We propose a novel ecohydrological model for the invasion of inland waters by the zebra mussel Dreissena polymorpha and test it against field data gathered within the Mississippi-Missouri river system in North America. This biological invasion poses major ecological and economic threats, especially due to the huge population densities reached by local zebra mussel colonies and the species' unparalleled dispersal abilities within fluvial systems. We focus on a quantitative evaluation, attempted here for the first time, of the individual roles and the mutual interactions of drivers and controls of the Mississippi-Missouri invasion. To this end, we use a multilayer network model accounting explicitly for zebra mussel demographic dynamics, hydrologic transport, and dispersal due to anthropic activities. By testing our results against observations, we show that hydrologic transport alone is not sufficient to explain the spread of the species at the basin scale. We also quantify the role played by commercial navigation in promoting the initial, selective colonization of the river system, and show how recreational boating may have determined the capillary penetration of the species into the water system. The role of post-establishment dispersal mechanisms and the effectiveness of possible prevention measures are also discussed in the context of model sensitivity and robustness to reparametrization.
 The zebra mussel Dreissena polymorpha (Pallas, 1771) is a freshwater bivalve native to Eurasia. Owing to its adaptability to a wide range of environmental conditions, combined with dispersal abilities within fluvial systems that are unrivalled by other freshwater invertebrates [Carlton, 1993], this invasive species managed to diffuse all over Europe and North America. After establishment, zebra mussel colonies can rapidly reach population densities in the order of tens (or even hundreds) of thousands of individuals per square meter [Bossenbroek et al., 2007] and inflict huge ecological and economic damages. In fact, zebra mussel colonies may deeply alter invaded ecosystems by filtering large volumes of water, thus removing phytoplankton and boosting nutrients, and by severely impairing the functioning of water works [MacIsaac, 1996; Higgins and Vander Zanden, 2010]. Although the economic impacts of the invasion have been poorly documented [Strayer, 2009], D. polymorpha is estimated to cost the United States about one billion dollars per year due to direct economic losses and related control efforts [Pimentel et al., 2005]. The zebra mussel has thus become a prototypical example of invasive species and has been included in the “100 World's Worst Invasive Alien Species” list drawn up by the International Union for Conservation of Nature [Lowe et al., 2000].
 One of the recurrent and most astonishing features of zebra mussel invasions is the speed at which the species can spread over river networks. The most notable example to date is the invasion of the Mississippi-Missouri river system (MMRS, see Figure 1a; data available online at http://fl.biology.usgs.gov/Nonindigenous_Species/ZM_Progression/zm_progression.html). The invasion started from the Great Lakes region (Michigan), where some specimens were first sighted in the late 1980s after being probably introduced via ballast water sheddings by boats sailing from Europe. By 1990 D. polymorpha had colonized the shores of the Great Lakes and made its way into the Illinois River through the Chicago Sanitary and Ship Canal, which connects the Great Lakes watershed with the MMRS. By 1991 the zebra mussel had invaded the Illinois River and started diffusing into the Mississippi River. Afterwards, the zebra mussel began its spread over the Mississippi River, reaching Louisiana along the backbone of the MMRS (i.e., along the Mississippi River), as soon as 1993 (Figure 1b). In the meanwhile, the species showed up also in selected reaches of the upper branch of the Mississippi River and in some of its most important tributaries (Ohio, Tennessee, and Arkansas Rivers), as well as in other North American river systems (St. Lawrence, Hudson, and Susquehanna). The rate of spread of D. polymorpha has decreased remarkably after 1994, primarily because the species did not expand west of the 100th meridian. However, the zebra mussel has been steadily infilling and colonizing new reaches of the MMRS during the last decade (Figure 1c). Quite surprisingly, a few connected river systems (most notably, the Missouri River) have not been invaded until recently. Nowadays, the zebra mussel steadily occupies much of central and eastern North America.
D. polymorpha invasion patterns result from the interplay between local demographic processes occurring over long time scales and basin-scale transport phenomena taking place over much shorter time spans. This is mainly due to the peculiarity of the species' life cycle, which can be roughly subdivided into two main periods: a short larval phase, lasting from a few days to a few weeks [Sprung, 1993; Stoeckel et al., 1997], and a relatively long adult stage (up to three years in North America, see Mackie and Schloesser ). Adults live anchored to a solid substratum, while larvae (also known as veligers) can be transported by the water flow, sometimes traveling for hundreds (or even thousands) of kilometers before settling [Stoeckel et al., 1997]. Therefore, rivers represent the primary and natural pathway allowing species spread. However, anthropic activities can often result in extrarange dispersal, i.e., in the movement of propagules from the current species range to a new area of habitat, thus in turn remarkably favoring both the speed and the extent of the biological invasion [Wilson et al., 2009].
 In the zebra mussel case this has been known for a long time, as any human activity that involves the movement of a mass of water can be a potential vector for the spread of D. polymorpha [Carlton, 1993]. Commercial navigation represents a major driver of movement for the zebra mussel [Carlton, 1993; Johnson and Carlton, 1996; Allen and Ramcharan, 2001]. For instance, large quantities of veligers are often shipped within the ballast water of commercial vessels. As ports are located even hundreds of kilometers apart from each other, connections among them allow the species to disperse over very long distances and to colonize stretches of the river network that could not have been reached otherwise. Furthermore, empirical evidence suggests that recreational boating may be an important determinant of medium-range mussel redistribution [Johnson and Carlton, 1996; Schneider et al., 1998; Buchan and Padilla, 1999; Bossenbroek et al., 2001]. A common mechanism associated with transient recreational boating is the transport of juveniles and adult mussels via macrophytes entangled on boat trailers [Carlton, 1993]. This mechanism has been proposed as the most likely cause of D. polymorpha interbasin range expansion due to touristic boating [Ricciardi and Carlton, 2001; Timar and Phaneuf, 2009]. Therefore, commercial navigation represents an efficient vector of long-distance dispersal, while touristic boating can provide a capillary mechanism for medium-range mussel relocation.
 Due to the importance of the zebra mussel as an ecosystem invader, significant modeling effort has already been devoted to the understanding of its demographic dynamics at a local scale [MacIsaac et al., 1991; Strayer and Malcom, 2006; Casagrandi et al., 2007], as well as to the description of the species spread along rivers [Stoeckel et al., 1997; Schneider et al., 2003; Morales et al., 2006; Mari et al., 2009], to the analysis of long-distance dispersal [Schneider et al., 1998; Bossenbroek et al., 2001, 2007; Leung and Mandrak, 2007] and to the study of relevant environmental correlates [Strayer, 1990; Drake and Bossenbroek, 2004; Leung and Mandrak, 2007; Whittier et al., 2008]. However, all these contributions have never been synthesized to produce a mechanistic, comprehensive analysis of the regional spread of D. polymorpha in North American inland waters. To single out the role of drivers and controls of the MMRS invasion, here we propose a novel spatially explicit, time-hybrid, multilayer network model that allows for the intertwining of hydrologic controls, acting through the ecological corridors defined by the river network, with long- and medium-range dispersal controlled by anthropogenic factors, which define secondary movement networks. We show that integrating multiple dispersal pathways is crucial to understand zebra mussel invasion patterns and, in particular, the role played by human activities in promoting the spread. Our analysis shows quantitatively that commercial navigation has been the most important determinant of the early invasion of the Mississippi-Missouri, and that recreational boating can explain the long-term, capillary penetration of the species into the water system.
2. Novel Multilayer Network Model for the Spread of the Zebra Mussel in the MMRS
 Based on the life cycle of D. polymorpha, we propose a new multilayer network model to describe the spread of the species in the MMRS. The primary, hydrologic network layer is an oriented graph made up of edges (channels) and nodes obtained via the spatial discretization of the river system (see below). Some of these nodes are special in that they represent closed water bodies connected to the water system. The nodes of the secondary layers are disjoint subsets of the nodes in the primary layer. In particular, the nodes of the second layer are the main fluvial ports of the MMRS (Figure 2a and Table 1), while the nodes of the third layer represent the largest lakes, impoundments, and ponds connected to the water system (Figure 2b). The edges of the secondary layers are either defined by commercial navigation (second layer) or recreational boating (third layer).
Table 1. The Most Important Fluvial Ports of the MMRS in Terms of Total Yearly Traffic According to the 2007 Report of the WCSCa
 The local-scale dynamics of riverine zebra mussel populations can be described by adapting the nonlinear, discrete-time model with density dependence and age structure proposed by Casagrandi et al. , which has already been used to study mussel demographic dynamics. Let nk(x, t) (mussels m−2) be the density of adult mussels in age class k (1 ≤ k ≤ 3, as the lifespan of D. polymorpha in North America does not typically exceed three years; see Mackie and Schloesser ) at location x on the river network and time t. The interannual dynamics linking the abundances of mussels in two subsequent years is thus defined by
where vs(x, t) (veligers m−2) is the density of settling veligers, is the survival from the veliger stage to the first adult class (thus accounting for all the mechanisms that can affect mussel recruitment like, e.g., the unavailability of suitable settling sites due to mechanical and/or chemical reasons), the (k = 1, 2) represent the yearly individual survival probabilities from age k to age k + 1, and the are mussel densities evaluated after relocation due to recreational transport has possibly taken place (see the third network layer below).
Equations (1) constitute the simplest ecologically well-founded model for local-scale dynamics of adult zebra mussels. However, they do not account for some other possibly relevant details of zebra mussel ecology at the local scale. In particular, local regulatory mechanisms such as competition for space or resources among adult mussels, and density-dependent larval settlement [Chase and Bailey, 1999] may be expected to play some role in determining the size and structure of D. polymorpha colonies, which can reach very high population densities. We refrained from introducing such mechanisms into the models, however, in order to keep complexity at a minimum. In the same way, Allee effects, whose relevance for alien species dynamics has been the subject of theoretical studies [Lewis and Kareiva, 1993; Kot et al., 1996], and that have sometimes been envisaged as possible determinants of zebra mussel's invasion success, are not included in the present analysis. In fact, no mechanistic demographic models including Allee effects have been proposed so far for D. polymorpha populations [Leung et al., 2004; Potapov and Lewis, 2008]. Because at large spatial and temporal scales the abundances of generated, transported, and settling veligers is likely to be strongly related to local population densities and age structures, we relied on tested demographic models specifically developed for the zebra mussel [MacIsaac, 1996; Strayer and Malcom, 2006; Casagrandi et al., 2007], none of which includes Allee effects (but see section 4.3).
2.2. Hydrologic Transport
 The local abundance of settling veligers vs(x, t) is the result of within-year processes occurring after reproduction, namely larval mortality, dispersal, and transport due to both river flow and human activities. In particular, the initial abundance of larvae v0(x, t) resulting from reproduction can be computed as
where fk (with 1 ≤ k ≤ 3) is the average numbers of eggs released (and successfully fertilized) per adult female in class k, the sex ratio at birth being typically balanced [Mantecca et al., 2003]. Larval mortality and transport can be described by means of a partial differential equation (PDE) approach [Schneider et al., 2003; Mari et al., 2009]. From a technical viewpoint, coupling PDE's with equations (1) makes the full model for the demography and spread of D. polymorpha a hybrid mathematical system [Pachepsky et al., 2008].
 Let (veligers m−2) be the abundance of veligers in the water column at spatial location x and year t, evaluated days after spawning. The abundance of settling larvae at the end of the dispersal period can be computed along each river stretch of the river network by integrating the following PDE between the time of spawning and that of settling:
is the total density of settled adult mussels. In equation (2)u (km d−1) represents the mean water velocity, which is reasonably assumed to be spatially constant for both landscape-forming and average events, owing to the self-tuning geometry of open channel flows and to the different leading-flow resistance mechanisms from source to outlet [Rodriguez-Iturbe and Rinaldo, 1997]. As for (km2 d−1), it represents longitudinal dispersion, which in this framework also accounts for processes other than hydrodynamic dispersion (say, the macroscopic shear-flow mixing processes typical of fluvial environments; e.g., Fischer et al. ), such as local-scale, along-stream larval dispersal operated by aquatic animal or small sailing boats (for a related discussion on active and passive transport in fluvial environments see, e.g., Bertuzzo et al. ). Note that for this reason no direct measurement of hydrodynamic dispersion (e.g., with tracers) might be simply taken at face value for veliger dispersion.
 Larval mortality may depend on both predation by filter-feeding adult mussels, which has been shown to be a key mechanism in the self-regulation of zebra mussel populations [MacIsaac et al., 1991; Strayer and Malcom, 2006; Casagrandi et al., 2007], and other natural causes not related to the presence and abundance of the population itself [Schneider et al., 2003; Stoeckel et al., 2004a]. In equation (2) (d−1) corresponds to the baseline larval mortality rate, while (m2 mussels−1 d−1) represents the adult per capita filtering rate leading to density-dependent mortality. The source term (veligers m−2 d−1) is an external flux of veligers, accounting for larval input from upstream reservoirs. In particular, it represents veligers that enter the system from Lake Michigan through the Illinois River. Quite interestingly, in absence of hydrological transport and input sources (u = 0, , and ) model (1−2) would converge to the spatially implicit density-dependent model proposed in Casagrandi et al. .
 The integration of equation (2) over the time horizon , where (day) is the average time span needed for a veliger to mature, gives the distribution of veligers that are ready to settle at the end of the larval phase in each year of the simulation. Solving equation (2) over the MMRS requires ad hoc numerical techniques (see below), and suitably defined initial and boundary conditions. For each year t in the simulation horizon we use the distribution of veligers released after reproduction as initial condition for intra-annual larval transport, i.e., v (x, t, 0) = v0(x, t), while we adopt no-flux and absorbing boundary conditions at the headwaters and the outlet of the river network, respectively.
2.3. Long-Distance Larval Dispersal due to Inland Commercial Navigation
 In order to give a straightforward, yet quantitative description of larval movement due to trading navigation, we estimate the fluxes of waterborne commercial traffic among the main fluvial ports of the MMRS (Figure 2a and Table 1) with a production-constrained gravity model [Erlander and Stewart, 1990]. Notice that port-to-port vessel traffic is not the only mechanism related to commercial navigation that may be responsible for the spread of the zebra mussel. Barge traffic can in fact be a viable driver of medium-distance, along-stream mussel dispersal [Keevin et al., 1992; Allen and Ramcharan, 2001]. Commercial barges are usually towed in small, variable groups, and can use smaller ports than individual vessels. As a result, they are less amenable to a quantitative analysis of mussel transport at a regional spatial scale than traditional commercial traffic, for which federal data are available. We have thus chosen port-to-port traffic as a proxy for the complex set of mechanisms related to commercial navigation.
 Stemming from transportation theory, gravity models can be used to describe human movement when actual mobility data are not available [Erlander and Stewart, 1990]. Models of this family have already been applied to describe the spread of the zebra mussel among neighboring lakes [Schneider et al., 1998; Buchan and Padilla, 1999; Bossenbroek et al., 2001; Leung et al., 2004; Bossenbroek et al., 2007]. Here we assume that the flux of boats departing from a port (say p) is proportional to the tonnage of goods passing through it, i.e., its mass Mp. The fraction of boats that travel to a destination port q increases with Mq and decreases with the distance Dpq (km) between p and q evaluated along the river network, so that commercial traffic directed from port p to port q is proportional to
where is a positive parameter describing the decay of port-to-port boat fluxes with increasing distance among ports. By assuming that the flux of transported veligers is proportional to that of vessels, larval transport due to commercial navigation can be evaluated with the following model:
where ip and iq are the nodes of the second network layer corresponding to ports p and q, respectively, (d−1) is the port-dependent rate at which veligers enter the ballast water of departing ships, np is the total number of ports, ub (km d−1) is the mean velocity of commercial boats, and H(·) is the Heaviside step function. We assume that veligers transported within boat ballast waters experience the same baseline mortality as those transported in the water flow, which translates into an exponential decay of larval abundance. We note that this assumption might lead to an overestimation of larval survival during transport, because we do not account for possible surplus mortality due to the noxious properties of ballast waters (but see below). We also note that in our model the effects of port-to-port larval shipping can be removed by simply setting , which allows for the evaluation of different invasion scenarios.
2.4. Medium-Distance Dispersal due to Recreational Boating
 Adult mussel transport due to recreational navigation can be evaluated by means of a gravity model accounting for the fluxes of boats that are trailered among neighboring lakes. Notice that boaters may occasionally move veligers as well [Carlton, 1993; Johnson and Carlton, 1996], yet the effects of incidental larval sheddings from recreational boats are far less noticeable, as veligers are expected to become diluted and suffer high mortality [Sprung, 1993].
 To estimate lake-to-lake boat fluxes, we extract the largest lakes, impoundments, and ponds connected to the MMRS from the GNIS (Geographic Names Information System) and the NHD (National Hydrography Data set; see Figure 2b). Specifically, we select closed water bodies with a minimum surface of 0.5 km2. Then, the flux Jlm of boats that are trailered from one lake (say l) to one of its neighbors (say m) is proportional to
where Sl and Sm are the surfaces (km2) of the two neighboring lakes, Bl is a measure of the number of recreational boats registered in the region (Table 2), dlm (km) is the Euclidean (i.e., aerial) distance between the two lakes, and is a shape parameter accounting for the decay of the number of travels with increasing distance among lakes. We assume that l and m are neighbors if they lie within a maximum Euclidean distance dmax (km). If we further assume that the flux of transported mussels is proportional to that of boats, and follow the same line of reasoning used to obtain the gravity model of equation (3), we obtain
where il and im are the nodes of the third network layer corresponding to lakes l and m, respectively, (d−1) is the lake-dependent rate at which mussels leave lake l, nl is the number of neighbors for lake l, (day) represents the time of boat trailering from one lake to another, is the survival probability during transport between the two lakes, H is the Heaviside step function, and (mussels m2 d−1) is an input term due to an external flux of recreational boating. In particular, we consider an incoming flux of mussels from Lake Michigan, Erie, and Ontario to nearby MMRS lakes (i.e., those located within a dmax radius from the Great Lakes). The effects of mussel relocation due to recreational activities can thus be estimated by integrating model (4) over the summer season, when most recreational trips occur. The solution of equations (4) has to be plugged into equations (1). As in the commercial network layer, the effects of recreational boating on mussel transport can be ruled out by setting .
Table 2. Total Number of Registered Boats in States With Lakes Connected to the MMRS According to the 2005 Report of the BTS
3.1. Numerical Simulation of the Multilayer Network Model
 In order to numerically integrate the multilayer network model just introduced, the MMRS can be usefully represented as an oriented graph [Rodriguez-Iturbe and Rinaldo, 1997; Muneepeerakul et al., 2008]. To this end, we derive the relevant discretized river network from the National Hydrologic Data set (NHD, available online at http://nhd.usgs.gov/). Following NHD classification, we consider the MMRS network at thinner level 1 (i.e., major streams) and discretize it into stream reaches of 2 km. A stream reach is represented in our graph as an edge connecting two network nodes. Edge lengths are properly measured following the actual river path. The discretization is performed with the constraint of having a node at each confluence of two or more tributaries, thus ending up with about 55,000 nodes.
 To apply equations (1) and (2) to the graph representing the MMRS, all variables have to be suitably discretized over space. This task is simple for the discrete-time component of the model, just requiring the introduction of a replica of equations (1) for each node in the graph. On the other hand, the study of the spatiotemporal evolution of the continuous-time processes described by equation (2) on the river network requires ad-hoc numerical techniques. Given the complexity and the size of the water system under study, here we adopt a straightforward numerical approach based on finite difference schemes. Specifically, we apply an upwind scheme to resolve advection and a centered stencil to approximate diffusion. All these numerical methods have been properly adapted to work with an oriented graph as computational domain [Bertuzzo et al., 2007; Mari et al., 2009]. We note that the use of upwind differencing may result in the appearance of artificial viscosity (whose value, depending on parameter values, may even be comparable to that of hydrological dispersion, see Strikwerda ) that may affect the estimation of parameter . However, the size of the spatial domain under analysis, the overall complexity of the model, and the need to limit computational costs dissuaded us from applying more sophisticated numerical techniques.
 The evolution of the process described by equation (2) on the river network can be thus numerically computed as the solution of the following system of differential equations (the so-called method of lines, see Wouwer et al. ):
where i is the node index, (km) is the spatial step used to discretize the river network (2 km), and is the indegree (outdegree) of node i, i.e., the number of inward (outward) edges, which also properly accounts for boundary conditions. As for the special nodes describing lakes, impoundments, and ponds of the MMRS, we assume that a fraction of veligers produced therein is locally retained after reproduction. Therefore, in any of these special nodes (say l) the initial condition for larval transport in each year t of the simulation reads as . In the same way, the abundance of veligers that are locally retained and survive to recruitment can be evaluated as [Casagrandi et al., 2007].
 With this approach the commercial network layer can be easily assembled together with the hydrological layer by properly adding up the right-hand sides of equation (5) with the right-hand sides of equation (3) in the nodes corresponding to the commercial ports of the MMRS. The description of intra-annual dynamics is completed by the concurrent integration of system (4), which describes adult mussel relocation due to recreational boating in the nodes corresponding to the closed water bodies of the MMRS. The time integration of the layered network system is accomplished by an explicit Euler method, with a time step chosen so as to ensure both accuracy of the numerical scheme and stability by Courant-Friedrichs-Lewy conditions [Roach, 1982].
3.2. Parameter Estimation
 Most model parameters can be directly estimated from field and laboratory data (Table 3). In particular, the adult survival parameters and are derived from a demographic study conducted in the Mississippi River [Akçakaya and Baker, 1998]. Fertility parameters fk (with 1 ≤ k ≤ 3) are evaluated by applying a species-specific allometric model [Sprung, 1991] to shell length data taken from the same demographic study used to estimate survivals. Because we do not have reliable estimates for fertilization probabilities, we make the assumption that all viable eggs are actually fertilized and become veligers. This assumption can be naturally relaxed by including fertilization probability into larval survival .
Table 3. Parameter Values Used for the Simulation of the Multilayer Network Model as Estimated From the Literature
3.6 × 104
5.2 × 105
2.4 × 106
km2 mussels−1 d−1
veligers m−2 d−1
 As for the larval stage, both the cannibalistic and the baseline mortality rates ( and , respectively) are estimated from laboratory data [MacIsaac et al., 1991; Schneider et al., 2003; Stoeckel et al., 2004a]. Baseline mortality can actually vary during the developmental cycle of zebra mussel veligers, thus introducing survival bottlenecks that may have some relevance for the maintenance of downstream populations that depend upon drifting larvae [Schneider et al., 2003]. However, given the spatial and temporal extent of our analysis, we opted for an aggregated measure of larval mortality. We also note that the use of laboratory instead of field data may lead to an underestimation of the baseline mortality rate [Schneider et al., 2003]. As for larval survival , it is well known that it does not exceed few percentage points, with mortality from the veliger stage to the first month of adult life being reported to be as high as 99% [Sprung, 1989]. The duration of the larval stage may be quite variable, with reported values spanning from a few days to a few weeks [Sprung, 1993; Stoeckel et al., 1997]. As this parameter has an obvious influence on the spreading of the species, we choose its value so as to fit approximately the initial speed of the invasion of the MMRS. The resulting value ( days) is well within the range reported in the literature [Sprung, 1993; Stoeckel et al., 1997].
 As for the parameters characterizing the hydrological network, the mean water velocity u is determined by averaging the mean annual velocity as estimated from the NHD over all stream reaches. The value of local larval retention within lakes, impoundments, and ponds does not seem to have a major impact on model outcomes, and has been set to 0.5 (but see below for a sensitivity analysis). The longitudinal dispersion , whose value is usually rather difficult to estimate from field data even in the relatively simple case of shear flow dispersion for passive scalars in rivers [Fischer et al., 1979], is used as a tuning parameter.
 The mass parameters used in the gravity model accounting for transport due to commercial navigation are derived from the 2007 Report of the Waterborne Commerce Statistics Center (WCSC; see Table 1, available online at http://www.iwr.usace.army.mil/ndc/wcsc/portton07.htm). Port-to-port travel times of commercial ships are computed using a mean reference boat velocity [Sidaway et al., 1995]. The numerical value of the intensity parameter cannot be determined from existing data. Therefore, is used as a tuning parameter. The value of the shape parameter cannot be derived from the literature either, so we choose (see sensitivity analysis below).
 As for the gravity model describing mussel transport due to recreational boating, the relevant mass parameter are retrieved from the Geographic Names Information System (GNIS, available online at http://geonames.usgs.gov/domestic/index.html) and the NHD. The number of registered recreational boats in each state is taken from the 2005 Report of the Bureau of Transportation Statistics (BTS; see Table 2, available online at http://www.bts.gov/publications/state_transportation_statistics/state_transportation_statistics_2006/html/table_05_06.html). Similar to the intensity parameter of the commercial gravity model, ε is used as a tuning parameter, while we set . The maximum Euclidean distance between neighboring lakes in the third network layer (dmax) is chosen so as to account for more than 95% of the typical lengths of recreational trips [Buchan and Padilla, 1999]. Notice that this choice does obviously prevent mussel movements exceeding the dmax threshold and might have some influence on the rate of spread of the population. Survival of adults during overland transport is estimated according to an experimental study on mussel tolerance to aerial exposure [Ricciardi et al., 1995]. Being on the order of 1–3 days, thus negligible compared to the duration of the recreational season, lake-to-lake trailering time has been safely set to [Buchan and Padilla, 1999].
 The external input of veligers is assumed to be constant and different from zero only at the headwaters of the Illinois River, thus representing the flux of veligers that enter the Mississippi-Missouri watershed from Lake Michigan through the Chicago Sanitary and Ship Canal. At the headwaters of the Illinois River, the order of magnitude of is derived from local field observations [Schneider et al., 2003]. The external input of mussels is also assumed to be constant and different from zero only in lakes lying within an aerial distance dmax from reportedly infested sites in the Great Lakes region. Therefore, mussel input represents the flux of mussels that are transported by recreational boats from Lake Michigan, Erie, and Ontario to nearby lakes connected to the MMRS. The numerical value of is used as a tuning parameter. Finally, as initial condition for our model simulations we assume n1(i0, 0) = n0 (with n0 = 100 mussels m−2) in the nodes i0 corresponding to the first sightings of zebra mussels in the MMRS (approximately 1990; see Figure 1a).
3.3. Model Calibration
 The longitudinal dispersion , the intensity parameters and ε of the gravity models, and the mussel input term in the Great Lakes region are used as tuning parameters. We separately fit (1) the model limited to its hydrological layer (model M1, , , and ), (2) the model with the hydrological and the commercial layers (M2, , and ), and (3) the full model (M3).
 For each model, fitting is performed via an iterated grid search method to find the set of parameter(s) that minimizes the weighted sum of three performance indices based on celerity, extent, and pervasiveness of the zebra mussel spread over the MMRS. Specifically, we estimate three quantities: (i) the downstream distance (DD) covered by D. polymorpha along the backbone of the river network (i.e., along the Mississippi River); (ii) the mean radius (MR) of the invasion, evaluated as the Euclidean distance between all infected nodes and the injection point; and (iii) the total fraction (IH) of invaded hydrologic unit codes (HUCs). To numerically compute these quantities we consider a node i as colonized at year t if the local mussel density N(i, t) is higher than a threshold that roughly corresponds to a colony of some hundreds of individuals in the reach represented by the node (i.e., N(i, t) ≥ 0.01 (mussels m−2)). In a similar way, we consider a HUC as invaded if at least 10% of the nodes included in the unit have been colonized. The three indices are defined as the residual sums of squares (RSSs) computed by summing the yearly square differences between the relevant quantities evaluated from data and from model simulations, i.e.,
where d and m refer here to data and model, respectively. Because downstream distances, mean radii of invasion, and invaded HUCs are measured in different units or can have very different numerical values, it is not trivial to aggregate the three indices in a unique indicator to be minimized. We have opted for a heuristic procedure in which weights are chosen so that each term of the sum providing the indicator is dimensionless and comparable in magnitude. The three weights must thus have the units of an inverse of the pertaining RSS. As each simulation would give a different RSS, we have to choose a statistical characteristic of the distribution of RSSs obtained while varying the parameter settings. Many options are available, like for example using the minimum (or the mean) value of each of the three RSSs. If we use the inverse of the minimum [Bertuzzo et al., 2008], the weights are obtained as
where h ∈ (i, ii, iii), K is the number of performed simulations (corresponding to K different parameter combinations), and k is the simulation index. Therefore, the aggregated objective function J that has to be minimized reads as
 We have verified that using the mean value instead of the minimum would not have significantly altered calibration results.
3.4. Model Selection
 To assess whether the complexity of the full multilayer network model is justified by its ability to reproduce D. polymorpha spatiotemporal invasion patterns, we rank the performances of the three fitted models according to a model-selection procedure that explicitly discounts model complexity, as in Akaike's information criterion (AIC, Akaike ). AIC-like methods can in fact account for the trade off between model accuracy and complexity, measured as the number of free parameters (respectively 1, 2, and 4 structural parameters for the three models, plus one residual variance parameter; see Burnham and Anderson ; Corani and Gatto ). For each candidate model we specifically compute
where J is the value of the aggregated objective function evaluated by computing weights on the basis of the RSSs of the three best-fit models and is the number of available data points (in our case corresponding to the time span of model simulations, i.e., ). The last term is a second-order correction for small sample size [Sugiura, 1978]. We also compute AICc differences as
where AICcmin is the minimum value of Akaike score found among all best-fit models.
 To achieve greater robustness, we also rank the models by separately considering each of the three indices (RSSs) defined above. For each candidate model we thus specifically compute
where h ∈ (i, ii, iii).
4.1. Model Selection
 We have separately calibrated (best fit according to min J, see section 3) a model confined to the first network layer (model M1), a model with the first and the second layers (M2), and the full model (M3). A simple visual inspection of the invasion patterns generated by the three models (Figures 3, 4, and 5) may suggest that model M3 be indeed the best candidate to explain the zebra mussel spread over the MMRS. More quantitatively, the full multilayer network model consistently produces lower RSSs and a smaller value of the weighted index J than the other candidate models (Table 4). However, as the three models differ in the number of free parameters, RSSs alone cannot be used as model selection indices. We have thus compared the performances of the three models discounting the effects of model complexity (see section 3 and Table 5) and found that the full model largely outcompetes the other two candidates with respect to overall performances (, meaning essentially no support for both model M1 and M2; see Burnham and Anderson ). Quite interestingly, though, the model confined to the hydrological network layer (M1) gives the lowest value for AICciii. In particular, since for M3, the full model would be considerably less supported [Burnham and Anderson, 2002] than M1 as a candidate to explain the pervasiveness of the zebra mussel invasion, evaluated as the fraction of occupied HUCs over time. However, model M3 is able to locate correctly 75% of the invaded HUCs, while the performance of model M1 is just limited to 40% (mean values averaged over the simulation time span). Therefore, M1 does better than M3 in reproducing the total fraction of invaded HUCs, yet it does significantly worse in reproducing local invasion patterns. Moreover, the large values of and evaluated for M1 (with both values larger than 10) indicate that there is essentially no support [Burnham and Anderson, 2002] for M1 as a viable candidate to explain the celerity and the geographic extent of the MMRS invasion. As for model M2, it turns out that it is always outcompeted by the full model. Notice however that the predictive ability of model M2 is quite similar to that of the full model when evaluated as the fraction of invaded HUCs located correctly (70% for M2). Therefore model M2 might still be appealing for its ability to describe the initial phase of the biological invasion, thanks in particular to its less demanding data requirements and its relatively easier parametrization.
M1 is the network model confined to the hydrological network layer; M2 takes also into account the commercial layer; M3 is the full model including also the recreational layer. The symbol (†) indicates parameter values that are not estimated via numerical fitting. The weights used to evaluate the aggregated objective function J for each of the three models are wi = 2.3 × 10−5, wii = 4.5 × 10−6 and wiii = 0.15, as resulting from the RSSs of the three best-fit candidate models.
2.0 × 10−4
1.9 × 10−4
d−1 km−2 boats−1
4.2 × 10−21
mussels m−2 d−1
1.0 × 10−16
1.9 × 105
9.7 × 104
4.3 × 104
2.8 × 106
4.1 × 105
2.2 × 105
(% invaded HUCs)2
Table 5. Selection of the Best Model According to AICc
 For all the reasons outlined above, we have concluded that the full model (M3) is the best candidate to explain the observed spatiotemporal patterns of zebra mussel spread at a regional spatial scale over the given 20 year time span. It is to be remarked, though, that there exist some occasional discrepancies between the full multilayer model and observations, such as the overestimation of the invasion of the Red River (one of the southmost left tributaries of the Mississippi River) and the Missouri River. While the Red River issue could be explained by abiotic factors (say, by insufficient calcium concentration in the water; Whittier et al. ), the Missouri anomaly is subtler, because the river is likely to satisfy both suitable abiotic conditions [Drake and Bossenbroek, 2004; Whittier et al., 2008] and anthropic transport requirements (presence of commercial and/or recreational navigation) to be colonized by D. polymorpha. A possible explanation of the failed invasion of the Missouri River is that it lacks locks and dams [Allen and Ramcharan, 2001], which provide potential retention zones that may facilitate establishment and long-term persistence of local zebra mussel populations [Stoeckel et al., 2004b]. An in-depth analysis of local hydrologic conditions and their variability [Di Maio and Corkum, 1995] could perhaps contribute to the explanation of the discrepancies found between simulated and observed invasion patterns.
4.2. Invasion Scenarios
 The main advantage of having subdivided ecological, hydrologic, and anthropogenic processes into separate network layers is that we can single out the effects of each mechanism driving the spread of D. polymorpha in the MMRS. To this end, while keeping fixed the parametrization obtained by fitting the full model, we have first excluded all human-mediated transport processes (second and third network layers, i.e., , , and ). This experiment has let us evaluate the role of hydrologic transport alone in the spread of the species. Our results (Figure 6) confirm that this primary driver cannot be responsible alone for the complexity of the spatiotemporal invasion pattern shown in Figure 1. In fact, hydrologic drift and dispersion are not sufficient to trigger in due time the invasion of remote downstream nodes and peripheral reaches of the river network. As a result, speed, geographic range, and pervasiveness of the invasion are all sensibly underestimated (Table 6). The model restricted to the first layer in fact predicts that, in absence of human-mediated dispersal mechanisms, the species would have spread along the upper and lower Mississippi only.
Table 6. Differences Among the Spatiotemporal Invasion Patterns Generated by the Three Invasion Scenariosa
Invasion Celerity 1990–1993 (km)
Invasion Radius 1993 (km)
Invasion Radius 1999 (km)
Invaded HUCs 1995 (%)
Invaded HUCs 2005 (%)
Accounting respectively for hydrological transport (H, Figure 6); hydrological transport and commercial navigation (H+C, Figure 7); hydrological transport, commercial navigation, and recreational boating (H+C+R, corresponding to the full multilayer network model M3, Figure 5). See text for details on numerical simulations.
 We have then repeated the same analysis excluding only recreational boating (third layer, and ). Comparing the patterns of spread produced by this invasion scenario against those produced by purely hydrological dispersal (Figure 6) shows that commercial navigation played a role of paramount importance in promoting the initial penetration of the zebra mussel deeply into the MMRS. The simulation reported in Figure 7 shows that the coupling between hydrologic transport and long-distance larval dispersal due to commercial navigation can reproduce several core features of the species spread at the basin scale. In particular, larval dispersal among ports can selectively trigger the invasion of tributaries lying far apart from the Mississippi River, which represents a characteristic feature of the MMRS invasion (see Figure 1a). As a result of the inclusion of port-to-port larval dispersal, the model also reproduces fairly well the timing of the spread along the main stem of the MMRS (Figure 7b) and the pervasiveness of the invasion in its early phases (up to mid 1990s, Figure 7c). However, as soon as the early explosive spread phase slows down, commercial navigation is no longer sufficient to explain the steadily increasing penetration of D. polymorpha into the MMRS, thus resulting in a 20% underestimation of the fraction of occupied HUCs in the final years of the simulation (Figure 7c and Table 6).
 On the other hand, comparing Figure 7 to Figure 5 shows that the inclusion of the third network layer provides a notable contribution to the understanding of the overall invasion process. In particular, recreational boating can help explaining the slow, yet continuing, spread of D. polymorpha from 1994 to date (contrast Figure 5d with Figure 7c). On a smaller spatial scale, the full multilayer network model also correctly reproduces the invasion of the Wabash River, the largest northern tributary of the Ohio River, and of several peripheral reaches of the upper Mississippi, Illinois, Arkansas, and Ohio Rivers, which might thus be ascribed to recreational boating.
4.3. Robustness of the Results
 To assess the robustness of our simulation results, we have performed a sensitivity analysis of the model predictions with respect to changes of the parameter set used in the reference simulation (Figure 5). In particular, we have let one or more parameters vary at large (±20% variations) and repeated model runs. Then, we have evaluated the same synthetic indicators of species spread used to compare different invasion scenarios. We have found that the spatiotemporal patterns of spread, while generally pretty robust, are indeed somewhat sensitive to variations of some selected parameters (Table 7), namely the duration of the larval stage , the mean water velocity (u), and the dispersion coefficient . Some of these quantities (in particular and ) may be characterized by ranges of variation that extend beyond the bounds considered in Table 7 (up to a few weeks for , Sprung ; Stoeckel et al. ; up to 130 km2 d−1 for in the Missouri River, Fischer et al. ; Rutherford ), thus remarking the importance of a careful estimation/tuning of the model parameters. We have also tested the effects of larger variations (one order of magnitude) of other quantities whose estimation is rather uncertain (i.e., , , n0, , ε, and ) and found that only survival from larval to adult stage and the intensity of the commercial gravity model ε produce remarkable results (more than 20% in at least one of our five indicators of species spread).
Table 7. Sensitivity Analysis of Model Outcomes With Respect to Parameter Valuesa
Invasion Celerity 1990–1993
Invasion Radius 1993
Invasion Radius 1999
Invaded HUCs 1995
Invaded HUCs 2005
All values are expressed as percentage variations with respect to the simulation of the full model (last row of Table 6). Variations whose absolute value is larger than 10% are marked in bold.
0.0 ÷ 0.0
−0.2 ÷ 0.1
−0.7 ÷ 0.2
−1.0 ÷ 0.4
−1.5 ÷ −0.7
f1,2,3 ± 20%
−0.4 ÷ 0.4
−1.4 ÷ 1.2
−1.0 ÷ 0.3
−2.4 ÷ 0.4
−3.9 ÷ 0.1
−0.4 ÷ 0.5
−1.8 ÷ 1.9
−1.0 ÷ 0.3
−2.8 ÷ 0.0
−8.8 ÷ −0.9
0.8 ÷ 0.4
2.7 ÷ −2.4
0.4 ÷ −1.3
4.2 ÷ −2.8
0.0 ÷ −9.6
0.2 ÷ −0.1
0.6 ÷ −0.4
0.2 ÷ −0.5
1.4 ÷ −2.8
00 ÷ −5.3
7.5 ÷ 7.5
−9.0 ÷ 8.7
−0.7 ÷ 0.0
7.3 ÷ 26.9
13.6 ÷ −3.1
u ± 20%
7.5 ÷ 3.2
4.7 ÷ 6.8
1.0 ÷ 0.8
36.6 ÷ 15.7
23.9 ÷ −3.9
ub ± 20%
−0.8 ÷ 1.1
−0.6 ÷ −2.6
−0.3 ÷ −0.7
−2.4 ÷ −1.0
−0.7 ÷ −1.5
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
−0.9 ÷ 0.0
−0.1 ÷ 0.1
−0.2 ÷ 0.0
0.0 ÷ 0.0
−2.8 ÷ 0.0
−0.9 ÷ 0.0
n0 ± 20%
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.1
0.00 ÷ 0.0
0.0 ÷ 0.0
0.4 ÷ 0.4
−4.7 ÷ −3.1
−0.1 ÷ 0.2
2.1 ÷ −1.3
0.2 ÷ 0.4
0.0 ÷ −1.4
−1.8 ÷ 1.8
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.4 ÷ −0.6
−0.7 ÷ −7.5
−9.9 ÷ 8.7
−0.3 ÷ 1.1
−12.2 ÷ 21.3
−9.5 ÷ 7.2
−0.2 ÷ 0.2
−0.8 ÷ 1.0
−0.8 ÷ 0.4
−1.4 ÷ 0.0
−4.4 ÷ −0.9
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
−0.9 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
0.0 ÷ 0.0
−3.5 ÷ 0.0
 Along these lines, we have also assessed how the connectivity structure of the commercial network layer may have influenced the spread of D. polymorpha. To this end, we have simulated the full network model by removing one commercial port at a time, so as to mimic the effects of clearing the ballast water of commercial boats departing from a selected fluvial port. From a technical viewpoint, to remove the pth port we have set Fpq = 0 for any port q in the second network layer, while keeping all other simulation details unchanged. Our numerical experiment suggests that the invasion patterns are quite robust to changes of the secondary network layer. However, preventing veliger shipping from some selected ports, most notably the port of St. Louis (MO and IL, marked as port G in Figure 2a), might have had important effects on large-scale invasion patterns (Table 8). Specifically, according to the model, the prevention of long-distance veliger dispersal from St. Louis would have remarkably delayed the invasion, in terms of both spread velocity along the Mississippi River and mussel penetration into the network. This is most likely due to the strategic position of St. Louis between the upper and lower Mississippi River, and between the Missouri and the Ohio Rivers. Such a connectivity makes this port a hub of the commercial network layer and, in turn, one of the best candidates for a targeted management action, as observed in studies on the resilience of complex networks [Albert et al., 2000]. This supposition is also confirmed by topological measures of node importance in weighted networks, such as eigenvector centrality [Estrada and Rodríguez-Velázquez, 2005]. We have in fact ranked MMRS ports according to this measure and found that the port of St. Louis comes third for centrality in the commercial network layer, just after the ports of St. Paul (MN, marked as M in Figure 2a) and Pittsburgh (PA, F in Figure 2a), and before the port of South Louisiana (LA, A in Figure 2a), the largest port in the MMRS.
Table 8. Robustness of the Results With Respect to Changes of the Connectivity Structure of the Commercial Network Layera
Invasion Celerity 1990–1993
Invasion Radius 1993
Invasion Radius 1999
Invaded HUCs 1995
Invaded HUCs 2005
See text for technical details on model runs. All values are expressed as percentage variations with respect to the simulation of the full model (last row of Table 6). Variations whose absolute value is larger than 10% are marked in bold. Letters refer to the ports in Figure 2a (see also Table 1).
 We have finally tested to what extent the simulated invasion patterns are sensitive to some of our simplifying modeling hypotheses. In particular, we have tested whether (1) relaxing the assumption of equal veliger mortality in ballast waters and in the water flow or (2) introducing inverse density dependence (i.e., Allee effects, see above) may lead to qualitative changes in the proposed results. To this end we have run additional control simulations of the full model, properly modified so as to account for (1) differentiated mortality rates or (2) a minimum local population density for reproduction to occur successfully [Potapov and Lewis, 2008]. We have found that these changes may have a quantitative effect on the estimation of the parameters of either the second or the third network layers (respectively , and ε and ), depending on the numerical values used for mortality rates or reproduction thresholds (for which there is no clear indication from the available literature, though), but do not produce important alterations of the simulated invasion patterns at the regional spatial scale, thus reinforcing our choice of a parsimonious approach.
5. Discussion and Conclusions
 In this paper we have proposed a novel mechanistic network model aimed at characterizing quantitatively hydrologic controls and anthropogenic drivers of the zebra mussel invasion of the MMRS. The scale of the modeling attempt and the ecohydrological interactions addressed are deemed noteworthy. Recent advances in the study of river networks as ecological corridors for species, populations or pathogens of water-borne diseases [Campos et al., 2006; Bertuzzo et al., 2008; Muneepeerakul et al., 2008; Rodriguez-Iturbe et al., 2009; Bertuzzo et al., 2010], as well as in the study of ecosystem patchiness [Rietkerk et al., 2004; van de Koppel et al., 2008], had indeed suggested the relevance of landscape heterogeneities, directional dispersal, and hydrologic controls for the formation of ecological and epidemiological patterns. Human-mediated dispersal processes are of great importance, though, in particular because they allow species to disperse beyond their normal ranges, thus eventually shaping global biogeographical patterns (i.e., global biodiversity, see Wilson et al. ). Our model thus accounts for the interplay among a few selected mechanisms, such as density-dependent larval mortality, basin-scale hydrologic transport, and human-mediated dispersal due to either port-to-port commercial navigation or recreational boating, that are deemed of key importance in D. polymorpha invasions. Despite our parsimonious approach, the model succeeds in reproducing the zebra mussel invasion patterns observed in the MMRS at a regional spatial scale and over a 20 year time span.
 Our simulations represent a hindcasting exercise for the MMRS, yet the proposed multilayer network approach could be easily applied to predict and control other potential invasions of the same or related alien species in the MMRS or other river systems, provided that a comprehensive analysis of extant ecological and hydrologic processes, as well as of relevant human-mediated transport mechanisms, is available. For instance, some recent sightings suggest that dreissenids are making their way into the western U.S. as well. The quagga mussel D. bugensis, a close relative of the zebra mussel, has been discovered in Lake Mead and in the effluent Colorado River, an event that has been termed the “unfolding western drama” [Stokstad, 2007]. Our finding that a local precautionary control action could have remarkably delayed the zebra mussel invasion of the MMRS may thus assume a particular relevance. In the context of biological invasions, in fact, taking early actions is extremely difficult, yet extremely valuable [Puth and Post, 2005]. In the zebra mussel case, in particular, the containment and/or eradication of established colonies are very difficult and costly. Prevention should thus be favored over control [Finnoff et al., 2007], but this is possible only if quantitative tools to predict the development of the spread are available.
 To that end, we point out the importance of a fully dynamical and spatially explicit approach to modeling biological invasions, for it allows combined predictions of both temporal and spatial patterns of species spread. The impossibility of deriving such predictions represents in fact the main limitation of both spatially implicit models aimed at reproducing local mussel dynamics (e.g., MacIsaac et al. ; Strayer and Malcom ; Casagrandi et al. ) and static multivariate approaches to infer the geographic distribution of the species (e.g., Strayer ; Drake and Bossenbroek ; Whittier et al. ). In addition, we deem our multilayer approach an effective and general framework to model the spread of species, like D. polymorpha, that are characterized by multiple dispersal pathways, as usually found in biological invasions [Wilson et al., 2009]. More in general, as the coexistence of multiple propagation pathways is not unique to harmful invasive species, the idea of a multilayer network approach could be profitably extended to other contexts, such as the spread of waterborne infectious diseases.
 We conclude by noting that biological invasions are inherently complex stochastic processes for which only a single realization is available for observation. As such, there exist obvious limits to our ability to actually predict them from models that necessarily need estimation of parameters from actual data [Melbourne and Hastings, 2009]. However, forecasting the main (or mean, in some ensemble-averaging sense) patterns of spread of alien species is becoming increasingly important, though more and more difficult, with anthropic activities rapidly coming abreast of (or even overwhelming) natural invasion pathways [Clark et al., 2001]. For all these reasons, a mechanistic approach to modeling invasions of the type proposed herein, that is, including important details about the role of ecological corridors and of related anthropogenic drivers, is deemed very valuable. A thorough, quantitative understanding of the different processes that boost the spread of alien species is in fact our only hope to prevent and, to some extent, control biological invasions.
 The authors wish to thank three anonymous referees for their extensive and valuable comments on the previous version of this manuscript. L.M., E.B., and A.R. gratefully acknowledge funding from ERC Advanced grant RINEC 22,761 and SFN grant 200021_124930/1. I.R.-I. acknowledges the support of the James S. McDonnell Foundation. Part of this work has been performed during L.M.'s visit to S.A.L.'s laboratory under MIUR grant Interlink II04CE49G8 and E.B.'s visit to I.R.I.'s Laboratory funded by Scuola di Dottorato SICA, Università di Padova, at Princeton University.