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Appendix S2. Calculation of the long run probabilities of being in a given patch type and
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theoretical value of the RSF coefficient for patch type H1 under simulation scenario 1.
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We simulated 200 virtual animals travelling in a completely random landscape (1000 
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1000 cells) characterized by 15% of patch type H1, 15% of H2, and 70% of H4. Under
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scenario 1, virtual animals were assigned random and independent starting locations.
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Their 50 subsequent moves were simulated independently from one another and
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according to the following rules: When the forager was in H4, its next move had a 1/6
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probability of ending up in H1 and a 5/6 probability of finishing in H2. When the forager
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was in H1 or H2, the direction of its next move was drawn from a uniform distribution (0,
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359o) and the distance was drawn from a normal distribution with mean 10 and variance
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3 (i.e., N[10, 3]).
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With Znt denoting the patch type at the location of animal n at time step t, the
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processes {Znt, t = 0, 1, 2, …} are Markov chains (Grimmett & Stirzaker 1992) with the
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transition matrix
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H1
H 1  0.15

P
H 2  0.15
H 4  1 / 6
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where the value in row i and column j of P gives the probability of being in patch type j
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at time t, given that the animal occupied patch type i at time t-1, i, j = H1, H2, H4.
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Because the matrix P yields a chain that is irreducible and aperiodic, if we observe {Znt, t
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= 0, 1, 2, …} for a large number of time steps, then the proportion of the time that steps
H2 H4
0.15 0.7 
,
0.15 0.7 
5 / 6 0 
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are spent in each patch type converges towards the long run (steady state) probabilities π
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= (πH1, πH2, πH4) that solve πP = π. For the matrix P given above, we obtain π = (0.157,
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0.431, 0.412).
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Now the theoretical value of the RSF coefficient for H1 under scenario 1 corresponds
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to the log of the following odds ratio: odds that a location is visited when it is of habitat
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type H1 over the odds that a location is visited when it is of habitat type H2 (baseline
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habitat type). If Y = 1 denotes the event that a given location chosen at random in the
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landscape is visited and Y = 0 is the event that it is not visited, while X = 1 denotes that a
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given location chosen at random is associated with a patch of type H1 and X = 0 with a
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patch of type H2, then the theoretical value of the regression coefficient is given by
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 .
 H 1  ln 
 Pr[Y  1 | X  0] / Pr[Y  0 | X  0] 
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Assuming that the probabilities π1 (long run probability that the animal is in H1) and π2
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(long run probability that the animal is in H2) derived from P above are good
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approximations of Pr[X=1|Y=1] (probability that a location is in H1 given that it is
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visited) and Pr[X=0|Y=1] (probability that a location is in H2 given that it is visited),
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respectively, then we can use Bayes’ rule to compute all four probabilities involved in
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βH1:
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Pr[Y  1 | X  1] 
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where Pr[Y =1]=10,000/1,000,000 (number of visited locations / total number of
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locations), Pr[Y=0]=1-Pr[Y=1] and Pr[X=1|Y=0]=0.15 (locations chosen at random in the
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landscape are of type H1 with probability 15%). Note that the value of the probability
 Pr[Y  1 | X  1] / Pr[Y  0 | X  1] 
eqn S1
Pr[ X  1 | Y  1] Pr[Y  1]
,
Pr[ X  1 | Y  1] Pr[Y  1]  Pr[ X  1 | Y  0]P[Y  0]
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Pr[Y =1] is completely irrelevant because it cancels out in the odds ratio of interest.
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Repeating a similar argument for all four probabilities implied in equation S1 yields
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 H 1  ln 
 0.0947 / 0.905 
  1.01 .
 0.223 / 0.777 
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The long run probabilities π = (0.157, 0.431, 0.412) provide a slight underestimation
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of the ratio of the odds of being in H1 over the odds of being in H2, because the virtual
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animals were distributed randomly across the landscape at time 0, i.e., with probabilities
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of starting in H1, H2, and H4 of 0.15, 0.15 and 0.70, respectively. As a result, it takes a
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few steps before the simulations sample from the steady state distribution (0.157, 0.431,
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0.412) and therefore the value of -1.01 is a slight underestimation of the true value of the
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regression parameter. The calculations in this appendix would have yielded the true value
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of the regression parameter if the virtual foragers had begun (time 0) their path in the
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various patch types according to the steady state distribution π = (0.157, 0.431, 0.412).
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Reference
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Grimmett, G.R. & Stirzaker, D.R. (1992) Probability and random processes, 2nd edn.
Oxford University Press, New York.