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Appendix S1. Detailed description of the four simulation scenarios used to assess the
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effect of heterogeneous habitat selection among animals.
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We use computer simulations to investigate the effect of departures from the
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assumption of homogeneous habitat selection among individuals. Deviations from the
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assumption were induced by imposing inter-individual variations in movement rules and
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by forcing movement decisions that violated the IIA assumption. Individual-based,
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spatially explicit modelling was conducted using the Spatially Explicit Landscape Event
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Simulator (Fall & Fall 2001). We simulated the movements of 200 virtual foragers, with
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each individual starting (time zero) at a random location within the landscape (1000 
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1000 cells), and followed for 50 consecutive moves. Landscapes were comprised of four
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types of randomly distributed habitat patches: Patch type H1 offered the most food,
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followed by H2. Neither H3 nor H4 offered any food. H1 was risky, unless located <15
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cells from H3, in which case H1 became safe. H2 was always safe.
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We tested four scenarios differing in the movement rules of individuals, with distinct
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statistical implications. Movements for scenarios 1 and 2 were both consistent with the
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IIA hypothesis; scenario 1 assumed a homogeneous movement rule, whereas scenario 2
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involved inter-individual variation in the rules. Scenarios 3 and 4 both led to violation of
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the IIA hypothesis at the individual level; a homogeneous movement rule was used for
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scenario 3 whereas inter-individual variation in movement rules characterized scenario 4.
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In scenario 1, the landscape was comprised of 15% H1, 15% H2, 0% H3, and 70%
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H4. Movement decisions depended on the current location. When the forager was in H4,
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its next move had a 1/6 probability of ending up in H1 and a 5/6 probability of finishing
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in H2. When the forager was in H1 or H2, the direction of its next move was drawn from
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a uniform distribution (0, 359°) and the distance was drawn from a normal distribution
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with mean 10 and variance 3 (i.e., N[10, 3]). With this simulation algorithm, the types of
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habitat visited at each time step form a Markov chain. A simple calculation (Appendix
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S2) shows that in the long run, each individual spends 15.7%, 43.1%, and 41.2% of its
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steps in H1, H2 and H4, respectively. Based on these probabilities, we derive an
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approximate lower bound of -1.01 for the true value of the regression coefficient for
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habitat H1 in a RSF with H2 as baseline patch type (Appendix S2).
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The same general movement rules were also applied to scenario 2, with the exception
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that the relative selection of H1 and H2 varied among individuals: When a forager was in
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H4 then its next move had a (1/6)-b chance of ending up in H1 and a (5/6)+b probability
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of finishing in H2, where b was fixed for a given individual throughout its path but varied
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among individuals according to N(0, 0.2). When the simulated probabilities for (1/6)-b or
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(5/6)+b were <0 or >1, they were truncated to 0 or 1, respectively. Nevertheless,
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individuals were much more likely to end up in H2 than H1.
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In scenarios 3 and 4, patches of refuge habitat H3 were added to the landscape which
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was then comprised of 15% H1, 15% H2, 0.04% H3, and 69.96% H4. According to
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scenario 3, when the forager was in H3 or H4, it had a 5/6 chance of moving to H1 if the
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nearest H1 patch was <15 cells of a H3 patch, and 1/6 of ending up in H2. In contrast,
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when the animal was in H3 or H4 but the nearest patch H1 was ≥15 cells from H3, then
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its next move had a 1/6 chance of finishing in H1 and a 5/6 chance in H2. When the
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animal was in H1 ≥15 cells from H3 or in H2, the direction of its next move was drawn
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from a uniform distribution (0-359o) and the distance from N(10, 3). When individuals
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occupied a patch H1 located <15 cells from H3, then the animal had a 4/5 chance of
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remaining in the current patch at the end of its next step, and 1/5 of moving according to
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a uniform direction (0-359o) with a distance N(25, 1). These general movement rules for
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scenario 3 were also used for scenario 4, with the exception that the latter scenario
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displayed inter-individual variations in the selection probabilities for H1 and H2. More
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specifically, when an animal was in H3 or H4, then the forager had a (5/6)+b chance of
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making a move ending in H1 if that patch was <15 cells of a refuge H3, and a (1/6)-b
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chance of finishing in H2, where b was fixed for a given individual but varied among
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individuals according to N(0, 0.2). To assess the effect of varying patch availability on
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inferences, scenario 3 was applied to 5 additional landscapes, where the proportions of
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H1 and H2 remained unchanged but those of H3 and H4 varied according to Landscape
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1: 0.01%, 69.99%, Landscape 2: 0.02%, 69.98%, Landscape 3: 0.03%, 69.97%,
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Landscape 4: 0.05%, 69.95%, and Landscape 5: 0.06%, 69.94%, respectively.
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In all scenarios, each observed location was matched to 10 locations randomly drawn
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within a 30-cell radius. Patch type (H1-H4) was identified at all observed and random
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locations. Fixed and mixed-effects conditional logistic regressions were used to build
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RSFs. Mixed-effects RSFs allowed the coefficient of H1 to vary among individuals
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according to a N(β1,σ2). In all models, H2 was used as the baseline patch type.
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Reference
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Fall, A. & Fall, J. (2001) A domain-specific language for models of landscape dynamics.
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Ecological Modelling, 141, 1-18.