point of
view and with the implementation of mathematical models for the analysis of
real cases. In particular the following chapters report the results obtained in the
study of the colonization process (i.e. the growth movement of a single species)
in river networks, the spreading of waterborne diseases and the riverine biodiversity. All these processes develop in an environmental matrix constituted by
the ecological corridors defined by river networks. In particular we investigate
how this geomorphological substrate could affect patterns and dynamics of the
process itself.
All the models presented have a common framework. The River network is
modelled by an oriented graph constituted by nodes and edges. An oriented
graph is a directed graph (i.e., a graph where the edges have a direction) having
no symmetric pair of directed edges. Edge directions are chosen accordingly to
the flow direction. Models are obtained by coupling two sub-models: i) a local reaction model at the nodes scale and ii) a transport model between nodes
through the edges. The particular choice of the sub-models, along with the temporal and spatial scales involved, is suited to the process at hand time by time.
Models are applied to different types of networks like real rivers and Optimal
Channel Networks (OCNs) [Rodriguez-Iturbe and Rinaldo, 1997]. OCNs are artificial networks obtained through a specific selection process from which one
11
obtains a rich structure of optimal scaling forms that are known to closely conform to the scaling of real networks. To derive exact results, instead, we resort
to Peano’s network which is a deterministic fractal whose main topological and
scaling features have been determined analytically.
We start our research with the analysis of invasion, an ecological process
that describe the growth and the spreading of a single species in a new territory where it was not present before. The simplest and best known colonization
model is the Fisher equation. It is obtained by adding a logistic reaction term,
that describe how the organisms reproduce and grow, to a simple diffusive equation that models the movement and spreading of the organisms. Moving from
a recent quantitative model of the US colonization in the 19th [Campos et al.,
2006] century that relies on analytical and numerical results of reactive-diffusive
transport on fractal river networks, we considers its generalization to include an
embedded flow direction which biases transport [Bertuzzo et al., 2007]. Organisms, in fact, can either move by their own energy (active dispersal) or be moved
by water (passive dispersal). Most likely, movements along the flow direction
would be favoured. Referring to the common framework described above, in this
case the local model is a logistic growth, whereas the transport model is a discrete time biased random walk on oriented graph. The geometrical constraints
imposed by the networks imply strong corrections on the speed of the invasion
process that can be enhanced or smoothed by the bias. Even if the results are
tied to the logistic growth model used, we speculate that the hydrologic controls found are general and may provided a null model for the comparison with
more complex systems. In order to analyse real cases, we need to substitute
the simplified logistic equation with a model suitable to describe the population
dynamics of the species studied. This type of model can be applied to examine
the invasion of river systems by non-indigenous species. The relevance of this
problem is highlighted by the case of the Dreissena polymorpha, a mussel that
has colonized the whole Mississippi basin in less than five years, leading to serious
ecologic and economic problems. Moreover, the environmental matrix defined by
river networks can play a role in the spreading of infective agents that cause wa-
12
CHAPTER 1. INTRODUCTION
ter borne diseases. We analyse the case of cholera, an intestinal disease caused
by the bacterium Vibrio cholerae. The spreading of the diseases is controlled
by two mechanisms: i) a human-environmental contamination through infected
stools and ii) an environmental-human transmission caused by the ingestion of
contaminate water or food. V. cholerae can survive in the aquatic environment
in associations with chitinaceous zooplankton like copepods, shellfish and fish.
Therefore it can spread, along with the disease, through waterways. We propose
a model that explicitly accounts for the role of the river network in transporting
and redistributing V. cholerae between several human communities. In analogy
with the previous case, the model is obtained by coupling a complete local epidemic model of the SIR (Susceptibles, Infected and Recovered) type at nodes and
a biased random walk scheme to implement the transport of vibrios. The model
will be tested against a very well documented epidemic in the KwaZulu-Natal
province of South Africa. We show that the spatial distribution of different communities, along with the distribution of their population size, and how they are
interconnected by the river network, could indeed affect the spatial and temporal
patters of the epidemic [Bertuzzo et al., 2008].
The last part of this research work deals with river biogeography: the study of
the species’ spatial distribution of a riverine biota. We study how the dispersion
and growth of several species that compete for the same resources control the
river biodiversity. This is the natural generalisation of the models previously introduced that consider only one species (e.g. Dreissena polymorpha,V. cholerae).
We adopt a neutral metacommunity model. Every node of the network is a local
community constituted by a certain number of individuals that depends on the
habitat capacity. The whole system is constituted by many local communities
interconnected by the network. The model captures basic ecological processes
at the individual level: birth, death, dispersal, colonization, and diversification;
the transport through the network is modelled with a dipersal kernel. The competition dynamic is assumed to be neutral (i.e. all the species are considered
equivalent) [Hubbel, 2001]. We implement the model in the Mississipi-Missouri
river system and compare the resulting biodiversity patterns with those derived
13
from a large database of freshwater fishes diversity. We find that a neutral metacommunity model with a proper description of the habitat capacity distribution
and river system structure is able to simultaneously reproduce several biodiversity patterns of α and β diversity and geographic range [Muneepeerakul et al.,
2008a]. Moreover we find a hydrological signature in the spatial distribution of
local species richness that increases going downstream.
The thesis is organised as follows. Each of the three main chapters are self
explanatory and can be read separately. They summarize the findings in the
study of the three main topics described above: invasion processes (chapter 2),
spreading of diseases (chapter 3) and fluvial biogeography (chapter 4). Each
chapter, included this introduction (chapter 1), ends with the list of references
cited.
14
CHAPTER 1. INTRODUCTION
Bibliography
[1] Bertuzzo, E., S. Azaele, A. Maritan, M. Gatto, I. Rodriguez-Iturbe, and A.
Rinaldo (2008), On the space-time evolution of a cholera epidemic, Water
Resour. Res., 44, W01424.
[2] Bertuzzo, E., A. Maritan, M.Gatto, I. Rodriguez-Iturbe and A. Rinaldo
(2007), River Network and ecological corridors: reactive transport on fractal, migration fronts, hydrocory, Water Resources Research, 43, W04419.
[3] Botter, G., E. Bertuzzo, A. Bellin, and A. Rinaldo (2005), On the Lagrangian formulations of reactive solute transport in the hydrologic response, Water Resources Research, 41, W04008.
[4] Campos, D., J. Fort, and V. Méndez (2006), Transport on fractal river
network: Application to migration fronts, Theor. Popul. Biol., 69, 88-93.
[5] Hubbell, S. P. (2001), The Unified Neutral Theory of Biodiversity and
Biogeography, Princeton University Press.
[6] Muneepeerakul, R., J. Weitz, S. Levin, A. Rinaldo and I. Rodriguez-Iturbe
(2007), A neutral metapopulation model of riparian biobiversity, Journal
of Theoretical Biology, 245, 351-363.
[7] Muneepeerakul, R., E. Bertuzzo, H. Lynch, W. B. Fagan, A. Rinaldo, and
I. Rodriguez-Iturbe (2008a), Neutral Metacommunity Models Predict Fish
Diversity Patterns in Mississippi-Missouri Basin, in press Nature.
15
16
BIBLIOGRAPHY
[8] Muneepeerakul, R., E. Bertuzzo, A. Rinaldo, and I. Rodriguez-Iturbe
(2008b), Patterns of Vegetation Biodiversity: The Roles of Dispersal Directionality and River Network Structure, under review Journal of Theoretical
Biology.
[9] Rinaldo, A., E. Bertuzzo, and G. Botter (2005), Nonpoint source transport models from empiricism to coherent theoretical frameworks, Ecological Modelling, 184, 19-35.
[10] Rinaldo, A.,G. Botter, E. Bertuzzo, A. Uccelli, T. Settin, and M. Marani
(2005a), Transport at basin scale: 1.Theoretical framework, Hydrology and
Earth System Sciences, 2, 1613-1640.
[11] Rinaldo, A.,G. Botter, E. Bertuzzo, A. Uccelli, T. Settin, and M. Marani
(2005b), Transport at basin scale: 2. Applications, Hydrology and Earth
System Sciences, 2, 1613-1640.
[12] Rodriguez-Iturbe, I., and A. Rinaldo (1997), Fractal River Basins: Chance
and Self-Organization, Cambridge Univ. Press, New York.
[13] Rinaldo, A., A. Marani, and R. Rigon (1991), Geomorphological Dispersion, Water Resources Research, 27 (4), 513-525.
Chapter 2
River networks and ecological
corridors: reactive transport on
fractals, migration fronts,
hydrochory 1
2.1
Summary
Moving from a recent quantitative model of the US colonization in the 19th century that relies on analytical and numerical results of reactive-diffusive transport
on fractal river networks, this chapter considers its generalization to include an
embedded flow direction which biases transport. We explore the properties of
biased reaction-dispersal models, in which the reaction rates are described by a
logistic equation. The relevance of the work is related to the prediction of the
role of hydrologic controls on invasion processes (of species, populations, propagules or infective agents, depending on the specifics of reaction and transport)
1
The contents of this chapter has been published in: Bertuzzo, E., A. Maritan, M.Gatto,
I. Rodriguez-Iturbe and A. Rinaldo (2007), River Network and ecological corridors: reactive
transport on fractal, migration fronts, hydrocory, Water Resources Research, 43, W04419.
17
18
CHAPTER 2. INVASION MODELS
occurring in river basins. Exact solutions are obtained along with general numerical solutions, which are applied to fractal constructs like Peano basins and real
rivers. We also explore similarities and departures from different one-dimensional
invasion models where a bias is added to both the diffusion and the telegraph
equations, considering their respective ecological insight. We find that the geometrical constraints imposed by the fractal networks imply strong corrections
on the speed of travelling fronts that can be enhanced or smoothed by the bias.
Applications to real river networks show that the chief morphological parameters affecting the front speed are those characterizing the node-to-node distances
measured along the network structure. The spatial density and number of reactive sites thus prove to be a vital hydrologic control on invasions. We argue
that our solutions, currently tied to the validity of the logistic growth, might
be relevant to the general study of species’ spreading along ecological corridors
defined by the river network structure.
2.2
Introduction
The role of the structure of river networks in modelling human-range expansions, i.e., predicting how populations migrate when settling into new territories,
has been recently recognized through interesting quantitative models of diffusion
along fractal networks coupled with logistic reaction at their nodes [Campos et
al., 2006]. An essential ingredient therein is the fact that settlers did not occupy
all the territory (isotropically, in the language of homogeneous continuous models), but rather followed rivers and lakes and settled near them to exploit water
resources. It was thus quite interestingly argued in a quantitative manner that
landscape heterogeneities must have played an essential role in the process of migration [Ammermann and Cavalli-Sforza, 1984; Fort and Mendez, 1999; Campos
et al., 2006].
One interesting byproduct of the analysis of migration fronts is the important role attributed to the structure of the network acting as the substrate for
wave propagation. This indeed calls for specific structural models to be invoked.
2.2. INTRODUCTION
19
One must observe that mathematical models of natural forms as fractals involve
non-trivial assumptions [e.g. Rodriguez-Iturbe and Rinaldo, 1997], in particular
concerning the (relative) independence of results from the seeding point chosen
for spreading material and species along the network where reaction and diffusion
occur. This is seen as a corollary of the type of self-similarity shown by trees,
although entailing somewhat complex issues in cases where loops are observed
[Rinaldo et al., 2006]. The type of self-similarity observed for trees needs also
proper finite-size corrections because upper and lower cutoffs in the aggregation
structure reflect respectively the drainage density defining where channels begin,
and the loss of statistical significance of areas close to the overall basin size [e.g.
Maritan et al., 1996; Rodriguez-Iturbe and Rinaldo, 1997]. For general numerical
calculations we shall adopt here: i) the topology and geometry of real rivers (e.g.
Figure 2.1a); and ii) those of Optimal Channel networks (OCNs)(Figure 2.1b)
[Rodriguez-Iturbe et al., 1992 a,b; Rinaldo et al., 1992, 1993; Rodriguez-Iturbe
and Rinaldo, 1997]. They hold fractal characteristics that are obtained through
a specific selection process from which one obtains a rich structure of optimal
scaling forms that are known to closely conform to the scaling of real networks.
To derive exact results, instead, we shall resort (as is usually the case in this context [Marani et al., 1991; Colaiori et al., 1997; Campos et al., 2006]) to Peano’s
network (Figure 2.1c), which is a deterministic fractal [Mandelbrot, 1983] whose
main topological and scaling features have been determined analytically [Marani
et al., 1991; Colaiori et al., 1997].
Our starting point is the analysis of Campos et al. [2006] concerning a reaction random-walk (RRW) process through a Peano construct and OCNs. It is
based on the following model. A particle, at an arbitrary node of the network,
jumps, after a waiting time τ , to one of its z nearest neighboring nodes with
probability 1/z. During the waiting-time τ , the particles ’react’ following the
logistic equation. The determination of the wave front speed that this process
develops along a network path [e.g. Mendez et al., 2004 a,b; see also, for the case
of discrete comblike structures Campos et Mendez, 2005] is the starting point for
our extensions. Figure 2.2 illustrates the main result of Campos et al. [2006].
20
CHAPTER 2. INVASION MODELS
It shows that the isotropic diffusion-reaction front (Fisher’s model) propagates
much faster than the wave forced to choose a treelike pathway. This proves that
geometrical constraints imposed by a fractal network imply strong corrections on
the speed of the fronts. It should be noted that it is not surprising that Peano
and OC networks lead to similar results, because the speed of the front depends
on topological features that are indeed quite similar for all the (rather different
otherwise) networks shown in Figure 2.1 [see e.g. Rodriguez-Iturbe and Rinaldo,
1997; Rinaldo et al., 1999, 2006]. In fact,the wave speed is affected mostly by the
gross structure encountered by the front while propagating along the network,
chiefly the bifurcations. Hence topology, rather than the fine structure of the
sub-paths, dominates the process.
The model proposed by Campos et al. [2006] assumes simple diffusive transport to purportedly describe migration fluxes. This seems indeed reasonable in
the case of human population migrations – the need for water resources should
drive settlers regardless of the direction of the flow. We wonder whether adding a
bias to transport properties would basically alter this interesting picture. This is
done on purpose: in fact, any other ecological agent (be it a an aquatic organism
or an infective agent of water-borne disease) would likely be affected by the flow
direction to propagate within the network. Organisms can either move by their
own energy (active dispersal) or be moved by water (passive dispersal). Most
likely, movements along the flow direction would be favored, although movements
against flow direction are completely admissible because of various ecological or
physical mechanisms [Bilton et al., 2001; Muneepeerakul et al., 2007]. All this is
of great interest for the problem of hydrochory, i.e., the transport of species along
the ecological corridors that are shaped by the river network [see e.g. Power and
Dietrich, 2002].
This chapter is organized as follows. An introductory section (Section 2.3)
recalls the properties of the biased Fisher model and presents a generalization
of the less-known reaction-telegraph model [Holmes, 1993] which is more apt
to describe biased transport. Section 2.4 illustrates technical aspects of biased
random walks on oriented graphs, which shall be of use in Section 2.5 where the
2.3. ISOTROPIC INVASION MODELS AND MIGRATION FRONTS
21
role of reaction described by a logistic equation is dealt with. Our main results
are collected in a specific chapter (Section 2.6). A set of concluding remarks
closes the chapter.
2.3
Isotropic Invasion Models and Migration Fronts
Travelling waves are a common mathematical byproduct of reaction-diffusion
(RD) models of transport where density-dependent reaction terms are employed.
Fisher’s model [Murray, 1993] is the basic one-dimensional RD model; it is obtained by adding a logistic reaction term to the classical one- dimensional diffusion equation:
∂ρ(x, t)
∂ 2 ρ(x, t)
= aρ(x, t)(1 − ρ(x, t)) + D
∂t
∂x2
(2.1)
where ρ is the reagent density ([L]−1 ) as a function of space (x) and time (t),
a the logistic growth rate ([T ]−1 ) and D the diffusion coefficient ([L]2 [T ]−1 ). If
the initial condition ρ(x, 0) has compact support, the solution ρ(x, t) evolves to
√
a travelling wave solution with minimum speed v = vmin = 2 aD [Kolmogorov
et al., 1937]. This occurs because the above equation admits two spatially homogeneous steady states, one stable (ρ = 1) and one unstable (ρ = 0), and a phase
plane analysis of the boundary value problem via an equivalent ODE problem
shows that a heteroclinic connection between the two states exists only when
v ≥ vmin [e.g. Murray, 1993]. Details about the derivation of the front speed are
given in Appendix A (Section 2.8).
Biased transport could be modelled by simply adding at the left-hand side of
equation (2.1) a term taking into account the advective flux (i.e., u∂ρ(x, t)/∂x,
where u is the advection velocity). By the change of variables z = x−ut, one can
√
re-obtain equation (2.1) and therefore the front speed as v = u+2 aD. Equation
(2.1) could also be obtained starting from a discrete time-space reaction random
walk (i.e., the particle makes steps of length δx at every time step δt) and taking
the limit δx, δt → 0 while keeping δx2 /δt = constant = 2D. Therefore, the
22
CHAPTER 2. INVASION MODELS
actual velocity of the particles during a jump (δx/δt) tends to infinity.
To avoid this assumption, it is possible to resort to a generalization of the
one-dimensional reaction-telegraph model [Holmes, 1993] in which we include
biased transport. Consider a large number of particles moving, reproducing
and dying on a line. Particles make steps of length δx and duration δt moving
at finite velocity γ = δx/δt. The walk is random, but correlated: a particle
continues in its previous direction with probability p and reverses its direction
with probability q. The process is supposed to be Poisson, so that for small δt
we have p = 1 − λδt and q = λδt where λ is the rate of reversal. We assume
that there is indeed a preferential direction, say from left to right. Therefore, the
rate of reversal λR of particles arriving from the right is bigger than the rate of
reversal λL of particles arriving from the left. We also assume that new particles
produced at a certain location (resulting from the difference between natality
and mortality) move with equal probability (1/2) to the right or to the left. The
equations of the model are then obtained by taking the limit δx, δt → 0 while
keeping δx/δt = constant = γ. Technical details for the derivation of the speed
of the front that this process yields are described in Appendix B (Section 2.9).
The biased reaction-telegraph model is more realistic with respect to Fisher’s
because it assumes a correlated walk rather than a completely random one. Moreover, the model assumes that organisms move with constant finite velocity, rather
than infinite as in Fisher’s model. The key consequence of this basic conceptual
difference is that the front speed admits an upper bound, as it cannot exceed the
particles’ velocity γ. This model is a much better approximation to the discrete
time-space RRW which we will describe in the next sections. In fact, in Appendix
B we have compared the reaction-telegraph model with a Reactive Random Walk
in a one-dimensional lattice, showing that for this simple network the two models
lead to similar front speeds in a range of reaction rates most likely to cover the
range of practical interest [Holmes, 1993].
2.4. BIASED RANDOM WALKS ON ORIENTED GRAPHS
2.4
23
Biased Random Walks on Oriented Graphs
In this section we shall first illustrate, and adapt to the problem at hand, the
discrete time-space biased random-walk process through an oriented graph constituted by edges of equal length. This analysis will be useful in the next section
where we add the reaction term and study the properties of a RRW on oriented
graphs.
An oriented graph is a directed graph (i.e., a graph where the edges have a
direction) having no symmetric pair of directed edges. At every time step, τ , a
walker can move with some probability from a node to one of the adjacent nodes,
which are all the nodes that are connected to it through an inward or outward
edge. Consider first a particular case of the graph describe above, in which
every node has only one inward and one outward edge (i.e., a one-dimensional
lattice). We define as Pout (Pin ) the probability that a particle moves from a node
to another along an outward (inward) edge. We analyze the case in which all
particles move at every time step, hence Pout + Pin = 1. In greater generality, for
a random-walk process on a generic oriented graph in which every node can have
an arbitrary number of inward and outward edges, we assume that a particle can
move (following either an outward or inward edge) with a probability proportional
to Pout and Pin respectively. In this case, the probability Pij for a particle to
jump from node i to one of its neighbors j could be expressed as:





Pout
if i → j
dout (i)Pout + din (i)Pin
(2.2)
Pij =

Pin



if i ← j
dout (i)Pout + din (i)Pin
where dout (i) and din (i) are, respectively, the outdegree and indegree of node i
(i.e., the number of outward,or respectively inward, edges of node i). Obviously
Pd(i)
j=1 Pij = 1, where d(i) = dout (i) + din (i) is the total degree of node i. We
define b = Pout − Pin = 2Pout − 1 the bias of the transport.
We particularize the results of equation (2.2) to the case of the Peano network,
because its deterministic spanning tree structure allows the derivation of exact
24
CHAPTER 2. INVASION MODELS
results. Figure 2.3 shows the Peano basin at the third level of its construction
process. Owing to the purported topological similarity between the Peano basin
and a real river basin [Marani et al., 1991; Colaiori et al., 1997; Rodriguez-Iturbe
and Rinaldo, 1997], we choose to assign to each edge of the graph a direction
from the leaves to the root (namely the outlet: point B in Figure 2.3). In this
way, we can also define cumulative flow through aggregated area, say the number
of upstream links connected to the current node. The nodes of this graph could
be classified, on the basis of their total degree, in first- and fourth-degree nodes.
Every fourth-degree node has three inward edges and one outward edge. In the
following we define:
Pout
Pin
, P− =
(2.3)
Pout + 3Pin
Pout + 3Pin
the probabilities that a particle, starting from a fourth degree node, jumps to
P+ =
its downstream neighbor, and the probability that it jumps to one of its three
upstream nodes respectively. The expressions for P+ and P− derive straightforwardly from equation (2.2). Also, a particle starting from a first degree node
jumps to its downstream node with unit probability.
2.4. BIASED RANDOM WALKS ON ORIENTED GRAPHS
25
Figure 2.1: Examples of networks on which transport is considered herein: a) A
real river network, the Dry Tug Fork (CA), suitably extracted from digital terrain
maps; b) a single-outlet optimal channel network (OCN); c) Peano’s network
26
CHAPTER 2. INVASION MODELS
1.0
front speed
0.8
Fisher’s model
0.6
Peano (analytical)
0.4
O.C.N.
0.2
Peano (numerical)
0
0
0.1
0.2
0.3
0.4
0.5
growth rate
Figure 2.2: Front speed as a function of the growth rate of the logistic equation
[redrawn from Campos et al., 2006]. Solid line is the exact solution of the continuous isotropic Fisher model. The dashed line and the dots represent exact and
numerical values for propagation along the backbone of Peano and OC networks.
2.4. BIASED RANDOM WALKS ON ORIENTED GRAPHS
A
27
B
1
2
1
3
1
2
1
Figure 2.3: a) Peano fractal basin at the third level of its construction process.
The arrows show the edge direction. The line indicated as AB is the backbone of
the network. The numbers indicate the order of the secondary branches emerging
from the backbone.
28
CHAPTER 2. INVASION MODELS
2.5
Reaction Random Walks on Oriented Graphs
In this section we focus on the analysis of a biased reaction random-walk (RRW)
process through a Peano network. At every time step τ a particle moves from a
node to one of its neighbors following the rules (2.3). During the waiting-time
τ , the particle density at every node grows following the logistic equation. If
we observe the process only at the nodes of the backbone (line AB in Fig 2.3),
the secondary branches emerging from one of these nodes yield a waiting-time
distributions of jumping to adjacent nodes of the backbone that depends on the
structure of the branches. The process along the backbone can be handled like
a one-dimensional reactive continuous time random walk. The Hamilton-Jacobi
formalism [Mendez et al., 2004a,b] allows the analytical computation of the speed
of the travelling wave generated by a one-dimensional continuous time random
walk (CTRW), with a density-dependent term of reaction. We first present here
the general case in which the distribution of jump lengths, Φ(x), and the waitingtime distribution, ϕ(t), are continuous and independent; then, in the second part
of this section, we particularize to the case at hand using distributions suitable
to describe discrete space-time random walks on networks.
The evolution of the probability density field ρ(x, t) of being at position x
at time t (assuming that ρ(x, t) = 0 ∀t < 0) is described by the relationship
[Campos et al., 2006]:
Z
t
ρ(x, t) =
Z
0
Z
0
0
dt ϕ(t )
0
0
0
t
0
dx Φ(x )ρ(x − x , t − t ) +
R
dt0 φ(t0 )f (ρ(x, t − t0 )), (2.4)
0
where φ(t) is the probability distribution of waiting (at least) a time t between
two consecutive jumps, i.e.
Z
∞
φ(t) =
dt0 ϕ(t0 ).
(2.5)
t
The first term at the right hand side of equation (2.4) accounts for all dispersal
processes, while the second term takes into account the reactive character of
the process: the density of particles at position x grows according to the rate
2.5. REACTION RANDOM WALKS ON ORIENTED GRAPHS
29
of increase f (ρ(x, t)). In the following we assume, as in the original context, a
logistic reaction term of the type:
f (ρ) = aρ(1 − ρ) ,
(2.6)
where a is the intrinsic population growth rate, assumed constant. For details
about how equation (2.4) could be derived directly from master equations see
e.g. Mendez et al. [2004a].
The process described by equation (2.4) yields a travelling wave which connects the unstable state (ρ = 0) to the stable state (ρ = 1) and moves with
constant celerity. If the initial condition ρ(x, 0) has compact support, for example:
(
ρ(x, 0) =
1
x ≤ x0 ,
0
x > x0 .
(2.7)
the front selects its minimal propagation speed v. Following Mendez et al. [2004a]
and Campos et al. [2006], this speed could be derived as:
v = min
s
s
,
p(s)
(2.8)
where p(s) is the solution of the Hamilton-Jacobi equation
1
a 1
= Φ̄(p) + (
− 1).
ϕ̂(s)
s ϕ̂(s)
(2.9)
In equation (2.9) the function ϕ̂(s) is the Laplace transform of the waiting-time
distribution ϕ(t), while the function Φ̄(p) is defined by the transformation:
Z
+∞
Φ̄(p) =
epx Φ(x)dx ;
(2.10)
−∞
Thus, the knowledge of the jump distribution Φ(x) and the waiting-time
distribution ϕ(t) allows the direct derivation of the speed v of the travelling
wave.
30
CHAPTER 2. INVASION MODELS
We apply this method (Equations (2.8) and (2.9)) to the study of a biased
RRW through a Peano drainage network. To derive the speed of the front via
equation (2.8) we first derive the expression of the waiting-time distributions
due to the emerging branches of different order N . The order N is defined in
this way: the cardinality of the subtree rooted in the emerging branch is 4N −1 .
A particle at a backbone node from which two 1st-order branches emerge, can
reach an adjacent backbone node after 2i + 1 jumps, with i = 0, 1, ..., ∞, namely
after waiting a time (2i + 1)τ , where i is the number of movements away from
the backbone and back. The probability that a particle waits for a time t along
one of infinite possible paths is a convolution of (2i + 1) distributions, i.e.,
(i)
ϕ1 (t) = (P+ + P− ) ϕ0 ∗ (2P− ϕ0 ∗ ϕ0 ) ∗ (2P− ϕ0 ∗ ϕ0 ) ∗ . . .
(2.11)
with i convolution terms of the type (2P− ϕ0 ∗ ϕ0 ). The function ϕ0 (t) = δ(t −
τ ) is the waiting-time distribution for the original random walk through the
entire network. Note that 2P− is the probability of stepping into one of the two
emerging branches, whereas (P+ + P− ) is the probability of the last jump being
to the left or to the right. Finally, the waiting-time distribution for all possible
paths is given by the sum of all the waiting-time distributions of each single
path weighted by the path probabilities. For a first-order branch the Laplace
transform of the waiting-time distribution is:
ϕˆ1 (s) =
∞
X
i=0
ϕˆ1 (i) (s) = (P+ + P− ) ϕˆ0 (2P− ϕˆ0 2 )i =
(P+ + P− )ϕˆ0
,
1 − 2P− ϕˆ0 2
(2.12)
where ϕˆ0 (s) = exp(−τ s) is the Laplace transform of ϕ0 (t).
Following rules already discussed in other contexts [Van der Broeck, 1989;
Campos et al., 2006], the waiting-time distribution for the backbone nodes from
which a pair of branches of higher order emerges, can be obtained exactly. We
illustrate here the case of second order branches. We first derive the waiting time
distribution ϕI2 (t) of jumping from the node adjacent to the backbone (point C
in Figure 2.4) to the backbone (point D) induced by the three first-order edges
2.5. REACTION RANDOM WALKS ON ORIENTED GRAPHS
31
connected to it. Following the procedure and the rules illustrated before, the
Laplace transform of this distribution becomes:
I
ϕˆ2 (s) = P+ ϕˆ0
∞
X
(3P− ϕˆ0 2 )i =
i=0
P+ ϕˆ0
.
1 − 3P− ϕˆ0 2
(2.13)
Using the distribution in equation (2.13), it is possible to derive the Laplace
transform of the waiting-time distribution induced by a pair of second-order
branches as illustrated before:
ϕˆ2 (s)
=
(P+ + P− )ϕˆ0
∞
X
(2P− ϕˆ0 ϕˆ2 I )i
i=0
=
(P+ + P− ) ϕˆ0
1 − 3P− ϕˆ0 2
.
1 − P− ϕˆ0 2 (2P+ + 3)
(2.14)
With this method we can calculate the Laplace transform of the waiting-time
distributions up to fifth order branches. The complexity of the expression ϕˆN (s)
for N > 5 (being N the branch order) prevents us from using this method
any further. Note that the Hamilton-Jacobi method assumes the waiting time
distribution ϕ(t) to be space invariant. Following Campos et al. [2005,2006],
we approximately assume that all the branches encroaching the backbone have
the same waiting-time distribution given by ϕ(t) = ϕ5 (t). The validity of this
approximation will be tested numerically.
The length distributions for the jumps of a particle moving along the backbone of the network is given by:
Φ(x) = Pout δ(x − ∆x) + Pin δ(x + ∆x) ,
(2.15)
where x increases toward the graph root and ∆x is the constant length of the
graph edges. Note that equation (2.15) allows us to apply the CTRW framework,
which is usually employed to study continuous systems, to a discrete lattice
process. Applying the transformation (equation (2.10)) to equation (2.15), one
has:
32
CHAPTER 2. INVASION MODELS
C
ϕ0
ϕ2Ι
D
Figure 2.4: A pair of symmetric second order branches, typical of the bifurcation structure of the Peano construct. The function ϕ2 I is the waiting-time
distribution for jumping from node C to node D.
Φ̄(p) = Pout ep ∆x + Pin e−p ∆x ,
(2.16)
Substituting the expression (2.16) in equation (2.9), and solving it for the variable
p, we can particularize the expression (2.8) to the case of the oriented Peano
graph:
v = min
s
ln[ 2P1out (c(s)
s
;
+ (c(s)2 − 4Pout Pin )0.5 )]
(2.17)
where the function c(s) has the following expression:
1
a
c(s) =
−
ϕ̂(s) s
µ
1
−1
ϕ̂(s)
¶
.
The minimum in equation (2.17) is then computed numerically.
(2.18)
2.6. COMPUTATIONAL RESULTS
33
Results for the Peano network are shown by solid lines in Figure 2.5, where
the dimensionless speed of the front v τ /∆x is reported as a function of the
dimensionless growth rate a τ , for different values of the bias b. Note that for b =
0 (Pout = Pin = 1/2) we recover the case of the unbiased process already obtained
by Campos et al. [2006]. For b > 0 (i.e., the walkers move preferably downstream)
the speed of the front increases. However, one should remark the difference
with the simple Fisher model with advection-diffusion-reaction: the speed of the
front is not simply given by the sum of the advection velocity in the backbone
u = (b∆x/ τ ) plus the speed of the front of the unbiased process (i.e., with b = 0).
In fact, there are two important new phenomena accounted for by our model:
(1) the front speed cannot physically exceed the particle velocity ∆x/τ ; (2) the
bias affects the transport not only on the backbone, but also in the secondary
branches, thus affecting the waiting-time distribution ϕ(t). The results shown in
Figure 2.5 are obtained with ϕ(t) = ϕ5 (t) by taking into account branches only
up to the fifth order. However, as already noticed in Campos et al. [2006], the
speed of the front converges rapidly as the order of the branches increases, and
then it is not particulary sensitive to the details of the self-similarity at all scales
(N → ∞) typical of fractal structures like the Peano basin. Note, from Figure
2.5, that the trivial result v = ∆x/τ for b = 1 is recovered as the upper bound
for the front speed.
2.6
Computational Results
In this section we illustrate simulations of the biased RRW process on the
unabridged structure of the Peano basin and on real river networks. We start
every simulation from an initial condition with ρ = 1 for the node A and its
neighbor (see Figure 2.3) and ρ = 0 for all the other nodes. At every time step τ
we first update the density ρ of all nodes through the master equation that obeys
the rules of equations (2.2) and (2.3), and then we let the node density grow for
a time τ following the solution of the local logistic equation dρ/dt = aρ(1 − ρ):
34
CHAPTER 2. INVASION MODELS
1.0
b=1
0.8
0.8
v τ /∆ x
0.6
0.4
0.6
0.2
0.4
0
0.2
0
0
0.2
0.1
0.3
aτ
Figure 2.5: Dimensionless wave front speed (vτ /∆x) obtained by the HamiltonJacobi method (lines) and by the analogous numerical simulations (circles) as a
function of the dimensionless growth rate a τ . Different lines from the bottom to
the top refer to values of the bias b from 0 to 1 with step 0.2 (shown as insets).
ρ(t + τ ) =
ρ(t)
.
ρ(t) + (1 − ρ(t)) e−aτ
(2.19)
We observe the density only at the backbone nodes and start measuring the
front speed only when it has reached a stable waveform that moves from A to
B. In accord to theory [Murray, 1993; Mendez et al., 2004a ], we find that
the front travels with constant speed along the backbone maintaining its shape.
Obviously, the front speed is not affected by the order of growth and movement.
2.6. COMPUTATIONAL RESULTS
35
In our simulations we recover the same results when the particles first grow and
then move. Note that it is not strictly necessary to consider the process along the
backbone. In fact, owing to the exact self-similarity of the Peano construct, one
obtains the same results by assigning an initial condition (with compact support)
at any site and following the process along the drainage path connecting that site
to the outlet (point B in Figure 2.3). In fact, by further simulation we find that
the front speed measured along any drainage path is the same, provided that
the chosen path is long enough to develop a stable travelling wave. The relative
independence of the characters of a travelling wave from the particular flowpath
is somewhat reasonable, in retrospect, because the front speed along a path
depends only on the sequence of the orders of secondary branches encountered
along the path, which, for the Peano network, is the same by construction. Note
that this is not an unreasonable assumption also for real rivers [Rodriguez-Iturbe
and Rinaldo, 1997], partly explaining the importance of the insight derived from
the study of deterministic fractals.
Four sketches of the simulation of the colonization process are shown in Figure
2.6 for two values of the bias (b = 0 and b = 0.4 respectively) and for two different
simulation times. Just from the observation of these sketches one notes how the
bias stretches the colonization cloud enhancing the front speed. Figure 2.7 shows
the computed density at backbone nodes for three different time steps; a front
can be recognized that travels with constant speed maintaining quite nicely its
shape. The continuous flux of particles from the secondary branches into the
backbone may indeed lead to locally stable states with ρ > 1. Different values of
the stable part of the front (peaks and troughs in Figure 2.7) depend on different
order of the secondary branches flowing into the backbone.
Figure 2.5 shows also a comparison between numerical (circles) and analytical
(solid line) results. The exercise goes beyond the trivial numerical control. In
fact, while the reaction process is applied at every node in the simulations, the
analytical Hamilton-Jacobi solution assumes that all the dispersing particles are
concentrated in the backbone nodes, and thus the branches affect the waiting
time, not the particles density. It is heartening that, despite this difference, the
36
CHAPTER 2. INVASION MODELS
b=0
t=55τ
t=25τ
b=0.4
t=25τ
t=55τ
Figure 2.6: a) Sketches of the numerical simulation of the colonization process
through a 5th order Peano basin. The gray color shading is obtained by space
interpolation of the nodes density. The dimensionless growth rate employed is
aτ = 0.5; all other parameters are reported in the figure.
simulations exhibit a good agreement with the analytical results for a wide range
of values of the growth rate a τ [as in Campos et al., 2005].
We have also studied front speed propagation throughout the geometry of
real river networks. Figure 2.8a shows the drainage network of the Tanaro river
basin, a 8000-km2 catchment in North-western Italy, extracted via suitable geo-
2.6. COMPUTATIONAL RESULTS
37
1.4
ρ
1.0
0.6
0.2
0
20
40
60
80
100
120
x/∆x
Figure 2.7: b) Travelling front of the density wave along the backbone. The
different profiles are taken at the following time steps: 100 (dots), 150 (dash)
and 200 (solid). The other parameters employed are: aτ = 0.3 and b = 0.2.
morphological criteria [e.g. Rodriguez-Iturbe and Rinaldo, 1997]. Initially, we
have only accounted for its topological structure, thereby neglecting its geometrical properties like edge lengths (i.e., we assume that all the edges have the
same length ∆x), to investigate how departures from the topological structure
of Peano’s network affect the ensuing reaction-diffusion processes. We note (Figure 2.9c) that at equal values of the dimensionless growth rate aτ , the speed of
the front developing through the topological structure of the real river (empty
circles) is larger than that computed for Peano (solid circles). This is due to the
fact that at every junction along the drainage path in a real river, a travelling
particle generally meets only one secondary branch instead of two as in the case
of the Peano basin. The effect is that of increasing the speed of an approximately
constant amount.
We have also investigated how the distribution of edge lengths (that is, the
distance between subsequent reactive sites) affects the speed of the front. One
38
CHAPTER 2. INVASION MODELS
a)
b)
Figure 2.8: a) Extraction of the River Tanaro network. Reactive nodes are
imposed at the junctions. b) The same network of case a) with the same number
of reactive nodes distributed randomly.
should notice that in our model it is possible to consider networks constituted
by edges of different length by subdividing each edge into sub-edges of equal
2.6. COMPUTATIONAL RESULTS
39
0.8
v τ /∆ x
0.6
0.4
Peano (analytical)
Peano (numerical)
Tanaro (1)
Tanaro (2)
Tanaro (3)
Tanaro (4)
0.2
0
0
0.1
aτ
0.2
0.3
Figure 2.9: Dimensionless front speed (vτ /∆x) as a function of the dimensionless growth rate a τ computed analytically for the Peano basin (solid line) and
numerically for: the Peano basin (full circles); the River Tanaro taking into account only the topological structure of the network (Tanaro (1), empty circles);
the River Tanaro taking into account the distribution of the edge lengths with
reactive nodes as displayed in Figure 2.8a (Tanaro (2), squares), with reactive
nodes as displayed in Figure 2.8b (Tanaro (3), diamonds) and with one half of
the reactive nodes displayed in Figure 2.8b (Tanaro (4), stars). All cases are
computed with b = 0.4.
length. Technically, we label as reactive only the nodes at the endpoints of the
channel (see Figure 2.8a), so that at inner nodes there occurs transport only.
This method introduces a new key parameter for the system: the mean ratio of
the distance between two subsequent reactive nodes and the characteristic jump
length of the travelling particle (hLi/∆x). Note that this ratio is equal to one in
40
CHAPTER 2. INVASION MODELS
all previous cases. Squares in Figure 2.9c show simulation results for a mean ratio
of about 3. It turns out that the speed is smaller (and closer to the Peano case)
for large values of the population growth rate. Diamonds (Figure 2.9c) show the
results obtained by placing the same number of reactive nodes at random, not
at network junctions (see Figure 2.8b). We find that the spatial distribution of
the reactive nodes does not produce a meaningful variation of the front speed.
Stars in Figure 2.9c show instead the front speed developing through the Tanaro
river network when one half of the reactive nodes is used. In the latter case the
ratio hLi/∆x is twice as big as that of the previous case. Differences are indeed
noteworthy. The front speed is greatly decreased specially for large values of
the population growth rate. We thus suggest that the number of reactive sites,
and hence the distribution of node-to-node network distances, places a combined
hydrologic-demographic control on the front progression.
Other cases have been studied numerically. Of particular interest is the case
in which the bias is assumed to depend on a state variable, notably the bias
increases with the numbers of node drained (i.e., total cumulative area). The
bias thus increases downstream, as do migration front speed. This determines a
non-stationary behavior of the system that would require extra machinery for a
proper description.
In a space-unbounded domain, all the models analyzed (Fisher’s, reaction
telegraph, and biased RRW on networks), yield, if they start from a compact support initial condition, two fronts that travel with velocity v1 and v2 respectively
(see the sketch in the upper left inset of Figure 2.10). For unbiased processes the
two fronts travel along opposite directions, whereas for a large enough bias they
may travel along the same direction. The analysis presented up to this point
refers to the computation of the speed v1 of the first front that moves along the
bias direction. For the telegraph and RRW model the speed v2 of the second front
can be obtained as v2 (b, a) = −v1 (−b, a). Note that for the derivation of v2 for the
backbone of a Peano network through the Hamilton-Jacobi formalism (equations
(2.8) and (2.9)) one has to consider an opposite bias only for the transport in the
backbone, not in the branches (i.e., the Laplace transform of the waiting-time
2.6. COMPUTATIONAL RESULTS
41
distribution ϕ̂(s) in equation (2.9) remains the same for the derivation of both
v1 and v2 ). For Fisher’s model the speed of the second front is symmetrically
√
given by v2 = u − 2 aD (where u is the advection velocity). Figure 2.10 shows
the the speeds v1 and v2 of the two fronts as a function of the logistic growth
rate, for: i) Fisher’s model (dash-dotted line); ii) telegraph model (dashed line);
and iii) RRW on Peano’s network (solid line). The bias for the three cases is
equal to 0.6. The comparison between the Fisher and RRW model is obtained
by using the approximation D = ∆x2 /τ . The comparison between the telegraph
and the RRW models follows the comparative rule discussed at the end of Appendix B. For the range of parameters for which v2 is positive for the RRW on
the Peano network (and then v1 (−b, a) is negative because v2 (b, a) = −v1 (−b, a))
we compute the speed numerically (dots in Figure 2.10) because the HamiltonJacobi formalism allows us to compute only positive speeds. It is noticeable that
with biased transport the telegraph and the network RRW models lead to an
asymmetric behavior of the two fronts (i.e., (v1 (a) + v2 (a))/2 6= v1 (a = 0)). This
is to be contrasted with the classical Fisher model in which the two fronts are
perfectly symmetrical. However, the asymmetry is less remarkable in the Peano
RRW model because the particles can disperse away from the backbone thus
slowing down the front progression.
We have derived a possibly important statistical measure related to the propagation of reactive particles throughout river networks. Knowing the speeds v1
and v2 of the two fronts characteristic of a network, it is possible to compute the
first colonization time, that is the time that the front needs to propagate from
one node to any other along the shortest path available in the river network.
Every such path consists of a downstream and an upstream part of length Ld
and Lu respectively. If we assume that the speeds do not depend on the particular path, i.e., the bias is roughly the same, the travel time is simply given
by: T = Ld /|v1 | + Lu /|v2 |. Note that the front can reach all the nodes of the
network only if v2 is negative. Figure 2.11 shows the distribution of the times
of first colonization from three different starting nodes to all the others for the
Tanaro river network. It is interesting to note that this statistics is related to the
42
CHAPTER 2. INVASION MODELS
ρ
2
v2
v1
front speed
x
1
Fisher
Telegraph
Peano RRW
0
−1
0.5
1
growth rate
Figure 2.10: Speeds of the two fronts v1 and v2 , for any value of the dimensionless
logistic growth rate; Fisher model: dash-dotted line, telegraph model: dashed
line, and reaction random-walk on Peano network: solid line. The bias in all
cases is equal to 0.6. Dots of the Peano RRW are computed numerically.
dispersal kernel of Muneepeerakul et al. [2007]. Colonization times are shortest
for a central node (e.g. 1) and largest for the outlet, node 3. The initial condition seeded at any one node will therefore result in a specific distribution of
invasions throughout the network, which is affected by geomorphology as well as
by the nature of the reaction imposed by the logistic growth. There is a subtle
interplay of structural and dynamic controls that may operate at the network
2.6. COMPUTATIONAL RESULTS
43
level. Indeed if the travelling particles in this example were infective agents or
organism propagules, thereby accounting for reactions different from the logistic
growth considered here, our results would imply a definite hydrological role in the
process of disease spreading or ecological colonization. This seems an important
quantitative insight into the understanding of the susceptibility of a territory to
invasions.
44
CHAPTER 2. INVASION MODELS
3
x 10
−3
1
3
2
1
1
2
0
3
x 10
−3
3
e
x 10
400
t/τ
800
−3
3
2
2
2
1
1
0
400
t/τ
800
1200
1200
0
400
t/τ
800
1200
Figure 2.11: First colonization time distributions of all network sites starting
from three different nodes. The position of the three nodes in the network is
reported in the upper-left inset. The speeds v1 and v2 employed correspond to
values of b = 0.4 and a τ =0.3.
2.7. CONCLUSIONS
2.7
45
Conclusions
The following main conclusions can be drawn from our results:
-Adding a constant bias to a reactive transport model along river networks
jointly acts with morphological effects in drastically modifying the speed of the
migrating front of the travelling wave that describes the invasion process. Although our analytical and numerical results are derived by using logistic growth
as the reactive component, we suspect that this might be a result of general
nature and of ecological interest even for other, more complex demographies.
- We have obtained exact solutions for the speed of the migrating front with
biased transport for different invasion models in one-dimensional and recursive
comb-like space settings. Topological similarities among real rivers, optimal networks and exact recursive constructs lead to similar behaviours in the spreading
of populations, thus strengthening the predictive power of our analytical results.
- If heterogeneous distributions of reactive sites occur, we find that the characteristic distance relative to a basic network length places further hydrologic
controls on the invasion process.
- First colonization time distributions characterizing the spreading of reactive
agents throughout the entire network can be computed within our framework.
Their importance for invasion predictions seems noteworthy because they are
site-specific.
- Overall, we have found a significant number of hydrologic controls, in addition to known structural effects, on migrating fronts of species that move along
the ecological corridors defined by the river basin.
46
CHAPTER 2. INVASION MODELS
2.8
Appendix A: Fisher Fronts and the ReactionDiffusion Equation
In this section we provide a brief description of the classic Fisher model. For
a more detailed discussion the reader is referred to Murray [1993]. The Fisher
equation describing the continuous one dimensional reaction diffusion process is:
∂ρ(x, t)
∂ 2 ρ(x, t)
= aρ(x, t)(1 − ρ(x, t)) + D
∂t
∂x2
(2.20)
where ρ is the reagent density ([L]−1 ), a the logistic growth rate ([T ]−1 ) and
D the diffusion coefficient ([L]2 [T ]−1 ). Rewriting equation (2.20) in term of the
p
dimensionless variable t∗ = at and x∗ = x a/D, one has:
∂ρ(x, t)
∂ 2 ρ(x, t)
= ρ(x, t)(1 − ρ(x, t)) +
,
∂t
∂x2
(2.21)
where we omit the asterisks for notational simplicity.
Equation (2.21) admits two spatially homogeneous steady state: one stable
(ρ = 1) and one unstable (ρ = 0). Therefore we look for travelling wave which
connect these two state with 0 ≤ ρ ≤ 1, because negative density has no physical
meaning. A travelling wave solution, provided that it exists, can be written in
the form:
ρ(x, t) = η(z),
z = x − vt ,
(2.22)
where v is the wave speed. Since (2.21) is invariant if x → −x, v can be positive
or negative, in the following we look only for v ≥ 0. Substituting (2.22) into
(2.21), the last becomes a second-order, nonlinear differential equation:
η 00 + vη 0 + η(1 − η) = 0 ,
(2.23)
where primes denote differentiation with respect to z. Equation (2.23) could be
studied through a phase plane analysis of the equivalent ODE system:
2.8. APPENDIX A: FISHER EQUATION
(
47
η 0 = µ = f (η, µ)
(2.24)
µ0 = −vµ − η(1 − η) = g(η, µ)
The phase plane trajectories of (2.24) have two singular points ((ηs , µs ) → f =
g = 0): (1, 0) and (0, 0) that represent the two steady states discussed before.
Expanding f and g in Taylor series around the singular points and retaining only
first-order terms, the system (2.24) could be approximated by:
Ã
η0
µ0
!
Ã
=
fη fµ
gη gµ
!
Ã
(ηs ,µs )
η − ηs
!
µ − µs
(2.25)
The eigenvalues of the matrix of system (2.25) computed for the steady state
(1, 0) are:
√
1
λ1,2 = (−v ± v 2 + 4) ,
(2.26)
2
they are both real and of oppositive sign for any positive v. Then (1, 0) is a
saddle point. For the state (0, 0), instead, the eigenvalues are:
For v ≥ vmin
√
1
λ1,2 = (−v ± v 2 − 4) .
(2.27)
2
= 2 the eigenvalues (2.27) are both real and positive and then (0, 0)
is a stable node; for v ≤ vmin = 2, instead, they are complex with negative real
part. In this case (0, 0) is a stable spiral, this implies that η oscillates around the
origin assuming also negative values. Therefore v ≤ vmin provides travelling wave
solution physically unrealistic since η < 0 (i.e. ρ < 0). In term of the original
√
dimensional equation (2.20), the range of wave speed is: v ≥ vmin = 2 aD. If
the initial condition ρ(x, 0) has compact support, the solution ρ(x, t) evolves to a
travelling wave solution with minimum speed v = vmin [Kolmogorov et al., 1937].
48
CHAPTER 2. INVASION MODELS
2.9
Appendix B: Wave Front in the Reaction Telegraph Model with Bias
In this Section we report technical details for the derivation of the front speed in
the biased reaction telegraph model presented in Section 2.3.
Let F be the rate of demographic growth. Let α(x, t) be the density at
coordinate x and time t of particles that arrived from the left and β(x, t) be
the density of particles that arrived from the right. Then we can stipulate the
following equations:
α(x, t + δt) = (1 − λL δt) α(x − δx, t) + λR δt β(x − δx, t) + 21 δt F (x − δx, t)
(2.28)
β(x, t + δt) = (1 − λR δt) β(x + δx, t) + λL δt α(x + δx, t) + 21 δt F (x + δx, t).
(2.29)
Being δt, δx, λR , λL the parameters introduced in Section 2.3. Expanding in
Taylor series with respect to time and space and neglecting second and higher
order terms we obtain
1
1
α + δtαt = (1 − λL δt)(α − δxαx ) + λR δt(β − δx βx ) + δtF − δtδxFx
2
2
(2.30)
1
1
β + δtβt = (1 − λR δt)(β + δxβx ) + λL δt(α + δx αx ) + δtF + δtδxFx ,
2
2
(2.31)
where the subscripts x and t indicate the partial derivatives with respect to space
and time, respectively. Taking the limit as δt and δx go to zero with δx/δt = γ
we have:
1
αt + γαx = λR β − λL α + F
2
1
βt − γβx = λL α − λR β + F .
2
(2.32)
(2.33)
2.9. APPENDIX B: BIASED TELEGRAPH MODEL
49
It is convenient to introduce the total density of particles at location x: S(x, t) =
α(x, t) + β(x, t) and the difference: R(x, t) = α(x, t) − β(x, t). The rate F of
demographic increase is supposed to be a unimodal, nonnegative function of
the total density S, such that F (0) = F (K) = 0, where K is the "carrying
capacity" or equilibrium population size. From equations (2.32) and (2.33) we
finally obtain:
St + γRx = F (S)
(2.34)
Rt + γSx = ∆λ S − 2λR ,
(2.35)
where ∆λ = λR − λL > 0 is the directional bias and λ = (λR + λL )/2 is the
average rate of reversal. Note that ∆λ/2λ ≤ 1 in any case. We now search
for a travelling wave moving from left to right with velocity c. To this end we
introduce the moving coordinate system z = x − ct, c > 0, into equations (2.34)
and (2.35) thus obtaining:
−cSz + γRz = F (S)
(2.36)
−cRz + γSz = ∆λ S − 2λR .
(2.37)
and then:
cF (S) + γ∆λS − 2γλR
γ 2 − c2
γF (S) + c∆λS − 2cλR
Rz =
γ 2 − c2
Sz =
(2.38)
(2.39)
The singular points of equations (2.38) and (2.39) are (S, R) = (0, 0) and (K, ∆λK/2λ).
The Jacobian is:
1
J(S) = 2
γ − c2
"
cF 0 (S) + γ∆λ −2γλ
γF 0 (S) + c∆λ −2cλ
#
(2.40)
50
CHAPTER 2. INVASION MODELS
where the superscript 0 indicates differentiation with respect to S. We must see
whether it is possible to have a heteroclinic connection going from (K, ∆λK/2λ)
to (0, 0) and such that S ≥ 0. This is equivalent to requiring that (K, ∆λK/2λ)
is unstable and (0, 0) is a saddle or a stable node. Let us consider (0, 0) first.
We must have either detJ(0) < 0 (saddle) or trJ(0) < 0 and 0 < 4detJ(0) <
(trJ(0))2 (stable node). Introduce the intrinsic rate of demographic increase
a = F 0 (0), the dimensionless rate of increase ρ = a/2λ, and the dimensionless
bias b = ∆λ/2λ. Notice that both a and ρ are greater than zero and 0 < b ≤ 1.
It turns out
2aλ
γ 2 − c2
ca + γ∆λ − 2cλ
trJ(0) =
γ 2 − c2
detJ(0) =
(2.41)
(2.42)
There are two cases:
First: c > γ; then detJ(0) < 0 and (0, 0) is a saddle for any a or equivalently for
any ρ.
Second: c ≤ γ; then we must verify the conditions for a stable node.
First condition: trJ(0) < 0 if and only if c > ĉ = bγ/(1 − ρ). On the other
hand, we must have 0 < c ≤ γ. Therefore, this imposes the further condition
that ρ ≤ (1 − b).
Second condition: 4detJ(0) < (trJ(0))2 if and only if G(c) = (1 + ρ)2 c2 + 2b(ρ −
1)γc + (b2 − 4ρ)γ 2 > 0. The roots of G(c) are:
c± =
b(1 − ρ) ±
p
4ρ[(1 + ρ)2 − b2 ]
γ.
(1 + ρ)2
(2.43)
As the dimensionless bias b is ≤ 1 then b ≤ 1 + ρ, hence both roots are real.
Therefore G(c) > 0 if and only if c < c− or c > c+ . Note that for ρ = 0 it turns
out that c± = ĉ = bγ, while for ρ = 1 − b it turns out that c+ = ĉ = γ. More
generally, it is easy to verify that c± ≤ γ. Also, we have for ρ < 1
2.9. APPENDIX B: BIASED TELEGRAPH MODEL
c− =
b(1 − ρ) −
p
4ρ[(1 + ρ)2 − b2 ]
b(1 − ρ)
b(1 − ρ)
γ≤
γ≤
γ = ĉ .
2
2
(1 + ρ)
(1 + ρ)
(1 − ρ)2
51
(2.44)
A cumbersome analysis also shows that c+ ≥ ĉ for ρ ≤ 1 − b. In fact, for ρ < 1
the inequality c+ ≥ ĉ is equivalent in succession to:
b(1 − ρ)2 +
p
4ρ[(1 + ρ)2 − b2 ](1 − ρ) ≥ b(1 + ρ)2 ,
¡
¢
(1 − ρ)2 [(1 + ρ)2 − b2 ] − 4ρb2 = (1 + ρ)2 (1 − ρ)2 − b2 ≥ 0
The last inequality is true for ρ ≤ 1 − b. Therefore, c− ≤ ĉ ≤ c+ .
We can thus state that the singular point (0, 0) is a stable node provided
c ≥ c+ The possible range of values for the dimensionless rate of demographic
increase is 0 ≤ ρ ≤ 1 − b.
We can conclude that the minimum value cmin of the velocity c for which the
singular point (0, 0) is the limit of a heteroclinic orbit for z → +∞ is given by:



cmin =
c+ =
b(1 − ρ) +


p
4ρ[(1 + ρ)2 − b2 ]
γ for ρ < 1 − b
(1 + ρ)2
γ
for ρ ≥ 1 − b
(2.45)
Moreover it is easy to verify that the singular point (K, ∆λK/2λ) is unstable.
In fact
2F 0 (K)λ
γ 2 − c2
cF 0 (K) + γ∆λ − 2cλ
trJ(K) =
γ 2 − c2
detJ(K) =
(2.46)
(2.47)
Remembering that F 0 (K) < 0 and b = ∆λ/2λ, we have that detJ(K) ≤ 0 for
c ≤ γ and trJ(K) > 0 for c > γ.
Our final conclusion is that the minimum velocity for the existence of a heteroclinic connection between the two singular points is cmin given by equation
52
CHAPTER 2. INVASION MODELS
(2.45). Therefore cmin is the travelling wave speed for a front moving from left
to right. Note that the wave speed without demographic growth (ρ = 0) is bγ,
which is the advection velocity. In fact, a particle in a certain position that arrived from the left has probability λL /(λR + λL ) of moving left and probability
1 − λL /(λR + λL ) of moving right, while a particle that arrived from the right
has probability λR /(λR + λL ) of moving right and probability 1 − λR /(λR + λL )
of moving left. So the average velocity is:
· µ
¶
¸
·
µ
¶¸
λL
γλL
1
γλR
λR
∆λ
1
γ 1−
−
+
1−
=
γ = bγ
2
λR + λL
λR + λL
2 λR + λL
λR + λL
2λ
On the other hand, there is a threshold of the dimensionless growth rate ρ = 1−b
beyond which the wave speed is equal to γ, the absolute velocity of an particle
moving to a nearby location. Introducing the normalized wave speed v = cmin /γ
we have:

p
2
2

 b(1 − ρ) + 4ρ[(1 + ρ) − b ]
for ρ < 1 − b
v(b, ρ) =
(1 + ρ)2


1
for ρ ≥ 1 − b
(2.48)
So the normalized wave speed depends on just two parameters: b, which is the
normalized advection velocity, and ρ which is the dimensionless rate of increase.
Note that 1/a is the average time for the production of a new particle and 1/2λ
is the average time between two direction inversions. ρ is just the ratio of the
latter time to the first. Also the threshold condition ρ > 1 − b is equivalent to
1/a < 1/2λL , namely the average time for the production of a new individual
must be smaller than the average time between two inversions in the direction
opposite to the wave direction. Needless to say, the case λL = λR = λ yields
Holmes’ [1993] results.
It can be proved that the velocity v2 of the retrogressive travelling wave is
simply v2 (b, ρ) = −v(−b, ρ). Note that v2 becomes negative for ρ > b2 . Also it
saturates to −γ for ρ = 1 + b.
Figure 2.12 shows the dimensionless front speed v/γ as a function of the
2.9. APPENDIX B: BIASED TELEGRAPH MODEL
53
growth rate a for a reaction telegraph model (solid lines) and for a RRW in a one
dimensional lattice (dashed lines). The latter computed through the HamiltonJacobi formalism described in section 2.5 by setting to zero the order of the
branches (i.e., ϕ(t) = δ(t − τ )). Different couples of lines (solid and dashed)
refer to different values of the bias b. The best fit between the two models is
obtained taking τ = 1/λ, where τ is the waiting time for the random walk and
λ the average reversal rate for the telegraph model. This could be explained as
follows: in the discrete random walk model, τ could be thought of as the time
spent by a particle moving, from left to right (from x to x + ∆x) or likewise from
right to left (from x to x − ∆x), maintaining its direction. In the differential
continuous biased telegraph model the mean times of preserved direction, from
left to right and vice versa, are 1/λR and 1/λL respectively. We can then take τ
equal to the average of 1/λR and 1/λL weighted through the probability of going
right (λR /(λR + λL )) and left (λL /(λR + λL )) respectively:
λR
1
λL
1
1
+
= ,
(2.49)
λR + λL λR λR + λL λL
λ
which is exactly the relation used. The two models lead to similar results for a
τ=
large range of values of the growth rate a.
54
CHAPTER 2. INVASION MODELS
1.0
v/γ
0.8
0.6
0.4
reaction-telegraph model
1-D RRW
0.2
0
0
0.2
0.4
0.6
0.8
1
dimensionless growth rate
Figure 2.12: Dimensionless front speed v/γ as a function of the dimensionless
growth rate (a/λ or aτ ) for a reaction-telegraph model (solid lines) and for a
RRW in a one-dimensional lattice (dashed lines). Different couples of lines (solid
and dashed) from the bottom to the top refer to different values of the bias b
from 0 to 0.8 with step 0.2.
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BIBLIOGRAPHY
Chapter 3
On the Space-time Evolution of a
Cholera Epidemic 1
3.1
Summary
We study how river networks, acting as environmental corridors for pathogens,
affect the spreading of cholera epidemics. Specifically, we compare epidemiological data from the real world with the space-time evolution of infected individuals
predicted by a theoretical scheme based on reactive transport of infective agents
through a biased network portraying actual river pathways. The data pertain to
a cholera outbreak in South Africa which started in 2000 and affected in particular the KwaZulu-Natal province. The epidemic lasted for two years and involved
about 140,000 confirmed cholera cases. Hydrological and demographic data have
also been carefully considered. The theoretical tools relate to recent advances in
hydrochory, migration fronts and infection spreading described in the previous
chapter, and are novel in that nodal reactions describe the dynamics of cholera.
Transport through network links provides the coupling of the nodal dynamics
of infected people, who are assumed to reside at the nodes. This proves a re1
The contents of this chapter has been published in: Bertuzzo, E., S. Azaele, A. Maritan,
M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo (2008), On the space-time evolution of a cholera
epidemic, Water Resour. Res., 44, W01424.
59
60
CHAPTER 3. SPREADING OF DISEASES MODELS
alistic scheme. We argue that the theoretical scheme is remarkably capable of
reproducing actual outbreaks, and indeed that network structures play a controlling role in the actual, rather anisotropic propagation of infections, in analogy
to spreading of species or migration processes that also use rivers as ecological
corridors.
3.2
Introduction
Cholera is an intestinal disease caused by the bacterium Vibrio cholerae, which
colonizes the human intestine. The dynamics of cholera epidemics have been
studied since the 1800s when John Snow established the link between cholera
cases and exposure to contaminated water of a well in London. V. cholerae is
also a natural member of the aquatic microbial community [Colwell, 1996; Lipp
et al., 2002]. Thus the spatial and temporal patterns of cholera epidemics are
strongly related to the ecology of the bacterium in the environment which is itself
driven by meteorological and climatic variability. Time series analyses of cholera
cases in endemic regions, such as Bangladesh, show a time variability with subannual, annual and interannual components. The low frequency variability has
been established to be related to long term climatic oscillations [Pascual et al.,
2000; Koelle et al., 2005]. In non-endemic regions, the annual component is much
more important. Also the spatial distribution of the disease plays a fundamental
role, which is, however, usually neglected in the existing cholera models. The
aim of this paper is to understand the space-temporal evolution of cholera by
explicitly accounting for the environmental matrix within which the disease can
spread.
The model will be tested against a very well documented case. In 2000, after
several years without cholera outbreaks, a new epidemic spread in South Africa
affecting in particular the KwaZulu-Natal province. The epidemic lasted for
two years with only a few cases recorded during the third year, and ultimately
involved about 140,000 confirmed cholera cases. The epidemic was caused by
the 01 el Tor strain [Mugero and Hoque, 2001], which is more easily transmitted
3.2. INTRODUCTION
61
through contamination of aquatic environments than the classical biotype of
Vibrio cholerae.
Models of cholera dynamics are relatively recent. Capasso and Paveri-Fontana
[1979] proposed a mathematical model to describe the 1973 cholera epidemic in
Bari (Italy) with two equations describing the dynamics of the infected population and the free-living pathogens. Codeço [2001] extended Capasso and PaveriFontana’s model adding an equation for the dynamics of the susceptible population and studied the role of the aquatic reservoir in the endemic-epidemic
dynamics of cholera. In the model proposed by Pascual et al. [2002], another
equation is added to describe the temporal evolution of the volume of water hosting the free-living bacteria. Recent laboratory findings suggest that passage of
the bacterium through the gastrointestinal tract results in a short-lived hyperinfectious state that can enhance the human-to-human vs environmental-to-human
transmission of cholera. Hartley et al. [2006] incorporate the hyperinfectious
state into Codeço’s model to achieve a better explanation of explosive cholera
outbreaks.
All the above models do not consider space explicitly. They assume a unique
community of people who interact and share the same resources. We believe,
however, that the spatial distribution of the communities and how they interact
is crucial to understanding the spatial spreading of the epidemic in a diseasefree region, particularly if travel times of pathogens are comparable with the
characteristic time of the virulent infection. The spatial distribution of different
communities, along with the distribution of their population size, and how they
are interconnected, could indeed affect the dynamics of the process, especially in
the case of a non-endemic region.
V. cholerae can survive in the aquatic environment in associations with chitinaceous zooplankton like copepods, shellfish and also with the aquatic vegetation
[Colwell, 1996]. Therefore V. cholerae (and the disease) can spread from the
coastal region, where it is autochthonous, to the inland area through waterways
and river networks. In the same manner the infection can spread from inland
regions with epidemic outbursts into the surrounding areas.
62
CHAPTER 3. SPREADING OF DISEASES MODELS
We propose a model that explicitly accounts for the role of the river network in
transporting and redistributing V. cholerae between several human communities
and we proceed to apply the model to the real conditions of the 2000 cholera
epidemic in the KwaZulu-Natal province. The model explicitly recognizes a role
for network structures acting as support for the infection, in analogy to recent
studies on migrating fronts constrained by landscape heterogeneities or spreading
of species along riparian ecological corridors [Campos et al., 2006; Bertuzzo et
al., 2007; Muneepeerakul et al., 2007].
The chapter is organized as follows. Section 2 describes the theoretical approach and the model used in detail. The complete data set and the case study
are presented in a specific chapter (Section 3.4). The main results and the discussion are collected in Section 3.5. A set of conclusions closes the chapter.
3.3
Theoretical Approach
Spreading of epidemics in networks is addressed by viewing the environmental
matrix as an oriented graph (i.e., a directed graph having no symmetric pair of
directed edges). Nodes represent human communities (cities, towns, villages) in
which the disease can be diffused and grow. The edges represent links between the
communities, typically hydrological links. Edge direction is chosen accordingly
to the flow direction. The model is assembled by coupling two models: i) a local
epidemic model at nodes of the graph; and ii) a transport model for the spreading
of the disease vector through the edges of the support. Details of the two models
follow.
As for the local dynamics we use a continuous model of the SIR (i.e., susceptibles, infected, recovered) class with a reservoir of free-living infective propagules.
It is obtained by a slight modification of the cholera epidemic model introduced
by Codeço [2001]. The model has three state variables: the number of susceptibles S, the number of infected I and the concentration of V. cholerae in the
aquatic environment B, whose respective temporal dynamics are described by
the following system of first order differential equations:
3.3. THEORETICAL APPROACH
B
dS
= n (H − S) − a
S
dt
K +B
dI
B
= a
S − (r + m + n) I
dt
K +B
dB
p
= nB B +
I
dt
W
63
(3.1)
The meaning of the parameters is explained in Table 3.1. The first equation
describes the dynamics of susceptibles in a community of size H. Susceptible
individuals are born and die on average at rate n. Newborn individuals are
considered susceptible. Susceptible people become infected at a rate a B/(K+B),
where a is the rate of contact with contaminated water and B/(K + B) is a
logistic dose response curve that links the probability of becoming infected to
the concentration of vibrios B in water. Infected people (whose dynamics is
described by the second equation) die at a rate which is the sum of natural
mortality n and disease-caused mortality m, and recover with rate r. The third
equation describes the dynamics of the free-living infective propagules in the
reservoir. Infected people contribute to the concentration of vibrios at a rate
p/W , where p is the rate at which bacteria are produced by one infected person
and W is the volume of the contaminated water body. The growth rate nB of
the free-living bacteria in the water body is usually negative because bacteria
mortality in natural environments exceeds reproduction. If nB is positive, the
model would predict an exponential growth of the vibrios concentration and all
the susceptible population would be affected by the disease. The hidden equation
for the recovered is (dR/dt) = rI − nR. People recovered from cholera are
considered immune. The model does not take into account any loss of immunity
(i.e., a flux from recovered to susceptibles) because immunity usually lasts for
a period longer than the two years of the epidemic we consider here [Koelle et
al., 2005]. Nevertheless the immunity loss could play an important role in the
dynamics of cholera in regions where it is endemic [Koelle et al., 2005].
As long as we are not interested in the numerical value of the concentration
B of V. cholerae, we can introduce the dimensionless concentration B ∗ = B/K,
64
CHAPTER 3. SPREADING OF DISEASES MODELS
thus obtaining from (3.1) the system of equation:
dS
B∗
= n(H − S) − a
S
dt
1 + B∗
dI
B∗
= a
S − (r + m + n) I
dt
1 + B∗
dB ∗
p
= nB B ∗ +
I
dt
KW
(3.2)
Notice that system (3.2) has the advantage of merging the parameters K, p and
W (which can hardly be directly estimated) into a unique ratio. This ratio will
be the control parameter of the process jointly with the V. cholerae growth rate
nB . On the contrary, the mortality rates, both natural (n) and due to cholera
(m), the recovery rate (r), and the exposure rate (a) can reasonably be estimated
from demographic and epidemiological studies as we show below.
A linear stability analysis shows that, given an initial condition of the type
S(0) = H; I(0) > 0; B ∗ (0) = 0, the model predicts an epidemic outbreak only
if the population size is greater than a certain critical threshold SC given by
[Codeço, 2001]:
H > SC =
−(r + n + m)KnB W
;
ap
(3.3)
otherwise the infected population decreases to zero. It is important to remark the
dilution effect: the larger the volume of the water body, the higher the critical
threshold.
We model the spreading of V. cholerae through the network with a biased
random-walk process on oriented graph [Bertuzzo et al., 2007]. For a detailed
discussion of the process see also Johnson et al. [1995]. An infectious propagule
can move with some probability from a node to one of the adjacent nodes, which
are all the nodes that are connected to it through an inward or outward edge. We
assign to each edge of the graph an orientation according to the flow direction.
Consider first a particular case of the network in which every node has only one
inward and one outward edge (i.e., a one dimensional lattice). We define as Pout
3.3. THEORETICAL APPROACH
65
(Pin ) the probability that a propagule leaving a node moves to another node
along an outward (inward) edge. We have then Pout + Pin = 1.
We now turn to the analysis of a random-walk process on a generic oriented
graph in which every node can have an arbitrary number of inward and outward
edges. We assume that a propagule can move following an outward or inward
edge with a probability proportional to Pout and Pin respectively. In this case,
the probability Pij for a propagule to be transported from node i to node j can
be expressed as follows:


Pout





dout (i)Pout + din (i)Pin



Pin
Pij =


dout (i)Pout + din (i)Pin







0
if i → j
if i ← j
(3.4)
if i = j
where dout (i) and din (i) are, respectively, the outdegree and indegree of node
i (i.e., the number of outward, respectively inward, edges of node i). Since
P
Pout + Pin = 1, one has N
j=1 Pij = 1, where N is the total number of nodes. We
term b = Pout − Pin = 2Pout − 1 the bias of the transport.
When we apply the local epidemic model at each node of the network, we
have 3N state variables Si , Ii , Bi∗ , where the subscript i identifies the nodes.
We assume that vibrios are removed at every node with a certain rate l (day −1 )
and transported through the network following the transition probabilities (3.4).
Then, the equations that describe the coupled process are:
Bi∗
dSi
= n (Hi − Si ) − a
Si
dt
1 + Bi∗
dIi
Bi∗
= a
Si − (r + m + n) Ii
dt
1 + Bi∗
N
X
p
dBi∗
Wj
∗
∗
= nB Bi +
Ii − l Bi +
l Pji Bj∗
dt
KWi
Wi
j=1
(3.5)
for i = 1, 2, . . . , N . Note that all the parameters are node-independent except for
66
CHAPTER 3. SPREADING OF DISEASES MODELS
the population size Hi and the water volume Wi . The latter represents the whole
set of water supplies available for that community, not only the one provided by
the river. The network acts as a link through which different sets of water supplies
of different communities can be connected and contaminated. In order to further
minimize the number of parameters, we assume that the water volume is a nondecreasing function of the population size: Wi = f (Hi ). Different choices of the
function f can lead to different scenarios of the epidemic. In fact, consider the
ratio Vi between the population size and the critical threshold (equation (3.3)):
Vi =
Hi
Hi a p
Hi
=
∝
.
SC i
−(r + n + m) K nB f (Hi )
f (Hi )
(3.6)
This is an index of the node vulnerability to an epidemic. Let us first analyze the
case in which the water availability is constant for all the nodes (Wi = const).
This corresponds to assuming that the communities utilize water resources that
are quite uniformly distributed in space. In this case the vulnerability of the
nodes increases linearly with the population size, namely Vi ∝ Hi . In this scenario
then, if an epidemic occurs, the most affected communities would be the most
populated. Another different scenario derives from assuming that larger communities manage to increase their own water supply so that the per-capita available
water is kept constant. This is equivalent to assuming that the water volume of
a node is proportional to the population size: Wi ∝ Hi and then Vi = constant.
Under such a scenario an epidemic would affect, even if with different dynamics,
all the communities regardless of their size. This assumption seems reasonable
for large and developed communities, but it is unsatisfactory for the nodes with
small population size, because it would imply a small water body associated with
the node regardless of the natural presence of water. A more general and realistic
hypothesis could derive from the combination of the two assumptions described
above. In particular, we assume that the nodes with population size Hi smaller
than a certain threshold HT have a constant water volume associated with them,
whereas the nodes with Hi > HT have a constant per-capita water availability.
Summarizing, we have Wi = max(c HT , c Hi ) = c · max(HT , Hi ), where c is the
3.4. THE CASE STUDY
67
per-capita volume of water resource.
Substituting the last relationship into the second term of the right-hand-side
of the third equation of system (3.5), we get: pIi /(K c · max(HT , Hi )). Thus, the
parameters to be separately estimated are the ratio p/(K c) and the population
threshold HT . These are parameters that depend on social and environmental
factors, like hygiene and health conditions, eating habits and lifestyle, and on how
these variables vary with population density. Nevertheless, the particular choice
of parameters that link vulnerability to the actual distribution of population has
to be evaluated for each case.
3.4
The Case Study
We apply the model to a well-documented case of cholera epidemics that occurred
in KwaZulu-Natal province of South Africa (see Figure 3.1). The data were
provided by the KwaZulu-Natal Health Department and consist in a record of
each single cholera case specified by the date and health subdistrict where it
occurred. The spatial representation of the districts is shown in Figure 3.2. The
record starts from August 2000 and runs continuously until present time. Our
analysis focuses on the two largest epidemic outbreaks occurred during the 20002001 and 2001-2002 summers which involved 135,000 cholera cases. The data set
also provides the population size of each district. The total population of the
province is about 8.5 million inhabitants. The temporal evolution of the weekly
cholera cases is reported in Figure 3.3. The data exhibit a clear seasonality with
the outbreaks occurring during the warmest months of the austral summer. This
is probably due to the increased growth rate of vibrios at warm temperatures
in association with plankton blooms. Evidence for this phenomenon comes from
the higher rates of isolation of the bacterium in the environment during warm
periods [Lipp et al., 2002].
68
CHAPTER 3. SPREADING OF DISEASES MODELS
Figure 3.1: Map of the KwaZulu-Natal province of South Africa.
3.4. THE CASE STUDY
69
cholera cases/
population size
Figure 3.2: a) Spatial representation of the health districts of the KwaZulu-Natal
province (South Africa). Colours represent the percentage incidence of cholera
cases over the population size of each district. b) Temporal evolution of the new
weekly cholera cases for the whole province.
70
CHAPTER 3. SPREADING OF DISEASES MODELS
8000
6000
4000
2000
0
Oct00 Jan01 Apr01 Jul01
Oct01 Jan02 Apr02 Jul02
Figure 3.3: Temporal evolution of the new weekly cholera cases for the whole
province.
3.4. THE CASE STUDY
71
In order to apply the model we need to build the network along which infection
is transported. First of all, we have derived the mathematical model of the
river networks from the hydrological GIS data provided by the South Africa
Department of Water Affairs and Forestry shown in Figure 3.4.
rivers
Thukela river
basin
km
Figure 3.4: Hydrographic map of KwaZulu-Natal province with the Thukela river
basin evidenced. The dot reports the location of the first epidemic outbreak in
the basin studied
.
All the channels of perennial rivers are considered edges and all the endpoints of these channels are considered as nodes. Second, we had to transfer the
72
CHAPTER 3. SPREADING OF DISEASES MODELS
information from districts to network nodes. This is done by assigning the population and the cholera cases of a subdistrict to the nearest network node; the
distances being computed from the centroid of the subdistrict to the node. The
results of this interpolation for the population and the total cumulated cholera
cases are shown in Figure 3.5 where the color coding is obtained by spatial linear
interpolation of the node values.
Comparing the spatial distribution of population sizes and total cases in the
nodes, one can note that the high population density areas recorded few cases
of cholera, as well as the low density ones. The most affected nodes are those
with intermediate population size. This clearly appears by plotting the cholera
average incidence (i.e., the total number of cases divided by the population size)
as a function of the population size (see Figure 3.6). The highest incidence was
recorded for population sizes between 2, 000 and 30, 000. This is probably due to
the fact that the highest population density regions correspond, in this particular
case, with the most developed ones. These cities can then rely on waste-water
treatment and treated water supply that help to reduce cholera transmission.
The framework described in this paper addresses the spread of an epidemic
in a single river basin, and, for the time being, we avoid any modelling of the
flux of bacteria across different catchments. For this reason, in order to test
the validity of the model, we have applied it to the basin of river Thukela, the
largest of the region (see Figure 3.4). The 29,000 km2 area drained by this river
is populated by 1.5 million people and cholera cases recorded there amounted to
29,000 (21% of the total cases of the whole province) during the two epidemic
outbreaks considered.
We estimated the birth and mortality rate of the population as the inverse
of the average lifetime for this region (about 60 years), so n ' 5 · 10−5 day −1 .
Because the average duration of the cholera disease in an infected person is
approximately 5 days [Codeço, 2001; Hartley et al., 2006], we set the recovery
rate at r = 0.2 day −1 . The deaths due to cholera for the epidemic analyzed
was the 0.2% of the cholera cases. Thus, we can estimate the cholera mortality
rate m by assuming that after the duration of the disease 99.8% of the infected
3.4. THE CASE STUDY
73
population survive, i.e. exp(−m/r) = 0.998, and then m = 4 · 10−4 day −1 .
From this simple analysis we conclude that, given the order of magnitude of the
parameters involved, we can simplify the model setting r + m + n ' r. Following
Codeço [2001], we assume that people ingest contaminated water or food once a
day (a = 1 day −1 ).
To model the seasonality of the bacterium ecology as discussed in Section 3.4,
we let the net growth rate of V. cholerae in the aquatic environment vary periodically in time according to nB (t) = nB (1 + sin(2πt/365)) (with t in days and
t = 0 corresponding to October 1st) following the water temperature cycle [Jury,
1998]. We implicitly assume that the free-living bacteria are in demographic
equilibrium (nB (t) = 0) during the warmest season. After the above considerations, the model parameters to be calibrated are the mean net V. cholerae
growth rate in the aquatic environment nB , the rate at which V. cholerae are
removed and transported l, the ratio p/(Kc), the threshold HT and the bias of
the transport b.
74
CHAPTER 3. SPREADING OF DISEASES MODELS
3
2
1
log10 cholera cases
0
5
4
3
log10 population
Figure 3.5: Spatial linear interpolation of network nodes value of a) cholera cases
and b) population size.
3.4. THE CASE STUDY
75
cumulated cases / population size
0.06
0.05
0.04
0.03
0.02
0.01
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
log10 population size
Figure 3.6: Average cholera incidence (i.e. total cholera cases over population
size). Nodes have been grouped in a logarithmic bin on the basis of their population size.
6
76
CHAPTER 3. SPREADING OF DISEASES MODELS
3.5
Results and Discussion
In order to calibrate the model we subdivided the set of parameters into two
groups: the first contains the parameters related to the transport and spatial
distribution of cases (l, b and HT ), while the second set groups the parameters
closely related to the local model and the temporal dynamics (nB and p/(Kc)).
For every randomly chosen combinations of the three parameters of the first
group, we calibrated the two parameters of the second group by minimizing the
mean square error between the data of the temporal evolution of cumulated cases
in the whole basin and the simulations. Note that the temporal evolution of the
cumulated cases for each node can be obtained via the equation:
dCi
Bi∗
=a
Si ,
dt
1 + Bi∗
(3.7)
which takes into account only the flux of susceptibles that actually become infected. By adding the cumulated cases at time t throughout all the nodes, we get
the cumulative evolution for the entire catchment. Every simulation starts with
the initial conditions of one infected person in a single node where, according to
data, the first case of cholera was recorded, and runs continuously for 2 years
(the location of this node is reported on the map of Figure 3.4). Then, for each
combination of the five parameters (the triplet l, b, HT and the corresponding
calibrated pair nB , p/(Kc)) we computed the mean square error between the
total cumulated cases of cholera at each node after two years obtained from the
simulations and from the data. Note that the error from the first calibration
measures the likelihood of the temporal patterns, while the second calibration
compares simulated and recorded spatial patterns of the epidemic. Finally, we
chose the combination of parameters that minimizes the weighted sum of the
two errors. The weight for each error is computed as the inverse of the minimum value of the error itself found among all the realizations. The values of the
calibrated parameters are listed in Table 3.1.
Figure 3.7 shows the comparison between data (dots) and model simulation
(solid line) of the temporal dynamics of the weekly (3.7a) and cumulated (3.7b)
3.5. RESULTS AND DISCUSSION
77
cholera cases in the whole Thukela river basin.
weekly cases
1600
a)
data
simulated
1200
800
400
0
cumulated cases
3
x 10 4
b)
2
1
0
Oct00
Jan01
Apr01
Jul01
Oct01
Jan02
Apr02
Jul02
Figure 3.7: Comparison between the data (dots) and simulated (solid line) temporal evolution of (a) weekly cholera cases and (b) cumulated cases for the
Thukela river basin of the KwaZulu-Natal province.
Figure 3.8 compares the spatial distribution of data with that obtained via
simulation. It shows the distribution of the cumulated cholera cases after the
first and second epidemic outbreak. As in Figure 3.5 the colours are graded via
78
CHAPTER 3. SPREADING OF DISEASES MODELS
spatial linear interpolation of nodal values.
data
simulated
after
1 year
after
2 years
2
1
0
3
log10 cumulated cases
Figure 3.8: Comparison between data and simulated spatial distribution of the
cumulated cholera cases after the first and second epidemic outbreak for the
Tukhela river basin. Colours are obtained via spatial linear interpolation of the
node values.
The model does well in reproducing the distribution of the cholera cases
during the two outbreaks as well as their spatial spreading. It is interesting to
note that cases from the second outbreak are mainly located in new regions with
3.5. RESULTS AND DISCUSSION
79
respect to the first one. This is related to the spread of cholera from the regions
involved in the first epidemic outbreak into disease-free ones. This supports
our hypothesis that the dynamics of cholera epidemics in non-endemic regions
depends on the spatially anisotropic spreading along an environmental matrix
defined by river corridors, as well as on inner local dynamics. Similar results in
a different context were obtained by Campos et al. [2006]; Bertuzzo et al. [2007]
and Muneepeerakul et al. [2007].
Finally, we have checked whether our model is able to reproduce the relation
discussed in Section 3.4 between cholera incidence and population size of each
node. The comparison for the Thukela river basin is reported in Figure 3.9. It
demonstrates that the model results regarding the incidence distribution agree
quite well with the data.
cumulated cases / population size
80
CHAPTER 3. SPREADING OF DISEASES MODELS
0.05
simulation
data
0.04
0.03
0.02
0.01
0
1
2
3
4
5
log10 population size
Figure 3.9: Comparison between data and simulated cholera incidence (i.e., the
ratio of total cholera cases to population size) for the Tukhela river basin. Nodes
have been grouped in a logarithmic bin on the basis of their population size.
Bars represent the mean incidence for each bin, while the error bars represent
the standard error of the mean. The Kolmogorov-Smirnov Goodness-of-fit test
value is 0.0781 that leads to accept the hypothesis that the model fits the data
with a significance level of 0.05.
6
3.6. CONCLUSIONS
3.6
81
Conclusions
The following main conclusions can be drawn from our results:
- Our model of cholera dynamics explicitly accounts for the spatial distribution of the communities and their interconnections, proving capable to reproduce the main spatial and temporal patterns of the spreading for a welldocumented epidemic into a disease-free region in the KwaZulu-Natal Province
of South Africa.
- A significant role emerges for the ecological corridors defined by waterways
and river networks. Such hydrologic control derives from the transportation
and redistribution of the free-living infective propagules. In particular, because
vibrios can spread alongstream both upstream and downstream with a slightly
biased propagation downstream (which is to be expected, of course), the infection
patterns are anisotropic;
- Despite its satisfactory performance, the model is not completely reliable in
reproducing secondary peaks of infections in the tail of both annual outbreaks.
We speculate that they might be the combined results of seasonality in the epidemiological parameters and the presence of short-lived hyperinfectious bacteria.
This hypothesis remains to be tested and will be the object of future work.
In conclusion, we suggest that this approach represents a first step towards
understanding how hydrology and population distribution along the water network control the spreading of water-borne diseases.
Symbol
Description
Value
Note
Si
number of susceptibles at node i
eq. (3.1)
Ii
number of infected at node i
eq. (3.1)
Bi
concentration of V. cholerae in aquatic
eq. (3.1)
3
environment (cells/m )
Ci
number of cumulated cases of node i
eq. (3.7)
Hi
total human population size of node i at the
input data
disease-free equilibrium
n
population natality and mortality rate (day −1 )
a
rate of exposure to contaminated water (day
−1
K
concentration of V. cholerae in water that yields
)
5 · 10−5
estimated
1
estimated
50%chance of being infected with cholera (cells/m3 )
r
rate at which people recover from cholera (day −1 )
−1
m
mortality rate due to cholera (day
)
nB
net growth rate (usually negative) of V.cholerae
in the aquatic environment (day −1 )
p
0.2
estimated
−4
4 · 10
estimated
−0.228
calibrated
rate of production per infected person of V. cholerae
that reach the water body (cells · day −1 · person−1 )
Wi
volume of water (m3 ) at node i
SC i
critical threshold of node i
b
transport bias (Pout − Pin )
0.08
calibrated
l
V. cholera mobility (day −1 )
3.5
calibrated
Vi
vulnerability of node i: Hi /SC i
c
HT
p/(Kc)
eq. (3.3)
eq. (3.6)
3
per-capita water volume (m · person
threshold on the node population
combined parameters ratio (day
−1
−1
)
29000
)
−6
4.76 · 10
Table 3.1: Description of the symbols used in the text
calibrated
calibrated
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Chapter 4
Neutral Metacommunity Models
Predict Fish Diversity Patterns in
Mississippi-Missouri Basin 1
4.1
Summary
River networks, seen as ecological corridors featuring connected and hierarchical
dendritic landscapes for animals and plants, present unique challenges and opportunities for testing biogeographic theories and macroecological laws [Muneepeerakul et al., 2007]. While local and basin-scale differences in riverine fish diversity
have been analyzed as functions of energy availability and habitat heterogeneity [Guégan et al., 1998], scale-dependent environmental conditions [Angermeier
et al., 1998], and river discharge [Oberdorff et al., 1995; Xenopoulos and Lodge
1998]; a model that predicts a comprehensive set of system-wide diversity patterns remains elusive. Here we show that fish diversity patterns throughout
the Mississippi-Missouri River System (MMRS) are well described by a neutral
metacommunity model coupled with an appropriate habitat capacity distribu1
The contents of this chapter has been published in: Muneepeerakul, R., E. Bertuzzo, H.
Lynch, W. B. Fagan, A. Rinaldo, and I. Rodriguez-Iturbe (2008a), Neutral Metacommunity
Models Predict Fish Diversity Patterns in Mississippi-Missouri Basin, in press Nature.
85
86CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
tion and dispersal kernel. River network structure acts as an effective template
for characterizing spatial attributes of fish biodiversity. The estimates of average
dispersal behaviour and habitat capacities, objectively calculated from average
runoff production, are shown to yield reliable predictions of large-scale spatial
biodiversity patterns in riverine systems. The success of the neutral theory in
two-dimensional forest ecosystems and here in dendritic riverine ecosystems suggest the possible application of neutral metacommunity models in a diverse suite
of ecosystems.
4.2
Introduction
The Mississippi-Missouri River System (MMRS) is an invaluable resource of great
biotic diversity, including freshwater fish. Its vast extent spans across diverse
habitat types operating under varying environmental conditions (e.g., climate,
hydrologic regime, primary productivity, human disturbance); these diverse habitats are connected to one another by one river network. An analysis that adequately captures major spatial biodiversity patterns in such a system is therefore
noteworthy. A new theory for the explanation of communities organization, the
"Neutral Theory", has recently emerged [Hubbel, 2001]. The basic assumption
of neutrality is that to understand issues such as the relative abundances of
species, species-area relationships, and spatial and temporal turnover in species
community composition, one can assume that all species are the same. In neutral
theories, relative abundances change by chance, rather than because one species
is superior to another. This theory is seemingly in antithesis with the classical
Niche Theory (formulated first in the book "On the Origin of Species by Means
of Natural Selection" ,[Darwin, [1859]); however recent ecologist studies has tried
to unify them [Tilman, 2004]. In fact, even if species show niche differentiation
and tradeoffs in functional traits, the dynamical consequences of such differences
might be obscured because of stochastic processes. Neutral theory at the very
least provides the appropriate null model for evaluating patterns in comparative
data sets. In recent years, the neutral theory of biodiversity, with its minimal set
4.3. DATA
87
of assumptions and parameters, has proven both influential [Condit et al., 2002;
Purves and Pacala 2005; Holt, 2006; Alonso et al., 2006; Condit et al., 2002] and
controversial [McGill, 2003; Chave, 2004; Dornelas et al., 2006] as an explanation of biodiversity patterns. The theory, however, has been mainly tested with
ecosystems in two-dimensional (2-D) landscapes or a mean-field context, where
spatial aspects play little or no roles [Hubbel, 2001; Volkov et al., 2003; Condit
et al., 2002; Volkov et al., 2003; Dornelas et al., 2006; Etienne and Olff, 2005;
Condit et al., 2002; Walker and Cyr, 2007; Volkov et al., 2007]. Only recently
the roles of landscape spatial structure [Economo and Keitt, 2007], e.g., in river
networks [Muneepeerakul et al., 2007], have been investigated. Furthermore, implications of hydrologic controls placed by river networks as ecological corridors
have recently been explored [Campos et al., 2007; Bertuzzo et al., 2007].
Here we analyze a large database of fish diversity in the MMRS to compare
empirical biodiversity patterns against those predicted by a neutral metacommunity model. The data analysis provides important insights in its own right,
while the comparison with model results allows us to investigate the extent to
which a neutral model captures observed patterns and extends inferences from
the database. We demonstrate that by implementing the neutral model in the
Mississippi-Missouri river system and incorporating the effect of average annual
runoff production (AARP) on fish habitat capacities, we can effectively reproduce
a wide spectrum of observed biodiversity patterns.
4.3
Data
The data used in the analysis were obtained from the NatureServe database of
U.S. freshwater fish distributions, which summarizes museum records, published
literature, and expert opinion regarding fish species distribution in the entire
U.S. a. Due to the current lack of data availability, the Canadian portions of
the MMRS are not included in the analysis, but we do not expect this to affect
the key results and conclusions reported herein. The database consists in the
presence-absence data of 433 freshwater fish species in the 824 direct tributary
88CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
areas in which the Mississippi.Missouri basin is divided. Here, a direct tributary
area (DTA) is a geographic region directly draining to a group of streams, i.e., not
including areas upstream of it; the DTAs correspond to the USGS HUC8-scale
subbasins defined by National Hydrography Database Plus. Occurrence data and
river network structure can be combined and analyzed for several biodiversity
patterns. We consider three patterns:
1. local species richness (LSR) (or α diversity), the number of species found
in a randomly selected DTA;
2. species ranges, the area of the region populated by a species;
3. between-community (β) diversity. To characterize β diversity, we consider
the overall spatial decay of Jaccard’s similarity index [Condit et al., 2002]
(JSI). JSI of any pair of DTAs is defined as Sij /(Si + Sj − Sij ), where Sij
is the number of species present in both DTAs i and j, and Si is the total
number of species in DTA i. To achieve reliable statistics, we consider only
topological distances for which more than 500 DTA pairs exist.
The Mississippi-Missouri river basin with the size and shape of the DTAs
is presented in Figure 4.1a. Different colours represent different values of local species richness. The DTA with the maximum LSR is Pickwick Lake (156
species) at the borders of the states of Alabama, Mississippi, and Tennessee
(NHDPlus subbasin number 06030005). The map can roughly be divided into
the western, species-poor half and the eastern, species-rich half. The sharp drop
in the species richness occurs around the 100◦ W meridian, which is also known
to be the location of sharp gradients of annual precipitation [Dingman, 2002]
and runoff production [Gebert et al., 1987]. Figure 4.1b highlights a correlation
between species richness and runoff production. More runoff production implies
more resources and then more habitat capacity for the freshwater fishes. Finally,
following a basic ecological law, the greater is the habitat capacity, the higher is
the biodiversity. Although these gradients partly explain the spatial variations on
the LSR [Master et al., 1998; Matthews, 1998] we will show that the α diversity is
4.3. DATA
89
also controlled by the network structure. Downstream DTAs in fact might receive
enough water supply and have access to a larger species pool from their upstream
sub-networks to maintain high fish diversity. This determines an increasing trend
of α diversity going from upstream to downstream as shown in Figure 4.2 where
the local species richness is reported as a function of the topological distance
from the network outlet. The topological distance is a measure of distance along
the network. An increment in the topological distance occurs when one travels
along the network and crosses from one DTA to another. In the present case,
one unit of topological distance corresponds to a distance in the range of 100200 km. The distance zero corresponds to Atchafalaya, Louisiana (NHDPlus
subbasin number 08080101). The LSR profile shows a significant increase in
the downstream direction, except at the very end in Louisiana, where we hypothesize that the freshwater fish habitat capacities are significantly reduced by
salinity, co-occurrence/intrusion by some freshwater-tolerant estuarine or coastal
fish species, human disturbance and pollution. The overall downstream increase
in richness results from the converging character of the river network [Fernandes
et al., 2004] and is steepened by the dry-wet climatic gradient mentioned above.
Figure 4.3 presents the frequency distribution of LSR, whose two peaks at
low and high values reflect the difference between the western and eastern halves
of the Mississippi-Missouri river system.
The species ranges are presented in Figure 4.4 as a rank-range curve, in which
the fish species are ranked by their range sizes. The rank-range curve (akin to
the familiar rank-abundance curves [Hubbel, 2001]) yields a straight line on a
semi-log scale, a pattern reminiscent of the rank-abundance curves predicted by
the neutral theory [Hubbel, 2001] [ Purves and Pacala 2005][Walker and Cyr,
2007].
Figure 4.5 shows that the JSI decreases as the topological distance between
DTA pairs under consideration increases, an expected trend for diversity. Interestingly, however, the JSI does not vanish even for DTA pairs that are very far
apart. Such long-distance similarity in species composition is likely maintained
by species with extremely large ranges, e.g., Ictalurus punctatus (channel cat-
90CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
a)
0
70
≥140
b)
0
25
≥50 cm
Figure 4.1: The maps of (a) local species richness (LSR), or diversity, of the
freshwater fish, and (b) the average annual runoff production (AARP) in the
Mississippi-Missouri River System (MMRS).
fish), Ameiurus melas (black bullhead), and Ameiurus natalis (yellow bullhead).
4.3. DATA
91
Local species richness
100
80
60
40
20
0
30
25
Upstream
20
15
10
5
0
Outlet
Topological distance to the outlet
Figure 4.2: Local species richness (LSR) profile as a function of the topological
distance from the outlet; squares (average values) with error bars (ranging from
the 25th to 75th quantile) represent the empirical data, whereas the solid line
represents the average values of the model results.
92CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
100
Frequency
80
60
40
20
0
0
50
100
150
Local species richness (LSR)
Figure 4.3: Frequency distribution of local species richness (LSR); bars and the
solid line represent the empirical data and the average values of the model results
respectively.
4.3. DATA
93
3
10
2
Range
10
1
10
0
10
0
100
200
300
400
500
Ranked species
Figure 4.4: Rank-range curve of freshwater fish species in the Mississippi-Missouri
River System (MMRS); squares represent the data and the line represents the
model result. Here, the range of a species is simply the number of DTAs in which
that species is reported as present.
94CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
Jaccard’s similarity index
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
Topological distance between DTA pairs
Figure 4.5: Jaccard’s similarity index (JSI) as a function of topological distance
between pairs of DTAs, whose overall decay represents diversity; the squares with
error bars represent the average values with the range between the 25th and 75th
quantiles of the empirical data and the line represents the average values of the
model results.
4.4. METHOD
4.4
95
Method
Our model is of structured metacommunity type. The neutral theory of biodiversity is implemented in the Mississippi-Missouri river system, using its network
as the structure of the metacommunity. Each DTA is a local community in that
metacommunity and has a different fish habitat capacity, H, defined as the number of ’fish units’ sustainable by resources in that particular DTA, a fish unit can
be thought of as a sub-population of fish of the same species. H is assumed to
be proportional to the product of the DTA watershed area and average annual
runoff production (AARP), an indicator of the amount of resources available
for fish [Guégan et al., 1998; Oberdorff et al., 1995]. Runoff is the portion of
precipitation that is drained by the river network. It depends on precipitation,
evapotranspiration and infiltration. The map used is estimated from the streamflow data of small tributaries collected from about 12,000 gaging station averaged
over the period 1951-80. For details, see Gebert et al. [1987]. Habitat capacity
of DTA i, Hi , is determined by:
Hi = CH
AARPi ∗ WAi
PN
k=1
AARPk ∗WAk
N
(4.1)
rounded to the nearest integer. WA denotes watershed area, N (= 824) the total
number of DTAs, and CH the estimate (due to rounding) of average habitat
capacity.
The model captures basic ecological processes: birth, death, dispersal, colonization, and diversification. Every DTA is assumed to always be saturated at its
capacity, i.e., no available resources are left unexploited. At each time step, a fish
unit, randomly selected from all fish units in the system, dies and the resources
that previously sustained the unit are freed up and available for sustaining a new
fish unit. With probability ν, the diversification rate, the new unit will represent
a new species (the diversification is a rate per birth and is due to speciation, or
external introduction of non-native species, or immigration (and re-immigration)
of a new species from outside the MMRS); with probability 1 − ν, the new unit
96CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
will be colonized by a species already existing in the system (the MMRS). In the
latter case, the probability Pij that an empty unit in DTA i will be colonized by
a species from DTA j is determined as follows:
Kij Hj
Pij = (1 − ν) PN
k=1 Kik Hk
(4.2)
where Kij is the dispersal kernel (see below), Hk is the habitat capacity of DTA
k, and N is the total number of DTAs (N=824). All the fish units in DTA j have
the same probability of colonizing the empty unit in DTA i where the death took
place. Therefore the probability of successful colonization, as the neutral theory
assumes, depends only on the relative abundance of the species. A dispersal
kernel determines how the fish units move within the river network. The model
uses the topological, rather than Euclidean, distances between DTAs as they are
representative of how far fish travel. Here, it is assumed to take the form of a
combination of back-to-back exponential and Cauchy distribution; note that a
combination of several theoretical dispersal kernels has been used to achieve a
good representation of real dispersal kernels. The dispersal kernel in this model
can be expressed as:
h
Lij
Kij = C a
b2 i
+ 2
Lij + b2
(4.3)
where Kij is the probability that a fish unit produced at DTA j arrives at DTA i
after dispersal; C is the normalization constant; Lij is the effective distance, defined as N Dij + wu N Uij , where N Dij and N Uij are the numbers of downstream
and upstream steps comprising the shortest path from DTA j to DTA i, and wu is
the weight factor modifying the upstream distance; wu > 1 implies downstreambiased dispersal, thereby characterizing dispersal directionality; a(< 1) and b
characterize the exponential and Cauchy decays, respectively. Here, C is deterP
mined numerically such that, for every DTA j, i Kij = 1; that is, no fish can
travel out of the network. At the upstream ends of the system, this is obvious;
it is also true at the downstream end of the system, i.e., the outlet to the Gulf of
Mexico, a marine body that acts as a barrier to freshwater fish. Finally, the dis-
4.4. METHOD
97
persal kernel of every species is assumed to be the same; this is perhaps a strong
assumption as fish species obviously differ in their dispersal abilities. However, a
’functional equivalence’ among species is a key way in which the neutral theory
of biodiversity departs from classical ecological models. We assume the species
equivalence to study just how good a fit the neutral metacommunity model can
make to our data in the absence of detailed, species-specific information. The
simulations are run until the system reaches a steady state and the biodiversity
patterns of interest are determined and compared to the empirical patterns.
The best-fit parameter set (see table 4.1) has been chosen following this procedure. We ran several simulations with different sets of parameters distributed
over a meaningful wide range. For every simulation, we computed the error between data and simulation for each of the four biodiversity patterns analyzed:
(1) mean LSR as a function of topological distance to the outlet, (2) frequency
distribution of LSR, (3) the rank-range curve, and (4) Jaccard’s similarity index (JSI) as a function of the topological distance between direct tributary area
(DTA) pairs. The error for pattern k, Ek (k = 1, 2, 3, 4) is estimated by the
mean square deviation between data and predicted values normalized by the
data variance; this can be expressed as follows:
PNk
(xk,i − x̂k,i )2
Ek = PNi=1
k
2
i=1 (xk,i − hxk i)
(4.4)
where Nk is the number of data points used in fitting pattern k, xk,i and x̂k,i are
data point i of pattern k and its predicted value, respectively, and hxk i the mean
value of the data points of pattern k. Note that for the rank-range curve, the
error, E3 , is estimated by the normalized mean square deviation between base-10
log values of empirical and predicted range of the first N3 ranked species, where
N3 is the minimum between empirical and modelled total number of species. We
then define the total error, T E, of each parameter set as E1 + E2 + E3 + E4 . The
parameter set with the minimum T E is selected. Then, the procedure is repeated
with increasingly narrow ranges of parameters. We tried the fitting procedure
with different types of dispersal kernels, and the best results were obtained from
98CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
the exponential-Cauchy kernel. In this case, there are a total of five parameters,
P
namely the system’s total habitat capacity ( i Hi ) (or equivalently the estimate
of average habitat capacity, CH ), diversification rate (ν), dispersal directionality
(wu ) and two parameters characterizing the shape of dispersal kernel (a and b).
We find that the assumption of no dispersal directionality, i.e., wu = 1, generally
yields better results. Therefore, wu is fixed at 1, and we search for the best values
of the remaining four parameters. Let us stress here that as we are dealing with
multiple patterns of a system with underlying stochastic and nonlinear processes,
the significance of the error measures should not be taken in an absolute sense,
especially when the differences are small. Rather, they should be used as helpful
guidelines to aid our judgment of the parameters, not strict criteria. Indeed, a
number of nearly best parameter sets give essentially the same results and do
not alter any of our conclusions. Furthermore, while we have clearly shown that
the neutral metacommunity model is capable of producing a wide spectrum of
spatial biodiversity patterns found in the data, similar good fits are possible for
more than one set of parameters in some cases [Etienne et al., 2006].
With such caveat in mind, we summarize in table 4.1 the parameters corresponding to the results reported. CH of 530 corresponds to the total number of
P
fish units, 824
i=1 Hi , of 436,731, with individual values ranging from 4 to 3,174.
The dispersal parameters imply that on average, about 70% of the fish offspring
stay within the DTAs where they were born, and that those that travel outside
of their birth DTAs cross, on average, 1.8 DTAs. This average distance is deceptively small: as will be discussed below, the heavy-tail character is essential for
long-distance similarity in species composition, despite the small average value.
The diversification rate may seem relatively high; this may reflect the dynamics
of rare species that appear to be new to the system any time they are introduced
and re-introduced from outside the MMRS.
4.5. RESULTS AND DISCUSSION
Parameter
99
Symbol
Value
CH
530
Diversification rate
ν
1.5 × 10−4
Exponential parameter
a
0.26
Cauchy parameter
b
0.03
wu
1
Habitat capacity proportionality constant
Directionality
Table 4.1: Best fit set of parameters
4.5
Results and Discussion
The model is able to well reproduce all the biodiversity pattern analyzed. In addition to the general trend and magnitude, the model also captures fine-structured
fluctuations of the LSR profile (Figure 4.2). The fits to the LSR frequency distribution and diversity pattern are also very good (Figure 4.3 and 4.5). The
straight-line character of the rank-range curves is quite apparent for both the
data and the model result (Figure 4.4).
We begin our discussion by emphasizing the advantages of the neutral metacommunity model. From Figure (4.1) in the main text, one may argue that a
regression model against the average annual runoff production (AARP) is sufficient to predict the spatial patterns of local species richness (LSR), or α diversity,
and one would in fact obtain reasonable predictions. Figure (4.6) shows that the
linear regression against AARP in a log-log scale can explain up to 58% of the total variance of LSR (r2 = 0.58). However, such a regression approach, as well as
its more sophisticated counterparts [Oberdorff et al., 1995; Guegan et al., 1998]
while providing valuable information, is incapable of simultaneously investigating other patterns, e.g., species ranges, and α diversity and thus yields a rather
incomplete picture of the biodiversity patterns of the river network [Grimm et
al., 2005]. Here we have shown that the neutral metacommunity model, which is
a process-based approach, can circumvent such limitations. What is particularly
important in this model is the small number of parameters it requires, which in
100CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
turn reduces the amount of fieldwork necessary to estimate them.
2
Local species richness
(α diversity)
10
1
10
0
10 −1
10
0
10
1
10
2
10
Average annual runoff production (cm)
Figure 4.6: The empirical relationship between average annual runoff production
(AARP) and local species richness (LSR), or α diversity, in a log-log scale.
The LSR profile in Figure 4.2 is attributable to the converging character of
river network structure and is steepened by the climatic gradient. Here we verify
the statement via both the empirical data and our model. In Figure 4.7, we
compare the LSR of a DTA against that of its immediately downstream DTA.
Due to the proximity of each DTA pair, it is reasonable to assume that the climate
is essentially the same for each comparison and that the emerging patterns result
from the network structure. That 76% of the data points lie above the diagonal
line indicates that LSR is enhanced in the downstream direction [Fernandez et
al., 2004].
We also explore the effects of network structure by removing the climatic effect
4.5. RESULTS AND DISCUSSION
101
160
LSR at the immediately downstream DTA
140
120
100
80
60
5
6
7
8
10U
10L
11
40
20
0
0
20
40
60
80
100
120
140
160
LSR at a given DTA
Figure 4.7: The pairwise comparison between local species richness (LSR) of a
given DTA and LSR of its immediately downstream DTA. The legend shows the
region numbers as defined by NHDPlus. That 76% of the points lie above the 1:1
line indicates that the converging nature of the network indeed enhances LSR in
the downstream direction. Note that as the climatic conditions do not change
dramatically over the distance between two adjacent DTAs, the climatic control
on LSR is eliminated in this pairwise comparison.
102CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
from the model; that is, the habitat capacity is assumed constant throughout
the system (i.e., with the same total habitat capacity). The resulting trend is
compared to the trend in Figure (4.8). The result supports our earlier argument:
the converging character of the network structure alone is sufficient to give rise
to the increasing trend in the downstream direction of the LSR profile, and the
climatic effect steepens the trend, in this case via reducing the habitat capacity,
and thus LSR, in the upstream portions of the Mississippi-Missouri River System.
Downstreambiased dispersal can also steepen the trend, although as mentioned
in the main text, the model results suggest that the dispersal kernel of fish in
the MMRS is relatively symmetric.
The decaying pattern of Jaccard’s similarity index (JSI) (Figure 4.5), or β
diversity, proved difficult to fit as it is sensitive to the shape of the dispersal
kernel. Attempts with a number of dispersal kernels suggest that the average
dispersal behavior of fish exhibits a heavy-tailed character. Figure 4.9a shows
the general shapes of two dispersal kernels: exponential and Cauchy. The Cauchy
kernel decays much more slowly than the exponential kernel at great distances but
drops more sharply at short distances. This difference of the kernels is reflected
in their ability to capture the empirical β diversity pattern (Figure 4.9b).
The exponential kernel can reproduce the β diversity pattern over short distances well but always underestimates the JSI at great distances (i.e., the tail
of β a diversity). Conversely, the Cauchy kernel can reproduce the tail well but
always underestimates the JSI at short distances. These observations motivate
us to employ a combination of the two kernels, which features both exponential
local dispersal and Cauchy long-distance dispersal. Note that it has been recognized that factors affecting the dispersal processes of plants vary across distance,
and that the best representation of the dispersal may be a combination of several dispersal kernels [Levin at al., 2003; Higgins and Cain, 2002], our analysis
suggests that such is also the case for fish. The good fit of the combined kernel
(Figure 4.5) implies the average dispersal behavior of fish: the majority of fish
travel rather locally, but a non-negligible fraction travels very great distances.
This should have important implications on large-scale conservation campaigns
Average local species richness
4.5. RESULTS AND DISCUSSION
103
100
80
60
40
20
0
30
Upstream
25
20
15
10
5
0
Outlet
Topological distance to the outlet
Figure 4.8: The effects of network structure and climatic gradient on the spatial
profile of local species richness (LSR): squares represent the mean empirical LSR
at a given distance to the outlet, the solid line the model results with the habitat
capacity distribution that incorporates the climatic gradient, and the dash line
the model results with constant habitat capacity at every DTA. The climatic
gradient clearly steepens the LSR profile.
and resource management; for example, correct characterization of the dispersal
kernel is essential to estimate the speed of the spread of invasive species [Kot at
al., 1996].
Here, a caveat is in order: the exponential-Cauchy combination is by no means
unique. There are in fact a number of kernel combinations (properly parameterized, of course) that can yield good fits to the JSI and other patterns. Specifically,
104CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
these combined kernels must maintain the key features that a majority of fish
travel locally and a fraction travels very far. For example, a combination of
two exponential kernels one with a small mean and the other with a very large
mean-yield an equally satisfactory fit. Consequently, from the point of view of
theoretical progress, the soundness of these combined kernels will be judged by
their mechanistic explanation and analytical tractability.
4.6
Conclusions
Simultaneous fits of these diverse patterns are a very stringent test for a model
[Grimm, 2005], especially a model with only four parameters in the present case.
Importantly, the model also allows for insightful inferences. The parameters
corresponding to the best fits imply that the spread of the average fish species
is quite symmetric, i.e., significantly biased in neither upstream nor downstream
directions (wu = 1). The model results also suggest that on average, the majority
of fish disperse locally (i.e., to nearby DTAs), but a non-negligible fraction travels
very long distances.
Given the diverse environmental conditions covered by the MMRS, our demonstration that a simple neutral metacommunity model coupled with an appropriate
habitat capacity distribution and dispersal kernel can simultaneously reproduce
several major observed biodiversity patterns has far-reaching implications. These
results suggest that only parameters characterizing average fish behaviour, as
opposed to those characterizing biological properties of all different fish species
in the system, and habitat capacities and connected structure suffice to obtain
reasonably reliable predictions of large-scale biodiversity patterns. The neutral
metacommunity model also provides a null model against which more biologically
realistic models may be compared, and further developments in our understanding of riverine networks and fish movement will allow for continued improvement
between model and data. Indeed, although this modelling approach has been
shown here to be useful for investigating key spatial patterns, it is crucial to
recognize that "neutral pattern does not imply neutral process". Therefore, dif-
4.6. CONCLUSIONS
105
ferent approaches will be necessary to predict transient dynamics of the system
or understand patterns and dynamics of specific species.
Finally, as mobile fish in a river network differ drastically from sessile trees in a
forest, it is remarkable that the neutral theory can reproduce key biodiversity patterns of both sets of organisms quite well. This suggests that patterns predicted
by the neutral metacommunity model-with appropriate habitat capacity distribution and dispersal kernel-may be broadly applicable across diverse ecosystems.
Importantly, it also offers a general, parsimonious modelling approach that acts
as a coherent framework to study several large-scale spatial biodiversity patterns
simultaneously. This framework allows for direct linkages from various environmental changes to biodiversity patterns. For example, changes in precipitation
patterns, perhaps due to global climate change, can now be mapped to changes
in habitat capacities in the model; changes in connectivity among local communities, e.g., flow rerouting or damming in the case of fish, can be characterized
by modifying the dispersal kernel. These linkages in turn enable us to make reliable predictions of a rather comprehensive set of altered biodiversity patterns,
with important implications for conservation campaigns and large-scale resource
management.
106CHAPTER 4. NEUTRAL METACOMMUNITY MODELS ON RIVER NETWORKS
0
10
(a)
−2
10
−4
10
K(x)
K(x) = b2 /(x2 + b2)
Cauchy
−6
10
−8
10
x
K(x) = a
Exponential
−10
10
0
5
10
15
20
25
x
0.7
(b)
Jaccard’s similarity index
0.6
Data
Exponential
Cauchy
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
Topological distance between DTA pairs
Figure 4.9: The dispersal kernels and their associated decaying patterns of Jaccard’s similarity index (JSI) (β diversity): (a) comparison between the shapes
of the exponential and Cauchy kernels and (b) their predicted β diversity patterns. The exponential kernel underestimates the empirical JSI at great distances, whereas the Cauchy kernel underestimates it at short distances.
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