Robert Schick, University Program in Ecology, Box 90328, Levine Science Research Center, Duke University, Durham, NC 27708–0328, USA (tel: + 1919 6138021; fax: + 1919 6848741; e-mail: firstname.lastname@example.org).
1The addition of large water storage dams to rivers in California's Central Valley blocked access to spawning habitat and has resulted in a dramatic decline in the distribution and abundance of spring-run chinook salmon Oncorhynchus tshawytscha (Walbaum 1792). Successful recovery efforts depend on an understanding of the historical spatial structure of these populations, which heretofore has been lacking.
2Graph theory was used to examine the spatial structure and demographic connectivity of riverine populations of spring-run chinook salmon. Standard graph theoretic measures, including degree, edge weight and node strength, were used to uncover the role of individual populations in this network, i.e. which populations were sources and which were pseudo-sinks.
3Larger spatially proximate populations, most notably the Pit River, served as sources in the historic graph. These source populations in the graph were marked by an increased number of stronger outbound connections (edges), and on average had few inbound connections. Of the edges in the current graph, seven of them were outbound from a population supported by a hatchery in the Feather River, which suggests a strong influence of the hatchery on the structure of the current extant populations.
4We tested how the addition of water storage dams fragmented the graph over time by examining changing patterns in connectivity and demographic isolation of individual populations. Dams constructed in larger spatially proximate populations had a strong impact on the independence of remaining populations. Specifically, the addition of dams resulted in lost connections, weaker remaining connections and an increase in demographic isolation.
5A simulation exercise that removed populations from the graph under different removal scenarios – random removal, removal by decreasing habitat size and removal by decreasing node strength – revealed a potential approach for restoration of these depleted populations.
6Synthesis and applications. Spatial graphs are drawing the attention of ecologists and managers. Here we have used a directed graph to uncover the historical spatial structure of a threatened species, estimate the connectivity of the current populations, examine how the historical network of populations was fragmented over time and provide a plausible mechanism for ecologically successful restoration. The methods employed here can be applied broadly across taxa and systems, and afford scientists and managers a better understanding of the structure and function of impaired ecosystems.
Effective management of species requires knowledge of population structure, because this is key to understanding how local impacts may affect the larger entity at both ecological and evolutionary time scales (Kareiva & Wennergren 1995; Wennergren, Ruckelshaus & Kareiva 1995; Tilman & Lehman 1997). For example, a metapopulation may have quite different dynamics than a panmictic population of the same aggregate size, depending on factors such as the dispersal rates among populations and internal dynamics of the metapopulation components (Levins 1969; Kareiva 1990; Hanski & Gilpin 1991). Ignoring spatial structure, especially immigration from nearby populations, can impair the management of protected species, such as incorrectly diagnosing population status or the response to habitat restoration (Cooper & Mangel 1999). At longer time scales, the relationship between the structure and dynamics of populations and landscapes may determine the degree to which populations adapt to local conditions (Sultan & Spencer 2002) and how they respond to disturbance (Pickett & White 1985).
In many cases, species conservation problems can be framed in terms of problems with spatial structure, because impacts to species often take the form of lost habitat patches or dispersal corridors. Restoration is aided with a ‘guiding image’ (Palmer et al. 2005), and the virgin state of the system is often used as such. To be most effective, the guiding image should be in the form of a conceptual model that can show system function, system impairment and restoration strategies (Jansson et al. 2005). We propose that graph theory provides the tools needed to construct conceptual models for spatially explicit problems in conservation that allow quantitative comparisons of historical, contemporary and potentially restored population structures.
While graph theory carries with it its own terminology (Harary 1969), many of the terms have direct ecological interpretations. Nodes can represent a range of things, from individuals to populations to patches on a landscape. Edges are the connections between nodes. Construction of a landscape graph typically requires at least two data structures (Urban & Keitt 2001). The first structure includes information about the node's spatial location and some indicator of size. The second structure is a distance matrix between all of the nodes. The degree of a node is the number of edges incident to it. A regular graph is one where the edges are bi-directional, i.e. for nodes a,b the connection is a↔b (Fig. 1a). In contrast, a digraph's edges (also called arcs) have direction, i.e. a→b (Fig. 1b). For a digraph, degree is slightly different: outdegree of a point v is the number of points adjacent from a node; and indegree is the number adjacent to a node. Logically, outdegree and indegree correspond to familiar source–sink dynamics with which most ecologists are familiar (Pulliam 1988). The connection between a pair of nodes in a given graph G is based on an adjacency matrix. The adjacency matrix is comprised simply of 0s and 1s, where 0 indicates no connection between a pair of nodes and 1 indicates that a connection, or edge, exists. [To help avoid confusion, we note that nodes can be adjacent (connected) in a graph theoretic sense even if they are not adjacent in a geographical sense.] Lastly, in most instances, populations and metapopulations can be represented realistically as weighted digraphs (Fig. 1c) with different population sizes and the asymmetric connections between them (Barrat et al. 2004; Bascompte, Jordano & Oleson 2006). These cartoon graphs serve as the conceptual basis for the connections in larger, more complicated, and in our case, spatially explicit graphs.
While the role of ecological connectivity in regulating and maintaining population distribution and population persistence has been documented in both the terrestrial (Fahrig & Merriam 1985; Taylor et al. 1993) and aquatic realms (Wiens 2002), the direction of the connectivity can have important impacts on a given system (Gustafson & Gardner 1996). Therefore, because regular graphs may not capture completely how connectivity influences population structure, we use weighted digraphs (Barrat et al. 2004; Bascompte et al. 2006) to examine how directed connectivity and asymmetrical dispersal elucidate population structure. Although directed connectivity has been mentioned previously (Gustafson & Gardner 1996; van Langevelde, van der Knaap & Claassen 1998; Urban & Keitt 2001; Schooley & Wiens 2003), its importance for fish populations has not been fully explored. Furthermore, the influence of the dendritic riverine structure on metapopulation persistence and population vulnerability for fish has only been noted relatively recently (Dunham & Rieman 1999; Gotelli & Taylor 1999; Fagan 2002) and no attempt has been made, to our knowledge, to use graphs to represent fish populations in a riverine setting. The representation of river/stream fish populations as a graph is notably different from most terrestrial graphs, because the dispersal corridors (rivers and streams) are generally fixed and immutable at ecological time scales, i.e. the fish already live in a network.
Endangered salmonid populations are managed as evolutionarily significant units (ESU), which are defined as a salmon population or group of salmon populations that is substantially isolated reproductively from other populations and that contributes substantially to the evolutionary legacy of the species (Waples 1991, 1998). Typically, ESUs are structured internally (Gharrett, Gray & Brykov 2001; Olsen et al. 2003; Guthrie & Wilmot 2004) due to the fact that salmon mainly return to their natal rivers after spending several years at sea, but there is some low level of dispersal among the populations that is probably important for ESU persistence. As salmon return to their natal rivers they stray naturally at varying rates (Ricker 1972; Quinn 1993), which allows them to occupy new habitat (Milner & Bailey 1989; Wood 1995) and is the mechanism by which populations are connected. The rate at which salmon stray has proved difficult to quantify, although observed rates in the wild range from 0 to 67% (McElhany et al. 2000). Changing the spatial structure through population loss or increased straying must have effects on an ESU, but to date these have not been quantified.
We examine spring-run chinook salmon Oncorhynchus tshawytscha (Walbaum 1792) in California's Central Valley (Fig. 2), which are listed as threatened under the United States’ Endangered Species Act. Spring-run chinook salmon are high-elevation mainstem spawners that migrate into the watersheds under high flow conditions in springtime (Yoshiyama et al. 2001; Lindley et al. 2004). They over-summer in cool temperature pools before migrating out of the pools in the fall to spawn (Lindley et al. 2004). After spawning the cool water temperatures delay maturation, and juveniles often remain in the system for a full year (Lindley et al. 2004). Spring-run chinook salmon occupied much of the Central Valley, although the installation and continued presence of major dams has blocked and restricted access to much of their historical habitat (Yoshiyama et al. 2001; Lindley et al. 2004) (Fig. 2). The first of 10 ‘keystone’ dams in the Central Valley, i.e. the lowest-elevation dam that completely blocks upstream habitat, was installed in 1894. The addition of such keystone dams proceeded until 1968, removing a total of 19 populations from the ESU. Lindley et al. (2004) describe the putative historical structure of the ESU, which forms the basis for our analysis. We presume this was a viable ESU prior to 1894.
We build and test a dispersal model that accounts for directional connectivity between populations within the historic spring-run chinook salmon ESU, and use graph theoretic methods to test how connectivity influences the spatial structure of populations within the ESU. We focus on (a) the organisms’ ability to disperse through fixed edges, (b) on the importance of individual fish populations (nodes) and (c) how the installation and continued presence of dams impacted the ESU. In addition, we examine the structure of the current spring-run chinook salmon ESU. Lastly, we use these results – notably changes in graph metrics and in the role of populations – to discuss the persistence and survival of this threatened species. The graph theoretic methods presented herein have broad application across a variety of ecological systems, and can be used in data limited environments to predict population structure, persistence and synchrony.
Materials and methods
To populate the first graph data structure, we initially identified populations in the spring-run chinook salmon ESU that historically contained spawning groups (Lindley et al. 2004). The nodes in our graph represent populations; to identify these populations spatially, we located the intersection of the 500 m elevation contour and the mainstem of each river within the ESU. (Yoshiyama et al. 2001 identify 500 m as the approximate lower extent of the breeding range for spring-run chinook salmon.) This intersection is then the spatial representation of the node. To represent the size of the population (node) in the historical spring-run chinook salmon ESU, we used a habitat proxy: extent of the mainstem spawning range > 500 m elevation (Yoshiyama et al. 2001). For populations whose habitat was below 500 m, e.g. several small populations on the western side of the Central Valley, we used estimated ranges from Yoshiyama et al. (2001). Previous studies have shown that spawning habitat, as we have defined it here, correlates significantly with effective population size, Ne (Shrimpton & Heath 2003). To represent the size of the population (node) in the current spring-run chinook salmon ESU, we used the mean number of annual spawners since 1980 in lieu of the habitat proxy for the historical ESU (R. M. Kano, California Department of Fish and Game, Sacramento, CA., USA, unpublished data). [We note that these definitions of node size are different, and comparisons between the historic and current graph were made with an appropriate degree of caution. The correlation between habitat length and number of spawners was negative (–0·301); however, a plot of these revealed the relationship between the two was nonlinear and that this negative correlation was driven by an outlier (Butte Creek). Once Butte Creek was removed, the correlation between habitat length and number of spawners was positive (0·65).]
To create the second graph data structure, we used a network module of a commercially available geographical information system (GIS) package (ArcInfo® workstation version 9·0) to estimate ‘as the fish swims’ distance between all identified populations. By ‘as the fish swims’ we mean minimum straight-line distance along the river network, i.e. fish do not explore available tributaries. We used the ArcGIS Network module to estimate this distance between node locations along the river network of the Central Valley (1 : 100 k routed stream layer, version 2003·6, available from CalFish: http://www.calfish.org/DesktopDefault.aspx? tabId = 76, last accessed 18 August 2006). This yielded a full (upper and lower triangles) distance matrix, which served as the second input to our model.
Any two nodes in the graph were deemed connected by an edge if the proportion of incoming fish from one population exceeds a certain threshold level of the total recruitment (local + incoming) in the target population. The edges in the graph were developed from a migration matrix, N. To construct N we needed the following data structures: (1) a full distance matrix D of all the interpopulation ‘as the fish swims’ distances; (2) a dispersal kernel; (3) a matrix M of dispersal probabilities; and (4) a matrix X of population size.
We assumed in this analysis that a fraction of fish returning to spawn will stray from their natal stream and that the probability pij of a fish from node i migrating to node j is a function of the distance between the populations. While this inter-population distance may seem biologically counterintuitive, we repeated the same analysis using a model where salmon return to their natal watershed with some high fidelity, but make ‘wrong’ decisions with some small probability. Because the results were quite similar, we chose the more parsimonious model for interpopulation distance, because it rested on fewer unverifiable assumptions. (See supporting material for full characterization of this ‘wrong-turn’ model and results.)
where α is a dispersion parameter, c a shape parameter, and djj, an interpopulation distance measured along the stream network (from a full distance matrix D, described above). α is an estimate of a species dispersal capability, while c controls the shape of the tail in the kernel. To parameterize α we used two different studies on chinook dispersal from McClure et al. (2003, unpublished data, available at: http://www.nwfsc.noaa.gov/trt/col_docs/independentpopchinsteelsock.pdf, last accessed 23 August 2006). The first was a within-basin movement study of wild spring-run chinook salmon, which indicated an α = 31·6 km; the second was a cross-basin study of hatchery fish that indicated an α = 166 km. While the first data source is on wild fish, and probably represents a better source, it was limited to one river basin and does not account for basin-to-basin straying. The second estimate of α does account for basin-to-basin straying; however, it is probably biased upwards because of the reduced homing ability of hatchery fish. Therefore we chose the average of the two, or α = 98 km. The nature of the tail is controlled by c, whereby c = 1 and c = 1/2 correspond to a kernel with an exponential tail and a fat tail, respectively (Clark et al. 1998; Clark et al. 1999). We chose c = 1, where the shape of the kernel is exponential and dispersal probability is controlled by the value of α (personal communication, J. S. Clark, Duke University, Durham, NC 27708, USA).
Whether populations were deemed adjacent depended upon the magnitude of migration between them, the magnitude of total recruitment and a threshold for the ratio of the two. If the percentage of a population's total recruitment coming from immigrating fish from another donor population exceeded some value, these populations were deemed connected (Bjorkstedt et al. 2005). To find these connections we first created a dispersal probability matrix M comprised of a mixture model composed of two probabilities: (1) m, defined as straying probability and initialized at 5%; and (2) pij, as defined above. We then set the off-diagonal elements of M to mpijand the diagonal elements to 1 – m. Because pij represented a discrete interpopulation movement, we normalized the off-diagonal probabilities over all possible movements, i.e.
We then used the matrix of population sizes X (described in the previous section) in conjunction with M to define a migration matrix N = XM. The diagonal elements of N contained the number of fish resulting from self-recruitment, and the column sums of the off-diagonal elements contained the number of fish immigrating to the populations (sensu Bjorkstedt et al. 2005). The proportion of recruitment in population i that comes from population j was then calculated in order to examine pairwise directed dependence. If this ratio exceeded some threshold, then population i was dependent upon population j. The relationships among all populations were visualized as a directed graph. Independent populations in the graph were populations that are not dependent upon any others indicated either by populations with either no connections to other nodes, or only outbound connections. In our model, populations were adjacent (connected) if the donor population contributes more than 1% of total recruitment to the recipient. In addition, we preserved the strength of the connection to represent the weighted graph fully.
where X represents population size, and δjj is local recruitment. We assessed how the trajectory of population independence changed over time by recalculating ζ for the remaining populations after each dam addition.
We examined the source–sink structure (Pulliam 1988) of the ESU by evaluating the importance of individual populations to the historical graph at the ESU scale (Bunn et al. 2000; Urban & Keitt 2001). Specifically, we examined node sensitivity for outdegree and indegree of a given node. Outdegree and indegree correspond logically to a qualitative representation of source and sink structure (Pulliam 1988), while node strength provides a quantitative representation of this structure (Barrat et al. 2004; Bascompte et al. 2006). We calculated outdegree and indegree of a given node by summing the rows and columns of the adjacency matrix A(D), respectively. To calculate node strength, we summed the row and column sums of the off diagonal elements of N. Note that we assumed all populations have at least some local recruitment and may be more accurately termed pseudo-sinks (Watkinson & Sutherland 1995).
We combined methods from Bunn et al. (2000) and Urban & Keitt (2001) with our digraphs to examine the effect dam addition had on the structure of the ESU. In addition to observing the actual fragmentation of the ESU, we used our model of connectivity and a series of alternate node removal scenarios (random, removal by largest available habitat and removal by largest node strength) to observe what happened to the graph as populations in the ESU went extinct.
We tested the sensitivity of the model to our assumptions by perturbing each of five model parameters by 10% and tallying the percentage change in the total number of edges in the graph. These parameters included: (1) the α parameter in the dispersal kernel; (2) the percentage of fish straying; (3) that migration is proportional to the interpopulation distance; (4) that population size is proportional to historical spawning extent; and (5) that all fish arriving at a new population recruit into that population (i.e. fitness of natives vs. strays).
The historical digraph G1 based on the dispersal adjacency matrix outlined above exhibited unbalanced indegree and outdegree, and contained six entirely disconnected (independent) populations (Fig. 3). All these populations are in the San Joaquin system, where the geography of the river basins is such that the populations are quite far apart (Fig. 2). In addition, the geographically closest of these populations (23–25) are all small enough to preclude outbound/inbound connections (Fig. 2). There are several populations in the Sacramento River Basin whose connections (> 1) were all outbound: Upper Sacramento (5), McCloud (6), Pit (7), Yuba (18), North Fork Feather (15) and the North and South Forks of the American River (19, 21) (Fig. 3). These large source populations, like those in the San Joaquin, are also demographically independent. Stronger demographic connections, on average, exist between nearby populations in which the source population is larger than the pseudo-sink population; as expected, the strength of the connection tends to decay with distance (Fig. 3).
The current graph is smaller than the historic graph, because most spawning habitat for historical populations is now behind dams (Fig. 4). At the ESU scale there are 15 demographic connections above the 10% threshold, four of which are outbound from the Feather River Hatchery (14). Butte Creek (13), a net importer before dam construction (indegree = 6, outdegree = 1 in Fig. 3), is a net exporter (indegree = 0, outdegree = 7 in Fig. 4). Both Stony (1) and Beegum Creek (3) had an average of zero reported spawners, hence the lack of connections in either direction (Fig. 4). Lastly, there is only one independent population, Battle Creek (8), in the current graph, while there were nine such populations in the historical graph.
In addition to blocking habitat, dam addition affects the remaining nodes both by increasing demographic independence on average and reducing the strength of the connections between nodes (Fig. 5a–d). As large spatially proximate nodes are removed from the graph, edges with an initial high weight are lost and the weight of certain remaining edges increases as migrants have fewer possible destinations (Fig. 5a–d). For example, when Shasta Dam was constructed in 1945 it blocked access to several major rivers including the Pit, McCloud and the Upper Sacramento (located just above the northernmost dam in Fig. 2), and the results illustrate what a vital source these three rivers were to the overall graph (Fig. 5b). Each of these nodes (especially the Pit River) had a high outdegree, and the removal of these three nodes results in a loss of 12 edges (Fig. 5b). However, the loss also affected the context of populations such as Battle Creek (8), which had an increase in the number of outbound edges, as well as their weight (Fig. 5b–d). The last two panels depict the loss of the American River and the Feather River populations through the addition of Nimbus Dam and Oroville Dam, respectively (Fig. 5c,d). Any dam that blocked access to anadromous habitat in the San Joaquin system had little effect on the remaining populations, because these populations were all quite isolated (nodes not shown).
Independence of smaller populations increases with the loss of large source populations (Fig. 6), suggesting that recolonization rates are lower under the current structure than they were historically. Some losses are worse than others; the addition of Shasta Dam (1942) not only removed many edges (Fig. 5b), but it also caused a dramatic increase in population independence for many of the populations present in the ESU (Fig. 6). Consider, for example, Butte Creek (13), which progresses from ζ = 0·77 in 1850 to ζ = 0·87 in 1968 (Fig. 6).
The median ζ across the ESU shows markedly different patterns when exposed to different node-removal scenarios (Fig. 7). Under the scenario aimed at removing the nodes with the highest node strength, population independence ζ of the remaining populations increased the fastest. The random removal scenario has the next strongest effect, followed by removal based on the largest habitat size of the remaining populations (Fig. 7). The difference between the node-strength and the random removal scenarios is particularly evident after approximately one-third of the habitat has been removed. Population independence ζ increased faster for all of these scenarios, as compared to actual removal (Fig. 7).
Our model was most sensitive to two parameters: (1) uncertainty about the percentage of fish that stray; and (2) to percentage of straying fish that recruit into the recipient population (Table 1). The model was less sensitive to uncertainty in dispersal capabilities of chinook and interpopulation distance. The model was not sensitive to our definition of population size. While we present only results for the historical graph (Table 1), these results hold as the graph fragments and the relative impacts of dams are the same as the unaltered empirical graph.
Table 1. Results from sensitivity analysis. For each parameter listed, we implemented a 10% perturbation and tallied the absolute change in number of edges in the final historical graph. Noted are the number of edges and the absolute percent change. There were 35 edges in the base historic graph
No. of edges
% of fish that stray
% Strays recruiting
Weighted digraphs have enabled us to understand more clearly the population context in the spring-run chinook salmon ESU, because they have shown whether populations are importers, exporters, or functionally independent. The historical digraph had both source and pseudo-sink populations, and a range of demographic connections between populations. The current graph has fewer source populations and fewer independent populations. Additionally, the current graph has populations that switch context from their position in the historical graph, and has more and stronger demographic connections between populations. While the impact of dams on fish populations has long been known, our examination of the sequential dam addition in the Central Valley showed clearly how a single dam can impact almost the entire ESU. This impact meant a loss of source populations to the ESU, resulting in fewer edges and increased isolation for the remaining nodes. This translates to decreased opportunity for recolonization after extinction or disturbance events.
Previous graph theoretic attempts to model how organisms perceive their landscape have relied mainly on regular graphs (Bunn et al. 2000; Urban & Keitt 2001; Brooks 2006) (although see Fortuna et al. 2006 for a recent example of the utility and strength of a digraph application). Here we have accounted for the strength and directionality of the connections in the graph, and while this is an obvious and intuitive extension of graph theoretic applications that has been mentioned several times in the literature (Gustafson & Gardner 1996; van Langevelde et al. 1998; Urban & Keitt 2001; Fagan 2002), we stress its importance in this and future applications. Imagine, for example, the different interpretation of Butte Creek (13) in a regular graph. There Butte Creek might jump out as a stepping-stone population (sensu Urban & Keitt 2001); however, it is clear from the digraph that this, in fact, is a pseudo-sink population whose demographic trajectory is influenced by several populations in the graph. Lastly, by accounting for recruitment as a measure of connectivity (sensu Bjorkstedt et al. 2005), we have extended the purely spatial application of graph theoretic measures and have uncovered not only how nodes are connected spatially, but what that spatial positioning means for the trajectories of populations within the ESU.
Defining what comprises a population remains an active research area in ecology and evolution. Indeed, relatively little work has been conducted on ascertaining what fraction of incoming recruitment affects population trajectories enough to consider them linked (Waples & Gaggiotti 2006). Hastings (1993), in a theoretical system, has shown that the 10% threshold is sufficient to consider population trajectories as linked. However, Lande, Engen & Sæther (1999) showed that under certain circumstances, i.e. weak density regulation, even very small migration rates can help to increase the spatial scale of synchrony. At two extremes, therefore, we can assume independence for populations with no edges or only outbound edges in Fig. 3, and cannot assume independence for populations with inbound connections over 10% (Fig. 3). Even if absolute independence thresholds are not definitive, relative changes in population independence are clear from the pattern of dam addition that successively fragmented the ESU and isolated remaining populations (Fig. 6).
Graph theoretic applications are appealing from a conservation standpoint because they are relatively simple to implement and they offer critical insight at both the landscape and population level (Urban & Keitt 2001). The graphs herein show interpopulation connectivity across the ESU, population importance, and how the removal of populations over time fragmented the ESU. Because this is a riverine setting, edge removal between two populations means typically that there are no alternate edges between that pair of populations (Fagan 2002). This means that fragmentation events lower down in the trunks of a watershed (Fagan 2002) can have dramatic effects – witness the effect of two single such events (Shasta and Oroville Dams) in our ESU, which removed a total of seven populations from the ESU (Fig. 5b,d). Clearly, the Pit River (7) had a major impact on the ESU, and were it not for the considerable complexities involved with removing major dams like Shasta and Keswick (just downstream of Shasta and the one depicted in Fig. 2), this would be an obvious place to highlight conservation and restoration efforts. However, Shasta Dam holds much of Northern California's water and so its removal would have serious implications for both the amount of water and its flow regulation throughout Northern California.
Palmer et al. (2005) underscore the need for a guiding image when restoring river ecosystems, and our depiction of the historical graph (Fig. 3) provides such an image. Further, the simulation of node-removal under different scenarios provides information that could be key to managers, as it highlights which restoration methods would bring about a reduction in demographic isolation fastest. While one might assume naively that restoring large populations first would have the greatest affect, that is not the case here (Fig. 7). Clearly, a scenario centred around restoring populations with large node strength first would accomplish this by adding more connections back to the graph (Fig. 7). Somewhat counter-intuitively, ζ decreased initially under the actual removal scenario; however, this is due simply to the spatial arrangement and timing of dam removal in the Central Valley. Notably, the first populations to be removed were in the southern San Joaquin, which meant that while habitat was lost, the resulting graphs were initially more compact and less isolated. Lastly, and perhaps most importantly, our graph framework accomplished what Jansson et al. (2005) called for in terms of a conceptual model that shows system function, system impairment and restoration strategies that ‘will move the system back to the guiding image’.
Connections are the mechanism by which recolonization can occur following disturbance, and they add stability and resilience to a system. It is intuitive that with more connections the removal of any one edge has less effect on the overall stability of the graph. Given the historical level of connections, then the graph as of 1968 (Fig. 5d) suffers from a lack of connections, and must be viewed as less resilient. This is echoed by the demographic isolation seen in Fig. 6, and adding connections back into the system would decrease demographic isolation and increase stability. There is a limit to this, however, in that a graph can have too many connections. While an increase in connectivity increases the likelihood of rescue (Brown & Kodric-Brown 1977), it also increases both the likelihood of pathogen spread (Hess 1996a) and spatial coupling. Hess (1996a,b) has shown that intermediate levels of connectivity provide a balance between extinction and persistence. With increased spatial coupling, Keeling, Bjørnstad & Grenfell (2004) have shown that synchronous populations are increasingly vulnerable to a similar extinction trajectory. Connections should therefore be viewed in light of a balance between these two opposing forces; simulation and/or analytical studies could help to uncover an optimum level of connectivity for population and ESU persistence.
The conceptual model presented herein has highlighted at least two other areas of future research. First, we might ask what other types of migration models make sense for salmon. We experimented with other models of straying, including implementing a ‘wrong-turn’ model where returning fish are faced with a series of choices as they migrate back to their natal stream. While this model is potentially more representative of the actual process undergone by a returning adult salmon, its results were qualitatively quite similar (see Appendix S1 in Supplementary Materials) to the more parsimonious distance-based model presented here, and it was less extensible to other systems. Secondly, we might ask how representative this model is for salmon dynamics. It was our intention that this model serve as an illustrative model of salmon connectivity, not necessarily a usable model of metapopulation dynamics. While the sensitivity results indicate that the model is fairly robust to uncertainty, they point to areas of further research. Namely, we need additional information about the percentage of fish that stray and the percentage of strays that recruit into populations.
Remarkable progress has been made in graph theory in just the last 8 years. Ecologists willing to wade into this realm will find that much awaits them in the way of different network structures, rapidly advancing algorithms and a wealth of interesting applications (Proulx, Promislow & Phillips 2005). Here graphs have enabled us to accomplish the following: (1) to enhance our understanding of the overall ESU structure; (2) to examine how ESU structure changed through time; and (3) to understand the historical importance of individual populations. In a data-limited environment, this exercise has shed light on this system from both an ecological and conservation standpoint. Our model of directed connectivity can be extended to many other systems, riverine or otherwise, and we recommend graph theory as an attractive analytical tool for rapid assessment of critical landscapes and endangered populations.
We thank Dean Urban, Brad Lamphere, Masami Fujiwara, Eric Treml, Dalia Conde, Chris Brooks and two anonymous reviewers, whose insightful comments considerably strengthened this manuscript. We also thank Ben Best for programming assistance. This work was supported in part by a James B. Duke Fellowship to R. Schick.