Continuous versus binary representations of landscape heterogeneity in spatially-explicit models of mobile populations

Continuous versus binary representations of landscape heterogeneity in spatially-explicit models of mobile populations

Ecological Modelling 221 (2010) 2409–2414 Contents lists available at ScienceDirect Ecological Modelling journal homepage:

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Ecological Modelling 221 (2010) 2409–2414

Contents lists available at ScienceDirect

Ecological Modelling journal homepage:

Short communication

Continuous versus binary representations of landscape heterogeneity in spatially-explicit models of mobile populations Steven T. Stoddard ∗ Program in Ecology and Conservation Biology, 386 Morrill Hall, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States

a r t i c l e

i n f o

Article history: Received 29 April 2010 Received in revised form 23 June 2010 Accepted 24 June 2010 Available online 24 July 2010 Keywords: Individual-based model Persistence Spatial heterogeneity Habitat loss Neutral landscape Landscape representation

a b s t r a c t How a landscape is represented is an important structural assumption in spatially-explicit simulation models. Simple models tend to specify just habitat and non-habitat (binary), while more complex models may use multiple levels or a continuum of habitat quality (continuous). How these different representations influence model projections is unclear. To assess the influence of landscape representation on population models, I developed a general, individual-based model with local dispersal and examined population persistence across binary and continuous landscapes varying in the amount and fragmentation of habitat. In binary and continuous landscapes habitat and non-habitat were assigned a unique mean suitability. In continuous landscapes, suitability of each individual site was then drawn from a normal distribution with fixed variance. Populations went extinct less often and abundances were higher in continuous landscapes. Production in habitat and non-habitat was higher in continuous landscapes, because the range of habitat suitability sampled by randomly dispersing individuals was higher than the overall mean habitat suitability. Increasing mortality, dispersal distance, and spatial heterogeneity all increased the discrepancy between continuous and binary landscapes. The effect of spatial structure on the probability of extinction was greater in binary landscapes. These results show that, under certain circumstances, model projections are influenced by how variation in suitability within a landscape is represented. Care should be taken to assess how a given species actually perceives the landscape when conducting population viability analyses or empirical validation of theory. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The loss and fragmentation of habitat are implicated as principal drivers of biodiversity loss worldwide (Wilcove et al., 1998). Theory predicts that populations may respond to habitat loss in a non-linear fashion and go extinct at a threshold amount of habitat determined by the degree of habitat fragmentation and organism life history (Bascompte, 2003; Fahrig, 2002, 2003; Huggett, 2005; Lande, 1987). These ideas have critical implications for the conservation management of rare species, but their empirical validation and implementation are made difficult by the challenges of acquiring data for natural systems (especially for populations of concern; Swift and Hannon, 2010). Nevertheless, insights from simple, heuristic models could provide tools for improved land management even when data are very limited. By carefully introducing greater realism into simple models and evaluating the consequences of these additional complexities, model predictions may be made more appropriate for a particular management con-

∗ Current address: Department of Entomology, One Shields Avenue, University of California, Davis, CA 95616, United States. Tel.: +1 530 601 0479. E-mail address: [email protected] 0304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2010.06.024

text. Furthermore, such exercises help identify factors relevant to the development of specific models, such as those used for population viability analysis (PVA). Spatial population models have highlighted the potential for an abrupt, non-linear increase in the probability of population extinction as habitat is lost from a landscape and that fragmentation of habitat may enhance this effect (Bascompte and Sole, 1996; Dytham, 1995; Fahrig, 2001, 2003; Flather and Bevers, 2002; With et al., 2002). Important implications of these models are that extinction can occur even when substantial amounts of habitat remains in the landscape and that the risk of extinction can increase dramatically over a narrow range of habitat loss or fragmentation. Thus, these predictions could provide a metric for land managers by which to assess the quality of an entire landscape for sustaining populations. Assuming the habitat requirements of a species are known, then an assessment of the amount of habitat and its spatial arrangement could provide an indicator of that landscape’s capacity (quality of habitat and other factors aside). Management could then direct activities in a way to keep conditions away from the threshold (Swift and Hannon, 2010). Testing the predicted effects of habitat loss and fragmentation on populations of rare species, however, is not possible. The question for land managers, then, is whether the ‘extinction threshold’ idea is valid for a particular


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system. One way to begin to evaluate the validity of models is to introduce increased realism and assess model output (Holland et al., 2007). An important assumption in spatial population models is the representation of the landscape. For the sake of simplicity, most heuristic models represent the landscape as a binary mosaic of habitat and non-habitat matrix (Haila, 2002; Manning et al., 2004). Often, however, we might expect a continuous representation of quality to more accurately reflect how an organism experiences the landscape, because suitability is at least partly defined by continuous factors (e.g. food availability, soil pH, etc.; Fischer and Lindenmayer, 2002, 2006; Holland et al., 2009; Lindenmayer and Luck, 2005; Manning et al., 2004; Turner, 2005; Wiegand et al., 1999). In this study I use an individual-based model to compare outcomes in binary and continuous landscapes across a range of conditions of habitat availability and spatial heterogeneity to address two related questions: (1) how well might predictions from simple models apply to more complex scenarios and (2) is additional detail about habitat suitability important for deriving useful insights? 2. Methods 2.1. Overview I model habitat in terms of organism life history, which is assumed to vary as a function of where in the landscape an organism chooses to reside. Within a binary construct, habitat is either good or bad, while in the continuous representation, habitat quality varies from very poor to very good, where quality simply reflects reproductive output and/or survivorship. Depending on the quality of a site, it may either contribute to the population or take away, i.e., it may be a ‘source’ of new individuals or a ‘sink’ of no reproduction and heightened mortality. As done elsewhere (Flather and Bevers, 2002; With and King, 1999), I simulate populations in static landscapes varying in the amount and spatial arrangement of habitat. 2.2. Landscapes I generated square binary and continuous lattices with each grid-cell representing the home-range of an individual. Each cell in the landscape was assigned a suitability value that, depending on simulation conditions (see Section 2.3), would allow net production of individuals (i.e., source) or net loss of individuals (i.e., sink). These neutral landscape models were generated by spectral synthesis to represent two levels of spatial heterogeneity, from highly fragmented (Hurst exponent, H = 0.05) to aggregated (H = 0.95; Hargrove et al., 2002; Keitt, 2000). For each landscape created in this way, I then made binary and continuous derivations using mean suitability values of 0.8 for source habitat and 0.2 for sink habitat. In binary landscapes, these suitability values were assigned to all source and sink cells, respectively. In continuous landscapes, I used a normal distribution with the mean values indicated above and  = 0.1 to assign a unique value to each source or sink cell. Values that fell outside the {0, 1} interval or beyond the cutoff suitability value designating habitat, Sc (see Section 2.3), were assigned the closest relevant value (e.g., a value of 1.03 would be set to 1) in order to retain the spatial structure of the landscapes. Thus extreme values (0, Sc , 1) were over-represented, but these were few and had very little effect on mean suitability values (data not shown). Since binary and continuous landscapes were derived from the same continuous surface generated through spectral synthesis, they were identical in all ways except for the distribution of habitat quality. I created 10 replicate landscapes

of each landscape type (continuous and binary), level of fragmentation (low and high) and habitat amount, p (0.005 thru 0.545 in intervals of 0.06; higher levels were not considered as persistence was generally certain at p = 0.545 for the grid-size and demographic parameter values used). All landscapes were 99 × 99 cell lattices and were generated in GRASS-GIS 6.1 and NetLogo 3.1 (Wilensky, 1999). Larger landscapes were not considered for computational reasons. 2.3. Population model The population model was an individual-based, spatiallyexplicit model of a closed population of mobile organisms. Within the model, mortality, dispersal, and reproduction occurred with an annual time step and only females were considered. No agestructure was included. Regulation of the population in this model was by site-dependence (Rodenhouse et al., 1997), meaning that population growth was governed by site suitability and the density of individuals in a site. The model was implemented as an individual-based simulation (see supplementary information for model code). Habitat suitability of sites, qi , and density within sites, Ni (i.e., the number of individuals in a single grid-cell) influenced population dynamics by determining reproduction, where the subscript, i, refers to single cell within the lattice. Mortality, m, was constant, making expected per-cell fitness at time-point t (i.e. the product of survival and reproduction probabilities), wi,t , a linear function of suitability:

wi,t = (1 − m)

q r i Ni,t

This relationship enables comparison between binary and continuous landscapes. More complex relations, e.g., making both mortality and reproduction functions of habitat suitability, would make fitness a nonlinear function and prohibit comparison between landscape types because of Jensen’s inequality (Ruel and Ayres, 1999). ‘Habitat’ was simply designated as any cell with a suitability value greater than the minimum, Sc , where population growth was greater than or equal to 1 in the absence of stochasticity: Sc =

m r(1 − m)

Thus, Sc defines the cutoff suitability value that distinguishes ‘source’ cells (births ≥ deaths) from ‘sink’ cells (births < deaths). The birthrate, r, was set to 0.7 in all simulations, while m was set to 0.25 (low mortality) and 0.3 (high mortality) to assess the effect of life history on responses. These values were chosen for experimental convenience because a wider range of mortality would alter the distribution of suitability values within the landscape (See Landscapes, above) and make interpretation of comparisons difficult. In addition, I wanted conditions near to the threshold for extinction to ensure sensitivity to landscape conditions. The parameter values used give population growth rates of 1.092 and 1.17 when habitat suitability is 0.8, ensuring moderate growth when conditions are suitable. 2.3.1. Dispersal Individuals moved in a random direction across the landscape with a probability and distance defined by a dispersal kernel (Chesson and Lee, 2005; Kot et al., 1996) I assumed most dispersal events would be short and within the immediate neighborhood of a currently occupied cell, with the occasional long-distance jump (Supplementary information). This is a simple representation of dispersal that seems to apply for many organisms and is commonly used in population models (e.g., Armsworth and Roughgarden,

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2005; Flather and Bevers, 2002). I described the probability of dispersing x cells, P(dx ), with a negative-exponential kernel: P(dx ) =

1 exp J

 −x  J

where J is the mean dispersal distance. I set J to 2 and 4 cells, corresponding to dispersal probabilities of 0.77 and 0.85 (i.e., the probability of departing a site; see supplementary information). 2.4. Simulations Simulations were initiated with 1000 individuals randomly distributed throughout the landscape. Each simulation was allowed to run for 300 annual time steps or until the population went extinct. Preliminary runs indicated that in most cases populations came to equilibrium within 50 time steps. Population abundance from each simulation was recorded as the mean of the last 50 time steps. Extinctions were recorded as any run where population size dropped to zero. Percent occupancy of source habitat was reported at the end of each simulation run. Mean production for source and sink cells was recorded as the mean number of individuals produced in cells over the full simulation. Each combination of parameter values was replicated 40 times. All simulations were run with NetLogo 3.1 (Wilensky, 1999). 2.5. Analyses To evaluate the effect of landscape representation, habitat amount, and habitat fragmentation on population persistence, I used generalized linear models with a logit link function (GLMlogit) to estimate a persistence probability P(persist). Because all of the landscape was viable (because the probability of mortality was not dependent on the suitability of local conditions), I used mean habitat suitability for the landscape to represent the amount of ‘source’ habitat. To further evaluate the effects of the main factors assessed here, I used simple ANOVA models to explain variation in population abundances and occupancy rates at the end of simulation runs. Because of large sample and effect sizes, parameters were usually significant. Hence, only the percent sum of squares was used to evaluate the importance of different effects. Analyses were conducted in (R Development Core Team, 2008). 3. Results In simulations, populations were very sensitive to the amount of source habitat (mean overall suitability) and the spatial arrangement of habitat, as shown elsewhere (Fig. 1; Flather and Bevers, 2002). Extinction occurred at higher amounts of habitat in binary landscapes than continuous. Overall, the frequency of extinction was 21% in continuous and 30% in binary landscapes. Population abundance at the end of simulation runs also differed significantly between binary and continuous landscapes (Fdf=1 = 309, p  0.001, Fig. 1, Supplementary information), controlling for spatial structure, mean overall suitability, their interaction, and the interaction between landscape type and mean overall suitability. Mean production of new individuals from ‘sink’ habitat was higher in continuous landscapes (Fdf=1 = 104, p  0.001) and under high spatial dispersion (i.e., more fragmentation; Fdf=1 = 32, p  0.001; Fig. 2). Mean production in ‘source’ habitat was also higher in continuous landscapes (Fdf=1 = 65, p  0.001), but lower under higher dispersion (Fdf=1 = 1760, p  0.001; Fig. 2). Hence, the observed differences between binary and continuous landscapes is due to increased production from both source and sink sites in continuous landscapes. The difference between continuous and binary landscapes was greater when ‘source’ habitat was spatially dispersed (H = 0.05).


This was true for extinction thresholds (persistence), abundances, and occupancy rates (Fig. 1, Supplementary Table 1). In addition, the difference was sensitive to life history and mean overall suitability (habitat amount). To better illustrate the conditions under which continuous and binary landscapes differ, I calculated the mean difference between abundances in continuous (Nc ) and binary (Nb ) landscapes scaled to mean abundance in continuous landscapes ((Nc − Nb )/Nc ) for identical conditions (Fig. 3). I then compare the ‘best-case’ scenario (spatial aggregation, low mortality, short dispersal) to an alternative (spatial dispersion, high mortality, long dispersal; Fig. 3) based on persistence and abundance metrics. In all instances, the difference between continuous and binary landscapes declines as mean suitability for the whole landscape increases. Spatial dispersion, high mortality, and long dispersal all increase the difference with stronger effect at lower mean landscape suitability. 4. Discussion 4.1. Summary of results My results illustrate how an important structural assumption of spatially-explicit simulation models influences outcomes. When habitat suitability is modeled to vary ‘continuously’ – that is, suitability across sites is graded – then predicted population size and persistence probability is greater than in a binary landscape, all else being equal. Also, when habitat was spatially dispersed (fragmented), extinction thresholds occurred at lower amounts of remaining habitat and abundances were higher in continuous landscapes. The observed differences are entirely a consequence of local dispersal and local density dependence. At the relatively low population densities modeled here, only a proportion of source habitat was ever sampled by individuals. In continuous landscapes the sites used were of higher suitability than the mean for all source habitat. This is evidenced by the fact that in landscapes with more habitat and less ‘fragmentation’ where populations achieved higher abundances, there was little difference between binary and continuous landscapes. Effectively in the continuous landscapes there were enough individuals in enough habitat to sample mean conditions. Increasing dispersal as done here augmented the difference because it was enough to reduce the negative effect of local crowding, but still local enough to ensure the range of habitat sampled to be of higher quality than the mean. 5. General discussion A binary simplification of landscape structure improves the analytical tractability of models. Such models have proven exceedingly theoretically interesting and insightful regarding the influences of spatial heterogeneity on population dynamics (Swift and Hannon, 2010). How well they represent reality and their utility in applied contexts, however, is uncertain because they are difficult to test. Existing simulation studies do indicate that population responses in more complex landscapes differ from those predicted by simple binary models, such as dispersal success (Malanson, 2003) and population abundance (Wiegand et al., 2005) in fragmented landscapes (Bender and Fahrig, 2005; Debinski, 2006). Empirical studies corroborate that increasing landscape complexity (or its perception by organisms) reduces the projected influence of spatial pattern (Lindenmayer et al., 2005; Lindenmayer and Luck, 2005; Lindenmayer et al., 2003; McGarigal and McComb, 1995; Monkkonen and Reunanen, 1999), substantiating the importance of thinking about the way organisms may perceive a landscape (Fischer and Lindenmayer, 2006; Fischer et al., 2004; Law and Dickman, 1998; Lindenmayer et al., 2003; Manning et al., 2004;


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Fig. 1. Population dynamics. Probability of persistence (a and b), abundances (c and d) and occupancy rates (e and f) as a function of mean suitability for the landscape. First column (a, c, e), aggregated habitat (H = 0.95); second column (b, d, f), dispersed habitat (see Supplementary information). Life history values were, m = 0.25, J = 2 (low mortality and short dispersal). Abundances and occupancy rates were taken as the mean of runs persisting the full simulation period (300 time steps) and are plotted with 95% confidence intervals (missing error bars indicate a CI too small to display).

Sibly et al., 2009). Nevertheless, predictions from simple models are very attractive in application because of their generality, motivating the question of their validity in practice and whether additional complexity in models is warranted. More realistic models generally incorporate multiple levels of habitat quality (Tischendorf et al., 2003; Wiegand et al., 2005) to address specific questions, but their structural assumptions are not evaluated nor are they necessarily used to validate simpler models. In assessing structural assumptions we effectively ask how simplifying assumptions affect dynamics and whether additional complexity improves the model in some way. In applied use of models, assessing the influence of structural assumptions helps bridge results from very specific models whose complexity is usually driven in part by end-user expectation to more simple models favored by ecologists (for an alternative approach, see

Sibly et al., 2009). These concerns motivated the simple exercise presented here, which was focused on assessing how ‘extinction thresholds’ as predicted from binary models might apply in scenarios where more complex landscape representations may be more appropriate. Binary and continuous representations are different in important ways (Holland et al., 2009) and the results presented here show that these differences are relevant in population models – and more so as conditions become more challenging (less habitat, lower population growth rate, more spatial heterogeneity). Qualitatively, however, trends were very similar and, since all population metrics were lower in binary landscapes, it suggests that a simpler representation in an applied model may be more conservative, though the generality of this statement is unknown. Note that, if individuals were more selective when dispersing (but still limited to local dispersal), we would expect a

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Fig. 2. (a and b) Production. Box-plots of mean production (over the full length of the simulation and all levels of habitat) from ‘sink’ and ‘source’ habitat for each landscape type and level of spatial heterogeneity (cases of persistence only). Solid lines at the center of boxes indicate the median, hinges are equivalent to the first and third quartiles, and whiskers depict the range of extreme values. Differences between means across landscape type and degree of heterogeneity were all statistically significant (p  0.001; m = 0.25, J = 2).

greater discrepancy between binary and continuous cases because individuals would seek out the highest quality sites available (and avoid unsuitable sites; Armsworth and Roughgarden, 2005; Pe’er and Kramer-Schadt, 2008). 6. Conclusions This work uses a simple simulation model to attempt to bridge insights derived from simple, theoretical models to more realistic, applied scenarios where empirical validation is very difficult. Generally, as conditions become more difficult, the choice of continuous versus binary appears more important – a relevant result for rare species conservation. This suggests that more theoretical attention to alternative models of habitat and spatial heterogeneity would be useful (Fischer and Lindenmayer, 2006). Clearly, how to represent habitat in a model is a key question to ask during development and more extensive characterization of how common structural assumptions influence model predictions is needed. At the least, these results indicate that assumptions about ‘habitat’ should be tested either through sensitivity analysis or based on field data. New methods, for instance, are available to quantify continuous landscapes (Holland et al., 2009; McGarigal et al., 2009). Nevertheless, for management the results presented here actually suggest that additional details about the quality of habitat – i.e., how this explicitly varies in suitability across a landscape – may not be needed to establish management guidelines. Acknowledgements I thank Patrick Weatherhead for guidance and for improving the quality of this work, Rebecca Sharitz for support, and Scott Schlossberg for reading a previous draft. I am also grateful to Jeff Brawn, Jim Westervelt, Bruce Hannon, and Hal Balbach. This work was funded by a Strategic Environmental Research and Development Program (SERDP) grant (SI-1302) to Rebecca Sharitz, Savannah River Ecology Laboratory, University of Georgia and co-administered by the US Army Corps of Engineers Construction Engineering Research Laboratory (CERL), Champaign, IL. Appendix A. Supplementary data

Fig. 3. Comparisons between outcomes. Mean difference between abundances in continuous (Nc ) and binary (Nb ) landscapes scaled to the abundance in continuous landscapes ((Nc − Nb )/Nc ) and plotted as a function of mean suitability. A value of zero indicates no difference between continuous and binary landscapes: (a) effect of spatial structure (m = 0.25, J = 2); (b) effect of mortality rate (H = 0.95, J = 2); (c) effect of mean dispersal distance (H = 0.95, m = 0.25). 95% CIs were plotted where large enough to display.

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