Maintaining Habitat Connectivity for Conservation by Bronwyn Rayfield

Maintaining Habitat Connectivity for Conservation

Bronwyn Rayﬁeld

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Ecology and Evolutionary Biology University of Toronto

Maintaining Habitat Connectivity for Conservation

Bronwyn Rayﬁeld

Doctor of Philosophy

Department of Ecology and Evolutionary Biology

University of Toronto

2009

Conserving biodiversity in human-dominated landscapes requires protecting networks of ecological reserves and managing the intervening matrix to maintain the potential for species to move among them. This dissertation provides original insights towards

(1) identifying areas for protection in reserves that are critical to maintain biodiversity and (2) assessing the potential for species’ movements among habitat patches in a reserve network. I develop and test methods that will facilitate conservation planning to promote viable, resilient populations through time.

The ﬁrst part of this dissertation tests and develops reserve selection strategies that protect either a single focal species in a dynamic landscape or multiple interacting species in a static landscape. Using a simulation model of boreal forest dynamics,

I test the eﬀectiveness of static and dynamic reserves to maintain spatial habitat requirements of a focal species, American marten (Martes americana). Dynamic reserves

improved upon static reserves but re-locating reserves was constrained by fragmenta-

tion of the matrix. Management of the spatial and temporal distribution of land-uses

in the matrix will therefore be essential to retain options for re-locating reserves in the

future. Additionally, to include essential consumer-resource interactions into reserve

selection, a new algorithm is presented for American marten and its two primary prey

species. The inclusion of their interaction had the beneﬁt of producing spatially aggre-

gated reserves based on functional species requirements.

ii The second part of this dissertation evaluates and synthesizes the network-theoretic approach to quantify connectivity among habitat patches or reserves embedded within spatially heterogeneous landscapes. I conduct a sensitivity analysis of network-theoretic connectivity analyses that derive least-cost movement behavior from the underlying cost surface which describes the relative ecological costs of dispersing through diﬀerent landcover types. Landscape structure is shown to aﬀect how sensitive least-cost graph connectivity assessments are to the quality (relative cost values) of landcover types. I develop a conceptual framework to classify network connectivity statistics based on the component of habitat connectivity that they quantify and the level within the network to which they can be applied. Together, the combination of reserve design and network connectivity analyses provide complementary insights to inform spatial planning decisions for conservation.

iii Acknowledgements

My research on the protection of habitat networks would not have been possible without a very supportive social network of collaborators, friends, and family. I want to thank, first and foremost, my advisor and mentor Dr. Marie-JoséeFortin. Working with her has taught me so much about ecology, research, leadership, and life. I hope to be as good an advisor to my own students one day. I am also thankful to my com- mittee members, Dr. Don Jackson and Dr. Lisa Manne, for their thoughtful sugges- tions and ongoing encouragement. A special thank-you is reserved for my Aunt Jan, an inspirational wildlife biologist and naturalist, who showed me my first caribou and shared her passion for Canadian wilderness with me.

This research beneﬁted greatly from my collaborations with Dr. Andrew Fall, Dr.

Atte Moilanen, and Dr. Dean Urban. Andrew and I shared many inspiring discussions about my research, particularly at the formative stages, and his sharp thinking helped to clarify and reﬁne many ideas. Atte was a role model for eﬃciency in writing and he generously shared his insights into conservation planning and his conservation software

- Zonation. Dean kindly hosted me at Duke University to share with me his enthusi- asm and ideas about the many conservation applications of graph and network models.

Generous ﬁnancial support for my work was provided by the Natural Sciences and

Engineering Research Council of Canada through a Postgraduate Scholarship (PGS-

A) and a Canada Graduate Scholarship (CGS) awarded to myself and a Discorvery

Grant awarded to my advisor, Dr. Fortin. Additional funding was provided by The

Department of Ecology and Evolutionary Biology at the University of Toronto.

The Department of Ecology and Evolutionary Biology at the University of Toronto has been a wonderful place to do graduate work. The whole Information Technology

Group, and Stephen Smith in particular, were invaluable. I am enormously grateful to all members of Spatial Ecology Lab (LE Lab), especially Patrick James, Aleksan- dra Polakowska, Stephanie Melles, Pilar Hernandez, Alistair MacKenzie, Josie Hughes,

iv Jonathan Ruppert, and Allan Brand. It has been a pleasure to share the daily comings and goings of grad life along with major successes and challenges with these people.

Many thanks to my lifelong friends - Emily, Melissa, Aleks, Ryan, and Colin - who never let me take myself too seriously. Lastly, thank you to my loved ones: Mom, Dad,

Sarah, Marc, Monkey, and Jamie. No words can express what you each mean to me.

v Chapter Acknowledgements

This thesis is comprised of ﬁve co-authored manuscripts that are either published, in press, submitted, or in preparation for peer-reviewed journals (Chapters 1-5). Permis- sions to use published materials in this dissertation have been obtained from the publishers. Experimental design, analyses, and manuscript preparation were all carried out by the principal author and PhD candidate. Co-authors of the chapters contributed conceptual discussions, expertise in computer programming, and editing of written materials. Chapter 1 introduces and provides the motivation for the dissertation. Chapter

6 concludes the dissertation and provides a starting point for future research.

1. Rayﬁeld, B., Fortin, M.-J., Urban, D. (in prep.) A network approach for conser-

vation planning in dynamic landscape mosaics. (Chapter 1)

2. Rayﬁeld, B., James, P., Fall, A., and Fortin, M.-J. (2008) Comparing static ver-

sus dynamic protected areas in dynamic boreal ecosystems. Biological Conserva-

tion 141:438-449. Reprinted with permission from Elsevier. (Chapter 2)

3. Rayﬁeld, B., Moilanen, A., and Fortin, M.-J. (2009) Incorporating consumer-

resource spatial interactions in reserve design. Ecological Modelling. 220, 725-

733. Reprinted with permission from Elsevier. (Chapter 3)

4. Rayﬁeld, B., Fortin, M.-J., Fall, A. (accepted with minor revisions March 2009;

LAND-08-1820 ) The sensitivity of least-cost habitat graphs to relative cost sur-

face values. Landscape Ecology. (Chapter 4)

5. Rayﬁeld, B., Fortin, M.-J., Fall, A. (in prep.) Connectivity for conservation: A

framework to classify habitat network connectivity statistics. (Chapter 5)

vi Contents

1 Network approach for conservation planning 1

1.1 Introduction ...... 1

1.1.1 Static and equilibrated landscapes: Island Biogeography Theory . 2

1.1.2 Network models and dynamic landscape mosaics ...... 4

1.2 Network-theoretical insights into conservation planning and reserve design 5

1.2.1 Assessing the resilience of reserve networks ...... 5

1.2.2 Network robustness and network structure ...... 8

1.2.3 Connectivity and resilience in reserve networks ...... 10

1.3 Conclusion ...... 11

1.4 Dissertation overview ...... 12

1.4.1 Chapter 2: Dynamic reserves in a dynamic boreal forest . . . . . 12

1.4.2 Chapter 3: Consumer-resource interactions in reserves ...... 13

1.4.3 Chapter 4: Sensitivity of habitat network connectivity assessments 13

1.4.4 Chapter 5: Quantifying connectivity of habitat networks . . . . . 14

1.4.5 Chapter 6: Conclusions and future directions ...... 14

2 Comparing static versus dynamic reserves 15

2.1 Abstract ...... 15

2.2 Introduction ...... 16

2.3 Methods ...... 20

vii 2.3.1 Study area ...... 20

2.3.2 Boreal forest landscape dynamics model ...... 21

2.3.3 Protected areas ...... 25

2.4 Results ...... 32

2.5 Discussion ...... 35

3 Consumer-resource interactions in reserves 41

3.1 Abstract ...... 41

3.2 Introduction ...... 42

3.3 Methods ...... 44

3.3.1 Summary of the Zonation reserve-selection algorithm ...... 44

3.3.2 Introducing novel spatial consumer-resource interactions into Zona-

tion ...... 49

3.3.3 Case study: predator-prey interaction in the boreal forest of Qu´ebec

(Canada) ...... 51

3.3.4 Reserve-selection scenario comparison with diﬀerent combinations

of species-speciﬁc and interaction connectivity layers ...... 52

3.4 Results ...... 53

3.5 Discussion ...... 56

3.6 Conclusion ...... 62

4 Sensitivity of least-cost habitat graphs 64

4.1 Abstract ...... 64

4.2 Introduction ...... 65

4.3 Methods ...... 71

4.3.1 Generation of artiﬁcial landscape spatial patterns ...... 71

4.3.2 Cost values to quantify resistance to movement ...... 74

viii 4.3.3 Graph-theoretic representations of habitat connectivity using least-

cost links ...... 75

4.3.4 Measuring the sensitivity of graphs with least-cost links ...... 79

4.4 Results ...... 79

4.5 Discussion ...... 84

4.6 Conclusion ...... 88

5 Classiﬁcation of network-connectivity statistics 90

5.1 Abstract ...... 90

5.2 Introduction ...... 91

5.3 Background ...... 93

5.4 Development of methods to construct habitat networks ...... 97

5.5 Quantifying connectivity in habitat networks ...... 101

5.6 Classiﬁcation framework of habitat network connectivity statistics . . . . 102

5.6.1 Network levels of analysis ...... 103

5.6.2 Components of habitat connectivity ...... 105

5.7 Missing habitat-network connectivity statistics ...... 110

5.8 Selecting network connectivity statistics ...... 113

5.9 Spatio-temporal connectivity in dynamic habitat networks ...... 113

5.10 Conclusions ...... 115

6 Conclusions 128

6.1 Thesis summary ...... 128

6.2 Future research directions ...... 133

Bibliography 136

ix List of Tables

2.1 Description of alternative protected area (PA) scenarios ...... 26

3.1 Focal species’ habitat, home range, and dispersal parameter estimates . . 46

3.2 Diﬀerences among reserve-selection scenarios ...... 54

4.1 Chronological and alphabetical presentation of connectivity studies using

a cost surface to identify least-cost routes ...... 67

4.2 Parameters used to generate spatial patterns of landscapes and their fac-

torial combinations ...... 74

4.3 Sets of relative cost values used in factorial experiment ...... 76

4.4 Eﬀects of matrix composition (HM COV), habitat fragmentation (H FRAG),

matrix fragmentation (HM FRAG), and the relative cost values (COST)

on the mean spatial deviation of least-cost links ...... 81

5.1 Summary of the scale-speciﬁc classiﬁcation framework for network statis-

tics that quantify habitat connectivity ...... 104

5.2 Topological connectivity statistics that have been applied to quantify con-

nectivity in habitat networks ...... 117

5.3 Node- and link-weighted connectivity statistics that have been applied to

quantify connectivity in habitat networks ...... 121

x List of Figures

1.1 Schematic representation of fragmented landscapes under a) the Island

Biogeography Theory (IBT) and b) the dynamic landscape mosaic model 6

2.1 Location of the Vermillion study area in the province of Qu´ebec, Canada 22

2.2 Illustration of the two steps of the PA selection process ...... 30

2.3 Mean properties of all home ranges in the study area ...... 34

2.4 Mean properties of home ranges in the static, static-core, and dynamic PAs 36

3.1 Example of the two types of interaction layers used as input for the Zona-

tion reserve-selection algorithm ...... 48

3.2 Reserves selected under ﬁve diﬀerent reserve-selection scenarios ...... 57

3.3 Three cases illustrating varying degrees of co-occurrence between consumer

and resource ...... 59

4.1 Examples of the simulated landscapes at a resolution of 100 × 100 cells . 73

4.2 An illustration of a) least-cost links, b) paths, and c) habitat graphs. A

link is a route that directly connects two habitat patches (nodes) . . . . . 77

4.3 Interaction plots for the percentage of hospitable matrix (HM COV) versus

the set of relative cost values (C1 - C8) when habitat fragmentation is

either a) low (H FRAG = 0.05) or b) high (H FRAG = 0.5) ...... 82

xi 4.4 Interaction plots for the fragmentation of habitat (H FRAG) versus the

set of relative cost values (C1 - C8) plotted separately at each percentage

of hospitable matrix (HM COV; a - e) ...... 83

5.1 Number of published studies that use graph, network, or circuit theory to

quantify habitat connectivity ...... 94

5.2 Examples of diﬀerent levels of analysis within a habitat network . . . . . 106

xii Chapter 1

A network approach for conservation planning in dynamic landscape mosaics

1.1 Introduction

One of the goals of conservation planning should be to maintain or restore resilient landscapes that have the capacity to absorb disturbances (Bengtsson et al., 2003), thereby maintaining the full variety of biodiversity, at all levels of organization, into the long-term future (Margules and Pressey, 2000). Achieving this goal in multi-use, dynamic landscapes (e.g., managed boreal forests; Bergeron et al., 1999) necessitates landscape-level conservation strategies that create networks of protected areas or reserves while managing the landscape matrix surrounding those reserves (Lindenmayer and Franklin, 2002; Lindenmayer et al., 2006; Rayﬁeld et al., 2008; Brudvig et al.,

2009). Designing eﬀective conservation strategies requires knowledge of: (1) the natural dynamics of landscapes and communities (Bengtsson et al., 2003; Leroux et al.,

2007a,b; Rayﬁeld et al., 2008); (2) the response of individual species, interacting species,

1 Chapter 1. Network approach for conservation planning 2 and communities to landscape modiﬁcation (Tewksbury et al., 2002; Fahrig, 2007; Ray-

ﬁeld et al., 2009); and, (3) taxa-speciﬁc use of habitat fragments and the surrounding matrix (McIntyre and Wiens, 1999; Bossenbroek et al., 2005; Guilhaumon et al.,

2008; Rayfield et al., accepted). If conservation efforts are successful, then ecological resilience should emerge as a quantifiable property of these landscapes over time, measured as the capacity of the landscape to absorb disturbances and reorganize without transitioning to an alternate stable state (Holling, 1973; Gunderson, 2000).

The recent attention given to landscape dynamics (Cumming et al., 1996; Meir et al., 2004; Wimberly, 2006; Leroux et al., 2007a,b; Rayfield et al., 2008) and ecological resilience (Bengtsson et al., 2003; Folke, 2003; Lindenmayer et al., 2006) in conservation theory reflects an important paradigm shift. There has been a move away from a static view of ecosystems that were thought to be in equilibrium with local resources and environmental conditions, towards a dynamic view of ecosystems that are continually being perturbed and that may exist in alternate stable states contingent on historical factors (Pulliam and Johnson, 2002; Beisner et al., 2003). Concurrently, the definition of resilience has progressed from a measure of the time it would take a perturbed system to return to equilibrium towards a measure of the magnitude of disturbance that a system can absorb before changing its stable state (i.e., its function, structure, identity, and feedbacks; Folke et al., 2004; Walker et al., 2004). The implications of this paradigm shift for conservation methods are presented below.

1.1.1 Static and equilibrated landscapes: Island Biogeography

Theory

The equilibrium view of ecosystems led to one of the central theories in conservation

- the theory of island biogeography (IBT; MacArthur and Wilson, 1967; Pulliam and

Johnson, 2002). A key aspect of the IBT that holds relevance for conservation is that the local species richness on islands is held in equilibrium not only by the resources Chapter 1. Network approach for conservation planning 3 present but also by a balance between immigration and local extinctions. The emphasis that IBT placed on immigration and emigration as a predictor of local species richness was novel at the time of its publication (Pulliam and Johnson, 2002) and has since precipitated research into the causes and consequences of movement among stationary islands (and later discrete, static patches of habitat). Maintaining movement among spatially subdivided populations and resource patches has since been shown to promote the persistence of species by stabilizing demographic stochasticity (Hanski and Gag- giotti, 2004, but see Levins, 1969 for an earlier example), allowing for patch colonization (Taylor et al., 1993; Hanski, 1999a; Hanski and Ovaskainen, 2000), and sustaining allelic diversity and evolutionary potential (Barton, 1992; Couvet, 2002 but see Wright,

1943 for an earlier example).

The IBT presented a very simplistic model of movement through a uniformly inhospitable “sea” such that islands were considered more or less isolated based on their size and their straight-line distance from a source population (such as the mainland;

Fig. 1.1a). In terrestrial landscapes, IBT has had a large impact on conservation where reserves are “islands” in a landscape matrix (Fig. 1.1a; binary treatment of habitat).

This simplistic model of movement has proven to have limited applicability in terrestrial systems where animals (even birds; B´elisle,2000) rarely move along straight-line paths in a landscape (Lindenmayer and Franklin, 2002). Indeed, the vast majority of terrestrial landscapes consist of a mosaic of land-cover types and habitat patches that provide the physical and biological conditions required for the survival, growth, and reproductive success for a particular species (Hall et al., 1997; Yapp, 1922). Habitat patches are more or less eﬀectively isolated than straight-line distance would suggest depending on the intervening land-cover types within the matrix and the movement capability of the species (D’Eon et al., 2002). Chapter 1. Network approach for conservation planning 4

1.1.2 Network models and dynamic landscape mosaics

As most landscapes are dynamic, new paradigms were proposed to move beyond a binary and static description of landscapes to a heterogeneous landscape model comprised of dynamic habitat patches embedded within a mosaic of other dynamic landcover types of varying qualities (Fig. 1.1b; Wimberly, 2006; Wiens, 2008). The ability of a species to move among habitat patches in these dynamic landscape mosaics can be described as potential or functional habitat connectivity (Calabrese and Fagan,

2004; Fagan and Calabrese, 2006) rather than simply focusing on a structural assessment based on the physical conﬁguration of habitats (i.e., habitat isolation as deﬁned in IBT). Potential habitat connectivity in static landscape mosaics is measured as a function of the spatial structure of the landscape and estimates of the movement behaviour of the species in response to that spatial structure (Taylor et al., 1993; With et al., 1997; Tischendorf and Fahrig, 2000b; Calabrese and Fagan, 2004; Fahrig, 2007).

In dynamic landscape mosaics, habitat connectivity will depend on both spatial and temporal habitat structure, species’ movement speed or rate of spread (Matlack and

Monde, 2004; Wimberly, 2006), and how species movement is aﬀected by landscape composition and conﬁguration (Wiens, 1997; Fahrig, 2007).

A network modelling framework (also called a graph-theoretic framework) is well- suited to capture our contemporary understanding of dynamic landscape mosaics (Ur- ban and Keitt, 2001; Fall et al., 2007; Brooks et al., 2008; Minor and Urban, 2008; Ur- ban et al., 2009). A network is a set of nodes that are connected to varying degrees by links that join pairs of nodes (Brandes and Erlebach, 2005). In conservation applications, nodes represent either habitat patches or reserves and links represent the dispersal likelihood or frequency between nodes (Fig. 1.1b; Fall et al., 2007; Minor and

Urban, 2008). Weights are assigned to nodes and links to include additional information about their level of connectivity, such as the size of nodes or the length of links in a reserve network. Incorporating the spatial properties of nodes and links is partic- Chapter 1. Network approach for conservation planning 5 ularly important in conservation and landscape applications; nodes can be defined as two-dimensional patches with fixed spatial locations and links can be defined as geo- referenced links between nodes connecting patch perimeter to patch perimeter (O’Brien et al., 2006; Theobald, 2006; Fall et al., 2007; Rayfield et al., accepted) and traversing along least-cost routes based on the suitability of the intervening matrix (Halpin and

Bunn, 2000; O’Brien et al., 2006; Theobald, 2006; Fall et al., 2007).

1.2 Network-theoretical insights into conservation

planning and reserve design

Network theory explicitly focuses on networks that are either naturally-occurring or have arisen out of decentralized planning by independent decisions over long time periods (e.g., habitat and reserve networks; Newman et al., 2006). The network-theoretic approach recognizes that networks are constantly changing over time and analyzes their dynamic structural properties (e.g., resilience; Watts, 1999; Newman et al., 2006).

Ultimately, the goal of network analyses is to understand the relationship between the structural properties of networks, their function, and their dynamical behaviour (e.g., functioning ecological communities; May, 2006; Newman et al., 2006). Hence, network theory can provide valuable insights for conservation planning in dynamic landscape mosaics in which maintaining connectivity among protected habitat patches in a reserve network is vital to the persistence of biodiversity.

1.2.1 Assessing the resilience of reserve networks

The numerous ways in which network theory can inform diﬀerent aspects of conservation planning and reserve design have been extolled by many authors (Urban and

Keitt, 2001; Brooks, 2003; Chetkiewicz et al., 2006; Fall et al., 2007; Minor and Ur- ban, 2008; Urban et al., 2009). However, one aspect of reserve design which has been Chapter 1. Network approach for conservation planning 6

Figure 1.1: Schematic representation of fragmented landscapes under a) the Island

Biogeography Theory (IBT) and b) the dynamic landscape mosaic model. In a) black nodes represent protected areas and the surrounding white area represents a uniformly inhospitable matrix. Solid lines represent the potential for an organism to move between nodes. In IBT nodes are equally well connected if they are of equal size and separated by equal distance. Note the landscape and connections among patches are static through time. In b) black and white nodes represent intact and disturbed protected areas respectively. Patches in grey scale represent different land-cover types in the landscape with increasing resistance for darker colours. Dashed line represents a linear barrier to movement such as a road. Solid lines represent least-cost links between nodes. Note the quality and configuration of land-cover types change through time and least-cost links between nodes reflect those changes. Chapter 1. Network approach for conservation planning 7 under-emphasized in conservation applications of network theory and which is particularly important in dynamic landscape mosaics is reserve network resilience (but see

Urban and Keitt, 2001; Minor and Urban, 2008; Urban et al., 2009). Similar to ecological resilience (Walker et al., 2004), network resilience can be deﬁned as the capacity of a network to absorb disturbance and retain essentially the same function, structure, identity, and feedbacks. When assessing network resilience, one is not concerned about the level of connection between any two particular nodes but rather that the network as a whole continues to function despite some endemic level of node or link destruction

(average behaviour of the network; Newman et al., 2006). The implication for reserve design is that if connection patterns among reserves are chosen appropriately, the reserve network can be highly resilient to the loss of nodes and links (although it may be susceptible to certain forms of node loss such as disturbances which speciﬁcally target highly connected nodes; Albert et al., 2000).

Network resilience is often measured in terms of network robustness - the number of nodes or links that can be removed without altering the connectivity of the network

(Albert et al., 2000; Callaway et al., 2000; Jeong et al., 2001; Newman, 2003). This assumes that the ability of a reserve network to sustain its biodiversity relies at least in part on the maintenance of connectivity among individual reserves (Peterson, 2002).

Network robustness can be characterized in a number of ways depending on: (1) the way connectivity is measured; (2) whether nodes or links are removed; and (3) the order in which nodes or links are removed. Hence, reserve network resilience could be assessed in terms of how connectivity of the network responds to diﬀerent types of disturbances (i.e., random vs. targeted) and diﬀerent aspects of the disturbance regime

(i.e., spatial pattern of disturbances; Bengtsson et al., 2003).

There are many ways to measure the connectivity of a network (see Chapter 5), hence the choice of a connectivity measure should be tied to an explicit hypothesis about how the structure of the reserve network affects key ecological processes that Chapter 1. Network approach for conservation planning 8 will ultimately influence the persistence of biodiversity. For example, the characteristic path length (i.e., the mean shortest path length between any two nodes in the network) and the diameter (i.e., the maximum shortest path length between any two nodes in the network) are two related connectivity measures that have been used in robustness assessments of the World-Wide Web and habitat networks respectively (Albert et al., 2000; Urban and Keitt, 2001). A short characteristic path length relative to the number of nodes provides a good indication that a network has a few highly connected nodes that make it possible to access all information on the web with just a few clicks of the mouse (Watts and Strogatz, 1998; Albert and Barabási,1999). A short diameter in a habitat network relative to the number of nodes implies that the network can be traversed rapidly so that a dispersing individual can more easily access all habitat patches (Minor and Urban, 2008).

1.2.2 Network robustness and network structure

Network robustness depends on a variety of structural properties of the network (Al- bert et al., 2000; Jeong et al., 2001; Strogatz, 2001). To assess the eﬀects of network structure on robustness, node and link deletions can either be performed on speciﬁc real-world networks (e.g., the World-Wide Web) or on abstract networks that are stochas- tically generated to produce certain structural properties (e.g., scale-free networks).

The distribution of links among nodes, or node-degree distribution, is a structural property of networks that aﬀects the robustness of a network to the loss of random and highly connected nodes. Networks in which all nodes have a similar number of links

(i.e., Poisson node-degree distributions) will quickly become disconnected into small clusters when nodes are deleted in either a random or targeted fashion (Strogatz, 2001; but see Holland and Hastings (2008), for an example in which randomizing network structure is positively related to the dynamics and persistence of modelled predator- prey systems). However, networks in which nodes have a highly heterogeneous node- Chapter 1. Network approach for conservation planning 9 degree distribution with a tail decaying as a power-law (i.e., power-law node-degree distribution also referred to as a scale-free network) are relatively robust to the deletion of random nodes, but are very sensitive to the targeted deletion of highly connected nodes or “hubs” (Albert and Barab´asi,1999; Jeong et al., 2001). Based on these general results, Minor and Urban (2008) proposed that an ideal reserve network would be a scale-free network comprised of a few large “hub” reserves connected to many smaller reserves.

Ferrari and Lookingbill (2009) have called into question the degree to which hubs that represent habitat patches or reserves in a landscape would function as true hubs from the classical network perspective described above. Classical hubs generally have much higher node degrees than the mean node degree of the network (orders of magnitude higher) and therefore they signiﬁcantly decrease the average path length of the network and account for a disproportionately large portion of the overall connectivity of the network (Barab´asi,2002). Ferrari and Lookingbill (2009) point out that habitat networks typically have fewer than 103 nodes compared to networks such as the

World-Wide Web which have more than 106 nodes; hence, the classical deﬁnition of a hub may not apply to even the most connected nodes in reserve networks. Another important distinction to point out is that spatial constraints may eliminate the possibility of movement between very distant nodes in a reserve network whereas hyperlinks connecting webpages in the World-Wide Web have no constraints and hence links are possible between all pairs of nodes. The node degree heterogeneity in reserve networks is therefore limited due to fewer nodes and the presence of spatial constraints. Conse- quently, it may not be possible for habitat patch hubs to have node degrees that are orders of magnitude larger than the mean. Much more research is needed to characterize the structural properties of existing and proposed reserve networks. In particular it is important to examine the consequences of the limited node-degree heterogeneity of reserve networks on their robustness. Chapter 1. Network approach for conservation planning 10

1.2.3 Connectivity and resilience in reserve networks

Network robustness is one approach to study the resilience of networks which focuses on the structural consequences of disturbances that remove nodes and links. Another way to think about the resilience of a reserve network is to consider structural properties of the network that can help to balance the connectivity for desirable processes

(e.g., daily foraging movements among patches, inter-population movements to maintain genetic mixing and maintain source-sink dynamics, seasonal migrations, and range shifts associated with global change) and undesirable processes (Schafer, 2001; Janssen et al., 2006, e.g., spread of invasive species, spread of forest ﬁres, coherence of other types of environmental stochasticity). This approach recognizes that the consequences of the structural properties of networks depend on the function of desirable and undesirable ﬂows within the network (Janssen et al., 2006).

Conservation planners have struggled for some time with this trade-oﬀ between locating reserves close enough that they allow for occasional exchange of individuals among them and far enough that they reduce the potential that a single disturbance or disease will impact them all (Schafer, 2001). Network theory can contribute to this quandary by identifying structural network properties that promote connectivity for certain processes at certain scales while breaking connectivity for others. For example, a structural property of networks that may enhance resilience is high compartmentalization whereby nodes with many links are connected to nodes with few links and vice versa (Meli´anand Bascompte, 2002; Minor and Urban, 2008; Urban et al., 2009).

Compartmentalization creates connected sub-network groupings that are internally well connected but which are only loosely connected to other sub-network groupings.

Resilience is enhanced through compartmentalization because disturbances are con-

ﬁned within this network structure (Meli´anand Bascompte, 2002). Minor and Urban

(2008) speciﬁed that in their idealized reserve network highly connected nodes (“hubs”) should be spatially separated to create compartmentalization that would isolate distur- Chapter 1. Network approach for conservation planning 11 bances but permit species movements across the landscape. The property of compartmentalization should apply to reserve networks because it requires node-degree heterogeneity but does not require that the node-degree distribution follows a power law.

1.3 Conclusion

Conservation planning in dynamic landscape mosaics can beneﬁt from a network-based approach to modeling the consequences of natural and anthropogenic disturbances on the resilience of ecological communities persisting in reserve networks. A network perspective focuses on the structural properties of landscapes and reserve networks that may enhance resilience such as node-degree distribution and compartmentalization.

The resilience of networks is an active area of research in many disciplines and there are many other structural properties that should be explored in the context of reserve networks such as connectance (Dunne et al., 2002), centrality (Jeong et al., 2001), and redundancy (Urban et al., 2009). Analysis of structural robustness of reserve networks should consider alternative sequences of node and link deletion that more closely mimic realistic patterns of disturbances (e.g., Srinivasan et al., 2007). Functional considerations regarding the spread of the disturbance and the dispersal abilities of species of interest can be simultaneously accounted for in reserve network structure. Examining the response of processes spreading through a given network structure simultaneously will improve our ability to estimate functional consequences for biodiversity conservation

(Janssen et al., 2006). Land-uses in the matrix surrounding reserves should be incorporated into conservation planning whenever possible (Lindenmayer and Franklin, 2002;

Bengtsson et al., 2003; Brudvig et al., 2009). The matrix will be very important for maintaining dispersal routes among reserves (Tewksbury et al., 2002; Damschen et al.,

2008; Walker and Craighead, 1997; Adriaensen et al., 2003; Pinto and Keitt, 2009) and for maintaining areas that are capable of becoming viable habitat in the future (Ray- Chapter 1. Network approach for conservation planning 12

ﬁeld et al., 2008). Resilient ecological communities will only emerge if the dynamics and contingencies of landscapes are incorporated into reserve network design and management of the matrix, thereby maintaining connectivity among reserves for species of conservation concern while limiting connectivity for disturbances.

1.4 Dissertation overview

Designing reserve networks and quantifying connectivity among individual reserves are two research areas which are essential to inform the design of eﬀective conservation management actions. My dissertation contributes to both of these areas of research.

I examine how the structure and dynamics of landscapes inﬂuence reserves and habitat connectivity. Chapters 2 and 3 focus on reserve design strategies for dynamic and static landscapes respectively where planning is based on the habitat requirements of a single indicator species (Chapter 2) or interacting species (Chapter 3). Chapters 4 and

5 focus on quantifying habitat connectivity using network theory. I ﬁrst evaluate the sensitivity of the habitat network model to the deﬁnition of matrix quality (Chapter

4) and then develop a conceptual framework to classify network connectivity statistics based on their ecological interpretation (Chapter 5). Taken together, these four chapters provide a complementary suite of novel methods and approaches to meet biodiversity persistence goals in reserves and improve assessments of potential connectivity among reserves.

1.4.1 Chapter 2: Dynamic reserves in a dynamic boreal forest

Reserve selection based on static habitat distributions may not ensure habitat persistence and species survival in dynamic landscapes. I compare the effectiveness of static reserves which are fixed in space through time with that of dynamic reserves which are periodically re-located in response to dynamic habitat. Reserves are selected based Chapter 1. Network approach for conservation planning 13 on the spatial habitat requirements of American marten (Martes americana) which is considered an indicator species for mature boreal forest biodiversity. I use a spatially- explicit landscape-disturbance model for a boreal forest region in Québec (Canada) to show that dynamic reserves do not dramatically improve upon static reserves because logging and fire disturbances in the matrix constrain options for re-locating reserves.

1.4.2 Chapter 3: Consumer-resource interactions in reserves

Despite their ecological importance, species interactions have not been explicitly included in terrestrial reserve-selection methods. I present a novel method for selecting reserves based on maintaining the connectivity between the distributions of a consumer and its resources. I illustrate the method with a conservation planning case-study of a predator, American Marten, and its two primary prey species in the same boreal forest in Qu´ebec. Including consumer-resource interactions in reserve-selection resulted in more spatially aggregated reserves when compared to reserves selected based on species spatial habitat requirements with no interaction.

1.4.3 Chapter 4: Sensitivity of habitat network connectivity

assessments

Least-cost routes through landscape cost surfaces are commonly used to model species- specific landscape traversability in a spatially explicit manner. There is considerable uncertainty involved in estimating the cost values for each land-cover type within a landscape for a particular species. I investigate how sensitive connectivity assessments based on least-cost habitat networks are to relative differences in the cost values. I performed this sensitivity analysis in artificial landscapes that I generated to control the area and degree of fragmentation of each land-cover. I found that the least-cost habitat networks were sensitive to differences in relative cost values assigned to land-cover Chapter 1. Network approach for conservation planning 14 types and that the spatial structure of the landscape was an important determinant of the degree of network sensitivity.

1.4.4 Chapter 5: Quantifying connectivity of habitat networks

The number of connectivity statistics for habitat networks can easily overwhelm ecologists. I develop a conceptual framework to classify connectivity statistics according to the component of connectivity that they quantify (i.e., inter-patch dispersal likelihood, route redundancy, route vulnerability or habitat availability) and the level of the habitat network to which they apply (i.e., element, neighbourhood, cluster or network).

This framework provides much needed clariﬁcation of the ecological interpretation of network connectivity statistics and will be an invaluable resource for anyone undertak- ing a habitat network connectivity analysis.

1.4.5 Chapter 6: Conclusions and future directions

This dissertation provides novel insights into how spatial and temporal landscape heterogeneity can be incorporated into reserve design and habitat connectivity assessments. In Chapter 6, I summarize my findings and offer new research directions. The natural extension of this research is to focus on how the structure and connectivity of reserve networks is linked to the long-term maintenance of biodiversity and ecosystem services in human-modified landscapes. Chapter 2

Comparing static versus dynamic protected areas in the Qu´ebec boreal forest

2.1 Abstract

Conservation planning is often based on static mapping of species ranges or habitat distributions. Succession and disturbance alter, however, habitat quality and quan- tity through time especially under global climate and land-use change scenarios; hence, static protected areas may not ensure habitat persistence and species survival. Here, I examined the relative merits of static and dynamic (ﬂoating) protected areas for the conservation of American marten (Martes americana) habitat in a dynamic boreal forest of Qu´ebec (Canada). Forest dynamics were modeled using a spatially-explicit landscape-disturbance model and protected areas were selected based on the quality and compactness of marten home ranges using the conservation planning software

MARXAN. Static protected areas were ﬁxed in space during 200 year simulations of boreal forest dynamics, while dynamic protected areas were re-located every 50 years

15 Chapter 2. Comparing static versus dynamic reserves 16 to track dynamic habitat. Dynamic protected areas supported more high-quality home ranges through time than static protected areas. The locations of dynamic protected areas were constrained, however, by the highly fragmented forest patterns created through logging and ﬁre in unprotected areas. My ﬁndings emphasize the often-overlooked point that if dynamic conservation planning is to be successful in the long term, the landscape matrix quality surrounding protected areas must be managed in such a way that options remain when it comes to re-planning.

2.2 Introduction

Protected areas play a vital role in safeguarding against further losses of existing biodiversity and wildlife habitat that result from increased human encroachment on natural spaces (Williams et al., 2004). To fulﬁll this role, protected areas (hereafter PAs) must not only ensure that multiple elements of biodiversity are adequately represented but that they persist in their protective role through time in dynamic landscapes (Mar- gules and Pressey, 2000; Gaston et al., 2006). In the last two decades, numerous site- selection algorithms have been developed to identify sites to be included in one or more

PAs that maximize the amount of extant biodiversity represented given existing economic constraints. These algorithms typically rely on static distribution patterns of biodiversity surrogates such as species (Csuti et al., 1997), habitat types (Olson and

Dinerstein, 1998), and land cover types (Pressey et al., 1996) to identify, prioritize, and select candidate PAs. However, because distribution patterns of species and habitats change over time due to natural processes (e.g., population dynamics, succession, and disturbances) and human impacts (e.g., land-use and climate change), site-selection algorithms based on representation objectives alone do not necessarily select PAs that ensure the long-term biodiversity persistence (Margules et al., 1994; Rodrigues et al.,

2000). Chapter 2. Comparing static versus dynamic reserves 17

Promoting long-term persistence of biodiversity in PAs therefore requires that site- selection algorithms move beyond simply meeting initial representation targets towards incorporating information about the spatio-temporal dynamics of populations and landscapes (Cabeza et al., 2004). Efforts to include population dynamics in site- selection algorithms have treated species-persistence objectives either implicitly by prioritizing higher quality habitat (Williams and Araujo, 2000) in large, compact, and well-connected PAs (McDonnell et al., 2002) or explicitly by modeling species-specific persistence through population-viability analyses (Carroll et al., 2003) or metapopulation models (Moilanen and Cabeza, 2002). Landscape dynamics have only recently been incorporated into site-selection algorithms and have primarily taken the form of an implementation problem in which PAs must be implemented gradually over several time periods due to budget constraints (Pressey and Taffs, 2001; Meir et al., 2004; Oet- ting et al., 2006). These studies incorporate the possibility that unprotected sites may be lost or degraded before the next implementation period due to anthropogenic landscape changes such as development or resource extraction. The conservation implications of stochastic landscape processes such as succession, natural disturbance regimes, and climate shifts that operate on timescales longer than the typical PA implementation horizon have just begun to enter discussions about PA design and have remained fairly theoretical (Cumming et al., 1996; Bengtsson et al., 2003; Araujo et al., 2004;

Pyke et al., 2005; Williams et al., 2005, but see Pickett and Thompson, 1978). A few studies have proposed, however, methods to incorporate dynamic ecological processes that provide important ecosystem services such as specialist pollinator interactions, hy- drological regimes, trophic interactions, and ecological diversiﬁcation of species lineages into quantitative conservation targets (Pressey et al., 2003; Chan et al., 2006).

The Québec boreal forest is a good system to examine the implications of landscape dynamics for conservation planning because it is perpetually dynamic due to both stochastic natural disturbances and deterministic human activities. Wildfire is Chapter 2. Comparing static versus dynamic reserves 18 the dominant natural disturbance agent and is highly variable in terms of extent, frequency, and ecological impact (Johnson, 1992). Logging, the dominant human disturbance agent, eliminates or substantially reduces the mature portion of these forested landscapes (Bergeron et al., 1999), and this has motivated the creation of a system of PAs to protect mature forest habitat and act as constraints to the logging sched- ule (Baskent and Keles, 2005). Selecting PAs in landscapes driven by highly variable disturbance regimes may be difficult because representative areas may not exist even at very broad scales (i.e., there may not be a smaller part of the region that is similar to the whole region; Cumming et al., 1996). The lack of representational areas in boreal forests in terms of an appropriate compositional measure has led some researchers to propose an alternative to static PAs, in which representation and persistence criteria are re-assessed periodically through time to create a dynamic system of PAs that change locations to track dynamic habitat (Cumming et al., 1996; Bengtsson et al.,

2003). Dynamic protected areas may be able to overcome some of the difficulties presented by highly variable natural systems when planning for conservation by ensuring that adequate habitat is always available. An intuitive illustration of the dynamic PA concept occurred as early as 1983 in the Gifford Pinchot National Forest when a wind- storm damaged nesting sites in six out of forty-five PAs for spotted owls (Strix occiden- talis) resulting in the re-location of one PA and the merging of two others (Ruediger,

1985).

When considering the implementation of dynamic PAs, an important distinction must be made between PAs implemented over broad, regional scales (Carroll et al.,

2003) and those PAs implemented within multi-use forest landscapes subject to logging (Baskent and Keles, 2005). The scale of PAs will determine which criteria and processes should be considered during the selection procedure. At a regional scale, PAs may be large enough that they can maintain internal re-colonization sources under the natural disturbance regime (Pickett and Thompson, 1978); hence, dynamic PAs at this Chapter 2. Comparing static versus dynamic reserves 19 scale are unnecessary because species occurrences alone in these PAs should serve as a surrogate for their long-term persistence. Smaller PAs implemented at local scales may also play an important role in biodiversity protection by preserving species or habitats that are under-represented in regional-scale PAs, providing a source of propag- ules and oﬀspring for re-colonization of logged areas, and facilitating the movement of biota across managed landscapes (Lindenmayer et al., 2006). Species persistence in these local scale PAs will be highly dependent on landscape dynamics as well as spatio- temporal population dynamics; hence allowing these PAs to be re-located in response to changes in habitat quality and conﬁguration may improve their ability to conserve target species.

Although several conceptual papers have suggested the use of dynamic PAs in boreal forest ecosystems (Cumming et al., 1996; Bengtsson et al., 2003; Angelstam et al.,

2004), none have discussed the details of their implementation. In this study, I investigate the potential of using dynamic rather than static PAs to maintain old growth habitat within the boreal forest and explicitly describe the steps necessary to identify, prioritize, and select PAs through time. The total area of PAs was ﬁxed between 10% and 12% of the study area to allow for alternative forest uses such as timber harvest and to comply with conventional conservation targets (McNeely, 1993; IUCN, 1994).

Patches in the static PAs were ﬁxed in space throughout the course of a 200-year simulation of boreal forest dynamics, whereas patches in the dynamic PAs were periodically re-evaluated and re-located in response to changes in forest conditions due to

ﬁre, succession, and logging. The length of the re-planning interval for dynamic PAs was assumed to be constrained by operational costs associated with PA re-location and was therefore conservatively set as 50 years corresponding to half of the harvesting age (100 years). PAs were chosen based on habitat requirements of the American marten (Martes americana Rhoads), which is often used as an indicator species for mature boreal forest biodiversity (Thompson, 1991; Buskirk and Powell, 1994) because of Chapter 2. Comparing static versus dynamic reserves 20

its large home range size (maximum home range size of 1000 ha; Chapin et al., 1998;

Potvin et al., 2000) and its degree of habitat specialization (prefers coniferous/mixed

forest greater than 30 years old; Buskirk and Powell, 1994; Thompson, 1994). To se-

lect PAs, I ﬁrst identiﬁed all potential marten home ranges in the study area-based on

the spatial conﬁguration of forest types and ages (James et al., 2005). I then used the

simulated annealing procedure implementation in MARXAN to ﬁnd two spatially com-

pact patches of high-quality potential marten home ranges to form a PA (Possingham

et al., 2000). Boreal forest dynamics were simulated using an existing spatially-explicit

landscape-scale model (Fall et al., 2004) that includes sub-models for forest succession,

ﬁre, and logging. I hypothesized that dynamic PAs would outperform static PAs be-

cause the marten home ranges within the static PAs may deteriorate through time due

to losses from wildﬁre. To compare static and dynamic PAs, I assessed the number,

quality, and connectivity of potential marten home ranges remaining within PAs and

across the entire landscape every 50 years over the course of the 200 year simulations.

2.3 Methods

2.3.1 Study area

The Vermillion study area is a boreal mixed-wood forest management unit that covers

approximately 430,000 ha in south-central Qu´ebec (Fig. 2.1). The northern portion of

the study area is a coniferous boreal forest characterized by black spruce (Picea mar-

iana (Mill.) BSP.), balsam ﬁr (Abies balsamea (L.) Mill.), jack pine (Pinus banksiana

Lamb.) and white birch (Betula papyifera Marsh.). The southern portion is a transition zone between the boreal and temperate forest regions characterized by white spruce (Picea glauca (Monech) Voss), balsam fir, and yellow birch (Betula alleghaniensis Britt.). Lakes and wetlands are distributed throughout the study area to make up approximately 10% of the total area. Stand-replacing fire is the dominant natural Chapter 2. Comparing static versus dynamic reserves 21 disturbance (Hélyet al., 2000) and logging has been the dominant anthropogenic disturbance since the 1950s. Data for the Vermillion study were derived from the 1990 decadal forest inventory carried out by Systèmed’Information FORestièrepar Tesselle

(SIFORT) in collaboration with the Qu´ebec Ministry of Natural Resources and Abitibi

Consolidated. Spatial data of tree age and species composition were derived from supervised classiﬁcation of aerial photography and are represented in raster format at a resolution of 0.25 ha (50 × 50 m).

2.3.2 Boreal forest landscape dynamics model

I used a spatially-explicit stochastic raster simulation model that integrates succession, growth, yield, and fire regime information to simulate landscape-scale forest dynamics. The model was implemented in SELES modeling platform (Fall and Fall, 2001) and consisted of sub-models for fire, succession, and logging that interact by modifying the dynamic forest conditions within each 0.25 ha pixel. Fall et al. (2004), Didion et al. (2007), and James et al. (2007) used this model to investigate interactions among forest management strategies and fire. I used this model to simulate boreal forest dynamics over 200 years with 10 replicates for both the static and dynamic PA scenarios.

The number of replicates and simulation duration was limited by the data processing time required to move between the simulation model and PA selection software.

Description of sub-models

Fire was modeled as a stochastic process based on empirical estimates of mean fire size and fire return interval (Van Wagner, 1978). The mean number of fires per year was a function of the size of the study area, the mean fire size, and the fire return interval.

Mean Number Fires per Year=(Study Area Size)/[(Mean Fire Size)(Fire Return Inter- val)].

Fire sizes were selected randomly from an exponential distribution with a mean of Chapter 2. Comparing static versus dynamic reserves 22

Figure 2.1: Location of the Vermillion study area in the province of Qu´ebec, Canada.

The map of the study area shows the spatial arrangement of habitat types at a resolution of 0.25 ha/cell. Chapter 2. Comparing static versus dynamic reserves 23

1500 ha (Fall et al., 2004). The exponential model is a commonly made assumption for studies of fire in the boreal and boreal mixed-wood forest and necessarily assumes inde- pendence among fire events (Van Wagner, 1978; Johnson and Van Wagner, 1985; Berg- eron et al., 2002; Fall et al., 2004). Fire return interval, the number of years required to burn an area of size comparable to the study area, was estimated to be 250 years to reflect recent changes in climate (Bergeron et al., 2001). Fire ignition occurred randomly and fire events spread randomly from the point of ignition to any eight neighboring cells until the total area burned reached the pre-selected size. In this statistical

fire model, fires burned independent of forest age or species composition so that the effects of different PA scenarios were not confounded by uncertainty in the burn probabilities for different fuel categories. This simplifying assumption emphasizes the effects of stand-replacing fires in boreal forests which are strongly linked to extreme weather events during which fires burn young and old stands alike (Johnson et al., 2001; Lefort et al., 2003). Such fires account for more than 97% of the area burned, although they account for less than 5% of the total number of fires (Stocks et al., 2002).

Forest succession was modeled using probabilistic successional pathways derived as a function of site-speciﬁc soil and drainage conditions and previous vegetation types based on long-term, plot-level data on forest transitions in the Vermillion study area

(Fall et al., 2004). Forest stand age was incremented annually until a maximum of 300 years unless either a fire or logging event occurred, which reset stand age to zero. For- est stand composition likewise followed a probabilistic successional trajectory unless reset by a fire or logging event. In the absence of disturbance, cells continue to age to a maximum of 300 years and site-specific maximum volume. They remain in the maximum age and volume categories until they are either burned or logged, implicitly assuming that such stands are in a gap-phase old forest state.

To simulate the effects of logging, the logging sub-model used an area-based annual allowable cut (AAC) with a minimum harvest age of 100 years and site-specific Chapter 2. Comparing static versus dynamic reserves 24 road accessibility that reflected plausible management scenarios for the mixed-wood boreal region in Québec. The AAC was set at 3900 ha (1% of the productive forested study area; Fall et al., 2004; Bergeron et al., 2006; Didion et al., 2007) and volume was estimated for each cell based on yield curves from plot data, but the details are not relevant here because the AAC was based on area not volume. Using an area-based logging target allowed for greater control and transparency in the forest management process by providing a common currency (i.e., forest area) for the logging and conservation targets and the fire sub-model. Harvest block sizes were chosen randomly from a uniform distribution between 5 ha and 60 ha based on size ranges common in this region. Harvest blocks were placed preferentially within active “operating areas” and in regions that had high wood volume and road access to capture the tendency for sets of harvest blocks to be grouped for operational efficiency. An operating area is represented as a circle of area 25,000 ha on the landscape that is active for approximately 20 years, after which time it is deactivated and new operating areas are randomly created within productive forest. Up to five operating areas are active at any given time during a simulation. By preferentially harvesting within operating areas, harvest blocks that are close in time will also tend to be close in space. Simple road construction was also included in the harvest model to account for changing site accessibility through time.

Maps of existing roads and projected roads were used to identify the initial road network and simulate future development of main roads. Areas further than 2 km from active roads were not available for harvesting until road access expanded to encompass them. Harvest blocks were explicitly connected to current and future main roads by straight-line, spur road segments. Future roads were activated once a spur road connected to them or when they were required to bridge newly activated segments with the existing road network. This method of modeling road development allowed road access restrictions to inﬂuence the spatial harvesting pattern whereas road-building associated with harvesting reduced access constraints over time. Chapter 2. Comparing static versus dynamic reserves 25

2.3.3 Protected areas

PAs were selected to promote the persistence of marten within the boreal forest study area. The area of the PA was fixed at 10-12% (39,200-47,040 ha) of the forested portion of the study area and could potentially support between 47 (mostly low-quality habitat requiring larger home range sizes) and 94 (mostly high-quality habitat resulting in smaller home-range sizes) marten depending on the density of habitat (percentage cover) within their home ranges. Each PA system was made up of two patches to increase the chance that some portion of the PA would survive in the event of a catas- trophic fire and could serve as a source of re-colonization (Shafer, 2001). Without imposing a 2-patch constraint on the number of patches, the PA system would have become highly fragmented and scattered across the landscape in very small patches each comprising only one or a few marten home ranges. This type of PA system would be operationally unrealistic to implement. Furthermore, aggregated PA systems that may include lower-quality habitats have been shown to be biologically more valuable than fragmented ones given high levels of uncertainty concerning the effects of fragmentation on species persistence (Moilanen and Wintle, 2006).

Static and dynamic protected area scenarios

For the static PA scenario, both patches of the PA were selected with MARXAN once at the beginning of the 200 year simulation. This static PA was used as the initial PA for the other PA scenarios. For the dynamic PA scenario, both patches were reselected with MARXAN every 50 years during the 200-year simulation (Table 1). I also explored a hybrid of these two scenarios, the static-core scenario, where one patch of the

PA was fixed in space for the duration of the simulation and the other was reselected every 50 years. Two baseline scenarios were simulated for reference: a no-logging scenario in which only succession and fire were simulated to estimate the historical range of natural variability; and a no-PA scenario in which succession, fire, and logging were Chapter 2. Comparing static versus dynamic reserves 26

Table 2.1: Description of alternative protected area (PA) scenarios.

Scenario Protected Area Logging Example

(Year 0, 50, 100, 150, 200)

No PA None Yes

Static PA 2 static patches Yes

1 static “core” patch Static-core PA Yes 1 dynamic patch

Dynamic PA 2 dynamic patches Yes

No Logging Entire study area None

In the example maps, black represents protected areas (areas where logging is excluded)

and white represents areas that are potentially subject to logging. Succession and ﬁre

occur across the entire landscape. Chapter 2. Comparing static versus dynamic reserves 27 simulated without designating a PA. Each of the ﬁve scenarios was replicated 10 times using the boreal forest simulation model.

Protected area selection method

The PA selection process involved two steps. First, the study area was divided into planning units corresponding to nonoverlapping, potential marten home ranges based on the spatial conﬁguration of preferred habitat (Fig. 2.2a; James et al., 2005). Plan- ning units must be non-overlapping in order to divide up the study area into polygons for potential selection by the reserve-selection algorithm. Marten are intrasexu- ally territorial although male home ranges generally overlap with one or more female home ranges depending on habitat conditions and marten densities (Payer et al., 2004); therefore, using non-overlapping home ranges provides us with a conservative estimate of marten population size. This home-range delineation method assumes that an individual marten may occupy a home range of less than the maximum size provided there is a suﬃcient density of preferred habitat within that home range (Schultz and Joyce,

1992; Thompson, 2004). Preferred marten habitat is mature coniferous and mixed forest (Buskirk and Powell, 1994; Thompson and Harstad, 1994) and at least 50% of each marten home range should consist of this preferred habitat type (Schultz and Joyce,

1992; Potvin et al., 2000). Hence, with a maximum marten home-range size of 1000 ha

(Potvin et al., 2000), any area up to 1000 ha that contained at least 500 ha of habitat was considered a viable home range. To identify home ranges with a high density of habitat within them, an expanding moving window centered at each cell was applied iteratively across a raster layer of marten preferred habitat. The resulting map of the habitat densities within home ranges (Fig. 2.1a) was used to prioritize planning units (home ranges) for inclusion in the PA. I used this relatively coarse method of home-range delineation to reduce the analysis time and to make best use of the habitat information tracked in the boreal forest simulation model (forest age and com- Chapter 2. Comparing static versus dynamic reserves 28 position) while keeping assumptions about space-use and life history at a minimum.

This method allows us to go one step beyond simply considering the amount of habitat available to marten in PAs by also incorporating spatial aspects of habitat for species

(i.e. the spatial juxtaposition of preferred and non-preferred habitat types) and re-

ﬂected the variability of marten home range sizes (reviewed in Buskirk and McDonald,

1989).

The second step involved selecting planning units to comprise the PA. I formulated the PA selection problem as a “minimum set covering problem” (Cabeza and Moila- nen, 2001): find the set of planning units that minimizes the inclusion of low-density home ranges within a fixed area PA. Formulating the PA selection problem in this way allowed us to employ the existing site-selection software MARXAN (version 1.8.2; Ball and Possingham, 2000; Possingham et al., 2000) which uses simulated annealing to approximate a global optimum for large minimum set covering problems. MARXAN has traditionally been used to select a PA that minimizes the area of the conservation area while meeting fixed targets for the biodiversity surrogate (McDonnell et al., 2002). I simply fixed the target area instead and minimized the difference between the maximum value of the biodiversity surrogate and the actual value in each planning unit

(i.e. 100% actual density of habitat in home range). This was done by modifying the

MARXAN input files: the conservation-feature file was modified to describe the area- based conservation target for the PA (47,040 ha); the conservation feature distribution

file was modified to describe the size of each planning unit; and the planning unit file was modified to describe the difference between the actual density of habitat in the planning unit (home range) and the maximum possible habitat density (100%).

MARXAN solves minimum set covering problems by minimizing a multivariate objective function that gives a value for a collection of planning units as if that collection constituted a PA. Although I modiﬁed the objective function, it still consisted of two main sections: the ﬁrst was a penalty for not meeting the target area (47,040 ha) and Chapter 2. Comparing static versus dynamic reserves 29 the second was a measure of the ‘cost‘ of the PA based on the density of habitat within marten home ranges and the boundary length of the entire PA. The objective function

I implemented was: X X total cost=APF ×AreaPenalty+BLM BoundaryLengths+ (100-HabitatDensity), P Us P Us where PUs are planning units (marten home ranges), APF is the area penalty factor, a term that assigns a penalty for failing to meet the PA area specified and BLM is the boundary length modifier, a weight used to control the spatial aggregation of selected planning units. The AreaPenalty is a penalty for not adequately meeting the target area for the PA; it is proportionate to the difference between the PA area and the target area. The BLM is used to reduce fragmentation of the PA. If the BLM = 0, the algorithm performs without spatial constraints.

Values of the BLM >0 increase the cost of the boundary length and encourage solutions that aggregate planning units into fewer, compact clusters with shared boundaries. I assigned a value of 106 for the APF and adjusted the value for the BLM such that the PA comprised exactly two distinct patches. I used an adaptive annealing sched- ule with one million iterations and 10,000 ‘temperature’ decreases. The temperature is the probability of accepting a change in the set of planning units that comprise the

PA even if this change produces a higher objective function value. This probability decreases throughout the algorithm once the chance of prematurely ﬁnding a local minimum is reduced. MARXAN was run 10,000 times and I kept the best solution (i.e. the solution that produced the lowest value for the objective function) as the ﬁnal PA (Fig.

2.2b).

Comparing protected area scenarios

To compare the PA scenarios (static, static-core, and dynamic) with the baseline scenarios (no-logging and no-PA), I performed 10 replicates of each scenario and monitored two characteristics of marten home ranges both within PAs and across the entire Chapter 2. Comparing static versus dynamic reserves 30

Figure 2.2: Illustration of the two steps of the PA selection process. The first step was to convert potential marten habitat, based on forest composition and age, into potential marten home ranges, based on the spatial configuration of potential habitat and marten movement abilities. The second step was to select potential marten home ranges for inclusion in the PA using the PA selection software MARXAN. MARXAN selected home ranges based on the density of the habitat within them and the boundary length of the PA. Home ranges with high habitat density that were compact were favored for inclusion in the final PA. Chapter 2. Comparing static versus dynamic reserves 31 landscape every 50 years: the number of home ranges and the mean density of habitat within each home range (home-range habitat density). Within PAs I also monitored both the amount of habitat and the amount of area covered by home ranges as a percentage of the area of the PA. To measure home-range properties within static

PAs (and the core patch of the static-core PAs), I re-ran the home-range delineation algorithm strictly within the boundaries of protection. I tracked the boundary length- to-area ratio of PAs through time using the corrected perimeter-area ratio (Baker and

Cai, 1992). This ratio corrects for polygon size by dividing the perimeter of a polygon by the square root of the product of 4π and the area of the polygon. The corrected perimeter-area ratio for a PA was computed as the average of the perimeter-area ratios for both patches within the PA. Its value is always greater than or equal to 1 (with a value of 1.0 for perfect circles) and becomes arbitrarily large for long, skinny polygons.

At the landscape-scale, I also monitored home-range connectivity (the connectedness among home ranges) measured as the density-weighted minimum planar graph

(DWMPG; O’Brien et al., 2006). This type of graph-theoretic connectivity analysis summarizes the spatial relationships between landscape elements by building a “graph”

(Harary, 1969) consisting of a set of “nodes” that represent landscape elements and

“links” that represent the potential movement of an organism among them (Urban and Keitt, 2001). In this particular application, the nodes were marten home ranges and the links were Euclidean distances among them. DWMPG is based on a particular type of graph called a minimum planar graph which consists of the maximum number of noncrossing links that connect all nodes (O’Brien et al., 2006). To calculate DWMPG, link lengths are weighted by the density of habitat in the pair of home ranges that they connect and then a mean length is computed for all links in the landscape. Mean DWMPG measures both the structural connectivity and the habitat density (quality) of connected home ranges. Its value is large when home ranges are far apart and have low habitat densities, whereas it is small when home ranges are close Chapter 2. Comparing static versus dynamic reserves 32 together and have high habitat densities.

2.4 Results

None of the PA scenarios (static, static-core, and dynamic) dramatically outperformed the others based on any of the measured properties of marten home ranges across the entire landscape over the 200 years of simulation: number (Fig. 2.3a); habitat density

(Fig. 2.3b); or connectivity (Fig. 2.3c). Error bars representing ± two standard errors were highly overlapping among the three scenarios at this landscape-scale (Fig. 2.3).

However, the dynamic and static-core PA scenarios performed at least as well, if not slightly better than the static PA scenario at all time steps for all home-range properties. Not surprisingly, all three PA scenarios were always worse than the no-logging scenario and better than the no-PA scenario. The number of home ranges across the entire landscape decreased initially but then remained fairly constant through time for all PA scenarios (Fig. 2.3a). The density of habitat within home ranges across the entire landscape also remained fairly constant for all scenarios except for the no- logging scenario, which improved gradually, and the no-PA scenario, which declined

(Fig. 2.3b). The dynamic PA scenario produced the most consistent home-range habitat density across the landscape through time; however, even for the static PA scenario, mean home-range habitat density only ranged from 67.88% to 72.80%. The connectedness among all home ranges in the landscape decreased for all three PA scenarios, as illustrated by the increase in mean DWMPG (Fig. 2.3c). The dynamic PAs maintained connectivity better than the static and static-core PAs for the first 150 years of simulation. The dynamic and static-core PAs resulted in the highest levels of connectedness among home ranges after the full 200 years of simulation and differences among the three scenarios were most pronounced at this final time step; however, these differences were still small enough and error bars were large enough that they Chapter 2. Comparing static versus dynamic reserves 33 may have been due to stochasticity alone.

I also tracked home-range properties strictly within the PAs and found stronger evidence of the dynamic and static-core PA scenarios outperforming the static PA scenario (Fig. 2.4). I standardized all PA-scale analyses by the area of the PA, which

fluctuated between 10.34% and 12.13% of the forested portion of the study area. The percentage of PAs that contained habitat was higher in the dynamic PAs than the static-core or static PAs and was more consistent through time (Fig. 2.4a). Differences among scenarios were larger when considering either the percentage of PAs that contained home ranges (Fig. 2.4b) or the number of home ranges in PAs (Fig. 2.4c) rather than simply the amount of habitat. The dynamic and static-core PAs maintained significantly higher percentages of home ranges than the static PAs over time because they were re-selected by MARXAN at the time of analysis whereas the percentage of home ranges in static PAs declined due to stochastic losses from fire (Fig. 2.4b). The number of home ranges in the dynamic PAs was also consistently higher than the static

PAs but the static-core PAs were more variable (Fig. 2.4c). In contrast, habitat density of home ranges within PAs was best maintained by the static scenario; however, after 100 years mean values for all scenarios were very similar and error bars among all scenarios were highly overlapping (Fig. 2.4d). Variability in the habitat density of home ranges was highest in the static-core scenario, which ranged from 58% to 82%, and lowest in the static scenario, which only ranged from 72% to 81%. Taken together,

Figures 2.4b-d show that home ranges in static PAs were less numerous and cumulatively covered a smaller area of the PAs but had similar, if not slightly higher habitat density than home ranges in the static-core and dynamic PAs.

The boundary length-to-area ratio of the static, static-core, and dynamic PAs showed a different pattern (Fig. 2.4e). The ratio was constant for the static PA because both patches in the static PA were fixed in space. Similarly, the static-core PA scenario maintained a consistent ratio largely because the core patch (which was fixed in space) Chapter 2. Comparing static versus dynamic reserves 34

Figure 2.3: Mean properties of all home ranges in the study area (both inside and outside of the PAs) over 10 replicates for the alternative PA scenarios at initial conditions and at 50, 100, 150 and 200 years of simulation: (a) number of home ranges; (b) mean density of habitat within home ranges and (c) mean density-weighted minimum planar graph (DWMPG). Error bars represent ± two standard errors and are oﬀset to the left (static-core scenario) and right (static scenario) of mean values to facilitate visualization. Chapter 2. Comparing static versus dynamic reserves 35 was very compact and nearly circular. In contrast, the ratio sharply increased for the dynamic PAs, in the ﬁnal time step suggesting that high quality home ranges became increasingly discontinuous in the dynamic PA scenario.

2.5 Discussion

In this paper I compared the performance of static and dynamic PAs in a dynamic forest landscape based on their abilities to maintain through time important habitat characteristics of a model species, the American Marten, as it is an indicator of mature forest. Dynamic PAs maintained more home ranges than static PAs when implemented within the spatially-explicit boreal forest dynamics model with a clear-cut logging regime and an annual allowable cut of 1%. Home ranges within dynamic PAs were of comparable habitat density to home ranges in static PAs even though they were more numerous. The beneﬁts of dynamic PAs were only observable when home- range characteristics were considered within PAs and were not seen at the landscape- scale. All three PA scenarios were much more similar to the no-PA scenario than the no-logging scenario at the landscape-scale likely due to the area constraint on PAs of

10-12% of the forested area. These general patterns can be attributed to the PA selection methods because the annual allowable cut of the logging regime was kept constant in all scenarios (i.e. the logging rate was not decreased when a portion of the area was protected). The performance of dynamic PAs was constrained by the fragmentation of forest in unprotected areas due to interactions between ﬁre and harvest, which limited options for re-locating PAs during the 200 year simulations. The increasing boundary length-to-area ratio of the dynamic PAs through time reﬂected a reduction in the contiguity of high-quality home ranges remaining on the landscape.

Several conceptual papers have suggested the use of dynamic PAs in boreal forest ecosystems (Cumming et al., 1996; Bengtsson et al., 2003; Angelstam et al., 2004); Chapter 2. Comparing static versus dynamic reserves 36

Figure 2.4: Mean properties of PAs over 10 replicates for the static, static-core, and dynamic PA scenarios at initial conditions and at 50, 100, 150 and 200 years of simulation: (a) percentage of PAs that is habitat (i.e. the area of habitat within each PA divided by the total area of the PA); (b) Percentage of PAs identiﬁed as home ranges

(i.e. the cumulative area of home ranges within each PA divided by the total area of the PA); (c) Standardized number of home ranges (i.e. the number of home ranges within each PA divided by the maximum possible home ranges of 100% habitat density that would ﬁt in the area of the PA); (d) Mean density of habitat within home ranges in the PAs and (e) Corrected boundary length-to-area ratio of PAs (see text for details). Error bars represent ± two standard errors and are oﬀset to the left (static-core scenario) and right (static scenario) of mean values to facilitate visualization. Chapter 2. Comparing static versus dynamic reserves 37 however, they have not discussed the details of their implementation. My study is the

ﬁrst to have fully implemented a dynamic PA by selecting PAs every 50 years with

MARXAN and simulating landscape change in a boreal forest with a spatially-explicit stochastic landscape model. My results are undoubtedly inﬂuenced by the way in which

I defined focal species habitat, implemented the dynamic PAs, represented landscape processes and captured interactions between the disturbance regime and PA identification. Two critical implementation parameters likely affected the performance of dynamic PAs: the length of the re-planning interval and the logging regime applied in unprotected areas. The longer the interval between re-planning, the larger effect logging will have on unprotected areas which will, in turn, limit continuous tracks of forest that could serve as options for re-locating PAs. However, if the replanning interval is too short, then the probability of a PA burning would be too small to merit the ef- fort required to re-plan and re-locate the PA. Therefore, an intermediate replanning interval informed by both the logging and fire regimes is desirable.

This problem of choosing a re-planning interval is also of interest to studies examining the periodic acquisition of PAs over an implementation horizon comprising several time periods (Costello and Polasky, 2004; Meir et al., 2004; Snyder et al., 2004;

Drechsler, 2005; Rush O’Hanley et al., 2007). These studies recognize that budget constraints force PAs to be implemented over several planning periods and that uncertainty in site availability and habitat loss during the implementation horizon may compromise the eﬀectiveness and eﬃciency of the PAs selected. They have considered a range in the number of planning periods from two to ten (Costello and Polasky, 2004;

Snyder et al., 2004; Drechsler, 2005; Strange et al., 2006) without specifying the length of each planning period (with the exception of Meir et al., 2004, who recommended an annual re-planning interval over a 10 year planning horizon). Although the length of the planning periods would affect the probability of site availability in these studies, they have placed greater concern on the number of planning periods because adding Chapter 2. Comparing static versus dynamic reserves 38 more planning periods increases the computational load. Oetting et al. (2006) implemented a dynamic PA selection process by updating their broad-scale land acquisition plan in Florida at six-month intervals during a 10 year implementation horizon in response to land-use changes and conservation actions. Strange et al. (2006) showed that allowing planners to swap PAs that become degraded through time increases the overall efficiency of the PA system in terms of the number of species represented for a given budget. However, they did not specify the length of the re-planning interval and they assumed that degradation of PAs was irreversible and that PA selection was not spatially-explicit. Given that I simulated boreal forest dynamics over 200 years and selected spatially-explicit PAs using traditional single-planning-period site-selection methods, the planning periods used in these studies do not directly apply. In this system, a “dynamic” re-planning period may be appropriate, in which re-planning only occurs when the conservation value of a PA has dropped below some threshold (e.g., due to fires). I identify the issue of re-planning intervals as an area that needs more research, particularly as the dynamics of ecological and human systems are increasingly acknowledged as important drivers for the persistence of biodiversity in PAs.

Another implementation parameter that likely affected the performance of the dynamic PAs was the type of logging regimes implemented in unprotected areas. It is becoming increasingly recognized that conservation of forest biodiversity is not only dependent upon protected PAs, but also the management of the forest matrix surrounding them (Lindenmayer et al., 2006). Studies examining the consequences of destroying habitat that is not selected for inclusion in a PA have shown that the probability of occurrence of the species inside the PA is reduced when all habitat outside PAs is lost (Cabeza, 2003; Cabeza and Moilanen, 2003). In the case of multi-use boreal forests however, habitat outside PAs is not completely lost due to the arrangement of logging activities in space and time. The spatial and temporal distribution of logging can be managed to promote mature forest habitat retention in the unprotected areas of the Chapter 2. Comparing static versus dynamic reserves 39 landscape (Lindenmayer et al., 2006). Designing alternative logging regimes may involve modifying three variables (Bergeron et al., 1999): size of the harvest unit, internal composition of the harvest unit, or period of time between successive logging events in a harvest unit (i.e., rotation length). Imposing a spatial constraint on the minimum- harvest-unit size would help to reduce fine-scale fragmentation and if harvesting occurred in patterns more in line with marten home ranges, then it may be easier to find new, good protected areas during re-planning. Modifying the composition of harvest units under the clearcutting system of even-aged stand management would help maintain more structural complexity in the form of snags and coarse woody debris within harvest units. Yet, the internal structural heterogeneity of harvest units would not affect my assessments of dynamic PAs as it is at a finer scale than I have modeled here.

The rotation length determines the rate at which the landscape is disturbed and consequently the proportion of mature forest on the landscape. Longer rotation lengths would increase the options for re-location of the PAs to mature forest areas; however, it would also reduce the volume of timber ﬂowing from the forest (Bergeron et al., 2001).

Another option is to protect a training PA that is a large tract of land that will be suitable mature forest habitat during the next re-planning period even if it is not suitable during the current planning interval. This option suggests the need to conserve an area larger than “needed” to ensure sustainable “recruitment” of high quality PAs over time. This option may reduce negative effects on the logging industry while increasing the probability of a high quality option for re-location of the dynamic PA. Initial conditions in this study were created by real interactions among management and fire and represent a realistic set of spatial constraints with regards to forest age and species composition in this region of the boreal forest. Real options for management in general, and for PA selection specifically, are necessarily constrained by the history of disturbance and management within a study area (Gustafson, 1998) and will also depend on current management and anticipated future changes in disturbance. Although the Chapter 2. Comparing static versus dynamic reserves 40 results of this study are specific to the study region, disturbance regime, and stochastic model used, the general qualitative messages emerging from these analyses are broadly applicable and should be instructive for many different systems. One important mes- sage that can be extracted from this localized case study is that small, dynamic PAs will not necessarily solve the problem of our current unwillingness to set aside large

PAs that can accommodate landscape dynamics within their boundaries (Pickett and

Thompson, 1978). The ability of dynamic PAs to safeguard against further losses of biodiversity will undoubtedly hinge on the details of their implementation and a key component of their success will be to manage the matrix surrounding PAs in such a way that options remain when it comes to re-planning. Chapter 3

Incorporating consumer-resource spatial interactions in reserve design

3.1 Abstract

The persistence of species in reserves depends in large part on the persistence of functional ecological interactions. Despite their importance, however, ecological interactions have not yet been explicitly incorporated into conservation prioritization methods. I show here a general method for incorporating consumer-resource interactions into spatial reserve design. This method protects spatial consumer-resource interactions by protecting areas that maintain the connectivity between the distribution of consumers and resources. I illustrate this method with a conservation planning case study of a mammalian predator, American marten (Martes americana), and its two primary prey species, red-backed vole (Clethrionomys rutilus) and deer mouse (Per- omyscus maniculatus). The conservation goal was to identify a reserve for marten that comprised 12% of a forest management unit in the boreal forest in Québec, Canada. I compared reserves developed using analysis variants that utilized different levels of information about predator and prey habitat distributions, species-specific connectivity

41 Chapter 3. Consumer-resource interactions in reserves 42 requirements, and interaction connectivity requirements. The inclusion of consumer- resource interactions in reserve selection resulted in spatially aggregated reserves that maintained local habitat quality for the species. This spatial aggregation was not induced by applying a qualitative penalty for the boundary length of the reserve, but rather was a direct consequence of modelling the spatial needs of the interacting consumer and resources. This method for maintaining connectivity between consumers and their resources within reserves has been made available via public software and can be applied even under the most extreme cases of either complete spatial overlap or complete spatial segregation of consumer-resource distributions.

3.2 Introduction

Nature reserves can be designated for the protection of single species (Carroll and Miquelle,

2006; Xu et al., 2006), multiple species (Pressey, 1999; James et al., 2005; Holzkamper et al., 2006), or whole ecological communities (Noss, 1987; Hunter, 1991; Done and Re- ichelt, 1998; Chan et al., 2006). Ultimately, the persistence of species or communities in these reserves will depend on the maintenance of key ecological interactions such as mutualisms, competition, and trophic interactions (Sinclair and Byrom, 2006; Baskett et al., 2007). The local extirpation of any species in a community can impact these key interactions and could lead to a series of cascading changes through the community in terms of habitat for other species, foodweb structure, and nutrient cycling (de Roos et al., 1998; Green and Sadedin, 2005). Despite their importance however, ecological interactions have been incorporated into multi-species reserve-selection methods only implicitly (e.g., by focusing conservation efforts on strongly interactive species (Soulé et al., 2003); or by identifying a minimum reserve area that incorporates natural disturbance and ecological processes (Leroux et al., 2007b)), even when the need to consider interactions is known (Gerber et al., 2003). Data are often sparse or absent alto- Chapter 3. Consumer-resource interactions in reserves 43 gether, but when they are available, the persistence of ecological communities would benefit from the inclusion of key interactions into conservation prioritization.

Of the many types of known species interactions, consumer-resource (including predator-prey and host-parasitoid) interactions are the principal driver of structure and function in many ecological communities (Sih et al., 1998). Protecting consumer- resource interactions requires maintaining areas for consumers and their resources to encounter one another (Van de Koppel et al., 2005). The spatial co-occurrence of consumers and resources is influenced by the spatial configurations of both consumers and resources, limitations on their spatial movements (Van de Koppel et al., 2005), and adaptive habitat selection (Abrams, 2007). Protecting only a portion of the spatial distributions of consumers and resources in reserves creates a network of habitat fragments that can alter the spatial properties of consumer-resource interactions and may in turn, affect population dynamics and persistence (Hoopes et al., 2005; Ryall and

Fahrig, 2006). Consumer-resource models have been used to derive generalized guidelines for reserve design (Micheli et al., 2004; Baskett et al., 2007) such as the minimum reserve patch size required to protect consumers and resources (May, 1994; Holt, 1997;

Swihart et al., 2001; Prakash and de Roos, 2002). However, accounting for the spatial co-occurrence of consumers and their resources when selecting the location of reserves remains an open challenge with implications for species persistence in reserve networks.

I present a general method for incorporating spatial consumer-resource interactions into reserve-selection using the spatial conservation prioritization software, Zonation

(Moilanen et al., 2005; Moilanen, 2007; Moilanen et al., 2008). The proposed method prioritizes areas for conservation based on their contribution to the maintenance of high quality habitat and connectivity between the distributions of consumers and resources. Consumer-resource interactions are modeled via an interaction kernel which defines the probability distribution of foraging distances based on the movement abilities of the consumer. Objective ecological measures of species-specific habitat qual- Chapter 3. Consumer-resource interactions in reserves 44 ity and connectivities at the scale of individual home ranges can easily be adapted to the interaction kernel approach. I illustrate the use of this method with a conservation planning study from a boreal forest management unit in Québec (Canada) prioritizing areas for inclusion in a reserve based on the distribution of habitat for the American marten (Martes americana) and two of its prey, the Red-backed vole (Clethrionomys rutilus) and the Deer mouse (Peromyscus maniculatus). The habitat requirements of the marten and its prey do not completely coincide, and the home range sizes and population-level connectivities of the species are very different (Table 1), which makes quantitative conservation planning in this region nontrivial.

3.3 Methods

Before presenting a method that incorporates consumer-resource interactions into reserve- selection using Zonation, I brieﬂy summarize the key features of the Zonation reserve- selection method and algorithms. I conclude this section with a description of the conservation case study used to illustrate the method.

3.3.1 Summary of the Zonation reserve-selection algorithm

Zonation is a spatial conservation planning tool that calculates maps of conservation priority. As input it uses primarily GIS raster maps of distributions of biodiversity features (here species), with local occurrence levels described by presence/absence, abundance, density, or probability of occurrence in each grid cell. Zonation produces a spatial priority ranking of the grid cells through the landscape via iterative removal of the least important remaining grid cell (Moilanen et al., 2005). “Least important” is de-

fined as the cell contributing the smallest marginal value to the remaining conservation value, the definition of which includes considerations such as local occurrence levels of species, species priorities (weights), remaining range sizes, complementarity of species Chapter 3. Consumer-resource interactions in reserves 45 composition, connectivity requirements of species and uncertainty of species distribution information (Moilanen et al., 2005; Moilanen, 2007). On the basis of a priori, user- defined thresholds, the top“Z%” of cells (e.g., 12% as suggested by IUCN, 1994) that jointly contribute to the greatest conservation value can then be considered one of several possible near-optimal solutions for conservation planning.

Each species can be individually weighted in Zonation to reﬂect its conservation priority status, economic value, taxonomic value, or its past distribution loss (Arpo- nen et al., 2007; Early and Thomas, 2007; Moilanen, 2007; Kremen et al., 2008). Data sets having millions of grid cells can be analysed using Zonation (Kremen et al., 2008;

Moilanen et al., 2008). The user manual (Moilanen and Kujala, 2006; Moilanen et al.,

2008) and software are freely available. In my application, I used the additive-beneﬁt- function formulation (Arponen et al., 2005; Cabeza and Moilanen, 2006; Arponen et al.,

2007; Moilanen, 2007), which incorporates additivity across local occurrence levels and additivity across species when aggregating conservation value (Moilanen, 2007).

When connectivity requirements are not included in the prioritization process, individual grid cells are treated as eﬀectively independent from each other, which may lead to spatially fragmented reserve structures (Moilanen et al., 2005; Moilanen and Win- tle, 2006). Fragmented reserves are generally undesirable. For example, Gaston et al.

(2002) pointed out negative biological consequences in cells close to the edge of the reserve network (increased disturbance, predation, invasibility, and abiotic changes);

Hanski (1998b) pointed out negative biological consequences due to altered metapopulation dynamics; and Groeneveld (2005) pointed out negative economic and logis- tic consequences associated with a large number of small reserve sites. Consequently,

Zonation provides three options for including connectivity requirements into the analysis: (1) boundary-length penalty (Possingham et al., 2000; Moilanen and Wintle, 2007);

(2) distribution smoothing (Moilanen et al., 2005; Moilanen and Wintle, 2006; Moila- nen, 2007) and (3) neighborhood-quality penalty (Moilanen and Wintle, 2007), which Chapter 3. Consumer-resource interactions in reserves 46 c c Mixed c 60 60 Mixed Deciduous Coniferous a,b ha 2 ha 2 ha a,b d a,b 0.0010 0.025 0.025 2000m3162m 80m 141m 80m 141m 22136m 990m 990m 0.000090 0.0020 0.0020 Martes americana Peromyscus maniculatus Clethrionomys rutilus HRsize √ HRLinearDimension = × e HRradius MedianDispersalDist =7 e =2/ f =2/ HRsize/π f Table 3.1: Focal species’ habitat, home range, and dispersal parameter estimates. p Thompson and Harstad (1994). Potvin et al. (2000). Buskirk and Powell (1994). Bowman et al. (2001). Bowman et al. (2002). Moilanen et al. (2005). Home range linear dimension Median dispersal distance Home range scale alpha Population scale alpha Parameter Minimum stand agePreferred habitatSecondary habitatHome range sizeHome range radius= a 30 b Coniferous c Mixed d 1000 e f Chapter 3. Consumer-resource interactions in reserves 47

can also operate in a directed fashion in freshwater/riverine systems (Moilanen et al.,

2008). Of these, the latter two options allow inclusion of species-speciﬁc connectivity

requirements into conservation prioritization.

In this work, I employed the distribution-smoothing option, which converts distri-

bution maps into connectivity layers via application of species-speciﬁc dispersal kernels,

which is an approach structurally identical to that commonly used in the context of

metapopulation modelling (Moilanen and Nieminen, 2002; Moilanen et al., 2005). The

species-speciﬁc connectivity, sij, for species j in grid cell i is computed from original occurrence levels (aj) as:

N X sij = exp(−αdin)anj, (3.1) n=1,n6=i where N is the number of cells in the landscape, α is a parameter for the species-speciﬁc

dispersal capacity (e.g., Moilanen and Nieminen, 2002), and din is the distance between cells i and n. Dispersal capacity can refer to diﬀerent scales of movement behav-

ior, from routine movements associated with daily resource exploitation such as food

searching within a home range, to special movements associated with directed displace-

ment events such as interpopulation movements or migration on a regional scale (Ims,

1995; Van Dyck and Baguette, 2005). Therefore α can be scaled accordingly to either

a home-range scale or an inter-population scale to produce connectivity layers describ-

ing either ﬁne- or coarse-scale connectivity requirements (Fahrig, 2003). These species-

speciﬁc connectivity layers can be entered into the same Zonation analysis along with a

species’ habitat distribution maps to model diﬀerent ecological requirements. If several

species-speciﬁc connectivity layers are used (e.g., home-range and inter-population con-

nectivity layers), some attention should be paid to their relative weighting. For exam-

ple, an inter-population connectivity layer could be assigned a relatively lower weight

to indicate that it is less important than home-range level connectivity or local habitat

quality. Chapter 3. Consumer-resource interactions in reserves 48

Figure 3.1: Example of the two types of interaction layers used as input for the Zona- tion reserve-selection algorithm: a resource-use intensity layer for resource j (Rj; upper panel) and a resource connectivity layer for consumer k (Ck; lower panel). The resource distribution is transformed into a resource-use intensity layer by multiplying the resource-occurrence level of each cell by the connectivity between that cell and the consumer population (Eq. (3.2)). The consumer distribution is transformed into a resource connectivity layer by multiplying the consumer representation level in each cell by the connectivity between that cell and the resource distribution (Eq. (3.3)). Connectivity among resources and consumers is determined by the parameter β which models the foraging distances of the consumer. In my illustration, I assume that distributions are binary (presence is grey, absence is white). In the resulting Rj and Ck interaction layers, darker colors represent higher values, as a result of the combination of high local value and high connectivity. Chapter 3. Consumer-resource interactions in reserves 49

The Zonation reserve-selection methods summarized above deal with the requirements of individual species that are treated as independent from each other. Interac- tions among species have not been directly accounted for. One can easily enter the distributions of both a consumer and its resource into the analysis and overlapping areas will be possibly selected for protection, but only if locations of moderate to high levels of co-occurrence do exist. However, if only a small fraction of the landscape can be protected and there are many species in the analysis, it is in no way guaranteed that overlapping areas of consumer and resource will be selected. I next address this potential deﬁciency by introducing the use of interaction connectivity layers.

3.3.2 Introducing novel spatial consumer-resource interactions

into Zonation

I present a variant of the distribution smoothing technique to compute two diﬀerent interaction connectivity layers between a consumer and its resources (Figs. 3.1 and 3.2):

(1) a resource-use intensity layer for resource j (Rj); and (2) a resource connectivity layer for consumer k (Ck). The resource-use intensity layer, Rj, is a transformation of

the resource’s distribution map based on connectivity to the consumer. Rj describes those parts of the resource distribution that are accessible to the consumer (i.e., re-

source locations within the consumers foraging distance). The resource connectivity

layer, Ck, is a transformation of the consumers distribution based on connectivity to

resources scaled by foraging distances. Ck describes those parts of the consumer distribution that have access to the resource (i.e., consumer locations within foraging dis-

tance from resources). These two interaction connectivity layers can be weighted and

included in conservation prioritization along with the species distributions and species-

speciﬁc connectivity layers. Chapter 3. Consumer-resource interactions in reserves 50

The local occurrence levels of resource j and consumer k in grid cell i, are denoted

as rij and cik, respectively (Fig. 3.1). The resource use intensity, Rij, of resource j at cell i by consumer k is deﬁned as the local occurrence level of the resource multiplied by connectivity to the consumer distribution,

N X Rij = rij exp(−βkdin)cnk, (3.2) n=1 in which foraging distances of the consumer are modeled as a negative exponential in-

teraction kernel function; parameter βk is the foraging-distance capacity of consumer

k and din is the distance between cells i and n. Similarly, resource connectivity, Cik, of consumer k at cell i is computed as the local occurrence level of the consumer multi-

plied by connectivity to the resource distribution,

N X Cik = cik exp(−βkdin)rnj, (3.3) n=1

using the same parameter βk to model foraging distances. Thus grid cells with high Rj are locations where both the resource is well represented and a relatively high number of consumers are within foraging distance. Grid cells with high Ck are locations where both the consumer is well represented and a relatively high number of resources are within the foraging distance. Note that the connectivity between consumers and resources is calculated from cell to cell so it is desirable to have high-resolution distribution maps relative to the species dispersal abilities. Here I am using distribution maps with 50m × 50m (0.25 ha) resolution and the median dispersal of marten is 22,136m

(Table 1). These generic consumer-resource formulations could also represent a range of species interactions including herbivore-plant interactions, host-parasitoid interactions, and predator-prey interactions. Furthermore, functions describing the negative consequences of connectivity for some interacting species (Hof and Flather, 1996; Mi- nor and Urban, 2008) could easily be used to include negative species interactions. Re- gardless of the type of connectivity modeled, the interaction will maintain its general Chapter 3. Consumer-resource interactions in reserves 51

form, which is (local habitat quality of the resource/consumer) × f(connectivity to the

consumer/resource) (Koelle and Vandermeer, 2005).

3.3.3 Case study: predator-prey interaction in the boreal for-

est of Qu´ebec (Canada)

I illustrate this method with a conservation planning case study in a boreal forest man-

agement unit (approximately 430,000 ha) in south-central Qu´ebec focusing on the in-

teraction between the American marten (predator) and two of its prey species, the

red-backed vole and the deer mouse. The objective of the conservation plan was to

select 51,600 ha (12% of the management area IUCN, 1994) that would best protect

American marten as it is considered an old growth indicator species (Thompson, 1991;

Buskirk and Powell, 1994; Rayﬁeld et al., 2008). The case study area is character-

ized by a northern coniferous boreal forest, dominated by black spruce (Picea mari-

ana (Mill.) BSP.), balsam ﬁr (Abies balsamea (L.) Mill.), jack pine (Pinus banksiana

Lamb.) and white birch (Betula papyifera Marsh.), and a southern boreal-temperate transition forest, dominated by white spruce (Picea glauca (Monech) Voss), balsam

ﬁr, and yellow birch (Betula alleghaniensis Britt.). Spatial forest composition data

(stand age and species) were derived from supervised classification of aerial photography and plot-level ground surveys from the 1990 decadal forest inventory carried out by Systèmed’Information FORestièrepar Tesselle (SIFORT) in collaboration with the

Qu´ebec Ministry of Natural Resources and Abitibi Consolidated. Spatial forest data are represented in raster format at a resolution of 0.25 ha; hence, the study area consists of approximately 1.3 million grid cells.

To incorporate consumer-resource interactions into Zonation reserve-selection, I converted spatial forest composition data into binary habitat distribution maps (habitat and non-habitat) for marten, voles, and mice (see Table 1 for species-speciﬁc habitat requirements). I obtained estimates for species home-range sizes from the litera- Chapter 3. Consumer-resource interactions in reserves 52

ture and estimated species-speciﬁc median dispersal distances using the scaling factor

presented by Bowman et al. (2002, see Table 1). The home-range radius and median-

dispersal distance of each species were used to calculate species-speciﬁc estimates of

dispersal ability (α) at home-range and population scales to be used in connectivity calculations (Table 1). The estimate of home-range scale dispersal ability (α) for marten

was also used as the foraging distance capacity estimate (β) to be used for creating in-

teraction layers.

3.3.4 Reserve-selection scenario comparison with diﬀerent com-

binations of species-speciﬁc and interaction connectivity

layers

To evaluate the relevance of including a predator-prey interaction in reserve-selection,

I selected a reserve comprising 12% of the landscape using ﬁve diﬀerent combinations

of predator and prey habitat distribution maps, species-speciﬁc connectivity layers, and

interaction connectivity layers (hereafter referred to as Scenarios I-V; Table 2).

In Scenario I, reserves were selected without using Zonation by simply overlay-

ing potential home ranges of marten, mice, and voles to identify those marten home

ranges which had the greatest number of prey (mice and vole) home ranges within

them. Marten home ranges with the highest number of mouse and vole home ranges

making up 12% of the landscape were selected for inclusion in the reserve. Potential

home ranges were delineated by assuming that each individual requires a suﬃcient den-

sity of habitat within an area smaller than a species-speciﬁc maximum home-range size

(James et al., 2005; Rayﬁeld et al., 2008). For example, marten individuals require at

least 50% of their home range to consist of suitable habitat (Schultz and Joyce, 1992;

Potvin et al., 2000) and have a maximum home-range size of 1000 ha (Potvin et al.,

2000). An expanding moving window was passed across the binary habitat map and Chapter 3. Consumer-resource interactions in reserves 53 any area up to 1000 ha that contained at least 500 ha of habitat was identiﬁed as a potential home range (with an associated measure of home-range quality - habitat density). This deterministic home-range delineation algorithm was run for each species.

The resulting maps showed the density of habitat within potential home ranges and were overlaid to identify high-quality marten home ranges that contained the most high-quality prey home ranges.

Scenarios II through V used Zonation to identify top 12% fractions of the landscape using different local quality, connectivity and interaction components (Table 2). These scenarios were based on habitat distribution maps and estimates of species’ movement and foraging distances; the estimated home ranges described above were not used beyond Scenario I. When prey-habitat maps were included (Scenarios II through IV), they were cumulatively down-weighted to 0.5 so that they would be only half as important as the predator-habitat map. Reserves were targeted towards marten because they are indicators of boreal forest biodiversity (Buskirk and Powell, 1994) and their main prey, Red-backed voles and Deer mice, were down-weighted because they are abundant throughout their ranges. Scenario V did not include prey-habitat distributions or prey- connectivity layers in the reserve-selection, which is reasonable as the mice and voles and their habitat are not particularly endangered. Spatial reserve structures proposed by the five different reserve-selection scenarios were contrasted based on their spatial configuration and habitat composition.

3.4 Results

I present the conservation-area designs generated using different considerations of local habitat quality and connectivity in Figure 3.2. Candidate solutions were compared both in terms of habitat quality and connectivity. Using local habitat quality alone as an evaluation criterion would ignore the importance of reserve spatial configuration Chapter 3. Consumer-resource interactions in reserves 54 Home range Population scale Form of interaction Connectivity b Relative prey weights indicating the combined weight of all prey habitats. b Prey weight IRC 0.5 R&C Yes Yes Table 3.2: Differences among reserve-selection scenarios. pQHPpQHP 0.5IC N/A 0 C Yes Yes Yes Yes pQ 0.5 N/A No No a mQ mQHP mQHP mQHP Scenario names indicate which components were used during reserve-selection. Z indicates that Zonation was used. IIIZ IV Z VZ No. Name IIIZ Home range overlap N/A N/Aa mQHP and pQHP indicate that marten (m) orpopulation prey connectivity (p) (P) quality were of used. habitat IRC (Q), indicates home thatC: N/A range the resource connectivity interaction connectivity) (H) layers were and (R: used. resource-use intensity and In the example maps, black represents protected areas (areas N/A where logging is excluded). Chapter 3. Consumer-resource interactions in reserves 55 and could over-estimate the contribution of reserve patches with high local quality but which are too small and isolated for the marten to occupy (e.g., marten have been shown to use patches of residual forest within clearcut areas if the patches were greater than 15 ha; Buskirk and Ruggiero, 1994). A solution that has apparently high co- occurrence of marten and prey would not be ecologically justified if it is very fragmented; therefore, I also considered the composition of reserves in terms of the proportion of original connectivity they maintained at the scale of marten home ranges, populations, and predator-prey interactions.

The spatial conﬁgurations of the top 12% of cells (the reserve size) selected by the

five scenarios show general consensus on important locations to protect: three central patches, a patch in the northwest, and a patch in the south-east (Fig. 3.2a). Increasing the percentage of cells included in the reserve (e.g., 12% increased to 20%) would result in larger and more contiguous reserve patches that would eventually become connected based on proximity (results not shown). Even so, there are notable differences in the aggregation levels and details of the alternative scenarios; importantly, realistic inclusion of connectivity effects tends to result in much more aggregated reserve areas. Sce- nario I differed from all others in that the planning units it used for reserve-selection were potential marten home ranges rather than individual grid cells; hence, it did not exhibit any fine-scale fragmentation below the scale of individual marten home ranges

(Fig. 3.2a). Nevertheless, it achieved less home-range and population-level connectivity than did the Zonation analyses that included connectivity requirements quantitatively within the prioritization process.

When species-speciﬁc connectivity layers were used in reserve-selection (Scenarios

III, IV and V), selected habitat was very aggregated compared to when selection was based on local quality alone (Scenario II). Including the interaction-connectivity layers resulted in more contiguous reserve patches (Scenarios IV and V compared to Sce- narios II and III). Scenario V produced the most contiguous reserve patches because Chapter 3. Consumer-resource interactions in reserves 56 it included three types of connectivity for the marten, but did not give weight to the

ﬁnely fragmented prey species’ distributions, naturally leading to some loss of prey occurrence levels inside the reserves.

The composition of habitat types within reserves was altered by the inclusion of the predator-prey interaction during reserve-selection (Fig. 3.2b). The proportion of the original distribution of habitat locally suitable for the predator and one or more prey decreased slightly whereas the interaction connectivity increased signiﬁcantly.

This means that areas are selected where marten and prey occur in close proximity, although not necessarily in the same grid cells. Note that this increased interaction connectivity was observed in this case study despite a relatively high degree of spatial overlap among the habitat requirements of marten, mice, and voles (see Figure 3.3 for other cases of predator-prey spatial habitat matching). The degree of marten home- range connectivity and marten population connectivity in reserves did not notably differ among scenarios regardless of whether interactions were included (Fig. 3.2b), meaning that an increase in interaction connectivity could be obtained practically for very small losses of species-speciﬁc connectivity.

3.5 Discussion

In this paper I have showed a generalized method for incorporating consumer-resource interactions into spatial reserve-selection. This work was motivated by population- and community-level studies demonstrating that the persistence of single species and entire ecological communities depends on ecological processes that involve not only multiple species but also their interactions (Abrams, 1987; Carpenter et al., 1994; McCann et al., 1998; Polis, 1998; Klug et al., 2000; Sabo, 2008). Spatially explicit formulations of these interactions have been promoted in ecological models (Wu and Marceau, 2002).

Including information about the limited mobility of individuals in consumer-resource Chapter 3. Consumer-resource interactions in reserves 57

Figure 3.2: Reserves selected under five different reserve-selection scenarios using different combinations of species-specific and interaction-connectivity layers (Scenarios

I-V in Table 2). (a) Areas with the highest conservation value making up 12% of the landscape are shown in black and potential marten habitat is shown in grey in (I). (b)

Habitat composition of reserves was measured as the proportion of original distribution of diﬀerent habitat types derived from the preferred habitat of martens, mice, and voles (see Table 1). Habitat supporting marten alone was coniferous or mixed forest between 30 and 60 years. Habitat supporting marten and at least one prey was mixed or coniferous forest greater than 60 years. Habitat supporting marten and two prey was mixed forest greater than 60 years. Other compositional measures of the reserves included the proportion of original connectivity among marten and its prey species, marten home ranges, and marten populations. Chapter 3. Consumer-resource interactions in reserves 58 models has produced more stable consumer-resource dynamics than aspatial equiv- alents (Cuddington and Yodzis, 2000); therefore, it behooves conservation planners to incorporate spatially explicit interactions into conservation management decisions.

However, parameterizing Lotka-Volterra reaction-diffusion models (Svirezhev, 2008) or individual-based simulation models (Breckling et al., 2006) by experimentally tracking individuals remains time consuming, expensive and a difficult challenge (Koenig et al., 1996; Tischendorf, 1997). Until now, the pattern of interactions among species has mainly contributed towards deciding which suites of mutually dependent species should be prioritized for conservation (Witting et al., 2000; Souléet al., 2003, 2005;

Van der Heide et al., 2005; Moulton et al., 2007) rather than informing decisions about which locations should be protected to preserve these species. The spatial conservation prioritization method incorporates species-speciﬁc habitat-connectivity requirements and inter-speciﬁc connectivity requirements between predator and prey habitats.

It is a relevant question whether reserves selected with and without the consumer- resource interaction diﬀer in terms of species representations and spatial conﬁgurations.

The results from the reserve-selection scenario comparison show very minor tradeoﬀs between predator-prey co-occurrence within reserves and contiguity of reserve patches when predator-prey interactions are incorporated (Scenario III compared to Scenario

IV; Fig. 3.2). In fact, when reserves were selected based only on marten habitat requirements and the predator-prey interaction connectivity layer (Scenario V), both species co-occurrence and reserve patch contiguity increased (Fig. 3.2). The increased contiguity achieved by including species interactions in reserve-selection makes these reserves more likely to be feasible from a practitioner’s perspective (Possingham et al.,

2000; Gaston et al., 2002; Moilanen and Wintle, 2006, 2007). It is encouraging that such solutions have been suggested based on a quantitative analysis that includes species- speciﬁc connectivity considerations - I have used no qualitative method, such as the commonly used boundary length penalty (Possingham et al., 2000), to enforce struc- Chapter 3. Consumer-resource interactions in reserves 59

Figure 3.3: Three cases illustrating varying degrees of co-occurrence between consumer and resource: Case 1, no co-occurrence; Case 2, partial co-occurrence; and Case 3, complete co-occurrence. The corresponding resource-use intensity layer and resource- connectivity layer are shown with darker areas indicating higher connectivity between the consumer and resource. Chapter 3. Consumer-resource interactions in reserves 60 tural aggregation. And of course, a reserve network selected using connectivity and interaction components then by deﬁnition represents those components in the chosen areas, indirectly promoting species persistence in the reserve network.

Another relevant question is: did the proposed analysis produce a result which is likely to be better than the result obtained by the simple home-range overlay method of Scenario I? Although I have no reliable spatial population viability analysis (PVA) model available for rigorously evaluating the proposed reserve structures using a common currency, such as extinction risk, I can make the following observations. First, I was able to ﬁnd a solution (Scenario IV) which was superior to Scenario I in all aspects of co-occurrence of marten and prey, home range, and population level connectivity for the marten and interaction between the marten and prey (Fig. 3.2). Second, whereas

Scenario V is comparable to Scenario I in terms of marten areas with or without prey,

Scenario V is far superior to Scenario I in terms of connectivity components and fea- sibility of implementation (as seen in the solution maps in Fig. 3.2). In Scenario V,

87% of the proposed reserve occurs as spatially contiguous areas that would support at least nine marten home ranges (9000 ha). For comparison, in Scenario I, only 37% of the proposed reserve occurs in continuous areas greater than 9000 ha. Furthermore, in

Scenario V the four largest patches include 95% of the area and the boundary length- to-area ratio of the reserve is low (BL/A = 0.11) compared to that of Scenario I, where the four largest patches include 64% of the area and BL/A = 0.22. Thus, Scenario V appears to be superior to Scenario I both in terms of expected regional population viability and implementability. In summary, it appears that the proposed method for incorporating consumer-resource connectivity has indeed been able to ﬁnd solutions that are likely to improve on Scenario I both with respect to conservation value and practicality of implementation (Scenarios IV and V).

The large home-range size and high dispersal ability of the marten in this case study smoothed out the ﬁne-scale distribution of the mice and voles when their habi- Chapter 3. Consumer-resource interactions in reserves 61 tat maps were included (Scenario IV). This prioritization of more contiguous reserve patches when predator-prey interactions are included in reserve-selection depends on the degree of similarity between predator- and prey-habitat requirements and dispersal abilities (Van de Koppel et al., 2005) as well as the initial distribution patterns of each. Modelling eﬀorts to determine the minimum reserve size required to protect predators and prey have also emphasized the importance of including parameters from both predators and prey, regardless of whether the predator is a specialist or generalist

(May, 1994; Holt, 1997; Swihart et al., 2001; Prakash and de Roos, 2002). My results highlight the importance of accounting for predation not only when setting minimum reserve patch sizes but also when deciding on reserve patch conﬁguration.

Although I illustrated the method with a simple predator-prey case study, it can be applied to very large data sets containing hundreds of habitat-quality, connectivity and interaction layers in multi-million element landscapes. Adding more species will inevitably complicate data acquisition and interpretation of planning outputs; therefore, I modelled the consumer-resource interaction in a simplified form to minimize data requirements and facilitate interpretation. Areas with the highest probability of successful consumption were identified based on the probability of occurrence of consumers and resources and their spatial connectivity. I did not explicitly model numeric responses of consumers and resources that would have involved assumptions about density dependence (Krebs, 2002), functional responses (de Roos et al., 1998), or adaptive dynamics (Abrams, 1999). My main assumption was that consumers preferentially uti- lize regions close to their current location for acquiring resources and I modelled this assumption with an exponential-decay curve describing the connectivity between consumers and resources. This assumption treats the matrix surrounding habitat patches as homogeneous when proximity is based on Euclidean distance; however, proximity could also be measured in terms of effective cost distance to incorporate information about dispersal barriers through a heterogeneous landscape matrix (Wiens, 1989; With, Chapter 3. Consumer-resource interactions in reserves 62

1994; Adriaensen et al., 2003).

3.6 Conclusion

A novel method for the inclusion of species interactions in reserve-selection was presented. Using a case study, I have demonstrated large inﬂuences of connectivity and interaction terms on ideal reserve structure. Further evaluation of this method could involve trophic models (Baskett et al., 2007) or metacommunity models (Guichard et al.,

2004) to track population sizes of consumers and resources in reserves selected with and without information about the interaction. Empirical evaluations of the reserves selected by this method would be more challenging because the scale of conservation planning for predatory mammals, such as the American marten, is prohibitive to experimental manipulation (Oksanen, 2001). Empirical investigations into the space-use behaviors of the focal predator would strengthen estimates of foraging distance capacity and would be of great beneﬁt to this method.

The case study I presented was fairly simple in that it considered only three species with highly overlapping habitat requirements. The predator-prey relationships between marten-vole and marten-mouse allowed us to examine the cases of partial (Case 2, Fig.

3.3) and complete (Case 3, Fig. 3.3) co-occurrence, respectively. Maintaining connectivity between consumers and their resources within reserves becomes more challenging as consumer- and resource-habitat requirements diverge; however, this novel algorithm can still be applied under the most extreme case of consumer-resource spatial habitat segregation to identify those areas that are most important for the maintenance of the interaction (Fig. 3.3). Furthermore, conservation planning often involves many more species and more interactions (Pressey, 1999). The discrepancies between reserves that include consumer-resource interactions vs. those that do not will likely increase as the number of species and interactions increase and as their co-occurrences decrease. Chapter 3. Consumer-resource interactions in reserves 63

The foraging capacities of the consumers will also aﬀect the degree to which including the interaction model improves the spatial cohesiveness of the proposed reserve.

Although the consumer in this case study had a large foraging distance capacity, the co-occurrence of the consumer and its resources likely produced a conservative estimate of the eﬀects of including species interactions in reserve-selection. Inclusion of connectivity and species interactions in spatial conservation planning should generate reserve networks that are comparatively resistant to extinctions, which is important as human land-use outside reserve borders intensiﬁes (Williams et al., 2004). Chapter 4

The sensitivity of least-cost habitat graphs to relative cost surface values

4.1 Abstract

Maintaining and restoring connectivity among high-quality habitat patches is recognized as an important goal for the conservation of animal populations. To provide an efficient measure of potential connectivity pathways in heterogeneous landscapes, least-cost route analysis has been combined with graph-theoretical techniques. In this study I use spatially-explicit least-cost habitat graphs to examine how matrix quality and spatial configuration influence assessments of habitat connectivity. I generated artificial landscapes comprised of three landcover types ranked consistently from low to high quality: inhospitable matrix, hospitable matrix, and habitat. I controlled the area and degree of fragmentation of each landcover in a factorial experiment for a total of 20 combinations replicated 100 times. In each landscape I compared 8 sets of relative landcover qualities (cost values of 1 for habitat, between 1.5 to 150 for hospitable matrix, and 3 to 10000 for inhospitable matrix). I found that the spatial location of least-cost routes was sensitive to differences in relative cost values assigned to

64 Chapter 4. Sensitivity of least-cost habitat graphs 65 landcover types and that the degree of sensitivity depended on the spatial structure of the landscape. Highest sensitivity was found in landscapes with fragmented habitat and between 20 to 50% hospitable matrix; sensitivity decreased as habitat fragmentation decreased and the amount of hospitable matrix increased. As a means of coping with this sensitivity, I propose identifying multiple low-cost routes between pairs of habitat patches that collectively delineate probable movement zones. These probable movement zones account for uncertainty in least-cost routes and may be more robust to variation in landcover cost values.

4.2 Introduction

Faced with the reality of ever increasing habitat fragmentation (Riitters et al., 2000;

FAO, 2006), wildlife conservation eﬀorts have recently focused on protecting and restoring habitat connectivity (Crooks and Sanjayan, 2006; Hilty et al., 2006). Habitat connectivity is a concept that describes the potential for an animal to move among high- quality habitat patches as a function of both the spatial structure of the landscape and the movement behavior of the animal in response to that spatial structure (Tay- lor et al., 1993; With et al., 1997, 1999; Tischendorf and Fahrig, 2000b; Goodwin and

Fahrig, 2002a,b; Brooks, 2003; Fahrig, 2007). Recent studies have emphasized the roles that the spatial structure and quality of the intervening matrix between high-quality habitat patches has on habitat connectivity (Ricketts, 2001; Tischendorf et al., 2003;

Bowne and Bowers, 2004; O’Brien et al., 2006). Ultimately however, the animal’s movement behavior will determine the extent to which inhospitable matrix structure in- hibits movement among habitat fragments (D’Eon et al., 2002; B´elisle,2005). Con- servation planners must therefore describe a landscape from an animal’s perspective in order to understand, measure, and conserve functional habitat connectivity (Wiens,

1989). Chapter 4. Sensitivity of least-cost habitat graphs 66

Animal perceptions of landscape spatial structure are thought to be primarily determined by the ﬁtness consequences, such as mortality and reproductive success, ex- perienced by individuals moving through diﬀerent landcover types (With et al., 1997;

Drielsma et al., 2007; Driezen et al., 2007). Physical resistance (Stevens et al., 2004) and energetic expenditures are two mechanisms that may influence fitness in a given cover type (Graham, 2001; Drielsma et al., 2007). A quantitative description of movement behavior within habitats involves assigning organism-specific “cost values” that reflect the quality of the habitat in terms of ecological costs incurred by an individual moving through them. These cost values are often a function of several environmental variables that are relevant for a particular species such as slope, elevation, water, vegetation cover type and structure, roads, and human settlements (Ferreras, 2001; Cle- venger et al., 2002; Schadt et al., 2002; Chardon et al., 2003; Verbeylen et al., 2003;

Beazley et al., 2005; Cushman et al., 2006; Kautz et al., 2006; Rouget et al., 2006;

Driezen et al., 2007; Epps et al., 2007; Gonzales and Gergel, 2007; LaRue and Nielsen,

2008, see Table 4.1).

Representing a landscape as a cost surface relevant to a particular species involves:

(1) identifying the number of perceptible habitat types in the landscape; (2) ranking all habitat types according to their cost for movement; (3) quantifying the relative cost of landcover types by assigning numerical cost values (note that cost values in some cells may need to be adjusted to account for additional elements such as slope or elevation).

Ideally, cost values should be assessed based on ﬁeld and experimental data; however, these data are diﬃcult and time consuming to collect, and this has resulted in a large number of studies relying entirely or in large part on expert opinion (Clevenger et al.,

2002; Schadt et al., 2002; Chardon et al., 2003; Verbeylen et al., 2003; Johnson and

Gillingham, 2004; O’Brien et al., 2006; Gonzales and Gergel, 2007). Some studies use

field data on occurrence patterns in order to delineate landcover preferences instead of landcover costs (but see Haddad and Tewksbury, 2005). They employ well-known Chapter 4. Sensitivity of least-cost habitat graphs 67 detected? No Yes Yes (only for very low cost values) Yes Yes “It appeared that withinrange the of resistance values used (1-20), no alarming differences occurred between the accessibility patterns obtained.” pg. 9 Half as many transitions were predicted when a higher relative cost value was assigned to the least-preferred landcover type (pasture). “Only when using verycost low values for matrixthe did connections change, but these values can beto assumed be below arealistic biologically range.” pg. 1479 “All resistance sets we explored had a significantly better fit to theEuclidean data distance.” than pg.Prediction 570 power varied among resistance sets. “the results of thisshow study sensitivity do to variation in these resistance values and allowed us to’most deduct likely’ a resistance set.” pg. 801 Response variable Conclusionbased on simulated dispersers Sensitivity per day among forest remnants under constant energy constraints location of least-cost routes presence/absence in the proximity of source patches presence/absence in the proximity of source patches Range of cost values 1 - 21 Patch accessibility landcover types 9, 11, 6 andrespectively 13 3 1 - 4 Number of transitions 3 1 - 1000 Identification and 6 1 - 100 Predict species’ 2 and 5 1 - 100000 Predict species’ Table 4.1: Chronological andtivity alphabetical studies presentation using of a connec- cost surface to identify least-cost routes. ) ) ) ) Ramphastos Lynx lynx Pararge Sciurus Large and small mammals, birds, and butterflies in the Netherlands Keel-billed toucan ( sulfuratus Eurasian Lynx ( Speckled wood butterfly ( aegeria Red squirrel ( vulgaris et al., 1992) 2001) et al., 2002) et al., 2003) et al., 2003) # Study1 (Knappen Study species Number of Table 4.1 continued on following page 2 (Graham, 3 (Schadt 4 (Chardon 5 (Verbeylen Chapter 4. Sensitivity of least-cost habitat graphs 68 detected? No Yes (only for coarse scale) Yes Yes “There were no discernable differences in the sensitivity analysis least-cost routes that were produced from versions of the costthat surfaces incorporated higher impedance scores for roads and water.” pg. 126 “Hence, error in thevalues cost should only haveminor a influence on the position of key thresholdswhich at large increases inoccur, ECS while the structureconnected of clusters will remain consistent.” pg. 77 “In our example with radiotracking data of hedgehogs, the average z-score of most resistance sets were negative and significantly different from zero, except for twoand sets 12) (11 where thisleast was expected.” also pg. 320 “For...both measures of gene flow, best-fitting models resulted when sloped terrain had 1/20th to 1/10thcost the of movement acrossterrain flat (Fig. 2), withslope the weight of 0.10often most favoured.” pg. 719 Response variable Conclusionleast-cost routes Sensitivity (ECS) of habitat patches in a network. Threshold distances at which large increases in ECS occurred empirical vs. least-cost routes (z-score) Genetic distance and gene flow (population pairwise FST values), Number of migrants per generation (Nm), and Migration rates (M) Range of cost values (1640 was applied to linear dispersal barriers) landcover types 10 and 11 1 - 20 Locations of 6 1.7 - 3.5 Expected cluster size 20 1 - 1000 Cost and location of 3 1 - 100 ) ) ) Ovis ) Puma concolor Rangifer Erinaceus Florida panther ( coryi Woodland caribou ( tarandus caribou Hedgehogs ( europaeus Desert bighorn sheep ( canadensis nelsoni et al., 2006) et al., 2006) et al., 2007) et al., 2007) # Study6 (Kautz Study species Number of Table 4.1 continued on following page 7 (O’Brien 8 (Driezen 9 (Epps Chapter 4. Sensitivity of least-cost habitat graphs 69 detected? Yes “We also found thatthe varying absolute values, and thus range (the incremental increase) of friction values, however, produced greater variation than expected... Finally, our expectation that output from thescenarios nine at the contiguous site would be moreto similar each other thanfrom outputs the fragmented sitesupported.” was pg. 1164 Response variableDistance and Conclusionaccumulated cost values of least-cost routes Sensitivity Range of cost values 0 - 160 (continuous landscape), 0 - 300 (fragmented landscape) landcover types 18 (continuous landscape), 22 fragmented landscape) ) Sciurus Eastern grey squirrel ( carolinensis and Gergel, 2007) # Study10 (Gonzales Study species Number of Chapter 4. Sensitivity of least-cost habitat graphs 70 methods such as compositional analysis (Aebischer et al., 1993), wherein vegetation cells are ranked according to species’ habitat preferences using a measure of time spent in each landcover type relative to its availability in the landscape (Ferreras, 2001; Gra- ham, 2001; Kautz et al., 2006; O’Brien et al., 2006). Resource-selection indices are also used to quantify landcover preferences by relating environmental variables with occurrence data through regression models (Boyce and McDonald, 1999; Ricketts, 2001;

Manly et al., 2002). The inverse (Ferreras, 2001; Graham, 2001; Schadt et al., 2002;

Chetkiewicz et al., 2006; Gonzales and Gergel, 2007; LaRue and Nielsen, 2008) or odds ratio (O’Brien et al., 2006) of these measures of landcover preference can then be taken as approximations of landcover cost values.

The resulting cost surface can serve as the basis for analyzing habitat connectivity using a combination of least-cost route analysis and graph-theoretic techniques (Bunn et al., 2000; Chetkiewicz et al., 2006; O’Brien et al., 2006; Theobald, 2006; Fall et al.,

2007). Least-cost route analysis estimates eﬃcient-movement routes and costs between pairs of habitat patches based on the suitability of the intervening matrix (Adriaensen et al., 2003). Graph-theoretic techniques derive holistic measures of habitat connectivity by using a dispersal-distance threshold as a surrogate of movement potential among all habitat patches in the landscape (Urban and Keitt, 2001; Fall et al., 2007).

By deﬁning the links of a graph using least-cost routes, spatial information about the habitat patches and the surrounding matrix can be incorporated into graph-based measures of overall habitat graph connectivity. The habitat graph model scales up from pairwise measure of eﬀective distances between patches, which are typically generated by least-cost route algorithms, to consider the connectivity of the entire habitat graph. The graph-theoretic approach also roots measures of habitat connectivity in a conceptual model derived from metapopulation theory whereby the importance of each habitat patch in maintaining overall connectivity of the graph can be attributed to its topological position and source/sink characteristics (Urban et al., 2009). Chapter 4. Sensitivity of least-cost habitat graphs 71

In this study, I addressed the difficulties and ensuing uncertainties that arise when deriving organism-specific descriptions of matrix structure. I examined the sensitivity of habitat-connectivity assessments, based on least-cost links (Adriaensen et al., 2003) and graph-theoretic methods (Urban and Keitt, 2001; Fall et al., 2007), to the way in which we commonly quantify species-specific perceptions of matrix quality. I expected that the spatial location of least-cost links would depend on both the spatial pattern of the habitat and matrix-landcover types as well as the relative cost values of each.

Hence, I ran a factorial experiment to generate artiﬁcial landscape spatial patterns and corresponding cost surfaces with three factors controlling landscape spatial pattern and one factor controlling cost values. Comparing habitat connectivity measures among these artiﬁcial landscapes allowed us to systematically test the importance of each of the factors.

4.3 Methods

4.3.1 Generation of artiﬁcial landscape spatial patterns

I generated landscape spatial patterns in maps of 100 × 100 cells using three land-

cover types: habitat (H), hospitable matrix (HM), and inhospitable matrix (IM). The

maps were generated based on a placement algorithm that arranged landcover types in

fragmented and clumped conﬁgurations (Fahrig, 1997, 1998; Tischendorf and Fahrig,

2000a; Tischendorf, 2001; Tischendorf et al., 2003, I follow the notation of Tischendorf

and Fahrig (2000) and Tischendorf et al. (2003) for consistency). The spatial pattern

of each landcover type was controlled by two parameters: COV, which deﬁned the pro-

portion of that landcover type in the landscape; and FRAG, which deﬁned the spatial

arrangement of cells assigned to that landcover type. Deﬁning these parameters for

H and HM constrains the placement of IM; hence the full landscape spatial pattern

can be controlled by the following four spatial pattern parameters: H COV, H FRAG, Chapter 4. Sensitivity of least-cost habitat graphs 72

HM COV, and HM FRAG.

The placement algorithm begins by assigning H to cells of the map. It repeats the

following two steps until the number of cells that have been assigned is equal to the

number of cells determined by H COV: (1) the algorithm randomly selects an unas-

signed cell; and (2) it decides whether or not H will be assigned to this cell based on

the assignment of its eight neighbor cells and the parameter H FRAG. The decision to

assign H to the selected cell is taken if it meets either of the following conditions: (1)

the value of a random number between 0 and 1 is smaller than H FRAG2 or (2) one or

more of the eight neighboring cells already has the assignment H. H FRAG is squared

in order to produce a linear relationship between H FRAG and the number of patches

comprised of cells assigned to H (Figure 1a in Tischendorf and Fahrig, 2000a). When

H FRAG is large, the random number will often be smaller than H FRAG2 and most

of the cells will be assigned to H based on the ﬁrst condition which will produce a ran-

dom distribution of habitat. When H FRAG is small, most of the cells will be assigned to H based on the second condition which will produce a more contiguous distribution of habitat with larger, less edgy patches. All groups of adjacent cells (based on eight neighbors) with value H are considered as habitat patches and paths are calculated based on all of these patches. The placement algorithm operates in an identical fashion to assign HM to cells and all remaining cells receive the assignment IM (see example landscapes in Fig. 4.1).

I deﬁned the value of H COV as 12% to represent the amount of habitat commonly

targeted for protection in reserves in Canada (Pressey et al., 2003, and references therein).

Evaluating the sensitivity of connectivity assessments, based on least-cost habitat graphs,

in the context of a typical conservation scenario is consistent with the promotion of

this habitat graph approach as a conservation planning tool for ecologists (Urban and

Keitt, 2001; Chetkiewicz et al., 2006; Urban et al., 2009). I manipulated the values of

the three other parameters in a factorial experiment with 20 combinations of the three Chapter 4. Sensitivity of least-cost habitat graphs 73

Figure 4.1: Examples of the simulated landscapes at a resolution of 100 × 100 cells.

Each landscape contains 12% habitat (black squares) and either a) 10% or b) 50% hospitable matrix landcover (HM COV; gray squares). Habitat area is either aggregated (H FRAG = 0.05) or more randomly distributed in (H FRAG = 0.5). Hospitable matrix area also either aggregated (HM FRAG = 0.05) or more randomly distributed

(HM FRAG = 0.5) in both landscapes. Chapter 4. Sensitivity of least-cost habitat graphs 74

Table 4.2: Parameters used to generate spatial patterns of landscapes and their fac-

torial combinations. Each cell in the artiﬁcial landscape was classiﬁed as habitat (H),

hospitable matrix (HM), or inhospitable matrix (IM). The amount and fragmenta-

tion of inhospitable matrix was completely constrained by parameters for habitat and

hospitable matrix. All factorial combinations result in 20 spatial patterns. Spatial

patterns were generated using the (Fahrig, 1997, 1998; Tischendorf and Fahrig, 2000a;

Tischendorf, 2001; Tischendorf et al., 2003).

Factor Description Parameter values No. levels

H COV Amount of habitat (percentage) 12% 1

H FRAG Fragmentation of habitat 0.05, 0.5 2

HM COV Amount of hospitable matrix 10, 20, 30, 50, 70% 5

HM FRAG Fragmentation of hospitable matrix 0.05, 0.5 2

parameters (Table 4.2). I examined H FRAG and HM FRAG values of 0.05 and 0.5

because they produced relatively extreme patterns of randomness and clumpiness re-

spectively for my 100 × 100 cell landscapes. I did not increase the parameters controlling the level of fragmentation, H FRAG and HM FRAG, beyond 0.5 to avoid hav-

ing too many single cell patches. To keep this a controlled experiment, I have assumed

that the cell size is the minimum habitat size required; ﬁltering out small habitat patches

would have compromised my ability to tie the results back to the H COV and H FRAG

parameter values. I generated 100 replicates of each given combination of parameters

for a total of 2 000 landscapes.

4.3.2 Cost values to quantify resistance to movement

Cost values were assigned to each of the three landcover types (H, HM, and IM) re-

flecting the ecological costs incurred by an individual of a generic species dispersing Chapter 4. Sensitivity of least-cost habitat graphs 75 through that cover type relative to the costs incurred by dispersing through the preferred habitat (H, which was always assigned a cost of 1). I performed a sensitivity analysis on these cost values to assess their effect on measures of habitat connectivity using least-cost links and graph-theoretic connectivity analyses. The ranking of landcover types was preserved such that preferred habitat (H) was always least-resistant to movement, followed by the hospitable matrix (HM), followed by the inhospitable matrix (IM; Table 4.3). Cost values were chosen to capture the range presented in the connectivity literature, which reported using fractional, twofold, and orders of magnitude differences between landcover costs (Table 4.1). Emphasis was placed on the relative differences of the cost values in different landcover types rather than absolute values (Chardon et al., 2003). I compared eight sets of cost values (C1 to C8) that systematically varied the relative cost between H and HM and between HM and IM. An illustration of the least-cost habitat graphs produced under different sets of cost values is provided in Figure 4.2. I examined four relative cost differences between H and HM

(1.5, 2, 5, and 100 fold) and two relative cost diﬀerences between HM and IM (1.5 and

100 fold; Table 4.3).

4.3.3 Graph-theoretic representations of habitat connectivity

using least-cost links

Graph-theoretic connectivity analyses summarize the spatial relationships between landscape elements. This is achieved by building a “graph” (Harary, 1969) consisting of a set of “nodes” that represent preferred habitat patches and “links” that represent the potential movement of an organism among them (Keitt et al., 1997; Calabrese and Fagan, 2004; Treml et al., 2008). When dealing with large landscapes that have many habitat patches and possible links, it is preferable to analyze a subset of all possible links that represents the connected core of the landscape for a given species (Keitt et al., 1997). Chapter 4. Sensitivity of least-cost habitat graphs 76

Table 4.3: Sets of relative cost values used in factorial experiment. H-Habitat, HM-

Hospitable matrix, IM-Inhospitable matrix.

Relative cost values

Cost set H HM IM

C1 1 1.50 2.25

C2 1 1.50 150

C3 1 2 3

C4 1 2 200

C5 1 5 7.50

C6 1 5 500

C7 1 100 150

C8 1 100 10 000

The minimum planar graph is one such graph as it contains the subset of all possible links that includes the maximum number of non-crossing links that are of least weight (therefore loops are allowed; O’Brien et al., 2006; Theobald, 2006). Least-cost routes are only delineated between topological neighbors (i.e., least-cost links) and will never cross habitat patches (see Fig. 4.2). The minimum planar graph allows for computational efficiency while including alternate pathways to reflect important redundancy required for population resiliency which is not included in other graph types such as the minimum spanning tree (O’Brien et al., 2006; Fall et al., 2007). In the minimum planar graph, nodes are 2-dimensional habitat patches that are fixed in space and links connect patches from edge-to-edge rather than from centroid-to-centroid (Fall et al., 2007).

A beneﬁt of the minimum planar graph is that it approximates the complete graph within a reasonable, bounded error (Keil and Gutwin, 1992). Empirical testing on real Chapter 4. Sensitivity of least-cost habitat graphs 77

Figure 4.2: An illustration of a) least-cost links, b) paths, and c) habitat graphs. A link is a route that directly connects two habitat patches (nodes). A path is route that indirectly connects habitat patches along a series of connected links and nodes in which no node is visited more than once. I used least-cost route analysis to identify least-cost links between habitat patches (nodes) to create least-cost habitat graphs.

Least-cost habitat graphs contain 12% habitat (black squares), 10% hospitable matrix landcover (HM COV; light grey squares) and the fragmentation parameter for both is

0.05 (H FRAG, HM FRAG). Habitat patches are connected by least-cost links (dark gray). The sets of cost values diﬀer with respect to the relative costs between the matrix landcover types: C1 - Habitat = 1, Hospitable Matrix = 1.5, Inhospitable Matrix

= 2.25; C2 - Habitat = 1, Hospitable Matrix = 1.5, Inhospitable Matrix = 150. Chapter 4. Sensitivity of least-cost habitat graphs 78 landscapes suggests that the mean diﬀerence between paths in a complete graph and in the minimum planar graph (Euclidean or least-cost) is much lower than the maximum deviation (and with a space eﬃciency that can scale to large landscapes; Fall et al.,

2007). Hence, minimum planar graphs could represent an ecological assumption that organisms move through the graph by using stepping-stone paths through topological neighbors (Urban et al., 2009), but this assumption is not necessary because they also approximate a complete graph.

The potential for movement between a pair of high-quality habitat patches (graph link) is based on the assumed movement behavior of the organism. If one assumes the organism follows straight-line movement behavior then links that minimize the geographical (Euclidean) distance of the route traveled between habitat patches will be identified. However, if one assumes that the organism follows least-cost movement behavior then least-cost links that minimize the cumulative cost along the route will be identified (Halpin and Bunn, 2000; O’Brien et al., 2006; Driezen et al., 2007). The length of both Euclidean and least-cost links can be measured in two ways: (1) the geographic length in metric distance units (e.g., meters or kilometers); or (2) the effective length as the cumulative sum of the cost values in cells traversed along the link multiplied by the length of the link segment crossing those cells.

In this study, I computed the minimum planar graph using Euclidean links for a given landscape. Then for each of the eight cost surfaces that corresponded to that landscape, I computed the least-cost links between all pairs of nodes that were present in the Euclidean minimum planar graph. This produced an additional ten graphs that all had identical node and link sets; however, the length, the eﬀective length, and the spatial location of the links could diﬀer depending on the cost surface. In total, for each of the 2 000 landscapes and 8 sets of cost values, I produced 18 000 graphs (2 000 using Euclidean links and 16 000 using least-cost links). Results were analyzed using

ANOVA (e.g., Li and Reynolds, 1994; Matter, 2003; Hufkens et al., 2008) in order to Chapter 4. Sensitivity of least-cost habitat graphs 79 examine main eﬀects, two-way interactions, and higher-order interactions among the four factors. No transformations of the data were required in order to meet the normal- ity assumptions of ANOVA.

4.3.4 Measuring the sensitivity of graphs with least-cost links

For each graph produced with least-cost links, I calculated a standardized measure of the spatial deviation of least-cost links attributable to the corresponding cost surface.

This standardized measure, referred to as the total spatial deviation of links, was calculated as the diﬀerence between the geographic length of the least-cost link and the

Euclidean link for each pair of connected patches, summed over all pairs of connected patches in the landscape graph. Large values indicate the presence of sinuous least-cost links that deviate from straight-line Euclidean links, whereas small values indicate that least-cost links closely follow the straight-line Euclidean links. The sinuosity of least- cost links will depend on both the spatial pattern of the habitat and matrix landcover types as well as the relative cost values assigned to each. This is why I ran a factorial experiment with three factors controlling landscape spatial pattern (degree of fragmentation of primary habitat, amount of hospitable matrix, and degree of fragmentation of hospitable matrix) and one factor controlling cost values (set of relative cost values) to test which factors are most important in determining total spatial deviation of least- cost links (Table 4.2).

4.4 Results

The results of the 4-factor ANOVA (Table 4.4) showed a significant three-way interaction between matrix composition (HM COV), habitat fragmentation (H FRAG), and the set of relative cost values assigned to different landcover types (COST). This significant interaction precluded any further significance testing for the two-way interactions Chapter 4. Sensitivity of least-cost habitat graphs 80

or main eﬀects; however, I was able to observe that the ANOVA clearly did not indi-

cate signiﬁcance for any two-way interactions including HM FRAG or its main eﬀect

(Table 4.4).

The three-way interaction can be examined visually by plotting the interaction be-

tween two of the factors at each level of the third factor (Figs. 4.3 and 4.4). In all of

the two-way interaction plots the most striking result was the consistent diﬀerentiation

between sets of cost values with odd numbers (C1, C3, C5, C7) versus even numbers

(C2, C4, C6, C8). Cost sets with odd numbers had small diﬀerences between the rel-

ative costs of hospitable and inhospitable matrix landcover types, whereas even num-

bered cost sets had large diﬀerences. This result is intuitive because larger diﬀerences

between matrix landcover cost values would expectedly cause least-cost links to deviate

more strongly from straight-line links. The diﬀerence in cost values between habitat

and the matrix was evidently not important despite relative values that diﬀered 100

fold. The interaction line for C2 lies between the odd- and even-numbered cost sets

and shows dampened peaks consistent with the odd-numbered cost sets.

The two-way interaction between HM COV and COST displayed a similar pattern for both low and high levels of H FRAG (Fig. 4.3); however, spatial deviation values

were ﬁve times larger where H FRAG was high (Fig. 4.3b) compared to where it was

low (Fig. 4.3a). Cost-value sets C1, C3, C5, C7 and sets C2, C4, C6, C8 diﬀered in

their response to increasing HM COV for both levels of H FRAG. Two thresholds of

HM COV are apparent in these interaction plots. First, there is an abrupt change in

the behavior of C2, C4, C6, and C8 when HM COV equals 30%. Below this level of

HM COV the spatial deviation of links increases with increasing HM COV and above

this level of HM COV the spatial deviation of links decreases or stays the same with

increasing HM COV. Second, when HM COV is above 50% all cost-value sets become consistent in their response to increasing HM COV.

The two-way interaction between H FRAG and COST also showed a diﬀerence be- Chapter 4. Sensitivity of least-cost habitat graphs 81

Table 4.4: Eﬀects of matrix composition (HM COV), habitat fragmentation

(H FRAG), matrix fragmentation (HM FRAG), and the relative cost values (COST) on the mean spatial deviation of least-cost links. Presented are F-ratios and P-values

from the four-factor ANOVA.

Source df F-ratio P-value

HM COV 4 12821.583 <0.001

H FRAG 1 46350.009 <0.001

HM FRAG 1 1.145 0.285

COST 7 14319.069 <0.001

HM COV*H FRAG 4 5616.703 <0.001

HM COV*HM FRAG 4 0.006 0.999

H FRAG*HM FRAG 1 1.265 × 10−05 0.997

HM COV*COST 28 2510.185 <0.001

H FRAG*COST 7 6923.999 <0.001

HM FRAG*COST 7 0.004 1.000

HM COV*H FRAG*HM FRAG 4 9.306 × 10−06 1.000

HM COV*H FRAG*COST 28 1135.699 <0.001

HM COV*HM FRAG*COST 28 2.652 × 10−05 1.000

H FRAG*HM FRAG*COST 7 2.033 × 10−05 1.000

HM COV*H FRAG*HM FRAG*COST 28 4.076 × 10−05 1.000

Residuals 15840 Chapter 4. Sensitivity of least-cost habitat graphs 82

Figure 4.3: Interaction plots for the percentage of hospitable matrix (HM COV) versus the set of relative cost values (C1 - C8) when habitat fragmentation is either a) low

(H FRAG = 0.05) or b) high (H FRAG = 0.5). Mean values of the total spatial deviation of least-cost links is plotted for each of the cost sets at each level of HM COV. Chapter 4. Sensitivity of least-cost habitat graphs 83

Figure 4.4: Interaction plots for the fragmentation of habitat (H FRAG) versus the

set of relative cost values (C1 - C8) plotted separately at each percentage of hospitable

matrix (HM COV; a - e). Mean values of the total spatial deviation of least-cost links is plotted for each of the cost sets at each level of H FRAG. Chapter 4. Sensitivity of least-cost habitat graphs 84

tween the responses of cost-value sets C1, C3, C5, C7 and sets C2, C4, C6, C8 for all

values of HM COV (Fig. 4.4). As H FRAG increased, the spatial deviation of least- cost links also increased; however, this increase was very slight for cost-value sets C1,

C3, C5, C7 and was dramatic for cost-value sets C2, C4, C6, C8. Diﬀerences among cost sets were in general much larger when H FRAG was high. The rate at which the

spatial deviation of links in C2, C4, C6, and C8 increased in response to H FRAG dif-

fered depending on the level of HM COV. When HM COV was 10%, their slopes were

visibly not parallel, indicating that they had diﬀerent responses to increasing H FRAG.

When HM COV was higher than 10% their slopes were generally parallel with the ex-

ception of C2.

4.5 Discussion

This study conﬁrmed that connectivity assessments based on least-cost routes do reveal

eﬀects of landscapes on potential movement pathways of organisms. Of interest, how-

ever, was the question of how sensitive these connectivity assessments were to the pa-

rameterization of the cost surface. Only a handful of connectivity studies that assume

least-cost movement behavior have performed sensitivity analyses on the landcover cost

parameter and most of these sensitivity analyses are performed on a single landscape

(Table 4.1). Although these studies employ a variety of connectivity measures and test

for their sensitivity with varying degrees of rigor, the majority did reveal some eﬀects

of the cost values on their results. My sensitivity analysis is not comprehensive but it

is the most rigorous of its kind in terms of examining the interaction eﬀects between

landscape structure and cost-surface parameterization on least-cost route analyses. I

found that indeed there were signiﬁcant interactions; hence, the sensitivity of least-cost

links to relative cost values changed depending on the level of habitat fragmentation

and the amount of hospitable matrix. Gonzales and Gergel (2007) also found that the Chapter 4. Sensitivity of least-cost habitat graphs 85 results of their least-cost route analyses diﬀered in fragmented and continuous landscapes.

My analyses clearly showed that the least-cost links were more sensitive to the difference in cost values between hospitable and inhospitable matrix landcover types than between habitat and hospitable matrix landcover. I restricted my analyses to the extreme differences in relative cost values between matrix landcover types that had been reported in the literature (1.5 and 100 fold; Table 4.1) so that I could look at interactions among landscape structure and cost sensitivity. I acknowledge that more work needs to be undertaken to define a potential range of cost values that could produce consistent results in least-cost connectivity assessments in a variety of landscapes.

Schadt et al. (2002) conﬁrmed that their ability to classify connections among habitat patches in terms of exchange of animals and mortality risk was only sensitive to cost values when habitats within the matrix had “very low” values. In their study, they varied the cost of a hospitable matrix landcover type with relative cost values of 4, 7, 10, 20, 30, 100, and 500 whereas inhospitable matrix was assigned a constant cost of 1000. Hospitable matrix cost values from 7 to 500 produced similar results and were deemed biologically plausible, but the smallest cost value of 4 changed their results and was dismissed as being below the biologically realistic range. Chardon et al.

(2003) and Verbeylen et al. (2003) also found that models assuming least-cost movement fitted patch-occupancy data better than the models assuming Euclidean movement for a wide range of cost values. From this result, they concluded that all chosen parameters and ratios were in the range appropriate for predicting the occurrence of their study species. However, they also found that the predictive power of their models were sensitive to the exact quantification of the cost surface. This supports the idea that a biologically-realistic range of cost values may produce satisfactorily consistent results for coarse measures of connectivity, although some sets of cost values will improve accuracy of precise predictions. In light of these findings and given the extreme Chapter 4. Sensitivity of least-cost habitat graphs 86 cost values assigned to matrix landcover types in my study, I expected to observe some differences in the spatial locations of least-cost links. Indeed, least-cost links did follow larger spatial detours from straight-line links where the difference in cost values between matrix cover types was large (e.g., Fig. 4.2). The weaker effect of the difference between cost values assigned to habitat and hospitable matrix was unexpected as it has not been explicitly tested in previous least-cost route sensitivity studies. This result is not entirely surprising however, given that my sensitivity analysis of least-cost routes was combined with a graph-analysis approach. In the graph approach least-cost links did not cross nodes; hence, I would expect that multiplying all cost values except for the cost of habitat (which is fixed at 1) by a positive integer would result in the same spatial pattern of least-cost links, but the effective distance of links would be correspondingly larger. Therefore, I reiterate that the most important relative cost values are those between landcover types in the matrix when evaluating the sensitivity of least-cost routes implemented within a habitat graph modelling approach.

My results also showed that the sensitivity of least-cost links to relative cost values was modified by the composition of the matrix. The spatial deviation of least-cost links from straight-line links peaked where the percentage of hospitable matrix in the landscape was 30%. For higher percentages of hospitable matrix, different sets of relative cost values produced more consistent least-cost links. Conversely, the degree of fragmentation of the matrix landcover types did not have an effect on the sensitivity of least-cost links to relative cost values. This result is consistent with results from empirical studies demonstrating that the effects of fragmentation are generally much weaker than the effects of habitat amount on various measures of species movement, distribution, and persistence (Fahrig, 2003, and references therein). I set the amount of habitat in all of my simulated landscapes as 12% to reflect common conservation targets

(Pressey et al., 2003) and found that for this proportion of habitat, the degree of fragmentation of the habitat did indeed aﬀect the sensitivity of least-cost routes. In this Chapter 4. Sensitivity of least-cost habitat graphs 87

respect, the least-cost habitat graph model is behaving consistently with the observa-

tion that the eﬀects of fragmentation increase with decreasing habitat amount in the

landscape (Andr´en,1994, 1999) because I have a low proportion of habitat in my land-

scapes. Indeed, the quality of landcover types has also been shown to modify the rela-

tionship between the size of habitat patches and population abundances (M˝onkk˝onen

and Reunanen, 1999). The complex interaction among habitat fragmentation, amount

of hospitable matrix, and relative cost values in my study further emphasize the impor-

tance of quantifying all three (pattern, amount, and quality) when making assessments

of potential movement through heterogeneous landscapes.

Although a wide variety of values have been used to quantify habitat cost values

for many diﬀerent species in the connectivity literature (see Table 4.1), very few stud-

ies have empirically estimated relative cost values. There have been two studies, how-

ever, which have addressed this question with very diﬀerent approaches (Ricketts, 2001;

Stevens et al., 2004). Stevens et al. (2004) conducted a manipulative, laboratory-based

experiment to compare physical movement ability through 5 landcover types corre-

sponding to sand, cement, field grass, and forest for the toad Bufo calamita. They estimated cost values of 1, 1.5, 2, and 2.5 for these habitats, but these cost values only measure a single aspect of fitness, which limits their direct applicability to connectivity analyses. Ricketts (2001) used field surveys to estimate the difference between willow and conifer costs for six butterfly taxa. He found that the willow landcover had a cost value of 1.7, 0.9, 1.4, 2.0, and 1.1 for five of the six taxa studied relative to the meadow cost of 1. For four taxa, conifer cost was between 3 and 12 times higher than the willow cost. The small differences in landcover cost estimates from these studies raise questions about the large relative cost values that are generally assigned by expert opinion (see Table 4.1); however, it is important to note that the studies were all examining different species with different ecological sensitivities which could account for the drastically different cost values. Manipulative methods for the empirical estima- Chapter 4. Sensitivity of least-cost habitat graphs 88 tion of landcover cost have also been suggested by Bélisle(2005), such as translocation and playback experiments, food titration experiments, and manipulating feeding and breeding-site locations. These types of empirical efforts coupled with my work on the sensitivity of results to the cost values will be essential for the wide-spread acceptance of graph-theoretic connectivity analyses that assume least-cost movement behavior as a tool to inform spatial planning decisions for conservation.

4.6 Conclusion

Despite their sensitivity to the parameterization of the cost surface, least-cost routes are increasingly being coupled with graph-theoretic techniques to assess connectivity in a diversity of applications. Easily accessible GIS software that allows least-cost routes to be computed rapidly has led to a proliferation in their use without examining key assumptions that may affect the ecological relevance of the conclusions based on their results (Gonzales and Gergel, 2007). In my systematic study I considered how the interactions between quality, amount, and fragmentation of habitat and matrix-cover types influence connectivity assessments that are based on least-cost routes and graph- theoretic analyses. I found that landscape structure affects how sensitive these connectivity assessments are to the quality (relative cost values) of landcover types, with the largest sensitivity occurring in fragmented landscapes. To overcome this sensitivity, I suggest identifying multiple low-cost paths between pairs of habitat patches that collectively delineate spatial zones (areas) accessible for probable movement within the intervening landscape (e.g., Sutherland et al., 2007). Pinto and Keitt (2009) have recently presented two methods for identifying multiple paths with similar costs, the Condi- tional Minimum Transit Cost method (see also Walker and Craighead, 1997; Halpin and Bunn, 2000; Theobald, 2006) and the Multiple Shortest Path method. These spatial movement zones may be more robust to variation in cost values because the uncer- Chapter 4. Sensitivity of least-cost habitat graphs 89 tainty in least-cost movement is incorporated into the delineation of the zones (Magle et al., 2009; Pinto and Keitt, 2009). Additional spatial properties of these movement zones such as their geometry and area can provide additional information about the quality of connections between habitat patches (Theobald, 2006; McRae et al., 2008).

Coupled with graph-theoretic analyses, these multiple-link movement zones will be particularly useful for guiding conservation planning decisions that must often be made quickly and with sparse data on disperser behavior. Chapter 5

Connectivity for conservation: A framework to classify habitat-network-connectivity statistics

5.1 Abstract

Graph theory, network theory, and circuit theory are increasingly being used to quantify multiple aspects of connectivity in habitat networks. There has been a proliferation of connectivity statistics for habitat networks resulting in over sixty statistics for ecologists to now choose from. Conceptual clariﬁcation on the ecological meaning of these network connectivity statistics and their interrelationships is overdue. I present a classiﬁcation framework that categorizes connectivity statistics based on the component of connectivity that they quantify (i.e., dispersal likelihood, route redundancy, route vulnerability, and habitat area connected) and the level of the habitat network to which they apply. The value of this framework lies in its ability to enable more

90 Chapter 5. Classification of network-connectivity statistics 91 informed choices and applications of network connectivity statistics. This will allow graph, network, and circuit analyses to vastly improve our ability to design and manage connected landscapes.

5.2 Introduction

The ability of individuals to move among habitat patches and populations is being un- dermined by ongoing habitat fragmentation, land-use intensification, and biotic homogenization (Saunders et al., 1991; Fahrig, 2003; Olden and Poff, 2003; Fischer and Lin- denmayer, 2007). Reduced and impeded movements have significant consequences for biodiversity conservation (Taylor et al., 1993; Kareiva and Wennergren, 1995; Moila- nen et al., 2005; Crooks and Sanjayan, 2006; Damschen et al., 2006; Minor and Ur- ban, 2008). Maintaining movements of individuals in the short- to medium-terms enables juvenile dispersal, recolonization of empty habitat patches, seasonal migration, and metapopulation persistence (Gilpin and Soulé,1986; Hanski, 1998a). Maintaining movements of individuals in the long-term enables range shifts in response to climate change and conserves genetic diversity required for evolutionary adaptation (Schemske et al., 1994; Minor and Urban, 2008). Quantifying the degree to which the landscape promotes or hinders movements among patches of habitat for a given species, hereafter habitat connectivity (Fischer and Lindenmayer, 2007), is therefore essential to inform conservation-management decisions (Calabrese and Fagan, 2004).

The graph-theoretic approach is a method for estimating habitat connectivity that is rapidly increasing in popularity in the disciplines of landscape ecology and conservation biology (Fig. 5.1 Urban and Keitt, 2001; Fall et al., 2007). This popularity can be attributed to three primary strengths of graph theory: (1) its eﬃciency in characterizing connectivity at broad spatial scales in landscapes with many habitat patches

(Urban et al. 2009); (2) its ability to balance data requirements with information con- Chapter 5. Classification of network-connectivity statistics 92 tent (Calabrese and Fagan, 2004); and (3) its ﬂexibility to incorporate additional information about relevant aspects of a species’ biology into connectivity assessments

(Fall et al., 2007; Minor and Urban, 2008). The graph-theoretic approach represents the connectivity of a set of habitat patches as a “habitat graph” - a collection of nodes

(habitat patches) and links that connect pairs of nodes (representing the potential or frequency of movement between habitat patches; Box 5.1). The way in which nodes and links are deﬁned will determine whether the habitat graph represents structural, potential, or functional connectivity among habitat patches (Calabrese and Fagan,

2004; B´elisle,2005; Fagan and Calabrese, 2006). Representing structural connectivity requires, for example, that links simply encode information about the physical adjacency of habitat patches or the physical distances among them. Additional information about the focal species’ dispersal abilities, such as its maximum dispersal-distance threshold, may be used to eliminate links that exceed that threshold distance and produce a representation of potential connectivity for the focal species (e.g., Brooks et al.,

2008; Pascual-Hortal and Saura, 2008). Finally, if data are available on actual movement patterns of individuals among habitat patches, then links can be deﬁned based on these observed movements and the corresponding habitat graph would represent functional connectivity. In the absence of observed movement data, only graphs representing landscape heterogeneity can provide some insights into potential connectivity.

However, not all species or individuals respond similarly to diﬀerent landscape con-

ﬁguration and composition. It is therefore important to evaluate habitat connectivity based on diﬀerent properties of the graph.

Over the past decade there has been extensive proliferation of connectivity statistics for habitat graphs emanating from the related ﬁelds of graph, network, and circuit theory (over 60 statistics; see Tables 5.2 and 5.3). These statistics were either developed speciﬁcally for the application to habitat-connectivity assessments (Urban and

Keitt, 2001; Pascual-Hortal and Saura, 2006; Saura and Pascual-Hortal, 2007, e.g.,) Chapter 5. Classification of network-connectivity statistics 93 or were adapted from other ﬁelds such as social sciences, transportation theory, communication theory, and epidemiology (e.g., Fortuna et al., 2006; Brooks et al., 2008).

In the absence of any selection guidelines, this proliferation of connectivity statistics poses a challenge for ecologists trying to select one or a few statistics for their graph- based connectivity assessments. The result is that most studies use different connectivity statistics making the results difficult to interpret and compare. A framework that classifies graph-based connectivity statistics is therefore necessary so that informed decisions can be made regarding the choice of statistics for any particular assessment of habitat connectivity.

My goal is to devise a classification framework of connectivity statistics for graph, network, and circuit analyses of habitat connectivity. I begin by reviewing the methods of graph, network, and circuit theories with an emphasis on the ways in which they incorporate spatial information about the habitat patches and the surrounding matrix in the definition of nodes and links. I then present a classification framework that categorizes connectivity statistics based on the level of the graph to which they apply (i.e., element, neighbourhood, cluster, and network levels) and the component of connectivity that they quantify (i.e., dispersal likelihood, route redundancy, route vulnerability, and habitat area connected). I conclude with a proposed extension of these static habitat graphs that explicitly incorporates temporal changes in habitat connectivity (i.e., spatio-temporal connectivity).

5.3 Background

“Habitat networks” (Keitt, 2003; Fortuna et al., 2006; Theobald, 2006; Brooks et al.,

2008; Economo and Keitt, 2008) and “habitat circuits” (McRae, 2006; McRae and

Beier, 2007; McRae et al., 2008) are special forms of habitat graphs with distinct properties that make them especially well-suited to represent potential and functional habi- Chapter 5. Classification of network-connectivity statistics 94

Figure 5.1: Number of published studies that use graph, network, or circuit theory to quantify habitat connectivity. The studies are ordered by their year of publication and include five review articles, three conference proceedings, and a user manual. Studies have been categorized based on their self-ascribed method of analysis: graph theory, network theory, or circuit theory. Note that these self-ascribed labels have been incon- sistently applied and do not necessarily correspond to my definitions (Box 5.1). Chapter 5. Classification of network-connectivity statistics 95 tat connectivity (Box 5.1). A “habitat network” is defined as a habitat graph with nodes and links that are assigned positive numeric weights that reflect their influence on movements (i.e., nodes could be weighted by their expected or observed number of emigrants and links could be weighted by their expected or observed number of migrants that successfully travel their length; Theobald, 2006; Brooks et al., 2008). A

“habitat circuit” is deﬁned as a habitat network with nodes that are connected by special links made up of multiple, alternative movement routes (McRae, 2006; McRae and

Beier, 2007; McRae et al., 2008). Hereafter, I use the term habitat network as an intuitive, general reference to a graph-based model of the potential or functional connectivity among habitat patches. I make speciﬁc reference to habitat graphs, habitat circuits, and their respective analytic methods as necessary (Box 5.1).

Once the nodes and links of a habitat network have been deﬁned, network connectivity statistics can be calculated to quantify patterns of compartmentalization and connectivity within the network or across the entire network (Table 5.1; Keitt et al.,

1997).

Box 5.1. Graphs, networks, and circuits Habitat connectivity can be represented by graphs, networks, and circuits that differ with respect to the ecological information they encapsulate. These different representations of the habitat system will be suited to different applications depending on the data available and the questions of interest. The distinction between graphs, networks, and circuits is not well defined and the terms are often used interchangeably in different disciplines. Some distinctions can be drawn however based on either the models themselves or on the methods of analysis. In general, networks and circuits can be thought of as special types of graph models with restricted definitions and applications. In its most basic form, a graph is a set of nodes connected to some degree by links that join pairs of nodes (Harary, 1969). The term “network” is used synonymously with the term “graph” (Borgatti et al., 2009) but it should in fact refer exclusively to weighted graphs (Gross and Yellen, 2004; Diestel, 2006, Box2;), in which nodes and links have positive numeric weights. Nodes and links may be assigned weights based on attributes that affect connectivity, such as the size of nodes and the length of links in the case of habitat networks. In general, networks are used in an applied context and represent real-world systems (Newman et al., 2006). A circuit (specifically an electrical Chapter 5. Classification of network-connectivity statistics 96

circuit) is a network or multigraph (Box 5.2) in which nodes are connected by special links made up of one or more resistors (electrical components that conduct current; McRae et al., 2008). Such parallel connections may correspond to alternative movement routes between pairs of connected nodes in a habitat networks. Just as networks and circuits are graphs with specialized properties or applications, network theory and circuit theory are both derived from the mathematical subdiscipline of graph theory which is broadly defined as the study of graphs (Harary, 1969). Graph theory has been the primary mathematical language used to describe the structural properties and behavioral characteristics of networks to date (Newman et al., 2006); however, its scope extends beyond applied networks and also includes a large body of work in algebraic analyses, geometric problems, and pure mathematics (Urban et al., 2009). Network theory began as an application of graph theory carried out in a number of applied research areas including social networks, electrical circuits, transportation systems, communication networks, epidemiology, and now ecological habitat networks (Brandes and Erlebach, 2005). One of the first applications of network theory was to model ecological food webs and, inturn, this application has greatly contributed to the development of network theory (Bascompte, 2007). It has recently set itself apart from graph theory by establishing a well-defined research agenda focused on properties of real-world networks, their structural dynamics, and the relationship between their structure and function (Newman et al., 2006). Hence, network theory provides a mathematical foundation for studying the effects of habitat network structure on dispersal processes and ultimately biodiversity persistence. Circuit theory (specifically electrical circuit theory) is a specific application of network theory dedicated to quantifying voltage, current, conductance, and resistance in circuits (Dorf and Svoboda, 2003). Circuit analyses are particularly useful for measuring habitat connectivity for species with dispersal behaviors that respond positively to increasing connections and redundancy. Measures of connectivity in circuit analyses incorporate all possible pathways and the cumulative flow between nodes (McRae, 2006; McRae and Beier, 2007; McRae et al., 2008).

Box 1 Figure 1. Representations of habitat connectivity that diﬀer with respect to the amount of ecological information that they incorporate. Habitat patches (black polygons) are connected by links (black lines) that cross hospitable (grey) and inhospitable (white) matrix cover types. a) A habitat graph connects patch centroids without incorporating a lot of spatial and ecological information about Chapter 5. Classification of network-connectivity statistics 97

nodes and links. b) A habitat network connects patch edges by least-cost links that incorporate information about matrix heterogeneity. Additional node and link attributes may also be included by assigning weights. c) A habitat circuit connects patches with multiple links thereby incorporating additional spatial information about the matrix. .

5.4 Development of methods to construct habitat

networks

Network theory belongs to the branch of mathematics that studies connections among discrete objects known as graph theory (see Box 5.1). Developments in the application of network theory to quantify habitat connectivity have been made primarily using the language of graph theory (Urban and Keitt, 2001; Fall et al., 2007; Pinto and Keitt,

2009). Initially, graph-theoretic approaches were proposed to quantify habitat connectivity based on very simple habitat graph models (Marcot and Chinn, 1982; Cantwell and Forman, 1993; Forman, 1995; Keitt et al., 1997; Marcot, 1998). These simple habitat graphs were non-directed, unweighted graphs (Box 5.2), in which habitat patches were represented as dimensionless nodes and links were represented as lines connecting nodes. Any information about the spatial location, length, shape, and quality of habitat patches and of links was disregarded. Therefore, estimates of habitat connectivity based on these simple habitat graphs considered only the presence or absence of connections between habitat patches (topology; Box 5.2) instead of the strength of connections.

Further developments of the network-theoretic application have resulted from the inclusion of more information about habitat suitability, landscape permeability, and movement behavior into habitat graphs (van Langevelde et al., 1998). The most straight- forward means of including additional information into simple habitat graph models is Chapter 5. Classification of network-connectivity statistics 98 to assign positive numeric weights to the nodes and links. Nodes can be weighted to reﬂect habitat-patch properties that may inﬂuence immigration and emigration such as patch area (e.g., Keitt et al., 1997; Bunn et al., 2000; Estrada-Pena, 2002; Miller and

Russell, 2004) or patch quality (e.g., Jord´an,2003; Minor and Urban, 2007; Saura and

Pascual-Hortal, 2007). Links can be weighted to reﬂect link properties that may affect dispersal such as the geometric length of the link or the eﬀective length of the link based on estimated movement costs of underlying landcover types (e.g., Halpin and

Bunn, 2000; Jord´an,2000; Urban and Keitt, 2001; Keitt, 2003; Rothley, 2005; O’Brien et al., 2006; Fall et al., 2007). Link weights have also been used to describe the dispersal probability between two nodes by assuming a negative exponential relationship between dispersal probability and link length (e.g., Urban and Keitt, 2001; Keitt, 2003;

Brooks et al., 2008; Treml et al., 2008). These weighted habitat graphs have been described using the language of both graph theory (Bunn et al., 2000; Urban and Keitt,

2001; Brooks, 2006, e.g.,) and network theory (Keitt, 2003; Fortuna et al., 2006; Bodin and Norberg, 2007; Brooks et al., 2008, e.g.,). More sophisticated means of incorporating additional information into habitat networks focus on maintaining spatial ref- erencing of nodes and links during the construction of habitat networks (Theobald,

2006; Fall et al., 2007). These more sophisticated methods explicitly account for habitat patch shape and landscape structure in two ways: (1) nodes are treated as two- dimensional patches with ﬁxed spatial locations; and (2) links between nodes connect patch perimeter to patch perimeter and follow geo-referenced least-cost routes.

Connectivity of these habitat networks has been measured in terms of the probability of movement among habitat patches and the area of habitat connected (Saura and

Pascual-Hortal, 2007, and references therein).

The most recent development of the network-theoretic approach has been to treat links as multiple routes instead of single, least-cost routes (McRae, 2006; Theobald,

2006; McRae and Beier, 2007; McRae et al., 2008; Pinto and Keitt, 2009, reviewed in Chapter 5. Classification of network-connectivity statistics 99

Urban et al., 2009). Two methods have been proposed for identifying multiple routes between habitat nodes: the conditional-minimum-transit-cost method (CMTC method;

Theobald, 2006; Pinto and Keitt, 2009) and the multiple-shortest-paths method (MSP method; Pinto and Keitt, 2009). Both of these methods identify a large set of routes between a pair of habitat patches but only retain the subset that are within a given distance of the length of the least-cost route. The CMTC method identiﬁes a set of routes such that each path minimizes the CMTC through a diﬀerent location on the landscape. The CMTC is measured as the least-cost distance between two habitat nodes contingent upon the least-cost path crossing a given location on the landscape

(i.e. a given grid cell in a raster map). The MSP generates a set of paths between two nodes by repeated permutation of the least-cost route. Route segments are randomly deleted and the least-cost route is re-calculated in an iterative fashion. These methods allow us to create multi-route habitat networks (or multigraphs or circuits; Box

5.2) which provide important spatial information about movement options through the matrix and allow for increased ﬂexibility in connectivity conservation planning (Urban et al., 2009).

Box 5.2. Definitions of Network Terminology Adjacency matrix - a binary matrix in which each entry is defined aij = 1 if nodes i and j are share a link, otherwise aij = 0. The matrix has one row and one column for each node. Circuit (electrical) - a network or multigraph in which nodes are connected by special links made up of one or more resistors (electrical components that conduct current). Circuit theory (electrical) - a specific application of network theory dedicated to quantifying voltage, current, conductance, and resistance in electrical circuits. Cluster/component - a group of interconnected nodes for which a path exists between every pair of nodes. Cost value - a value assigned to each habitat type in a landscape that reflects the quality of the habitat in terms of ecological costs incurred by an individual moving through that habitat type. The cost value of each habitat type is normalized by the cost value of the preferred habitat type. Degree of a node - the number of links incident to that node. Chapter 5. Classification of network-connectivity statistics 100

Degree distribution - the frequency distribution of node degrees for all nodes in a network. Directed graph/digraph - connected nodes in a directed graph have two links, one in each direction. Digraphs represent directional asymmetries. Dispersal probability matrix - consists of the probabilities that an individual disperser will move between pairs of nodes, for all node pairs in the network. Distance matrix - consists of the metric distances (either geographic or cost) between pairs of nodes, for all node pairs in the network. Geodesic distance - the length of the geodesic path measured as either the topological or the metric distance. Geodesic path - the shortest path between two nodes calculated as either 1) the path that traverses the shortest number of links between two nodes or 2) the path that minimizes the sum of the link weights of the links in the path. Graph - a set of nodes connected to some degree by links that join pairs of nodes. Graph theory - the study of properties of graphs. Incidence matrix - a binary matrix in which each entry is deﬁned as cij = 1 if node i is incident upon link j, 0 otherwise. The matrix has one row for each node and one column for each link. Least-cost surface - a map of the cost values associated with each habitat type in a landscape. Least-cost route - a link or path that minimizes the cumulative cost values along its length. Link/edge - a network element that connects nodes. Metric distance - distance between two nodes in a network can be measured in geometric distance units or cost distance units based on the estimated movement costs of landcover types that are traversed. Motif - a small repeating pattern (sub-graph) within a network. Multigraph - a graph which has multiple or parallel links between a pair of nodes. Neighborhood - the set of nodes that are within a given number of connections from a focal node (primary neighborhood corresponds to nodes within one connection, secondary neighborhood corresponds to nodes within two connections, etc.). Network - a weighted graph or digraph that carries all the information about the inter-connection of the elements in a real-world system. Network theory - the study of properties of networks with an emphasis on their structural and behavioral dynamics. Network static - single numbers or series of numbers that capture the relevant characteristics of a network. Node/vertex - distinct elements within a network that are connected by links if they interact in some way. Path - a route that indirectly connects nodes along a series of connected links and nodes in which no node is visited more than once. Planar graph or network - a graph or network is planar if it can be drawn on a plane so that the links only intersect at the nodes (i.e., they do not cross between Chapter 5. Classification of network-connectivity statistics 101

nodes). Redundancy matrix - consists of the number of alternative routes that exist between pairs of nodes, for all node pairs in the network. Topological distance - distance between two nodes in a network is the number of links between these nodes. Topology - the conﬁguration of nodes and links in a network. Undirected graph - connected nodes in an undirected graph have a single link representing symmetric connectivity in both directions. Produces a symmetric adjacency matrix. Unweighted graph - a graph in which links are binary: assigned a value of one if the link exists and zero otherwise. Vulnerability matrix - consists of the number of cut-nodes, cut-links, or cut-grid cells that exist along routes between pairs of nodes, for all node pairs in the network. Weighted graph - nodes and links are assigned positive numeric integers describing their contribution to connectivity of the graph. Sometimes used synonymously with the term “network”. .

5.5 Quantifying connectivity in habitat networks

Due to the large size and complexity of habitat networks, it is necessary to condense and summarize the relevant information via the use of graph and network statistics

(Box 5.2). A network statistic is either a single number or a series of numbers that quantiﬁes some property of a network (Brinkmeier and Schank, 2005). Through the years, a wide variety of network statistics have been proposed to quantify the connectivity of habitat networks. I reviewed the literature covered by the ISI Web of Knowl- edge Science Citation Index (www.isiknowledge.com) database using the following word combinations in a topic search: graph, network, or circuit in combination with habitat connectivity or landscape connectivity. I found 51 relevant articles dating from January

1982 to May 2009 that have cumulatively proposed 60 network connectivity statistics for quantifying habitat connectivity. These network connectivity statistics have been assembled in Tables 5.1 and 5.2.

Many of these statistics have been imported or adapted from other disciplines. For Chapter 5. Classification of network-connectivity statistics 102 example, the Gamma Index (i.e., the number of links in a network divided by the number of links in the corresponding planar network; Table 5.2) was first imported from transportation geography by Ricotta (2000) to quantify the connectivity of different vegetation types in a landscape and later modified by Acosta et al. (2003) to include qualitative differences in the conservation values of vegetation types. Other statistics have been developed specifically for ecological applications such as the Integral Index of Connectivity (Table 5.3; Pascual-Hortal and Saura, 2006) which was proposed as a means of integrating habitat area and habitat connectivity in a single measure.

Network statistics that quantify habitat connectivity can broadly be divided into topological indices (Table 5.2) and weighted indices (Table 5.3). Topological network statistics only consider the presence or absence of a link between nodes during their calculation. They characterize the basic structure of a network based on the qualitative pattern of connections among nodes (Table 5.2). Weighted network statistics consider the variation and strength of connections between nodes by including node and link weights during their calculation. They characterize connectivity patterns in the network based on ecological diﬀerences between nodes and links (Table 5.3).

5.6 Classiﬁcation framework of habitat network con-

nectivity statistics

To organize this large number of network-connectivity statistics, I have classified them based on two criteria: (1) the network level at which they can be applied (i.e., element, neighborhood, cluster, and network; Fig. 5.2); and (2) the specific components of connectivity that they measure (Table 5.1). To facilitate this classification, the components of connectivity are simplified into four primary categories: inter-patch dispersal likelihood, route redundancy, route vulnerability, and habitat availability. The purpose of this classification scheme is to illustrate the parallels that exist between com- Chapter 5. Classification of network-connectivity statistics 103 ponents of habitat connectivity that can be measured at different levels of the habitat network. This framework will allow researchers and practitioners to quantify habitat connectivity at multiple levels of analysis and make a more informed selection of habitat-network-connectivity statistics based on the component of connectivity that they measure.

5.6.1 Network levels of analysis

Distinct levels of analysis exist within networks (Fig. 5.2; Wasserman and Faust, 1994;

Brandes and Erlebach, 2005) and network connectivity statistics have been developed for each of the distinct levels. There are connectivity measures, for instance, pertaining exclusively to the elements of the network themselves (i.e., nodes and links). At this “element level”, connectivity statistics focus on properties such as the number of links per node (i.e., node degree) or the importance of a node based on its structural position within the network (i.e., node centrality). “Neighborhood-level” statistics assess the connectivity among a set of nodes that are within a given number of connections from a focal node (neighborhood; Box 5.2). For example, a neighborhood analysis could examine the relationship between the degree of a node and the degrees of its neighbors (i.e., connectivity correlation). “Cluster-level” statistics examine groups of interconnected nodes (cluster; Box 5.2), focusing perhaps on the number of nodes connected (i.e., cluster order) or the average length of the shortest-path connecting node pairs (i.e., characteristic path length). Finally, there are “network-level” connectivity measures that focus on the patterns of connectivity among all nodes and links in the network (e.g., the mean node degree or the number of clusters).

Higher levels of analysis, such as the cluster and network levels, often summarize the distribution of connectivity properties at lower levels with frequency plots or measures of central tendency and spread. These high-level network descriptors can be either single-valued statistics or distributions (Brinkmeier and Schank, 2005). Basic Chapter 5. Classification of network-connectivity statistics 104 Assymetry Connectance CV of cluster order Node degree distribution Diameter of largest cluster Expected dispersal flux Gamma index Mean cluster size Mean node degree Number of clusters Order Order of largest cluster Order of smallest cluster Recruitment Size Traversability Total number of cutTotal nodes number of cut links Area index Class coincidence probability Correlation length Expected cluster size (area) Integral index of connectivity Landscape coincidence probability Max. connected local pop.Probability of size connectivity Average path strength Characteristic path length Cluster order Cluster size Diameter Harary index Path strength Wiener index Number of cut nodes Number of cut links Cluster area Level of Network Analysis Clustering coefficientEcologically scaled connectivity Meshedness Betweeness centrality Closeness centrality Dispersal flux Eccentricity Node degree Node degree correlation Node depth Node in degree Node out degree Node influx Node outflux Reachability index Commute time Effective resistance Network flow Route (link) redundancy Current densityNode area Modified incidence Connectivity correlationfunction Reliability Quality-weighted area Element Neighbourhood Cluster Network Inter-patch dispersal Route redundancy Route vulnerability Habitat availability Component of Connectivity Table 5.1: Summary of the scale-specific classification framework for network statistics that quantify habitat connectivity. Chapter 5. Classification of network-connectivity statistics 105 transformations can be performed among single-valued statistics and distributions for different levels in the network (Brinkmeier and Schank, 2005). Transforming network statistics and distributions from low to higher levels of analysis involves globalization or scaling-up the dependence on elements to neighborhoods to clusters to the entire network. A local-level statistic pertaining to a single network element, such as the degree of a node, could undergo globalization to transform it to a network-level statistic, such as the maximum, minimum, summation, or average degrees of all nodes. To transform a low-level statistic into a higher-level distribution involves counting the number of elements, neighborhoods, or clusters that lie in a particular range of values (e.g., absolute, relative or cumulative degree distribution). Converting distributions to single- valued statistics involves parameter elimination whereby a single value is calculated from a sequence of values given by a distribution. An example of parameter elimination would involve calculating the power law exponent of a degree distribution. Clearly, researchers and practitioners must decide the level at which connectivity statistics are to be calculated for their particular study and their analysis may involve one or more of the aforementioned transformations.

5.6.2 Components of habitat connectivity

For the purpose of this classification framework, I have identified three important components of habitat connectivity that are measured by the different habitat-network- connectivity statistics: (1) inter-patch dispersal, (2) route redundancy, and (3) route vulnerability. These are not direct measures of functional habitat connectivity per se, but they provide important information on the different components of the species- landscape interaction that result in functional habitat connectivity. The final component of connectivity that I identified, habitat availability, is more of an outcome than a cause of connectivity, but I have included it nonetheless because it is commonly measured in connectivity assessments of habitat networks. Chapter 5. Classification of network-connectivity statistics 106

Figure 5.2: Examples of diﬀerent levels of analysis within a habitat network. Habitat patches (grey) are connected by multi-route links (black). Chapter 5. Classification of network-connectivity statistics 107

The ﬁrst component of connectivity is inter-patch dispersal likelihood which ac- counts for movements among habitat patches. Inter-patch dispersal depends on the source strength of donor habitat patches, the attraction strength of recipient habitat patches, and the probability of dispersal through the matrix (Hanski, 1999b; Tischen- dorf and Fahrig, 2000b). Network-connectivity statistics range from simple measures of habitat-patch source strength, such as “node out-degree” (Schick and Lindley, 2007;

Treml et al., 2008), to integrated measures of source strength and dispersal probability, such as ”dispersal flux” (Urban and Keitt, 2001). Inter-patch dispersal occurs across a multitude of spatial and temporal scales (Mueller and Fagan, 2008) which requires the use of scale-specific measures based on levels of analysis within the network. For example, at the scale of movement between a pair of habitat patches or the element level, “dispersal flux” focuses on the recruitment potential of each patch (estimated by their areas or quality-weighted areas) and the probability of dispersal between the two patches (based on a dispersal kernel and/or properties of the intervening matrix).

At the neighborhood level, “connectivity correlation” measures the degree of compartmentalization and identiﬁes neighborhoods with high levels of mixing. At the cluster level, “diameter” measures the inter-patch distance an organism would have to traverse to span the cluster. At network level, the “Gamma index” measures the number of directly connected pairs of habitat patches in the network.

The second component of habitat connectivity is route redundancy which measures the presence of multiple movement routes among habitat patches. Including multiple movement routes into connectivity analyses acknowledges that individuals rarely use a single optimal route due to variability in behavior and perception (Wiens, 2001;

B´elisle,2005; Driezen et al., 2007). Multiple routes may also be used either because the theoretical optimal route is unattainable or because there exist multiple ways to disperse optimally (or near optimally) among habitat patches in a landscape (Wiens,

2001). An example of the latter is provided by Farmer and Wiens (1998) who mod- Chapter 5. Classification of network-connectivity statistics 108 eled long-distance migration of shorebirds and showed variability in individuals’ migration patterns even when all individuals were making optimal decisions about when and where to refuel. Route redundancy is desirable to maintain in a habitat network from a conservation-planning perspective because it produces more stable connections among habitat patches in the event that natural or anthropogenic disturbances compromise one or more routes (Moilanen et al., 2006; Urban et al., 2009). Redundant movement routes between a pair of habitat patches may be quantiﬁed using statistics from circuit theory which specialize in measuring cumulative ﬂux between habitat patches distributed over multiple connections (routes). “Resistance distance” is one such measure

(McRae, 2006; McRae and Beier, 2007; McRae et al., 2008); it measures the opposition to movement of a set of dispersal routes between a pair of habitat patches and decreases with the addition of more movement routes. Redundancy at neighborhood level can be investigated with the “clustering coeﬃcient” which is a measure of the fraction of triangles present in the network. Triangular network topology means that for each pair of directly connected nodes, there exists at least one more indirect path connecting the nodes through a mutual neighbor. A similar measure at the network level is

“meshedness” which compares the actual number of links in the network to the number in the corresponding triangulated planar graph (to which no additional links can be added without creating a non-planar graph).

The third component of habitat connectivity is route vulnerability which describes the degree to which the landscape structure funnels or scatters the movements of a particular species (Lees and Peres, 2008; Pinto and Keitt, 2009). When movements are funneled through particular locations on the landscape then maintaining habitat connectivity hinges on protecting those areas as movement passageways. The presence of such dispersal bottlenecks makes the network more vulnerable to being disconnected due to habitat destruction, particularly if habitat is destroyed in a manner that preferentially destroys those areas (i.e., low attack tolerance; Albert et al., Chapter 5. Classification of network-connectivity statistics 109

2000). An element-level analysis of vulnerability can identify areas (grid cells) in the intervening matrix surrounding habitat patches through which dispersers are likely to move when traveling from one habitat patch to another (i.e., “current density”; Table

5.3). Pinto and Keitt (2009) showed that dispersal bottlenecks in the matrix emerge when the quality of diﬀerent landcover types in the matrix is spatially autocorrelated and hence there are large clumps of high-quality matrix landcover types to attract dispersers. At higher levels of analysis, vulnerability can be measured by the presence of

“cut nodes” and “cut links” which act as funnels for movements through the network.

If these nodes or links are removed they will disconnect a cluster to create two smaller clusters and thereby decrease connectivity at both the cluster and network levels.

The ﬁnal component of habitat connectivity, habitat availability, considers the area inside a patch of habitat as a space where connectivity occurs in addition to inter- patch movements (Hanski, 1999a; Pascual-Hortal and Saura, 2006; Saura and Pascual-

Hortal, 2007). The problem with measures of habitat connectivity based solely on inter- patch movements is that they estimate zero connectivity in landscapes with a single habitat patch (regardless of its size; Tischendorf and Fahrig, 2000b). The habitat availability component recognizes that habitat connectivity may result either from a spatial conﬁguration of habitat patches that promotes inter-patch movements or from large tracts of contiguous habitat (Ferrari et al., 2007). Pascual-Hortal and Saura

(2006) have proposed that conservation be guided by indices of habitat availability that integrate the amount of habitat and inter-patch connectivity into a single measure.

Network-connectivity statistics often include information about the area of habitat patches at diﬀerent levels of analysis such as node area and cluster area. At the network level, habitat availability can be measured as the maximum or expected area of habitat connected in clusters or through the use of more integrated measures such as

“integral index of connectivity” (Pascual-Hortal and Saura, 2006) or “probability of connectivity” (Saura and Pascual-Hortal, 2007). These network-level measures of habi- Chapter 5. Classification of network-connectivity statistics 110 tat availability will be low either if habitat patches are poorly connected by inter-patch movements or if habitat patches are well connected but the amount of habitat is low

(Saura and Pascual-Hortal, 2007).

5.7 Missing habitat-network connectivity statistics

This classification framework of network statistics (Table 5.1) reveals a striking im- balance in the distribution of measures across different levels of network analysis and among the different components of habitat connectivity. It is evident that most network- connectivity statistics apply at the level of the entire network or individual elements and indeed these are the network levels that are most commonly measured in habitat- connectivity assessments. Analyses at the level of clusters are also fairly well represented by connectivity statistics and have been identified as an important intermediate scale at which to measure connectivity (Urban, 2005; Urban et al., 2009). For example, the size of connected clusters of habitat patches has been correlated with the spatial distribution of Woodland caribou (Rangifer tarandus caribou; O’Brien et al., 2006).

Few connectivity statistics quantify habitat connectivity at the neighborhood level despite the prominence of this level of analysis in other disciplines applying network theory (dyadic and triadic levels in Wasserman and Faust, 1994; Newman, 2003; Proulx et al., 2005). The importance of neighborhoods in a particular habitat network will likely depend on the distances among habitat patches relative to the perceptual range of a species. The connectivity structure of neighborhoods will presumably be more important if the neighborhood level corresponds more closely to the scale of the sensory information window to which an animal can respond (Lima and Zollner, 1996). For- tuna et al. (2006) provide an elegant example of the importance of the neighborhood- level statistic “clustering coeﬃcient” (Table 5.2) in a dynamic network of temporarily

flooded ponds. In their system, a high clustering coefficient provided the opportunity Chapter 5. Classification of network-connectivity statistics 111 for an amphibian, which had moved from a dry to a flooded pond, to move again to another flooded pond if the conditions of the former were not suitable for reproduction.

Another means of quantifying neighborhood topology in networks is through the identification of “network motifs” (Milo et al., 2002; Proulx et al., 2005). Network motifs are small repeated patterns or subgraphs that occur significantly more often than expected from random networks. A wide variety of biotic and abiotic networks have been found to contain motifs, some of which are common across networks derived from biochemistry, neurobiology, ecology, and engineering (Milo et al., 2002, 2004). For example, seven separate food webs contained a four-node motif in which two species shared a common predator and prey; this motif was also found in C. elegans neuronal network and five technological networks (Milo et al., 2002). The presence, number, and distribution of network motifs has been linked to the functioning of biological networks such as gene regulatory networks (Becskei and Serrano, 2000; Shen-Orr et al., 2002) and transcription networks (Mangan and Alon, 2003). To identify motifs in a network, it is possible to search and count all possible configurations of subgraphs with a fixed number of nodes (e.g., all three- and four-node subgraphs; Milo et al., 2002) or to re- strict the search to specific motifs that are hypothesized to be important for function- ality.

Motifs have yet to be identified in habitat networks. I propose that the presence and number of source-sink motifs could indicate inter-patch dispersal at the neighborhood level. This motif is rooted in the Pulliam (1988) source/sink model which has also been invoked as a potential conceptual foundation for habitat networks (Urban et al., 2009). A source-sink motif would be defined as a connected pair of source and sink habitat patches in which source and sink habitat patches have positive and negative net reproduction respectively. To detect source-sink motifs, the habitat network must be delineated with directional links (i.e., as a digraph, Box 5.2) differentiating the outward dispersal from sources and the incoming dispersal into sinks. The long-term Chapter 5. Classification of network-connectivity statistics 112 persistence of populations occupying sink patches in the habitat network should also be linked to the presence of source-sink motifs.

The habitat-network-connectivity framework (Table 5.1) also highlights a dearth of network statistics that measure route redundancy at higher network levels. Recently there has been an emphasis placed on quantifying route redundancy at the element level via the use of circuit theory (McRae et al. 2008). These methods have focused on the redundancy of multi-route links that connect pairs of neighboring habitat patches with a series of routes through the matrix (Theobald, 2006; Pinto and Keitt, 2009).

Redundancy also exists at the level of clusters between nodes that are not neighbors.

Multiple paths of varying lengths indirectly connect non-neighboring nodes by traversing along a series of connected intermediate links and nodes (Box 5.2). I propose the use of “path redundancy”, which reﬂects the number of possible paths between pairs of nodes (Kim et al., 2003), as a measure of redundancy at the cluster level. The redundant paths may be calculated and counted by modifying any suitable network search algorithm such as Dijkstra’s well known shortest path algorithm (see Pinto and Keitt,

2009, for an example of how to modify Dijkstra’s algorithm to ﬁnd redundant routes).

An alternative measure of path redundancy is the sum of the ’redundancy degrees’ of intermediate nodes in the shortest path (Kim et al., 2003). A node’s redundancy degree is the number of links incident to that node apart from the incoming and outgoing links involved in the shortest path. A higher number of redundancy degrees would indicate a route consisting of intermediate nodes with relatively more neighboring nodes and hence the potential for more redundant paths toward the destination (Kim et al.,

2003). Chapter 5. Classification of network-connectivity statistics 113

5.8 Selecting network connectivity statistics

The network connectivity statistics discussed here (Table 5.1) range from reduction- ist measures of the basic features of a network to integrated measures that simultaneously measure multiple aspects of network structure. Some of these statistics may be correlated either because they represent the same basic aspect of network structure (e.g., class coincidence probability and landscape coincidence probability; Table

5.3) or because different aspects of the network structure are correlated for the particular landscape under investigation. Presumably, a relatively small subset of all possible network-connectivity statistics would suffice to quantify habitat connectivity for a particular study system, as is the case with other landscape pattern metrics (Riiters et al., 1995; Cain et al., 1997; Cushman et al., 2008). It is unlikely however that the exact same set of network-connectivity statistics will be appropriate for all studies that differ with regards to location, data models (vector/raster), scale (grain and extent), and objectives. The choice of statistics should explicitly reflect a hypothesis about the observed habitat network and how the structure of the habitat network affects key ecological processes. This classification framework provides researchers and practitioners who are faced with this choice much needed clarification on the ecological meaning of existing network connectivity statistics and the relevant levels of analysis at which they can be applied.

5.9 Spatio-temporal connectivity in dynamic habi-

tat networks

Most habitat network models to date have assumed that the composition and conﬁgu- ration of landscapes is static; hence, most network-connectivity statistics quantify habitat connectivity at a snapshot in time. In reality, the amount and spatial conﬁguration Chapter 5. Classification of network-connectivity statistics 114 of habitats and matrix landcover types change through time due to natural processes and anthropogenic pressures. However, these landscape dynamics are implicitly incorporated in network statistics that quantify route redundancy and vulnerability because they measure the presence of alternate movement routes through the network that can maintain overall connectivity in the event that habitat patches or links are destroyed.

A more explicit accounting of network dynamics involves assessing network “robustness” or the sensitivity of network connectivity to the removal of nodes or links (New- man, 2003; Minor and Urban, 2008). A small handful of studies have considered the implications of removing nodes and links on habitat-network connectivity measured as:

(1) network diameter, recruitment potential, and area-weighted dispersal ﬂux by Ur- ban and Keitt (2001); (2) clustering coeﬃcient and node-degree correlation by Fortuna et al. (2006); (3) probability of metapopulation extinction in 100 years by Bode et al.

(2008); (4) network size by Matisziw and Murray (2009). In these studies, nodes or links are removed either randomly or in a priority sequence to emulate natural or anthropogenic landscape disturbances. More robust networks can tolerate a higher fraction of nodes or links being removed before habitat connectivity signiﬁcantly decreases.

Robustness has been shown to be highly related to node degree (Albert et al., 2000) and is likely also measures route redundancy and vulnerability (see Missing habitat- network statistics).

Although these robustness assessments provide valuable insights into the consequences of completely destroying habitat patches or links, they do not capture the full range of variability in patch spatial pattern changes. Habitat spatial features may be categorized into primitive events: generation, disappearance, expansion, shrinkage, union, and division (Sadahiro and Umemura, 2001). These primitive events may hap- pen simultaneously across dynamic landscapes which makes the task of quantifying concurrent changes in habitat connectivity non-trivial. Characterizing habitat connectivity in these dynamic landscapes requires an explicit analysis of spatio-temporal Chapter 5. Classification of network-connectivity statistics 115 changes in landscape structure.

I propose the construction of spatio-temporal habitat networks to measure habitat connectivity between two maps of a single landscape at two diﬀerent times. Spatio- temporal links among the set of habitat patches at time one and the set of habitat patches at time two would have a temporal length equal to the time step between the two landscape maps and a spatial length equal to the spatial distance between end- points (patch centroids). Element- and neighborhood-level statistics could be calculated to detect regions of highest connectivity changes based on classiﬁed primitive events to produce corresponding “temporal link types”. Cluster- and network-level statistics could be calculated to estimate the degree of change based on the distribution of temporal link types or the mean spatial distance of the shortest link for each patch.

5.10 Conclusions

The number of network-connectivity statistics presents a daunting task in selecting among them for any particular assessment of habitat connectivity. This classiﬁcation framework provides a much needed description of the qualitative relationships among network-connectivity statistics. Categorizing statistics according to (1) their level of analysis within the network and (2) the component of habitat connectivity that they quantify allows for an ecologically meaningful choice of statistic grounded in hypothesis testing. One of the major advantages of network thinking is that it encourages ecologists to focus on multiple levels of analysis and ask multi-scale questions (Kotliar and

Wiens, 1990) about how habitat patches are embedded within a habitat network and how the network structure emerges from local connections between habitat patches.

The importance of local patterns on overall landscape connectivity has been stressed in static (Bascompte and Sole, 1996; With and King, 1999) and dynamic (Matlack and Chapter 5. Classification of network-connectivity statistics 116

Monde, 2004; Wimberly, 2006) landscapes. Ultimately the goal of network connectivity analyses is to determine how the structure of habitat networks influences the persistence of ecological communities. An important step towards this goal will involve determining how well measures of potential connectivity can predict actual functional connectivity. Separately quantifying the four components of habitat connectivity at different network levels can help to decompose the relationship between potential and function connectivity to move towards identifying the most important aspects of habitat network structure to maintain. Until these statistics have been tested empirically, composite measures in network theory (e.g., Reachability; Table 5.3) should be avoided. Chapter 5. Classification of network-connectivity statistics 117 (Bodin and Norberg, 2007; Bode et al., 2008;et Brooks al., 2008; Minorban, and 2008; Ur- Treml etUrban al., et 2008; al., 2009) (Brooks et al., 2008;et Urban al., 2009) (Bunn et al., 2000;Keitt, Urban 2001; and Estrada-Pena et al., 2006) (Cantwell and Forman, 1993; Forman, 1995; Marcot, 1998; Ricotta et al., 2000;2003; Keitt, Fortuna et al.,Jordánet 2006; al., 2007; MinorUrban, and 2007) (Keitt, 2003) (Jordán,2003; Fortuna et al., 2006; Minor and Urban, 2008) (Franc, 2004; Minor and Urban, 2008) (Ferrari et al., 2007;et Jordán al., 2007; EconomoKeitt, and 2008; Minor andban, Ur- 2008) Indicates habitat patches that serve as stepping stoneslights and routes high- that aredispersal important pathways Indicates habitat patches that occupy a central positionhabitat in network the due toimity their to prox- other habitatIndicates patches the maximum isolationa of habitat patch and influence of aa habitat habitat patch network in Indicates the potential fortat a patch habi- to linkpatches other habitat Indicates the degree tonode’s which neighborhood a is agraph complete Indicates compartmentalization in a network which mayspread reduce of the a cascading disturbance habitat patches in thedistinguishes cluster between and a single, large patchy populations (small value) and a metapopulationvalue) (large Table 5.2: Topological connectivityapplied statistics to that quantify have connectivity been in habitat networks. Number of nodes inbetween the a geodesic focal path nodeminal and node the nearest ter- Average degree of athe focal average node degree relative of to its neighbors Mean geodesic distance in a cluster Indicates the reachability of all Number (or proportion) ofgeodesic pairwise paths in thethrough network a that focal pass node Mean geodesic distance ofevery a other focal reachable node node and in the network Largest geodesic distance betweennode that and any other node Average fraction of adiately focal connected node’s neighbors imme- thatimmediately are connected also neighbors with each other Node degree Number of links incident on a focal nodeNode depth (topological) Indicates the potential accessibility Connectivity correlation path length (topological) centrality Closeness centrality (topological) Eccentricity (topological) coefficient ScaleElement Betweeness Name DefinitionNeighborhood Clustering Ecological Significance References Table 5.2 continued on following page. *Note that geodesic paths and distances are calculated and measured in topological distances. Cluster Characteristic Chapter 5. Classification of network-connectivity statistics 118 (Bunn et al., 2000;and Halpin Bunn, 2000; UrbanKeitt, and 2001; Fagan, 2002; Manseau et al., 2002;Hortal Pascual- and Saura, 2006; Jordánet al., 2007; Brooks et al., 2008) (Marcot and Chinn, 2003; 1982; Jordán, Pascual-Hortal and Saura, 2006; SauraPascual-Hortal, and 2007; Schick and Lindley, 2007; Saura, 2008; Treml et al., 2008) (Calabrese and Fagan, 2004; Brooks, 2006; Ferrari et2007) al., (Ricotta et al., 2000;2001, Jordán, 2003; Pascual-Hortal and Saura, 2006; Saura, 2008) (Marcot and Chinn, 1982; Cantwell and Forman, 1993; Ricotta et al., 2000;2001; Jordán, Urban and Keitt,Minor 2001; and Urban, 2007; Treml et al., 2008) (Bodin and Norberg, 2007; Treml et al., 2008) (Ricotta et al., 2000;2001, Jordán, 2003) (Fagan, 2002) patches that are mutuallyable reach- cluster based on thegeodesic maximum distance between anyof pair habitat patches inIndicates the smaller cluster geodesic distances between habitat patches Indicates the presence ofto barriers dispersal if habitatdestroyed patches are Indicates the presence ofthat key hold links clusters ofgether habitat to- Indicates smaller geodesic distances between habitat patches Indicates the variation inber the of num- habitat patches per cluster immediately connected habitat patches in the cluster Sum of the geodesicThe distances Wiener in index a may cluster. the be corresponding normalized chain by andworks planar (i.e. net- the leastnetworks and respectively most that connected areorder) of the same Maximum geodesic distance in a cluster IndicatesSum the of compactness of the inverse a tances of in the a geodesic cluster. dis- be The normalized Haray in index aWiener may similar index fashion to the Number of nodes whosenects removal the discon- cluster by creating a new cluster Number of links whosethe removal cluster disconnects by creating a new cluster Standard deviation of clusterby order mean divided cluster order Wiener index and Normalized Wiener index Cluster size Total number of links in the clusterDiameter (topological) Indicates theHarary number index of pairs of and Normalized Harary index Number of cut-nodes Number of cut-links variation (CV) of cluster order ScaleCluster Name Cluster order Number of nodes in the cluster Definition Indicates the number of habitat Ecological Significance References Table 5.2 continued on following page. *Note that geodesic paths and distances are calculated and measured in topological distances. Network Coefficient of Chapter 5. Classification of network-connectivity statistics 119 Jrń 2003) (Jordán, (Franc, 2004; Brooks, 2006; Fortuna et al., 2006;et Rhodes al., 2006; Minor2007; and Brooks Urban, et al.,Economo 2008; and Keitt, 2008; Minor and Urban, 2008) (van Langevelde et al.,Franc, 1998; 2004; Pascual-Hortal and Saura, 2006; Bodin and Norberg, 2007; Saura and Pascual-Hortal, 2007; Economo and Keitt, 2008;nor Mi- and Urban, 2008;et Treml al., 2008) (Forman, 1995; Ricotta et2000; al., Jordán,2001, 2003) (Marcot and Chinn, 1982; Fagan, 2002; Jordán,2003; Treml et al., 2008) (Marcot and Chinn, 2003; 1982; Jordán, Pascual-Hortal and Saura, 2006; SauraPascual-Hortal, and 2007; Schick and Lindley, 2007; Treml et al., 2008; Saura2009; and Matisziw Torne, and Murray, 2009) (Bunn et al., 2000;and Halpin Bunn, 2000; UrbanKeitt, and 2001; Jordán,2003; Pascual-Hortal and Saura, 2006; Saura and Pascual- Hortal, 2007; Saura, 2008) of immediately connected habitat patches standardized bynumber the of habitat patchesnetwork in the Indicates the distribution oftential po- source and sinkpatches habitat as well asof the the robustness network todestruction habitat patch Indicates the compactness ofhabitat a network based onimum the distance max- between anyhabitat pair patches of Indicates the number ofately immedi- connected habitat patches clusters of habitat patches patches in the habitat network immediately connected habitat patches Probability distribution of nodeover degrees the entire network Maximum geodesic distance incluster the largest the number of linksplanar in network the corresponding Total number of connected sub-graphs Number of distinct, unconnected Degree distribution Diameter of largest cluster (topological) Gamma index Number of links inNetwork network order divided by Total number ofNetwork nodes size within network Total number of links Indicates within the network number of habitat Number of Indicates theclusters number of pairs of ScaleNetwork Connectance Name Density of links in network Definition Indicates the number of pairs Ecological Significance References Table 5.2 continued on following page. *Note that geodesic paths and distances are calculated and measured in topological distances. Chapter 5. Classification of network-connectivity statistics 120 (Brooks et al., 2008) (Marcot and Chinn, 1982; Cantwell and Forman, 1993; Forman, 1995; Jordán,2003; Ferrari et al., 2006) (Forman and Gordon, 1986; Forman, 1995) (Bunn et al., 2000;and Halpin Bunn, 2000; UrbanKeitt, and 2001; Fagan, 2002; Manseau et al., 2002;2003; Jordán, Pascual-Hortal and Saura, 2006; Brooks et2008) al., (Marcot and Chinn, 1982; Cantwell and Forman, 1993; Ricotta et al., 2000;2001; Jordán, Urban and Keitt,Minor 2001; and Urban, 2007; Treml et al., 2008) (Bodin and Norberg, 2007; Treml et al., 2008) pairs of immediately connected habitat patches in theIndicates cluster the average orof variation the accessibility ofpatch each habitat Indicates the presence ofnate alter- pathways which may assist organisms to avoid disturbancepredation and Indicates the maximum ormum mini- number of habitatthat patches are mutually reachable Indicates the presence ofto barriers dispersal if habitatdestroyed patches are Indicates the presence ofthat key hold links clusters ofgether habitat to- Number of connected nodesor in smallest the cluster largest Number of nodes whosenects removal the discon- network byter creating a new clus- Number of links whosethe removal network disconnects by creating a new cluster Mean number of links per clusterMean or standard deviationof of links the per number node IndicatesActual the number average of number loops of by in number network of divided loopsmaximal/triangulated in planar the network corresponding Order of largest or smallest cluster Total number of cut-nodes Total number of cut-links Size Mean or standard deviation of node degree Meshedness or Alpha Index or Network Circuity ScaleNetwork Mean Name Cluster Definition Ecological Significance*Note that geodesic paths and distances are calculated and measured in topological References distances. Chapter 5. Classification of network-connectivity statistics 121 (McRae et al., 2008) (Urban et al., 2009) (McRae et al., 2008) (Urban and Keitt, 2001; Minor and Urban, 2008) (Bunn et al., 2000;and Urban Keitt, 2001; Urban2009) et al., Indicates the expected timeindividual for to an move fromtat one patch habi- to theagain other if and they back move randomly Indicates the areas inscape the which land- dispersers areto likely pass through whenone moving habitat from patch to another Indicates the amount ofbetween movement a pair of habitat patches Indicates the maximum isolationa of habitat patch Indicates habitat patches that occupy a central positionhabitat in network the due toimity their to prox- other habitat patches , and y , xy and 1 R x n is the resistance =1 y X xy n =1 R X x , v uv ˆ R = and : v u uv is the effective resistance be- CT and uv Table 5.3: Node- andhave link-weighted been connectivity applied statistics to that quantify connectivity in habitat networks. ˆ R u quality-weighted area of nodesthe adjacent link to are multipliedof by dispersal the between probability nodes Largest geodesic distance betweennode that and any other node of the link connectingn nodes is the number of nodes in the network resentation of a habitatmeasures network the that flow ofthrough current a (dispersers) network whencurrent one (a ampere fixed of numberinjected of into dispersers) one is nodeconnected and to the ground other (are is through forced the to network). move tween nodes A property of thenodes set of links link between where Mean geodesic distance ofevery a other focal reachable node node and in the network ) CT Dispersal flux A property of a link in whichEccentricity the area or (metric) Current density A property of each cell in a raster rep- centrality (metric) Commute time ( ScaleElement Closeness Name Definition Ecological Significance References Table 5.3 continued on following page. *Note that geodesic paths and distances are calculated and measured in metric distances. Chapter 5. Classification of network-connectivity statistics 122 (Magle et al., 2009) (Fortuna et al., 2006) (Schick and Lindley, 2007; Brooks et al., 2008;et Treml al., 2008) (Miller and Russell, 2004;nor Mi- and Urban, 2007;and Schick Lindley, 2007) (McRae, 2006; McRae and Beier, 2007; McRae et2008) al., Indicates the role ofpatch a in habitat connecting otherpatches habitat through itself Indicates the amount ofment move- into a habitatprimary patch neighbors from its Indicates the connectivity ofhabitat a patch to allpatches other in habitat the habitatassuming network, that patch sizespermeability and of the the intervening matrix are important ity Multiple links between anodes pair represent of alternative movement routes and effectivedecreases resistance with the additionlinks of more k A v and , u ) i , ¯ A and all other A ( uvi uvi k k R R A ik m m =1 =1 Y i i X w is the number of links in = n =1 X k m uv = : . ˆ R v v i k is the resistance (or opposition) T offers to movement between is the area of the focal node, i and and uvi i is the mean area of all nodes, and and R A u i u ¯ A is a term that describes the potential ik that link where nodes the set of links between nodes where A property of aof node the which area- is and aevery cost-weighted function other distance node to in the network: is the area ofarea, a single node in the study A property of thenodes set of links between Pearson product-moment correlation coefficient between the directedundirected degree degree and of the a node cent to a node patches based on ancosts estimate derived of from movement multiplenodes links between a w movement between patch ) ˆ R ) i T resistance/ Resistance distance ( Node degree correlation Node in-degree Total number of linksNode coming influx into a node Indicates Sum habitat of patch fluxes accessibil- for all incoming links adja- Modified incidence function model ( ScaleElement Effective Name Definition Ecological Significance References Table 5.3 continued on following page. *Note that geodesic paths and distances are calculated and measured in metric distances. Chapter 5. Classification of network-connectivity statistics 123 Jrń 2003,(Jordán, modified slightly in Jordánet al.,Vasas 2007; et al., 2009) (Bode et al., 2008) (Jordánet al., 2007; Minor and Urban, 2008) 2007; Brooks et al.,Treml 2008; et al., 2008) (Miller and Russell, 2004;nor Mi- and Urban, 2007;and Schick Lindley, 2007) (Minor and Urban, 2007) (Ferrari et al., 2007) (van Langevelde et al.,Bunn 1998; et al., 2000;Keitt, Urban 2001; and Brooks, 2006; Ferrari et al., 2007;and Minor Urban, 2008; Treml2008) et al., Indicates topological, link- weighted, and node-weighted properties that affect theof connectivity a node Indicates the rapidity withthe which average unoccupied patchnetwork in can a be recolonized habitat patches in thedistinguishes cluster between and a single, large patchy populations (small value) and a metapopulationvalue) (large that is accessible toin an that individual cluster Indicates the amount ofout movement of a habitatprimary patch neighbors from its Indicates the reproductive potential of a habitat patch Indicates the compactness ofcluster a based on themetric maximum distance geo- between anyhabitat pair patches of in the cluster ) i ( is i , )) i CC ( ) is its i , ( i LP Smax ) i davtgr CC LP Smax − i ) + i D ( ( davtgr ( is the degree of node i = i D I average geodesic distance, and is the maximal connectedsize local after population its deletion Mean path strength overpatches all in pairs a of cluster Mean geometric distance in network Indicates the reachability of all where its clustering coefficient, Total number of links leaving a nodecent to a node Indicates habitat patch influenceNode area (i.e. habitattiplied patch by size) node mul- quality (Schickaverage (defined and distance as Lindley, between the eacha pixel habitat in patch andvalue a ranges non-habitat from edge; 0-1) Maximum geodesic distance betweenin nodes a cluster ) i I strength Characteristic path length (metric) Cluster Area Sum of areas of all nodes in cluster Indicates the amount of habitat degree Node outflux Sum ofQuality- fluxes for all outgoingweighted links area adja- Reachability/ Importance index ( Diameter (metric) Cluster Average path ScaleElement Node Name out- Definition Ecological Significance References Table 5.3 continued on following page. *Note that geodesic paths and distances are calculated and measured in metric distances. Chapter 5. Classification of network-connectivity statistics 124 (Bode et al., 2008) 2000) (Jordán, (Ferrari et al., 2007) (Bode et al., 2008) (Saura, 2008; Pascual-Hortal and Saura, 2008; SauraPascual-Hortal, and 2007; Pascual- Hortal and Saura, 2006) Indicates the probability thatrandomly two located points (orviduals) indi- within the habitatto belong the same cluster Indicated the probability ofment move- between a pairpatches of habitat Indicates the proportion ofavailable habitat to an individualhaving without to traverse theIncreasing matrix asymmetry has aative neg- effect on themetapopulation probability persistence of Indicates the probability that habitat patch immigration and emigration will operate successfully in the habitat network , i i c , 2 ) i C c A ( =1 is the probability that i NC X P = CCP is the number of clusters, , where P is the sum of all node areas in the − NC C A = 1 network and with a path betweenhas two the patches maximum which associatedmovement probability of R the area of the largest cluster connectivity pattern and itsequivalent symmetric (the mean ofconnections the in asymmetric both directions) where the habitat network becomesinto disconnected two or more isolated clusters is the sum of the node areas in cluster ) The difference between the asymmetric ) Z R ) CCP Reliability ( Asymmetry ( Class Coinci- dence Probabil- ity ( ScaleCluster Name Path strength The probability of movement associated DefinitionNetwork Area index Area of largest habitat patch divided by Ecological SignificanceTable 5.3 continued on following page. *Note that geodesic paths References and distances are calculated and measured in metric distances. Chapter 5. Classification of network-connectivity statistics 125 (Keitt et al., 1997;Pena, Estrada- 2002; Calabrese and Fagan, 2004; Rothley, 2005) (Miller and Russell, 2004) (Manseau et al., 2002;et James al., 2005; O’Brien2006; et Fall al., et al., 2007) Indicates the average distanceindividual an is capable ofbefore dispersing reaching a barrier,randomly if on placed the landscape Indicates the probability ofcessful suc- dispersal among habitat patches based on themultiple presence dispersal of routes, variation in matrix quality, andbarriers dispersal Indicates the mean areathat of a habitat disperser hasthe access mean to cluster (i.e. size),a if random placed habitat in patchnetwork within the , 2 ) s are n . i s y < y > − and i i y x , are the centroid s + ( s 2 R ) s n n =1 m s X =1 m s X < y > < x > = − i and C x ( p is the number of clusters and n =1 i X < x > m 1 n = is the radius of gyration calculated as R R follows: where is the number of grid cells in cluster The area-weighted mean clusterthat size. size Note here refersnodes to rather the than number number of cluster of links in the the coordinates of theof surrounding the polygon patch. Thispatch corrects shapes for irregular where coordinates of the patch and Mean connectivity of allconnectivity nodes of where each the nodeof is its a area, function itsmovement effective route distances from of all each its other barriers nodes, along and all dispersal routes Expected Cluster Size Ecologically scaled connectivity length ScaleNetwork Correlation Name Definition Ecological Significance References Table 5.3 continued on following page. *Note that geodesic paths and distances are calculated and measured in metric distances. Chapter 5. Classification of network-connectivity statistics 126 (Bunn et al., 2000;Keitt, Urban 2001; and Estrada-Pena et al., 2006; Minorban, and 2007; Ur- Pascual-Hortal and Saura, 2007; SauraPascual-Hortal, and 2007; Saura, 2008) (Pascual-Hortal and Saura, 2006; Saura and Pascual- Hortal, 2007; Saura, 2008) (Jordán,2003; Bodin and Norberg, 2007; Jordánet al., 2007) (Pascual-Hortal and Saura, 2006, 2007; Saura and Pascual-Hortal, 2007; Saura, 2008) ) among them ij nl Indicates the maximum metapopulation size within thenetwork habitat Indicates the probability thatrandomly two located points (orviduals) indi- within the landscapepoints (i.e., can lie eithernon in habitat habitat areas) or belongsame to habitat the cluster Indicates total movement among habitat patches in thenetwork habitat Indicates the probability thatanimals two randomly placed within the landscape fall intoeas habitat that ar- are reachableother from (interconnected) each given aof set n habitat patchesconnections and ( the binary is i , and c i , ij , j 2 nl a ) i L i a c 2 L A 1 + ( A n =1 =1 j X i NC X n = =1 i X are the areas of the nodes is the number of links j = ij a LCP nl is the number of clusters, IIC and and i NC a j is the total landscape area is the total landscape area L L and Sum of population sizestional or areas areas in or the propor- largest network cluster where the sum of theA node areas in cluster Sum of flux associatedthe with network each multiplied link by in of the total proportion habitat area in its incident node(s) i where in the geodesic pathgeodesic (i.e. distance) the between topological nodesA i and j. ) ) IIC LCP Maximal connected local population size Expected dispersal flux or Area-weighted flux of connectivity ( Landscape Coincidence Probability ( Scale Name Definition Ecological Significance References Table 5.3 continued on following page. *Note that geodesic paths and distances are calculated and measured in metric distances. Network Integral index Chapter 5. Classification of network-connectivity statistics 127 (Pascual-Hortal and Saura, 2007; Saura and Pascual- Hortal, 2007; Saura, 2008) (Urban and Keitt, 2001; Estrada-Pena et al., 2006) (Bunn et al., 2000;Keitt, Urban 2001; and Estrada-Pena et al., 2006) ) among ij p Indicates the potential toindividuals add to new the populations living in habitat patchesnetwork in the Indicates the potential fordistance long rescue within thenetwork habitat them Indicates the probability thatanimals two randomly placed within the landscape fall intoeas habitat that ar- are reachableother from (interconnected) each given a set of n habitatweighted patches connections and ( the i , ij ? p j a i 2 L a A n =1 j X n =1 is the total landscape i are the areas of the nodes X is the maximum product j L = a ij A ) belonging to each step in ?p ij PC p and i , and a j j and where in the network graph i area. The product probabilityis of the a product path ofabilities all ( the dispersal prob- probability of all paths between patches that path. and ) PC Traversability Diameter of the largest component of the Probability of connectivity ( Scale Name DefinitionNetwork Recruitment Sum of quality-weighted area for all nodes Ecological Significance*Note that geodesic paths and distances are calculated and measured in metric distances. References Chapter 6

Conclusions and future research directions

6.1 Thesis summary

Natural habitats are rapidly being destroyed and fragmented at a global scale (Foley et al., 2005). Conservation measures are required for plant and animal species to survive in the face of this widespread habitat destruction and fragmentation. Contempo- rary conservation biology favours an approach of prioritizing habitats for protection in ecological reserves and conserving or restoring natural levels of connectivity (species’ movements) among them (Margules and Pressey, 2000; Crooks and Sanjayan, 2006).

New methods and strategies presented in this dissertation provide novel ways to integrate important ecological considerations, such as landscape dynamics, species interactions, and spatial heterogeneity of habitat types, into reserve selection and habitat connectivity assessments. My research has shown that explicitly accounting for these ecological complexities in conservation planning and assessment can result in a re-prioritization of habitats and spatio-temporal land uses to preserve biodiversity.

The need to incorporate stochastic landscape processes, such as succession and

128 Chapter 6. Conclusions 129 natural disturbance regimes, into terrestrial reserve-selection strategies has only recently been discussed in either theoretical (Cumming et al., 1996; Bengtsson et al.,

2003; but see Pickett and Thompson, 1978) for an early discussion on this topic) or practical terms (Rayﬁeld et al., 2008; Leroux et al., 2007a,b). One of the ﬁrst strategies proposed to select reserves in dynamic landscapes was to create a system of dynamic reserves which are re-located through time to maintain representation and persistence targets (Cumming et al., 1996). In Chapter 2, I coupled a reserve-selection procedure with a spatially-explicit boreal forest dynamics model to compare the ability of dynamic and static reserves to maintain spatial habitat requirements of Amer- ican Marten through time. This is an underused approach for evaluating alternative reserve selection strategies using a spatially-explicit dynamic simulation model (but see Leroux et al., 2007a,b) despite its prevalence in natural resource management (Fall et al., 2004). Simulation results showed that dynamic reserves moderately improved upon static reserves by maintaining more marten home ranges that were of comparable quality to those in static reserves. The performance of the dynamic reserves is thought to be highly contingent on implementation details such as the length of the re-planning interval and the logging regime applied in the surrounding matrix. This study provides a valuable illustration of why comprehensive conservation plans should not rely exclusively on the use of reserves but should also focus on the spatial and temporal arrangement of land uses in the surrounding matrix to provide opportunities for re-planning in the future (Lindenmayer and Franklin, 2002). The performance of dynamic reserves was constrained largely due to inappropriate matrix planning. A focus on complementary reserve and matrix planning will require a novel management practice that adopts a longer time frame. As time progresses, spatial legacies of previous management actions can “lock in” spatial patterns that limit options for managers in the future (James et al., 2007).

Conservation planning based on the requirements of an indicator species, such as Chapter 6. Conclusions 130 the American Marten in Chapter 2, implicitly assumes that by maintaining their habitat requirements the conditions will also be suitable for other species (Landres et al.,

1988). There has been considerable debate surrounding the eﬀectiveness of indicator species in conservation planning (Roberge and Angelstam, 2004) with many studies arguing in favour of planning based on the needs of multiple species (Landres et al.,

1988; McLaren et al., 1998; Carignan and Villard, 2002; Thompson, 2006). In Chap- ter 3, I present a multi-species, reserve-selection method that explicitly incorporates consumer-resource interactions. This novel method protects consumer-resource interactions by prioritizing areas that are important to maintain connectivity between their distributions. The inclusion of interacting-species connectivity requirements resulted in the spatial aggregation of reserves which is recognized as a desirable feature of reserve from the perspective of minimizing negative edge eﬀects (Fagan et al., 1999) and reducing implementation costs from ecological and economic perspectives (Ball and

Possingham, 2000). Many reserve-selection methods impose an arbitrary penalty (i.e., boundary-length penalty; Ball and Possingham, 2000) to force reserves to be more spatially aggregated; however, this was achieved in the reserves in Chapter 3 by modelling explicitly the spatial connectivity requirements of a consumer and its resources. This represents an important shift away from reserve-selection methods that impose an arbitrary aggregated geometry to reserve-selection methods that induce spatial aggregation via models of species use of space.

Connectivity among consumer and resource distributions was modelled using a 2- dimensional kernel smoothing with a species-speciﬁc foraging parameter for the width of the smoothing kernel. This modelling approach, commonly employed in metapopulation studies (Moilanen and Nieminen, 2002; Moilanen et al., 2005), assumes that connectivity is a function of geographic distances among habitat patches embedded within a homogeneous matrix. However, the landscape matrix is seldom homogeneous with respect to animal movement and species generally do not follow straight-line routes Chapter 6. Conclusions 131 to reach their destination (Adriaensen et al., 2003; Chardon et al., 2003; Verbeylen et al., 2003; Lindenmayer and Franklin, 2002). The ability of species to move among habitat patches (habitat connectivity) is therefore not simply a function of geographic distance but rather it is a function of the landscape heterogeneity and species-speciﬁc movement responses to that heterogeneity (Taylor et al., 1993; Tischendorf and Fahrig,

2000b). Quantifying species perceptions of landscape heterogeneity can be achieved by creating a cost surface that represents the ecological costs of movement through each landcover type. This cost surface can serve as the basis for ﬁnding least-cost routes between habitat patches and analyzing habitat connectivity of the landscape using a graph or network approach (Bunn et al., 2000; Chetkiewicz et al., 2006; O’Brien et al.,

2006; Theobald, 2006; Fall et al., 2007). There is no consensus on the best approach to estimate movement costs and there is considerable uncertainty associated with these cost surfaces (Cushman et al., 2006). Diﬀerent studies have performed restricted sensitivity analyses in which they examine a limited set of alternative cost values. In Chap- ter 4, I performed a comprehensive sensitivity analysis of least-cost habitat networks to the relative cost values covering the range of cost values reported in the literature.

Least-cost habitat networks proved to be sensitive to relative cost values of the matrix landcover types but the degree of sensitivity diﬀered among landscapes based on their spatial structure. Spatial deviation among routes was highest when the habitat structure was fragmented and when the amount of hospitable landcover in the matrix was in the range of 20-50

Recognizing this sensitivity of least-cost routes in habitat networks to cost values, novel methods have been developed to identify potential movement among habitat patches through low-cost zones comprised of multiple movement routes rather than single least-cost routes (Pinto and Keitt, 2009; Sutherland et al., 2007, Fall et al., in prep). Specifically, methods from graph, network, and circuit theory are currently being applied to quantify habitat connectivity in habitat networks constructed with mul- Chapter 6. Conclusions 132 tiple movement routes between patches (McRae, 2006; McRae and Beier, 2007; McRae et al., 2008; Urban et al., 2009). Over the last decade, these methods have proposed over sixty connectivity statistics that can be used to quantify different aspects of connectivity in habitat networks. To clarify the ecological interpretation of these network connectivity statistics, I developed a classification framework. The framework categorizes connectivity statistics based on four components of connectivity (i.e., dispersal likelihood, route redundancy, route vulnerability, and habitat area connected) and four network levels (i.e., element, neighbourhood, cluster, and network). The framework revealed a lack of statistics in the categories of “dispersal likelihood among neighbour- ing habitat patches” and “route redundancy at the level of network clusters”. Hence,

I describe network-motif and path-redundancy statistics that ﬁt these categories. This framework is a timely and important contribution because an overall assessment of network connectivity is needed rather than focusing exclusively on structural properties at local levels within the network. By following this framework, one can better understand network-connectivity statistics so that explicit hypotheses can be tested about the relationship between network connectivity structure and the persistence of biodiversity in habitat networks.

Throughout this dissertation, I have emphasized ways in which our conceptual understanding of ecological systems can be incorporated into conservation management and assessment. The methods and results that I have presented can help us act strate- gically to protect biodiversity at a time when the human footprint is rapidly expanding. My finding that dynamic reserves will benefit from a whole-landscape approach to conservation planning, emphasizes the need for a better understanding of the interaction between reserves and matrix management. I stress that within reserves one should account for the spatial habitat requirements of interacting species, such as consumers and resources, as they are the foundation upon which ecological communities are sus- tained. I also emphasize that within the matrix, heterogeneity in habitat and landcover Chapter 6. Conclusions 133 types has important consequences for habitat network connectivity assessments based on least-cost routes. The sensitivity that I found in these assessments calls for a more robust means of incorporating uncertainty about matrix quality into the delineation of multiple movement routes between habitat patches. I therefore explored the potential for graph, network and circuit analyses to provide a more complete characterization of the connectivity structures of habitat networks at many different network levels. Such a network-connectivity classification framework will improve the ability of conservation planners to select connectivity statistics that can identify habitat patches and zones within the landscape matrix for connectivity conservation or restoration.

6.2 Future research directions

Many important questions remain about how reserve design and the connectivity among reserves aﬀects broad-scale ecological processes that support the persistence of biodiversity. These processes include dispersal (e.g., demographic rescue, infections disease spread, range expansion, exotic species invasions), species interactions (e.g., metacommunity dynamics, predation, competition, mutualisms), and gene ﬂow (e.g., genetic drift, genetic bottlenecking, inbreeding depression)(Kareiva and Wennergren, 1995;

Moilanen et al., 2005; Crooks and Sanjayan, 2006). Answering these questions empirically is challenging, if possible, given the diﬃculties associated with manipulating whole ecosystems and replicating landscape-scale experiments (Schindler, 1998). In light of these challenges, I propose three complementary approaches that can be used to provide some insights into questions about the consequences of reserve structure for species persistence.

The ﬁrst approach would involve combining the spatially-explicit landscape simulation modelling method presented in Chapter 2 with spatially-explicit population dynamics models (Turner et al., 1995; Wintle et al., 2005). This modelling approach Chapter 6. Conclusions 134 allows for comparisons among alternative reserve structures and strategies by examining the range of possible persistence outcomes that are simulated under a given set of assumptions and initial conditions. Another approach would involve the use of natural microcosms or experimental model systems (Srivastava et al., 2004) to experimentally test the causal relationship between reserve network structure and the persistence of biodiversity. These experiments could involve manipulating the topological structure of micro habitat networks in a laboratory setting (e.g., Gonzalez and Chaneton, 2002) and relating those structural diﬀerences to community patterns of resident microfauna

(e.g. diversity, relative abundance, distribution of patch abundance and occupancy).

A final approach would be to use “mensurative experiments” (McGarigal and Cush- man, 2002) in which existing habitat networks are carefully selected based on contrasting structures. For example, a researcher might look for habitat networks that have markedly different levels of compartmentalization and node isolation, such as random and small-world topologies (Bode et al., 2008; Minor and Urban, 2008). While manipulative experimentation at a landscape scale is generally not possible, it should be noted that restoration ecology offers an excellent opportunity for landscape-scale, replicated mensurative (or even possibly manipulative) experiments (Palmer et al., 2006).

The vast majority of habitat connectivity assessments to date have measured connectivity at a single snapshot in time. Relating this snapshot of connectivity to observations of species’ distributions and occupancy patterns assumes that landscape structure is static and ignores temporal variation caused by disturbance and successional processes. Hence, static habitat network models may have limited predictive power (Wiens, 1997; Matlack and Monde, 2004; Wimberly, 2006). For example, Mat- lack and Monde (2004) and Wimberly (2006) showed that individuals use ephemeral habitat patches as stepping stones to facilitate dispersal, a phenomenon that would not be accounted for by static models. In Chapter 5, spatio-temporal habitat networks are introduced as a means of quantifying habitat connectivity between habitat networks Chapter 6. Conclusions 135 at successive time periods. These spatio-temporal habitat networks could improve our ability to better manage dynamic landscapes to improve long-term connectivity. This novel application of network theory provides fertile grounds for research into dynamic network structures and the consequences for biodiversity conservation. Bibliography

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