Model-based estimators of density and connectivity to inform conservation of spatially structured populations DANA J. MORIN,1, ANGELA K. FULLER,2 J. ANDREW ROYLE,3 AND CHRIS SUTHERLAND4 1
New York Cooperative Fish and Wildlife Research Unit, Department of Natural Resources, Cornell University, 211 Fernow Hall, Ithaca, New York 14853 USA 2 U.S. Geological Survey, New York Cooperative Fish and Wildlife Research Unit, Department of Natural Resources, Cornell University, 211 Fernow Hall, Ithaca, New York 14853 USA 3 U.S. Geological Survey, Patuxent Wildlife Research Center, 12000 Beech Forest Road, Laurel, Maryland 20708 USA 4 Department of Environmental Conservation, University of Massachusetts-Amherst, 118 Holdsworth Hall, Amherst, Massachusetts 01003 USA Citation: Morin, D. J., A. K. Fuller, J. A. Royle, and C. Sutherland. 2017. Model-based estimators of density and connectivity to inform conservation of spatially structured populations. Ecosphere 8(1):e01623. 10.1002/ecs2.1623
Abstract. Conservation and management of spatially structured populations is challenging because solutions must consider where individuals are located, but also differential individual space use as a result of landscape heterogeneity. A recent extension of spatial capture–recapture (SCR) models, the ecological distance model, uses spatial encounter histories of individuals (e.g., a record of where individuals are detected across space, often sequenced over multiple sampling occasions), to estimate the relationship between space use and characteristics of a landscape, allowing simultaneous estimation of both local densities of individuals across space and connectivity at the scale of individual movement. We developed two model-based estimators derived from the SCR ecological distance model to quantify connectivity over a continuous surface: (1) potential connectivity—a metric of the connectivity of areas based on resistance to individual movement; and (2) density-weighted connectivity (DWC)—potential connectivity weighted by estimated density. Estimates of potential connectivity and DWC can provide spatial representations of areas that are most important for the conservation of threatened species, or management of abundant populations (i.e., areas with high density and landscape connectivity), and thus generate predictions that have great potential to inform conservation and management actions. We used a simulation study with a stationary trap design across a range of landscape resistance scenarios to evaluate how well our model estimates resistance, potential connectivity, and DWC. Correlation between true and estimated potential connectivity was high, and there was positive correlation and high spatial accuracy between estimated DWC and true DWC. We applied our approach to data collected from a population of black bears in New York, and found that forested areas represented low levels of resistance for black bears. We demonstrate that formal inference about measures of landscape connectivity can be achieved from standard methods of studying animal populations which yield individual encounter history data such as camera trapping. Resulting biological parameters including resistance, potential connectivity, and DWC estimate the spatial distribution and connectivity of the population within a statistical framework, and we outline applications to many possible conservation and management problems. Key words: abundance; conservation planning; density-weighted connectivity; landscape connectivity; landscape design; landscape resistance; potential connectivity; spatial capture–recapture; spatially structured populations. Received 14 October 2016; accepted 24 October 2016. Corresponding Editor: Debra P. C. Peters. Copyright: © 2017 Morin et al. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. E-mail:
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INTRODUCTION
spatial and temporal scales including daily movement of individuals, seasonal migrations, dispersal events, and range shifts of populations (Crooks and Sanjayan 2006). Thus, quantifying resistance in heterogeneous landscapes has the potential to mitigate the negative impacts of isolation at multiple scales, including restricting individuals from reaching all the resources required for survival (Fahrig and Rytwinski 2009, Sawyer et al. 2009), or preventing individual dispersal resulting in decreased population size, inbreeding, and genetic drift (Hedrick 2001, Keller and Waller 2002). Despite being a multiscale process, modeling and predicting connectivity have focused largely on the scale of dispersal, at times interchanging dispersal ability and connectivity synonymously. Moreover, while the term “spatially structured” is often equated with metapopulations (Minor and Urban 2001, Calabrese and Fagan 2004), it can be readily applied to continuous populations that are heterogeneously distributed in space (Nicol et al. 2016). Metapopulation models and applications related to connectivity between patches have dominated recent conservation literature and applications (Hanski 1994, Moilanen and Hanski 2001, McRae et al. 2008, Rayfield et al. 2011, Cushman et al. 2013). One possible reason for the focus on patch-based connectivity models is that models and metrics that more realistically represent spatially structured populations have been too data hungry and unwieldy to produce useful metrics (Wennergren et al. 1995, Kindlmann and Burel 2008, but see Sutherland et al. 2014). However, focusing only on patches of suitable habitat without concern for connectivity can result in isolated populations, whereas a focus only on species distributions and conserving gene flow will not identify areas of changing abundance, degraded habitat, or areas important for seasonal resources (Spear et al. 2010, Balkenhol et al. 2015, Mateo-Sanchez et al. 2015, Veloz et al. 2015). As a result, conservation and management of spatially structured populations still often address density and connectivity as separate objectives (Pressey et al. 2007, Saura et al. 2014, Veloz et al. 2015), even when the goal is to conserve both (Williams et al. 2005). In this study, we assess the performance of two model-based estimators of connectivity using the spatial capture–recapture (SCR) ecological
Spatial structure occurs in animal populations that utilize, and are distributed across, multiple habitat types (Kareiva 1990, Dunning et al. 1992, Nicol et al. 2016). Whether part of the natural ecology of a species, or imposed on a population through disturbance or anthropogenic changes to the landscape, most species exhibit some form of spatial structure including classic metapopulations (Levins 1970, Hanski 1994), stepping-stone population structure (Kimura and Weiss 1964, Crowley 1977), or continuous populations distributed heterogeneously in space as a function of the characteristics of the landscape (Andow et al. 1990, Kareiva 1990). Successful conservation and management must therefore consider two critical components jointly: the spatial distribution of individuals in heterogeneous landscapes, that is, local abundance or density, and how individual space use is influenced by landscape structure, that is, connectivity (Dunning et al. 1992, Taylor et al. 1993). Estimation of the first component, abundance, provides a metric for evaluating viability of threatened populations (Shaffer 1981, Soule 1987, Mangel and Tier 1994), as well as management of abundant species, to promote coexistence with human populations (Decker and Purdy 1988, Riley et al. 2002). The ideal number of individuals varies by species and system, but the definition and objective is well defined and explicit. The definition of connectivity, the second component, is notoriously vague, both within the literature and across applications and disciplines (Calabrese and Fagan 2004, Kindlmann and Burel 2008, Nicol et al. 2016). Generally, connectivity is defined as “the degree to which the landscape facilitates or impedes movement among resource patches” (Taylor et al. 1993). Thus, connectivity represents an interaction between movement and the physical structure of the landscape through which movement occurs (Schumaker 1996, Tischendorf and Fahrig 2000, Calabrese and Fagan 2004). Landscape resistance (or the inverse, conductance) describes the degree to which landscape features influence the ability of an individual to move through a particular environment and is a quantifiable metric that fits the definition of connectivity conveniently well (Zeller et al. 2012). Resistance to movement can occur at many ❖ www.esajournals.org
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distance model (described below), under different levels of landscape resistance. Our objective was to evaluate whether the SCR model based on ecological distance could be used to reliably estimate connectivity metrics that could be useful in landscape design and conservation. We then apply the SCR ecological distance model to data collected on an American black bear (Ursus americanus) population in the Southern Tier of New York to estimate density and connectivity, and we discuss applications for the derived connectivity measures for animal conservation and management at the landscape scale.
point process model (Royle et al. 2013b). The parameter r is a spatial scale parameter that relates detection probability (per sample interval) at a location x to distance from home range center s. Such models based on Euclidean distance imply that space use is symmetric: circular and centered on the activity center (s), and stationary: identical regardless of location or surrounding landscape structure. Thus, the SCR encounter probability model has a heuristic interpretation as a simplistic model of space usage. The Euclidean distance model is probably only reasonable when the landscape is homogeneous in the vicinity of an individual’s home range (Royle et al. 2013b). Home ranges are often structured by habitat or other landscape features, and therefore, the Euclidean distance model may not be a realistic representation of an actual home range and animal movement. As an alternative to the Euclidean distance model, Royle et al. (2013a) proposed the ecological distance model that utilizes a least-cost path distance (dlcp ) based on a landscape covariate-specific resistance parameter (a2). The resistance parameter determines how a particular landscape covariate, Z, defined by a discretized surface of pixel-specific covariate values zg, influences space use which is informed by how much the observed spatial pattern of observations deviates from the symmetric expectation (see Fuller et al. 2016, fig. 2). For all possible paths (w ¼ 1; . . .; W paths) between t 0 and t0 , Lt;t consists of mw path segments conw necting mw + 1 pixels. The cost-weighted distance between pixels which we seek to minimize for least-cost path is the product of the number of segments (length of path) and the associated cost of the landscape surface:
SPATIAL CAPTURE–RECAPTURE MODELS AND DERIVED PARAMETERS The advent of SCR models provides a statistical framework for simultaneous inference about abundance and connectivity at the individual scale using individual encounter history data (e.g., a temporal sequence of encounters of each individual in spatially referenced sampling devices) that are regularly collected by methods such as camera trapping and hair snares, and are relatively efficient to obtain for many species (Royle et al. 2013b). In the SCR framework, density and space use are estimated from spatial encounter history data yijk , a record (binary or frequencies) of where individuals i were encountered in traps (having locations xj ) on one or more sample occasions k ¼ 1; 2; . . .; K. The spatial pattern of encounters is modeled to depend on distance deuc ðsi ; xj Þ between individual activity centers and trap locations. The basic SCR model specifies a model for the observed encounters of individuals yijk that is conditional on the activity centers si which are regarded as latent variables. For encounter data which are binary “detection/non-detection” at each trap, we assume the encounters are Bernoulli trials: yijk Bernoulliðpij Þ. Standard SCR models assume that detection, pij , is a decreasing function of the Euclidean distance between trap locations xj and individual activity centers si , and therefore, individuals are more likely to be detected at traps that are closer to an individual’s activity center. For example, a standard model is the “half nor2 2 mal” encounter model pij ¼ p0 eð1=2r Þdeuc ðxj ;si Þ , and density is estimated by dividing the estimated number of individuals by the area of the state space (S), defined as all possible locations of the ❖ www.esajournals.org
dlcp ðt; t0 Þ ¼ min L1 ;...;Lw
mþ1 X
costðvg ; vgþ1 Þ
p¼1
deuc ðvg ; vgþ1 Þ; where costðvg ; vgþ1 Þ ¼
expða2 zðvg ÞÞ þ expða2 zðvgþ1 ÞÞ 2
(see also Royle et al. 2013a, Sutherland et al. 2015, Fuller et al. 2016). Thus, by allowing for home range asymmetry that is explicitly linked to the surrounding landscape structure (Fig. 1), 3
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Fig. 1. Covariate surface and cost, home range shape, potential connectivity, and density-weighted connectivity simulated for three increasing resistance values (a2 = 0, 1.25, and 2.25). Values for each surface are scaled between 0 and 1 where purple represents high probability of pixel use (1) and white represents low probability of pixel use (0). The covariate surface shown in the top row can represent any continuous variable or suite of variables that may shape home ranges and provide resistance to individual movement (e.g., elevation, % forest, slope). The first column shows the covariate surface in the first row, and the estimated cost for each a2 value in rows 2–4 below. Cost is constant in the second row where resistance is 0 (i.e., Euclidean distance). The second column shows the location of a hypothetical activity center for a single individual (top), and the resulting changes in home range shape and size with increased resistance (a2). The contours represent 90%, 75%, 50%, and 25% home range kernels. The third column shows the covariate surface with 1600 pixels (40 9 40) at top, and the potential connectivity surface for each a2 value below. Potential connectivity is the sum of the connectivity for each pixel in the landscape, and can be thought of as an overlay of all possible home ranges if there was an activity center in every pixel. The last column shows the covariate surface with 100 random activity centers at top (realized density for a single realization of the point process model N(s)), and the corresponding density-weighted connectivity for each a2 value below. Density-weighted connectivity is the potential connectivity weighted by the number of activity centers in each pixel.
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MORIN ET AL. Table 1. Mathematical expressions and biological interpretation for density and connectivity parameters derived from the spatial capture–recapture ecological distance model. Parameter
Mathematical expression
Biological interpretation
Realized density
N(s)
Estimated realized density
^ EðNÞðsjYÞ
Potential connectivity Estimated potential connectivity
C(g) ^ CðgjYÞ
Density-weighted connectivity
DWC ¼ CðgÞ NðsÞ
Estimated density-weighted connectivity
d ¼ CðgjYÞ ^ ^ DWC EðNÞðsjYÞ
estimating a2 represents a model-based characterization of the degree to which one or more covariate surfaces affects space usage within individual home ranges, that is, local connectivity at the individual scale (Sutherland et al. 2015). In this way, the SCR ecological distance model allows for the estimation of where individuals locate their home ranges (second-order selection) in addition to how individuals use space within their home range (third-order selection; Johnson 1980). Thus, SCR provides model-based estimates of the two critical components of conservation and management: the spatial distribution of individuals across the landscape (abundance or density), and a description of how individuals utilize space in their immediate surroundings (functional connectivity at the individual scale). The SCR model allows for the estimation of three spatially explicit quantities that are promising for use in landscape-scale connectivity conservation (Table 1; Sutherland et al. 2015):
encounter history data, evaluated at the maximum-likelihood estimates of the model parame^ ters. We denote this quantity by EðNðsÞÞ and refer to it as a prediction or estimate of the realized density N(s).
Potential connectivity We define potential connectivity as the expected number of individuals that might use a pixel g where the expectation is taken with respect to the population distribution of activity centers: P CðgÞ ¼ si 2S Prðgjs; hÞEðNs Þ. For the homogeneous point process model in which EðNðsÞÞ ¼ P const, then CðgÞ / s2S Prðgjs; ^hÞ. This quantity can be used to characterize connectivity of a given landscape as experienced by a population. Estimates of encounter probability model parameters ^2 Þ) can be used to estimate the cell^; a (^h ¼ ð^ a0 ; r specific connectivity values C(g) for any pixel ^ / P Prðgjs; ^ hÞ. (g) in the landscape si 2 S: CðgÞ s Thus, potential connectivity is proportional to the expected number of individuals that may use each pixel in the landscape and is a function of movement in response to the landscape including biological processes that may affect individual movement.
Realized density First, we can estimate N(s), the number of activity centers in pixel s, which we call realized density (Table 1). This is an estimate of local population size (local density, unscaled by area). If we could observe the true underlying point process, the quantity N(s) would provide a complete description of the population distribution. This latent variable can be estimated in the usual manner that random effects are estimated, using the “best unbiased predictor” (Royle and Dorazio 2008:290), which is the estimated posterior mean of N(s), conditional on the observed ❖ www.esajournals.org
A single realization of the point process model describing the number of individual activity centers per state space pixel Estimated density of activity centers in pixels conditional on the data Expected use of a pixel based on known cost of movement Expected use of a pixel based on estimated cost of movement assuming a single activity center in each pixel in the landscape Expected use of a pixel based on known cost of movement and weighted by the number of activity centers in each pixel for a single realization of the point process model Expected use of a pixel based on estimated cost of movement and weighted by the estimated density of activity centers in each pixel
Density-weighted connectivity Finally, we can combine realized density and potential connectivity to produce densityweighted connectivity (DWC; referred to as “realized connectivity” in Sutherland et al. 2015). Density-weighted connectivity differs from potential connectivity in that instead of multiplying the 5
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contributions of focal pixels to the connectivity of surrounding pixels, it is weighted by the local population size. If we know resistance and the location of every individual homePrange center, then DWC is given by DWCðgÞ ¼ si 2S Prðgjsi ; hÞ Nðsi jYÞ. However, in practice, we have to estimate both resistance and the expected number of activity centers in each pixel. Therefore, to estimate DWC we multiply potential connectivity by the estimated pixel local population size (as noted in ^ i jYÞ, to produce the estimated (1) above), Nðs P d ^ i jYÞ). EstiDWC: DWCðgÞ ¼ si 2S Prðgjsi ; ^ hÞNðs mated DWC can be thought of as a composite of all possible home ranges over a region of interest weighted by the estimated local density. In this way, the estimated DWC surface is a representation of the two critical processes required for population persistence, landscape connectivity, and density, estimated from data. The measures of connectivity proposed above are based on models both of how individuals in the population use space (the encounter probability model) and of how individuals are distributed in space (the density model). Furthermore, these measures can be estimated directly from encounter history data using SCR models, thus allowing for formal inference in the context of conventional animal population studies based on, for example, camera trapping. Therefore, it is of practical interest to assess how well DWC and related metrics can be estimated from field efforts that obtain capture–recapture data.
Fig. 2. The spatially correlated covariate surface with 1600 (40 9 40) raster cells used in the simulation study to assess estimation of resistance, potential connectivity, and density-weighted connectivity. The covariate surface was scaled between 0 and 1 and included a 14 9 14 trap array spaced 1.5 units apart (196 traps, denoted by +’s).
number of activity centers per pixel, N(s) (Data S1). For each realization, we simulated space use kernels with a Euclidean distance r = 0.15 (1.59 pixel resolution to allow for movement among adjacent pixels) for six values of landscape resistance (a2 = 0, 0.25, 0.75, 1.25, 1.75, and 2.25) representing a wide range of possible resistance values. Increasing the resistance values results in different values of potential connectivity and DWC for each simulated population, and also in a reduction of home range area with increasing a2 if r is held constant. To control for the effect of resistance-related variation in home range size, we increased r for each setting of a2 to maintain a mean home range area of approximately 64 pixels (Data S2). Thus, our simulation settings resulted in 500 realizations of a population with 200 individuals with ~64 pixel home range areas distributed over a 1600-pixel landscape. We centered a stationary trap array on the landscape to cover >90% of the state space, leaving a state space buffer equivalent to >2r to 3r to yield unbiased density estimates (Royle et al. 2013b). Trap spacing
