**RStudio project**

Open the RStudio project that we created in the first session. I recommend to use this RStudio project for the entire course and within the RStudio project create separate R scripts for each session.

- Create a new empty R script by going to the tab “File”, select “New File” and then “R script”
- In the new R script, type
`# Session b3: SDM introduction`

and save the file in your folder “scripts” within your project folder, e.g. as “b3_SDM_intro.R”

This session will introduce you to simple species distribution models (SDMs). Species distribution models (SDMs) are a popular tool in quantitative ecology (Franklin 2010; Peterson et al. 2011; Guisan, Thuiller, and Zimmermann 2017) and constitute the most widely used modelling framework in global change impact assessments for projecting potential future range shifts of species (IPBES 2016). There are several reasons that make them so popular: they are comparably easy to use because many software packages (e.g. Thuiller et al. 2009; Phillips, Anderson, and Schapire 2006) and guidelines (e.g. Elith, Leathwick, and Hastie 2008; Elith et al. 2011; Merow, Smith, and Silander Jr 2013; Guisan, Thuiller, and Zimmermann 2017) are available, and they have comparably low data requirements.

SDMs relate biodiversity observations (e.g. presence-only, presence/absence, abundance, species richness) at specific sites to the prevailing environmental conditions at those sites. Different statistical and machine-learning algorithms are available for this. Based on the estimated biodiversity-environment relationship, we can make predictions in space and in time by projecting the model onto available environmental layers (Figure 1).

We distinguish five main modelling steps for SDMs: (i) conceptualisation, (ii) data preparation, (iii) model fitting, (iv) model assessment, and (v) prediction (Figure 2.). The last step (prediction) is not always part of SDM studies but depends on the model objective (Zurell et al. 2020). Generally, we distinguish three main objectives for SDMs: (a) inference and explanation, (b) mapping and interpolation, and (c) forecast and transfer. I recommend getting more familiar with critical assumptions and modelling decisions by studying the many excellent review articles (Guisan and Zimmermann 2000; Guisan and Thuiller 2005; Elith and Graham 2009) and textbooks on SDMs (Peterson et al. 2011; Franklin 2010; Guisan, Thuiller, and Zimmermann 2017).

I would also like to emphasise that model building is an iterative process and there is much to learn on the way. In consequence, you may want to revisit and improve certain modelling steps, for example improve the spatial sampling design. Because of that I like to regard model building as a cycle rather than a workflow with a pre-defined termination point (Figure 2.)(Zurell et al. 2020).

In this session, we will only work with Generalised Linear Models (GLMs) and concentrate on the first three model building steps (Figure 2.)

Why do we not simply use linear regression to fit our species-environment relationship? Well, strictly, ordinary least squares (OLS) linear regression is only valid if the response (or rather the error) is normally distributed and ranges (\(-\infty,\infty\)). OLS regression looks like this

\[E(Y|X)=\beta X+\epsilon\]

where \(E(Y|X)\) is the
*conditional mean*, meaning the expected value of the response
\(Y\) given the environmental
predictors \(X\) (Hosmer and Lemeshow 2013). \(X\) is the matrix of predictors (including
the intercept), \(\beta\) are the
coefficients for the predictors, and \(\epsilon\) is the (normally distributed!)
error term. \(\beta X\) is referred to
as the linear predictor.

When we want to predict species occurrence based on environment, then the conditional mean \(E(Y|X)\) is binary and bounded between 0 (absence) and 1 (presence). Thus, the assumptions of OLS regression are not met. GLMs are more flexible regression models that allow the response variable to follow other distributions. Similar to OLS regression, we also fit a linear predictor \(\beta X\) and then relate this linear predictor to the mean of the response variable using a link function. The link function is used to transform the response to normality. In case of a binary response, we typically use the logit link (or sometimes the probit link). The conditional mean is then given by:

\[E(Y|X) = \pi (X) = \frac{e^{\beta X+\epsilon}}{1+e^{\beta X+\epsilon}}\]

The logit transformation is defined as: \[g(X) = ln \left( \frac{\pi (X)}{1-\pi (X)} \right) = \beta X+\epsilon\]

The trick is that the logit, g(X), is now linear in its parameters, is continuous and may range (\(-\infty,\infty\)). GLMs with a logit link are also called logistic regression models.

In the conceptualisation phase, we formulate our main research objectives and decide on the model and study setup based on any previous knowledge on the species and study system. An important point here is whether we can use available data or have to gather own biodiversity (and environmental) data, which would require deciding on an appropriate sampling design. Then, we carefully check the main underlying assumptions of SDMs, for example whether the species is in pseudo-equilibrium with environment and whether the data could be biased in any way (cf. chapter 5 in Guisan, Thuiller, and Zimmermann 2017). The choice of adequate environmental predictors, of modelling algorithms and of desired model complexity should be guided by the research objective and by hypotheses regarding the species-environment relationship. We can divide environmental variables into three types of predictors: resource variables, direct variables and indirect variables (Austin 1980; Guisan and Zimmermann 2000).

We aim at assessing potential climate change effects on the Ring
Ouzel (*Turdus torquatus*) in UK (Figure 3), a typical upland
bird. First, we try to make ourselves familiar with Ring Ouzel ecology.
For example, the RSPB in UK (The Royal Society for the Protection of
Birds) as well as the IUCN (Internatinal Union for the Conservation of
Nature) provide some species details on their websites (RSPB,
IUCN).
Think about which factors could limit the distribution of Ring Ouzel in
UK.

In this step, the actual biodiversity and environmental data are gathered and processed. This concerns all data that are required for model fitting but also data that are used for making transfers. In the previous sessions, you have already explored some ways to retrieve species data and environmental data. To simplify matters a little bit, I provide the processed data for this example, which you can download from here (or from the data folder in moodle). The data frame contains ring ouzel presences and absences as well as bioclimatic data.

The ring ouzel occurrence records were obtained from the British breeding and wintering birds citizen science atlas (Gillings et al. 2019). The atlas contains breeding bird records in 20-year cycles (1968-1972, 1988-1991, 2008-2011 and) wintering bird records in 30-year cycles (1981/1982-1983-1984, 2007/2008-2010/2011) at a 10 km spatial resolution throughout Britain, Ireland, the Isle of Man and the Channel Islands. The entire atlas data are available through the British Trust of Ornithology (www.bto.org) and can be downloaded here. I already extracted the confirmed breeding occurrences of the ring ouzel for the most recent atlas period 2008-2011. As absence data I used those locations where no ring ouzel occurrence was reported. Afterwards I spatially thinned the data such that all presence and absence locations are at least 20 km apart. This is important to avoid problems with spatial autocorrelation. The 19 bioclimatic variables were extracted from the worldclim data base (see detail on the variables here).

Read in the data table:

```
library(terra)
bg <- terra::rast('data/Prac3_UK_mask.tif')
sp_dat <- read.table('data/Prac3_RingOuzel.txt',header=T)
```

Inspect the data, and plot the presence and absence locations:

`summary(sp_dat)`

```
## EASTING NORTHING Turdus_torquatus bio1
## Min. : 75000 Min. : 35000 Min. :0.00000 Min. : 5.274
## 1st Qu.:265000 1st Qu.: 235000 1st Qu.:0.00000 1st Qu.: 7.784
## Median :345000 Median : 405000 Median :0.00000 Median : 8.932
## Mean :358432 Mean : 466815 Mean :0.09901 Mean : 8.679
## 3rd Qu.:440000 3rd Qu.: 700000 3rd Qu.:0.00000 3rd Qu.: 9.660
## Max. :655000 Max. :1175000 Max. :1.00000 Max. :10.841
## bio2 bio3 bio4 bio5
## Min. :3.895 Min. :29.99 Min. :290.9 Min. :14.43
## 1st Qu.:6.404 1st Qu.:36.87 1st Qu.:402.9 1st Qu.:17.53
## Median :6.972 Median :37.67 Median :430.7 Median :19.01
## Mean :6.856 Mean :37.55 Mean :427.3 Mean :19.01
## 3rd Qu.:7.529 3rd Qu.:38.45 3rd Qu.:459.5 3rd Qu.:20.83
## Max. :8.453 Max. :41.14 Max. :491.7 Max. :22.91
## bio6 bio7 bio8 bio9
## Min. :-2.8670 Min. :12.35 Min. : 1.090 Min. : 5.019
## 1st Qu.: 0.1564 1st Qu.:17.11 1st Qu.: 4.531 1st Qu.: 6.796
## Median : 0.8018 Median :18.49 Median : 5.583 Median : 8.898
## Mean : 0.7861 Mean :18.23 Mean : 5.664 Mean : 9.150
## 3rd Qu.: 1.3992 3rd Qu.:19.83 3rd Qu.: 6.677 3rd Qu.:10.893
## Max. : 4.4728 Max. :21.40 Max. :13.413 Max. :14.624
## bio10 bio11 bio12 bio13
## Min. :10.75 Min. :0.2874 Min. : 552.9 Min. : 51.90
## 1st Qu.:12.99 1st Qu.:3.2112 1st Qu.: 717.6 1st Qu.: 73.81
## Median :14.33 Median :4.0166 Median : 946.5 Median :105.07
## Mean :14.15 Mean :3.8380 Mean :1076.8 Mean :121.29
## 3rd Qu.:15.48 3rd Qu.:4.5036 3rd Qu.:1340.5 3rd Qu.:156.52
## Max. :16.90 Max. :7.0155 Max. :2749.7 Max. :330.78
## bio14 bio15 bio16 bio17
## Min. : 33.41 Min. : 9.668 Min. :147.3 Min. :116.0
## 1st Qu.: 44.25 1st Qu.:15.414 1st Qu.:212.7 1st Qu.:146.0
## Median : 53.75 Median :22.271 Median :300.2 Median :174.1
## Mean : 59.95 Mean :21.785 Mean :349.2 Mean :193.9
## 3rd Qu.: 71.71 3rd Qu.:27.119 3rd Qu.:447.5 3rd Qu.:226.7
## Max. :127.20 Max. :36.253 Max. :966.7 Max. :407.8
## bio18 bio19
## Min. :140.6 Min. :128.7
## 1st Qu.:165.8 1st Qu.:188.7
## Median :203.7 Median :266.7
## Mean :223.5 Mean :310.7
## 3rd Qu.:266.7 3rd Qu.:393.9
## Max. :459.3 Max. :871.1
```

```
# Plot GB land mass
plot(bg,col='grey',axes=F,legend=F)
# Plot presences in red and absences in black
plot(extend(terra::rast(sp_dat[,1:3], crs=crs(bg), type='xyz'), bg), col=c('black','red'), legend=F,add=T)
```

Before we start into the complexities of the different model fitting
steps, let us look at a GLM in more detail. We fit our first GLM with
only one predictor assuming a linear relationship between response and
predictor. The `glm`

function is contained in the R
*stats* package. We need to specify a *formula* describing
how the response should be related to the predictors, and the
*data* specifying the data frame that contains the response and
predictor variables, and a *family* argument specifying the type
of response and the link function. In our case, we use the
*logit* link in the *binomial* family.

```
# We first fit a GLM for the bio1 variable assuming a linear relationship:
m1 <- glm(Turdus_torquatus ~ bio1, family="binomial", data= sp_dat)
# We can get a summary of the model:
summary(m1)
```

```
##
## Call:
## glm(formula = Turdus_torquatus ~ bio1, family = "binomial", data = sp_dat)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7195 -0.3760 -0.1635 -0.1003 2.6042
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 10.1088 1.8787 5.381 7.41e-08 ***
## bio1 -1.5579 0.2521 -6.178 6.47e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 195.68 on 302 degrees of freedom
## Residual deviance: 128.76 on 301 degrees of freedom
## AIC: 132.76
##
## Number of Fisher Scoring iterations: 6
```

Additional to the slope values, there are a few interesting metrics printed in the output called deviance and AIC. These metrics tell us something about how closely the model fits the observed data.

Basically, we want to know whether a predictor is “important”. We thus need to evaluate whether a model including this variable tells us more about the response than a model without that variable (Hosmer and Lemeshow 2013). Importantly, we do not look at the significance level of predictors. Such p-values merely tell us whether the slope coefficient is significantly different from zero. Rather, we assess whether the predicted values are closer to the observed values when the variable is included in the model versus when it is not included.

In logistic regression, we compare the observed to predicted values using the log-likelihood function:

\[L( \beta ) = ln[l( \beta)] = \sum_{i=1}^{n} \left( y_i \times ln[\pi (x_i)] + (1-y_i) \times ln[1- \pi (x_i)] \right)\]

\(L( \beta )\) is the *Likelihood
of the fitted model*. From this, we can calculate the deviance \(D\) defined as:

\[D = -2 \times L\]

In our model output, the deviance measure is called
`Residual deviance`

. This deviance is an important measure
used for comparing models. For example, it serves for the calculation of
*Information criteria* that are used for comparing models
containing different numbers of parameters. One prominent measure is the
*Akaike Information Criterion*, usually abbreviated as AIC. It is
defined as: \[AIC = -2 \times L + 2 \times
(p+1) = D + 2 \times (p+1)\]

where \(p\) is the number of regression coefficients in the model. AIC thus takes into account model complexity. In general, lower values of AIC are preferable.

We can also use the deviance to calculate the *Explained
deviance* \(D^2\), which is the
amount of variation explained by the model compared to the null
expectation: \[D^2 = 1 -
\frac{D(model)}{D(Null.model)}\] The model output also provides
the `Null deviance`

, so we can easily calculate the explained
deviance \(D^2\).

We can also fit quadratic or higher polynomial terms (check
`?poly`

) and interactions between predictors:

- the term `I()`

indicates that a variable should be
transformed before being used as predictor in the formula

- `poly(x,n)`

creates a polynomial of degree \(n\): \(x + x^2 +
... + x^n\)

- `x1:x2`

creates a two-way interaction term between
variables x1 and x2, the linear terms of x1 and x2 would have to be
specified separately

- `x1*x2`

creates a two-way interaction term between
variables x1 and x2 plus their linear terms

- `x1*x2*x3`

creates the linear terms of the three variables,
all possible two-way interactions between these variables and the
three-way interaction

Try out different formulas:

```
# Fit a quadratic relationship with bio1:
m1_q <- glm(Turdus_torquatus ~ bio1 + I(bio1^2), family="binomial", data= sp_dat)
summary(m1_q)
# Or use the poly() function:
summary( glm(Turdus_torquatus ~ poly(bio1,2) , family="binomial", data= sp_dat) )
# Fit two linear variables:
summary( glm(Turdus_torquatus ~ bio1 + bio8, family="binomial", data= sp_dat) )
# Fit three linear variables:
summary( glm(Turdus_torquatus ~ bio1 + bio8 + bio17, family="binomial", data= sp_dat) )
# Fit three linear variables with up to three-way interactions
summary( glm(Turdus_torquatus ~ bio1 * bio8 * bio17, family="binomial", data= sp_dat) )
# Fit three linear variables with up to two-way interactions
summary( glm(Turdus_torquatus ~ bio1 + bio8 + bio17 +
bio1:bio8 + bio1:bio17 + bio8:bio17,
family="binomial", data= sp_dat) )
```

**Test yourself**

Compare the different models above.

- Which model is the best in terms of AIC?
- What is the explained deviance of the different models?
- HINTS: you can extract the metrics using
`AIC(m1)`

,`deviance(m1)`

and`m1$null.deviance`

Model fitting is at the heart of any SDM application. Important aspects to consider during the model fitting step are:

- How to deal with multicollinearity in the environmental data?
- How many variables should be included in the model (without overfitting) and how should we select these?
- Which model settings should be used?
- When multiple model algorithms or candidate models are fitted, how to select the final model or average the models?
- Do we need to test or correct for non-independence in the data (spatial or temporal autocorrelation, nested data)?
- Do we want to threshold the predictions, and which threshold should be used?

More detailed descriptions on these aspects can be found in Franklin (2010) and in Guisan, Thuiller, and Zimmermann (2017).

GLMs (and many other statistical models) have problems to fit stable parameters if two or more predictor variables are highly correlated, resulting in so-called multicollinearity issues (Dormann et al. 2013). To avoid these problems here, we start by checking for multi-collinearity and by selecting an initical set of predictor variables. Then, we can fit our GLM including multiple predictors and with differently complex response shapes. This model can then be further simplified by removing “unimportant” predictors.

We first check for pairwise correlations among predictors. Generally, correlations below |r|<0.7 are considered unproblematic (or below |r|<0.5 as more conservative threshold).

```
library(corrplot)
# We first estimate a correlation matrix from the predictors.
# We use Spearman rank correlation coefficient, as we do not know
# whether all variables are normally distributed.
cor_mat <- cor(sp_dat[,-c(1:3)], method='spearman')
# We can visualise this correlation matrix. For better visibility,
# we plot the correlation coefficients as percentages.
corrplot.mixed(cor_mat, tl.pos='lt', tl.cex=0.6, number.cex=0.5, addCoefasPercent=T)
```

Several predictor variables are highly correlated. One way to deal with this issue is to remove the “less important” variable from the highly correlated pairs. For this, we need to assess variable importance.

Now that we have selected a set of weakly correlated variables, we can fit the full model and then simplify it. The latter is typically called model selection. Here, I only use the two most important variables and include linear and quadratic terms in the full model.

```
# Fit the full model:
m_full <- glm( Turdus_torquatus ~ bio1 + I(bio1^2) + bio8 + I(bio8^2),
family='binomial', data=sp_dat)
# Inspect the model:
summary(m_full)
```

```
##
## Call:
## glm(formula = Turdus_torquatus ~ bio1 + I(bio1^2) + bio8 + I(bio8^2),
## family = "binomial", data = sp_dat)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.44526 -0.33800 -0.11735 -0.03192 3.01275
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.4831 15.6143 0.223 0.823
## bio1 -0.5171 4.6625 -0.111 0.912
## I(bio1^2) -0.0746 0.2972 -0.251 0.802
## bio8 1.7453 1.5663 1.114 0.265
## I(bio8^2) -0.2169 0.1676 -1.294 0.196
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 195.68 on 302 degrees of freedom
## Residual deviance: 122.24 on 298 degrees of freedom
## AIC: 132.24
##
## Number of Fisher Scoring iterations: 8
```

How much deviance is explained by our model? The function
`expl_deviance()`

is also contained in the
`mecofun`

package. As explained earlier, we can calculate
explained deviance by quantifying how closely the model predictions fit
the data in relation to the null model predictions.

```
# Explained deviance:
expl_deviance(obs = sp_dat$Turdus_torquatus,
pred = m_full$fitted)
```

`## [1] 0.3753193`

We can simplify the model further by using stepwise variable
selection. The function `step()`

uses the AIC to compare
different subsets of the model. Specifically, it will iteratively drop
variables and add variables until the AIC cannot be improved further
(meaning it will not decrease further).

`m_step <- step(m_full) `

```
# Inspect the model:
summary(m_step)
```

```
##
## Call:
## glm(formula = Turdus_torquatus ~ I(bio1^2) + bio8 + I(bio8^2),
## family = "binomial", data = sp_dat)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.42982 -0.33815 -0.11284 -0.02967 3.00631
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.76080 1.66925 1.055 0.29150
## I(bio1^2) -0.10745 0.02917 -3.684 0.00023 ***
## bio8 1.61773 1.04632 1.546 0.12208
## I(bio8^2) -0.20389 0.11734 -1.738 0.08229 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 195.68 on 302 degrees of freedom
## Residual deviance: 122.25 on 299 degrees of freedom
## AIC: 130.25
##
## Number of Fisher Scoring iterations: 8
```

```
# Explained deviance:
expl_deviance(obs = sp_dat$Turdus_torquatus,
pred = m_step$fitted)
```

`## [1] 0.3752566`

The explained deviance of the final model is a tiny bit lower than for the full model, but the final model is more parsimonious.

**Exercise:**

- Load the second species data set for the Yellowhammer.
- Plot the presences and absences of the Yellowhammer.
- Check for collinearity and reduce the data set to only weakly correlated variables.
- Define a full model with all weakly correlated variables including their linear and quadratic terms.
- Run the full model.
- Simplify the model using stepwise variable selection
`step()`

- Compare the full model and the reduced model in terms of AIC and explained deviance.

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