The Triangle Network
Even by our conservative standards (1500 m dispersal distance, only patches > 25 ha), the Triangle was very connected for forest songbirds. Eighty-three percent of patches (1148 out of 1382 patches) were contained in a single component that stretched from the southwest corner to the northeast corner of the map, indicating that it was possible for a bird to move across the entire landscape in small dispersal bouts.
This connectivity has some positive implications for conservation in the study area. Few patches were isolated so there should not be problems with permanent patch extinction or lack of gene flow. Depending on network topology, however, this level of connectivity could also have negative consequences. For example, West Nile virus or Avian Flu could spread rapidly across the Triangle, leaving few (if any) patches unaffected. To better understand the likely rate of spread of any such disturbance, it was necessary to further characterize the topology of the Triangle network.
The Triangle network had a much larger diameter than either of the simulated networks. This implied there were few shortcuts, and rate of movement through the Triangle will be slow. In other words, the Triangle seemed to display the desirable characteristic of intermediate connectivity that would allow dispersal and gene flow while slowing the spread of disease or other disturbance. The large diameter of the Triangle network relative to both the random and scale-free networks suggested the possibility of planar network topology.
A lack of empirical data dictated that our description of movement be qualitative rather than quantitative, but we expected that bird dispersal and gene flow would occur at the same rate as spread of a bird-dispersed pathogen. Although this movement should be slow due to the relatively large diameter of the landscape, the landscape topology indicated that both birds and pathogens could cross the landscape given enough time. Nevertheless, the slow rate of pathogen spread may allow managers to intervene or, over longer time periods, it may allow bird species to adapt. Calculating the exact level of landscape connectivity that balances “desirable” movement (e.g., dispersal and gene flow) with “undesirable” movement (e.g., spread of pathogens or exotic species) is no small task. These ideas have been touched on, particularly in the context of spatially autocorrelated phenomena and reserve design (Hof & Flather 1996), but graph theory holds promise for additional insight or even an analytic solution to this problem.
Node-degree distribution was more skewed in the Triangle network than in the random network and less skewed than the scale-free network. The skewness showed a heterogeneous node degree, which is thought to provide resilience against node removal or disturbance. As long as hubs are protected from development, networks with heterogeneous node degree can sustain random loss of many nodes before connectivity is compromised (Urban & Keitt 2001; Barabasi & Bonabeau 2003). In addition, if management efforts are focused on hubs and disturbances are quickly identified and contained or eliminated, the network is not likely to succumb to disturbance.
The clustering coefficient was quite a bit higher in the Triangle than in either of the simulated networks, which indicated the presence of many redundant pathways across the landscape. Redundant or alternative pathways confer resilience to random patch removal because they help maintain connectivity through the landscape when routes are deleted. High clustering coefficients are characteristic of small-world networks and uncharacteristic of random networks. Planar and scale-free networks may or may not show clustering. The skewed node-degree distribution and long path lengths revealed that the Triangle network did not have small-world network topology, but it was not clear whether it resembled a planar or scale-free network.
The connectivity correlation in the Triangle network was positive, indicating the inverse of compartmentalization—hubs tend to be clustered next to each other on the landscape. High-degree nodes (larger circles) tended to be grouped together and separated from groups of low-degree nodes (smaller circles) (Fig. 6). These groups also tended to be in highly clustered areas (darker colors), resulting in many darkly colored large circles and pale-colored small circles. This occurred because there were large regions on the landscape with high forest cover and many patches (e.g., Chatham county—the southwestern portion of the map) and other regions with low forest cover and few patches (e.g., Wake and Johnston counties—the southeastern portion of the map). Conservation strategy in the Triangle might focus on these highly clustered regions. On one hand, they may be desirable areas for many forest interior species because of their high forest cover and densely connected patches, which facilitate dispersal. On the other hand, however, their highly interconnected nature makes them vulnerable to spread of disease. Furthermore, some might argue that because clustering confers resilience to patch removal, these regions can withstand development more than other regions in the Triangle and development would be better concentrated in these areas than in more vulnerable, less-clustered areas, where connectivity may be easily severed.
Figure 6. Node degree and clustering in the Triangle region of North Carolina (circles, habitat patches, size of which is proportional to node degree and intensity of shading is proportional to clustering; RTP, Research Triangle Park in Durham County).
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Figure 6 highlights some areas of conservation interest in the Triangle network. Research Triangle Park, a research park in southeastern Durham County, is a heavily forested area that is undergoing rapid development. It currently contains many hubs that are also highly clustered, perhaps making this a safe choice for additional well-planned development. Umstead State Park is located in Wake County adjacent to the international airport. The park is a large (2201 ha) protected area and appears to be more compartmentalized than clustered, potentially conferring some protection from disturbances in the outside landscape. Finally, Jordan Lake State Recreation Area is a large (1585 ha) protected area contained mostly in Chatham County that is composed of several smaller forest patches, two of which appear to be large hubs. The southern hub is much more clustered than the northern one, and this variation may provide an interesting opportunity to test some of the concepts discussed in this paper, such as the effect of clustering on population stability or spread of disease.
For forest birds, the Triangle network appears to be somewhere between a planar network and a scale-free network, although network topology might change completely if a different organism were under consideration. For example, if the focal organism was Norway Maple (Acer platanoides), an exotic tree commonly used in residential landscaping, the average edge might be much shorter than for birds because seed dispersal is more limited. Nevertheless, the network would also contain many shortcuts radiating from plant nurseries to locations across the Triangle. This network would more closely resemble a small-world network and would display the small diameter and fast spread of disturbance characteristic of this topology.
Applications and Future Work with Graph Theory
Graph theory may be well suited for selecting habitat reserves (Opdam et al. 2006; Pascual-Hortal & Saura 2006; Minor & Urban 2007). Our results, however, do not suggest a formula for reserve design because each landscape and conservation problem is different. Despite these differences, the importance of considering landscape connectivity when designing reserves is becoming widely acknowledged (e.g., Briers 2002; Nikolakaki 2004; Moilanen & Wintle 2007). Here, we presented some new tools for measuring aspects of connectivity that might be important to consider in conservation planning. At times a high level of connectivity is desirable (Jordan et al. 2003), whereas at others it can be detrimental (Condeso & Meentemeyer 2007). Difficult choices must sometimes be made, such as whether it is more important to have a network that is robust to spread of disturbance or one that maximizes population stability. If the former is preferred, a highly compartmentalized reserve might be appropriate. If the latter is preferred, a highly clustered reserve might be the better design. Graph theory cannot make these decisions, but it does provide tools that make these decisions easier.
The next step in developing graph-based conservation theory might be to use simulation models to identify the topological characteristics that are most important to network resilience and connectivity. For example, is a large diameter or network compartmentalization more likely to slow spread of a disturbance? What is the interaction between the two? Also of interest are the relative effects of clustering and node-degree heterogeneity on resilience to node removal, which can be tested with node removal simulations. It may be even more enlightening to relate empirical population trends to network topology across a variety of landscapes—data from the Breeding Bird Survey could be ideal for this kind of analysis. An increased understanding of the ecological consequences of network topology would allow managers and conservationists to make better decisions about land acquisition and reserve design and to make predictions about the consequences of a variety of anthropogenic or natural disturbances for a variety of species.