A Network Approach for Analyzing Spatially Structured Populations in Fragmented Landscape

Abstract

We extend the recently proposed graph-theoretical landscape perspective by applying some network-centric methods mainly developed in the social sciences. The methods we propose are suitable to (1) identify individual habitat patches that are disproportionally high in importance in preserving the ability of organisms to traverse the fragmented landscape, and (2) find internally well-connected compartments of habitat patches that contribute to a spatial compartmentalization of species populations. We demonstrate the utility of these methods using an agricultural landscape with scattered dry-forest patches in southern Madagascar, inhabited by the ring-tailed lemur, Lemur catta. We suggest that these methods are particularly suitable in landscapes where species’ traversability is not fully inhibited by fragmentation, but merely limited. These methods are potentially highly relevant in studying spatial aspects of resilience and in the design of natural reserves.

Appendices

Appendix 1

Agglomerative hierarchical decomposition methods

The methods, which we present here, are among numerous others in the field of Social Network Analysis, and differ from each other in the underlying criteria determining whether any two nodes belong to the same group. LS-Sets and Lambda Sets (see Borgatti et al. 1990 and references therein) are two group-assessment methods based on the relative frequency of (direct and indirect respectively) relations among group members as compared to non-members. In this way, a group would have a relatively high frequency of direct or indirect relations within the group compared to outside the group. On the other hand, k-Cores (Seidman 1983) only require a group member to have a minimum number of k direct links to other group members to be part of that group.

LS-Sets

LS-Sets use the frequency of direct links between group members to form groups. The definition of LS-Sets is rather strict: it requires all proper subsets of a LS-Set to have more connections to the remaining members of the LS-Set than to outsiders. It has been shown that LS-Sets capture many of the key concepts underlying the intuitive notion of groups, e.g. they are difficult to disconnect, have a limited diameter and so forth (Borgatti et al. 1990). Thus the LS-Set is very powerful as a method of assessing compartmentalization, but LS-Sets are seldom found in real data because of their strict definition.

Lambda Sets

Lambda Sets, on the other hand, focus on another aspect of grouping criterion, namely the frequency of independent pathways between members. A Lambda Set has more distinctive paths between members compared to non-members. This makes them especially interesting from a dynamic perspective: assuming that over time links are temporally disrupted or permanently removed, one can expect that the high frequency of independent pathways between members can still provide a relatively high level of interconnectedness between them. This interpretation of Lambda Sets makes them appealing for studying e.g. the robustness of interconnectivity within groups to habitat loss. In fact, Borgatti mentioned the assessment of gene-flows in spatially separated patches of habitats as a possible application of Lambda Sets (Borgatti et al. 1990). Compared to LS-Sets, Lambda Sets carry a lower number of intuitively appealing notions of a group. Lambda Sets are, however, easier to find in empirical data.

k-Core

A different approach to group-assessment is the definition of k-Cores (Seidman 1983). This method has e.g. been used in studying compartmentalization in food webs (Melian and Bascompte 2004). A k-Core is a subset of nodes in which each node is adjacent to no fewer than k nodes in the subset. Hence, the definition does not include an explicit requirement of internal versus external linkages; it focuses solely on the number of internal connections and exerts no requirements on the members’ links to outsiders. According to Seidman, groups assessed by this method are not necessarily that interesting in themselves, but they could be described as regions of a graph where one can expect other groups to be found. In the context of landscape ecology, such compartments can be seen as sets of habitat patches where one can expect to find stronger internal groupings, thus k-Cores divide the landscape into rather loosely defined compartments suitable for further analysis.

Appendix 2

Software tools

During the course of our study, we developed a java-based computer program named JMatrixNet (downloadable at URL http://www.ecology.su.se/JMatrixNet) to compute networks of habitat patches in landscape map images, which is accessible for free. This program was used to create Figs. 2 and 5(A/D). The program identifies patches in a landscape by searching for pixels adjacent to each other of a, user-definable, unique color. All distances between the identified patches can be calculated and, based on one of the link-assessment methods described earlier, a network of habitat patches can be created and saved to file for further processing using other network-oriented software tools, e.g. Netdraw or Ucinet (Borgatti et al. 2002). Netdraw was used to create Figs. 1, 3, 5(B/C) and 6.