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Ecography E7103 Godsoe, W. and Harmon. L. J. 2010. How do species interactions affect species distribution models? – Ecography 000: 000–000
Supplementary material Appendix 3 Simulated ecological dynamics across CR parameter values
We ran simulations in which we modeled the dynamics of individual populations using numerical methods. To do this we simulated 10 replicated sets of 200 environments with supply points sampled from uncorrelated uniform distributions over the interval (0,1). Within each individual environment we used the ode function in the deSolve package in R to compute the abundance of each species after 100000 years from equations 1 and 2 in the main text. We focus on a range of parameters producing stable coexistence starting with the parameters c1=0.001, a11=0.2, a12=0.5, f11=0.25, f12=0.07, d1=0.035, a21=0.4, a22=0.2, f21=0.025, f22=0.07 d2=0.0095. In each set of 10 replicated simulations we then varied one parameter using the value listed in table S1. We chose parameter values that explore a broad range of the outcomes possible in our CR model. So for example low values of a11 ZNGI2 is almost invariably below ZNGI1 and as a result species 2 nearly always out‐competes species 1. At high values for a11 species one almost always out‐competes species 2.
To evaluate the importance of biotic interactions we contrasted environments in which both species were present and environments in which only species 1 was present. In 100 environments we simulated the abundance of species 1 in the absence of biotic interactions by setting N2=0. In the other 100 we included biotic interactions by allowing N2>0. The abundance of each species was governed by equations 1 and 2 in the main text, using a step size of 32 years. The initial abundance of species 1 was 0.01, as was the initial abundance of species 2 when it was present. Species were deemed to be present at the end of this simulation if their abundance was greater than 0.0001.
We then determined whether biotic interactions altered the results of our model using two statistical hypothesis tests. First, we tested whether the AUC score changed significantly between environments with and without biotic interactions. To do this we fit two separate GLM models, one for the 100 observations that included biotic interactions and one for the 100 observations that excluded biotic interactions. We computed the AUC scores of each GLM and report the difference. We then used a t‐test to determine whether the scores were significantly different.
This is a test of whether the presence of biotic alters our ability to make predictions. A significant positive result of this test indicates AUC scores are higher in the presence of biotic interactions and a significant negative score indicates that AUC scores are lower in the presence of biotic interactions.
Second, we tested whether including biotic interactions improved our ability to model the presence of species 1. To do this we used a generalized linear model to predict the presence of species 1 across all 200 environments using two predictor variables 1) S1 and 2) whether species 2 was initially present. We then used a likelihood ratio test with one degree of freedom to determine whether considering initial presence of species 2 improved our model. As this test was repeated 10 times we report the number of tests out of 10 that were significant.
The likelihood ratio test is significant less frequently when species 2 is rarely present, for example at high values of a11, a12, f11 and at low values of d2. Extreme values of some parameters also produce conditions under which AUC score are lower in the absence of biotic interactions, for example at lowest value of a12, f12 and at high values of a22. This corresponds to conditions under which transition between suitable and unsuitable environments is steeper in the absence of biotic interactions. So for example at the fifth parameter value tested for a22 (0.5) the slope of ZNGI1 is steep (blue solid line Fig. A1, row 8, column 5), thus in the absence of competition the probability of presence for species 1 changes dramatically with changes in S1. In the presence of competition in this plot species 1 is only present at very high values for S1 and low values for S2. No value of S1 is sufficiently high to ensure a high probability of presence for this species and so it is difficult to use S1 to make strong inferences on whether species 1 will be present.
Table A1. Results of simulations including the parameter varied in each simulation, the value of the parameter used in each set of 10 replicates and two measures of the effect of biotic interactions. The first measure is the difference between AUC scores of 100 environments with biotic interactions (species 1 and 2 initially present) and 100 environments without biotic interactions (species 2 absent). A positive value indicates that AUC scores were higher in the presence of biotic interactions and a negative score indicates AUC scores were lower in the presence of biotic interactions. An * next to a score indicates p<0.05 while ** indicates that p<0.001. Note that in some parameter combinations (such as a11=0.15) species 1 was absent from some simulations. These parameter values frequently indicate that the presence of species 2 was influential enough to completely exclude species 1. Never the less, when species 1 is completely absent AUC scores are undefined and we do not report the difference in AUC scores. In each of the 10 replicated sets of environments that we simulated for each parameter combination we also tested whether considering the presence of biotic interactions improved model fit using a Likelihood Ratio test. We report the number of significant tests (out of 10).
Parameter Varied Parameter Value
Difference between AUC scores
Significant Likelihood Ratio Tests
c1 0.0005 0.08** 10/10 0.001 0.084** 10/10 0.002 0.087** 10/10 0.005 0.077** 10/10 0.01 0.078** 10/10 a11 0.15 10/10 0.2 0.074** 10/10 0.3 0.04** 2/10 0.4 0.015* 3/10 0.5 0.017** 2/10 a12 0.4 0.013 10/10 0.5 0.085** 10/10 0.6 0.11** 9/10 0.7 0.079** 3/10 0.8 -0.033* 1/10 f11 0.15 10/10 0.25 0.091** 10/10 0.35 0.046** 7/10 0.45 0.03** 2/10 0.55 0.017* 3/10 f12 0.01 -0.096** 10/10 0.03 -0.06** 10/10 0.05 -0.012 10/10 0.07 0.083** 10/10 0.09 0.123** 6/10 d1 0.019 0.01 0/10
0.027 0.061** 1/10 0.035 0.075** 10/10 0.043 10/10 0.051 10/10 a21 0.35 0.072** 10/10 0.4 0.073** 10/10 0.5 -0.016 10/10 0.6 10/10 0.7 10/10 a22 0.18 0.06** 10/10 0.2 0.074** 10/10 0.3 -0.001 10/10 0.4 -0.022 10/10 0.5 -0.039* 10/10 f21 0.005 0.047** 2/10 0.015 0.08** 5/10 0.025 0.085** 10/10 0.035 10/10 0.045 10/10 f22 0.05 -0.007 0/10 0.06 0.076** 6/10 0.07 0.082** 10/10 0.08 0.051** 10/10 0.09 0.043* 10/10 d2 0.0055 10/10 0.0075 -0.063* 10/10 0.0095 0.087** 10/10 0.0115 0.052** 1/10 0.0135 0.004 1/10
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Figure A1. Sample datasets for the parameter values used in our simulations. Each row is the result of simulations for one parameter varied in a set of simulations, ploting S2 versus S1. Each column represents a distinct parameter value corresponding the values denoted in the parameter value column in Table S1. Blue lines represent ZNGI1 (solid) and L1 (dashed) while red lines represent ZNGI2 (solid) and (L2) dashed. Dots represents a sample of 100 environments in which both species are present, with purple denoting joint absences, yellow denoting joint presences, black denoting presences of species 1 only and grey representing presence of species 2 only.
The effect of varying the distribution of supply points
Figure A2. An illustration of the probability of presence given a distribution of resource supply points that is uniform over (0,1). This plot is provided for reference Subsequent plots use similar values for ZNGI1, ZNGI2, L1 and L2. In Panel A species 1 is present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel high at 0.98.
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Figure A3. A graphical interpretation for the probability a species will be present given the supply point for resource 1, assuming that resource 1 varies from 0 to 2 and assuming that our focal species interacts with a competitor. In Panel A species 1 is present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel A. When S1 varies from 0 to 2 the AUC score remains high and essentially unchanged at 0.99 Figure.
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Figure A4. A graphical interpretation for the probability a species will be present given the supply point for resource 1, assuming that resource 2 varies from 0 to 2 and assuming that our focal species interacts with a competitor. In Panel A species 1 is present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel A. When S2 varies from 0 to 2 the AUC score is slightly lower 0.97. Note that in this example the probability of presence estimated from the GLM (lower panel blue) differs from the probability of presence derived in our original model (lower panel black).
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Figure A5. A graphical interpretation for the probability a species will be present given the supply point for resource 1, assuming that both resources are exponentially distributed with a rate parameter of 2. In Panel A species 1 is present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel A. The AUC score for the model displayed is 0.982.
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Figure A6. A graphical interpretation for the probability that species 1 is present given supply points for resource 1, assuming resource supplies cannot be below the line S2=0.6‐S2 . In Panel A species 1 is present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel A. The AUC score for the model displayed is 0.97504.
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Figure A7. A graphical interpretation for the probability that species 1 is present given the supply point for resource 1, assuming that the supply points of the two resources are uniformly distributed with a correlation coefficient of 0.7. In Panel A species 1 will be present in environments where it coexists with its competitor (purple), or environments where it is competitively dominant (blue). It is absent from environments where it loses to its competitor (red) or environments with insufficient resources for either species (white). Dots represent a sample of two hundred environments including absences (black) and presences (white). Panel B shows the marginal probability of presence conditioned on S1 (black line) along with the probability of presence estimated from a GLM fit using the presence/absence observations portrayed in panel A. The AUC score for the model displayed is 0.995. Correlations between resources are generated using an R script available at http://comisef.wikidot.com/tutorial:correlateduniformvariates
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