Sometimes we want to include the presence of another species in our occupancy model. For example, leopard occupancy might be lower at sites where tigers are present because of competitive interactions, or it might be lower at sites where a preferred prey species such as muntjac is absent.
These we can model as one-way interactions: leopards avoid tigers but tigers don’t care about leopards; leopards need muntjac but muntjac surely don’t need leopards. In these systems, the leopards are the subordinate species and tigers or muntjac are dominant. (It may seem odd to describe prey as dominant and predators as subordinate, but that’s often the ecological reality.)
If we had reliable information on the presence or absence of the dominant species, we could include that in our leopard model as a normal covariate. But usually we only have detection/nondetection data, and need to allow for sites where the dominant species is present but not detected. We thus need to model occupancy for both species simultaneously. We’ll see how to do it using JAGS on this page.
The example data set
Richmond et al (2010)1Richmond, O.M.W., Hines, J.E., & Beissinger, S.R. (2010) Two-species occupancy models: a new parameterization applied to co-occurrence of secretive rails. Ecological Applications, 20, 2036-2046 describe a study of rails at freshwater marshes in California, USA. They hypothesized that occupancy for the sparrow-sized Black Rail would be lower at sites where the larger Virginia Rail was present.
The data collected by Richmond et al are not available, but the wiqid
package has a set of simulated data based on their results. You can load and look at the data with the following code:
library(wiqid)
?railSims
data(railSims)
head(railSims)
# A1 A2 A3 B1 B2 B3 logArea reeds
# 1 0 0 0 1 1 1 -0.585 TRUE
# 2 1 1 1 0 1 0 -0.231 FALSE
# 3 1 0 0 0 1 1 1.686 FALSE
# 4 0 0 0 1 1 0 0.092 TRUE
# 5 0 0 0 1 1 1 0.155 FALSE
# 6 1 1 1 0 0 0 1.854 FALSE
# Separate the two detection histories
DHA <- as.matrix(railSims[, 1:3])
DHB <- as.matrix(railSims[, 4:6])
The first 3 columns in the data frame are detection/nondetection data for the dominant species (Virginia rail) for 3 surveys at each site, and the next 3 columns are the detection/nondetection data for the Black rail for the same 3 surveys.
logArea
and reeds
are site covariates which we will discuss later.
A model with no interaction
Without interaction, the model is equivalent to separate models for each species. The JAGS code below is the same as that used for salamanders, except that each line is repeated for each species:
# File name "2sps_noInt.jags"
model{
# Likelihood
for(i in 1:nSites) {
# biological model
zA[i] ~ dbern(psiA) # 1 if sps A present
zB[i] ~ dbern(psiB) # 1 if sps B present
# observation model
yA[i] ~ dbin(pA * zA[i], n)
yB[i] ~ dbin(pB * zB[i], n)
}
# Priors
pA ~ dbeta(1, 1) # Uninformative priors
pB ~ dbeta(1, 1)
psiA ~ dbeta(1, 1)
psiB ~ dbeta(1, 1)
}
Running this model is straightforward and produces the same output that we would get if we ran two single-species models. We will go on now to the more interesting model with interaction.
The model with occupancy interaction
Recall that we are interested in one-way interaction: occupancy of species B is affected by the presence of species A, but B has no effect on A. Thus, the code for species A is exactly as before.
For species B, we have 2 values for probability of occupancy, psiB[1]
for sites where species A is absent, psiB[2]
for sites where it’s present. Which to use at a particular site, i
, depends on the value of zA[i]
, or rather zA[i] + 1
.
# File name "2sps_Int.jags"
model{
# Likelihood
for(i in 1:nSites) {
# biological model
zA[i] ~ dbern(psiA)
zB[i] ~ dbern(psiB[zA[i] + 1])
# observation model
yA[i] ~ dbin(pA * zA[i], n)
yB[i] ~ dbin(pB * zB[i], n)
}
# Priors
pA ~ dbeta(1, 1) # Uninformative priors
pB ~ dbeta(1, 1)
psiA ~ dbeta(1, 1)
psiB[1] ~ dbeta(1, 1)
psiB[2] ~ dbeta(1, 1)
}
Now we prepare the data; as before we can aggregate the detections across the 3 survey occasions. Then we can run the model fit:
# Aggregate detection data
yA <- rowSums(DHA)
yB <- rowSums(DHB)
nSites <- nrow(railSims)
zA <- ifelse(yA > 0, 1, NA)
zB <- ifelse(yB > 0, 1, NA)
jagsData <- list(yA = yA, yB = yB, n = 3, nSites = nSites,
zA = zA, zB = zB)
str(jagsData)
wanted <- c("pA", "pB", "psiA", "psiB")
library(jagsUI)
out <- jags(jagsData, NULL, wanted, model="2sps_Int.jags",
n.chains=3, n.adapt=1000, n.iter=10000, DIC=FALSE,
parallel=TRUE)
mc2sps1 <- mcmcOutput(out)
summary(mc2sps1)
# mean sd median l95 u95 Rhat MCEpc
# pA 0.796 0.028 0.797 0.742 0.852 1.000 0.797
# pB 0.717 0.029 0.717 0.657 0.770 1.000 0.846
# psiA 0.467 0.040 0.467 0.389 0.543 1.000 0.564
# psiB[1] 0.818 0.045 0.821 0.729 0.902 0.999 0.690
# psiB[2] 0.391 0.057 0.390 0.279 0.502 1.000 0.605
diagPlot(mc2sps1, max=5)
postPlot(mc2sps1)
postPlot(mc2sps1, "psiB", xlim=c(0.25, 0.95))

The diagnostic plots and posterior plots (not shown) look fine. Plotting the posterior distributions of the two psiB
parameters with the same x axis clearly shows the difference in occupancy, with black rail occupancy higher where the Virginia rail is absent (psiB[1]
) than where it is present (psiB[2]
).
Interaction and detection
One species may behave differently when the other is present, and this may affect the probability of detection. Rails are secretive but have loud calls, so are mostly detected from calls. A change in the frequency of calling will affect detection probability.
We can model this by having 2 values for pA
and pB
, the first used for sites where the other species is absent and the second when it’s present. The observation model then becomes:
yA[i] ~ dbin(pA[zB[i] + 1] * zA[i], n)
yB[i] ~ dbin(pB[zA[i] + 1] * zB[i], n)
A further possibility is that such effects may be occasion-specific: in this case, that the small rails keep quiet (and are not detected) when the big rails are making a noise (and are detected). We then have 3 values for pB
depending on whether:
- the big rails are absent,
zA = 0
- the big rails are present but not calling (= not detected),
zA = 1, DHA = 0
- the big rails are present and calling (= detected),
zA = 1, DHA = 1
We can’t use the aggregated data for this, as we have to deal with the survey occasions one by one. Here is the JAGS code:
# File name "2sps_detect.jags"
model{
# Likelihood
for(i in 1:nSites) {
# biological model
zA[i] ~ dbern(psiA)
zB[i] ~ dbern(psiB[zA[i] + 1])
# observation model
for(j in 1:n) {
DHA[i, j] ~ dbern(pA[zB[i] + 1] * zA[i])
DHB[i, j] ~ dbern(pB[zA[i] + DHA[i, j] + 1] * zB[i])
}
}
# Priors
pA[1] ~ dbeta(1, 1) # spsB absent
pA[2] ~ dbeta(1, 1) # spsB present
pB[1] ~ dbeta(1, 1) # spsA absent
pB[2] ~ dbeta(1, 1) # spsA present, not detected
pB[3] ~ dbeta(1, 1) # spsA present, detected
psiA ~ dbeta(1, 1)
psiB[1] ~ dbeta(1, 1)
psiB[2] ~ dbeta(1, 1)
}
And here’s the code to prepare everything and run the model:
jagsData <- list(DHA = DHA, DHB = DHB, n = 3, nSites = nSites,
zA = zA, zB = zB)
str(jagsData)
wanted <- c("pA", "pB", "psiA", "psiB")
out <- jags(jagsData, NULL, wanted, model="2sps_detect.jags",
n.chains=3, n.adapt=1000, n.iter=10000, DIC=FALSE,
parallel=TRUE)
mc2sps2 <- mcmcOutput(out)
summary(mc2sps2)
# mean sd median l95 u95 Rhat MCEpc
# pA[1] 0.846 0.039 0.848 0.766 0.919 1.000 1.015
# pA[2] 0.729 0.053 0.732 0.623 0.829 1.000 1.129
# pB[1] 0.809 0.029 0.810 0.750 0.864 1.001 0.834
# pB[2] 0.679 0.104 0.685 0.482 0.881 1.000 1.156
# pB[3] 0.350 0.069 0.349 0.218 0.486 0.999 1.271
# psiA 0.469 0.040 0.469 0.392 0.547 1.001 0.579
# psiB[1] 0.802 0.044 0.804 0.716 0.885 1.000 0.531
# psiB[2] 0.481 0.081 0.476 0.331 0.646 1.000 1.403
diagPlot(mc2sps2)
plot(mc2sps2)
plot(mc2sps2, "pA", xlim=c(0.6, 0.95))
plot(mc2sps2, "pB", xlim=c(0.2, 0.95))


The model runs quickly and the diagnostic plots and posterior plots (not shown) look fine.
As we suspected, the little rails are more difficult to detect when the big rails are around, and especially when they are actively calling.
More surprising is that the big rails behaviour changes when the small ones are present, making them less detectable. But the difference is small and the posterior distributions overlap, so let’s check on the difference:
diff <- apply(mc2sps2$pA, 1, diff)
postPlot(diff, compVal=0)

So indeed, we can be 95% certain that detection does decline, based on our data and uniform priors.
We’ll include these observation process interactions in the models that follow.
Model with covariates
The area of the habitat patch may affect occupancy (especially for the big rail which has a larger home range), but the difference in occupancy depends on the ratio of the areas of the sites, rather than the difference in area. Hence we work with the logarithm of the area; logArea
is the covariate standardized to mean 0 and SD 1.
We incorporate area into the model with a logistic link: logit(psiA[i]) <- a0 + aArea * area[i]
Richmond et al (2010) also hypothesized that the presence of reed-beds in the wetland would give the little rails an advantage and allow them to coexist with the big rails.
Since the reeds
covariate has only 2 values, TRUE/FALSE
, we could implement this by having 3 values for psiB
: species A absent; A present with without reeds; A present with reeds. Picking out the right value to use for each is tricky, however.
reeds
is a 0/1 vector, so reeds+1
is a 1/2 vector. Multiply by zA
and we have a 0/1/2 vector, with 0 when species A is absent. Add 1 to get a 1/2/3 vector we can use as the index into a 3-valued psiB
vector.
Here’s the complete JAGS model:
# File name "2sps_covars.jags"
model{
# Likelihood
for(i in 1:nSites) {
# Ecological model
logit(psiA[i]) <- a0 + aArea * area[i]
zA[i] ~ dbern(psiA[i])
zB[i] ~ dbern(psiB[(reeds[i]+1)*zA[i] + 1])
# Observation model
for(j in 1:n) {
DHA[i, j] ~ dbern(pA[zB[i] + 1] * zA[i])
DHB[i, j] ~ dbern(pB[zA[i] + DHA[i, j] + 1] * zB[i])
}
}
# Priors
pA[1] ~ dbeta(1, 1) # as in previous model
pA[2] ~ dbeta(1, 1)
pB[1] ~ dbeta(1, 1)
pB[2] ~ dbeta(1, 1)
pB[3] ~ dbeta(1, 1)
a0 ~ dunif(-5, 5)
aArea ~ dunif(-5, 5)
psiB[1] ~ dbeta(1, 1) # when sps A absent
psiB[2] ~ dbeta(1, 1) # when sps A present, no reeds
psiB[3] ~ dbeta(1, 1) # when sps A present, reeds
}
Now we can prepare the data and run the model. In the version of wiqid
on CRAN, railSims$logArea
is a 1-column matrix instead of a vector; we can fix that with c()
. We also change railSims$reeds
from TRUE/FALSE to 1/0.
jagsData <- list(DHA = DHA, DHB = DHB, n = 3, nSites = nSites,
zA = zA, zB = zB, area=c(railSims$logArea),
reeds=as.numeric(railSims$reeds))
str(jagsData)
wanted <- c("pA", "pB", "a0", "aArea", "psiB")
out <- jags(jagsData, NULL, wanted, model="2sps_covars.jags",
n.chains=3, n.adapt=1000, n.iter=10000, DIC=FALSE,
parallel=TRUE)
mc2sps3 <- mcmcOutput(out)
summary(mc2sps3)
# mean sd median l95 u95 Rhat MCEpc
# pA[1] 0.834 0.041 0.837 0.752 0.912 0.998 1.064
# pA[2] 0.738 0.053 0.741 0.630 0.835 0.998 1.184
# pB[1] 0.809 0.029 0.810 0.751 0.862 0.999 0.813
# pB[2] 0.691 0.098 0.696 0.496 0.873 1.001 1.015
# pB[3] 0.345 0.070 0.343 0.211 0.484 0.999 1.409
# a0 -0.150 0.195 -0.150 -0.522 0.243 1.000 0.708
# aArea 1.477 0.266 1.467 0.954 1.998 1.000 0.708
# psiB[1] 0.803 0.044 0.806 0.718 0.888 1.001 0.574
# psiB[2] 0.313 0.086 0.306 0.155 0.487 1.000 1.068
# psiB[3] 0.701 0.116 0.698 0.492 0.950 1.001 1.442
diagPlot(mc2sps3)
plot(mc2sps3)
plot(mc2sps3, "psiB", xlim=c(0.1, 1))
plot(mc2sps3, "aArea")


As before, the runs quickly and converges nicely; the diagnostic and posterior plots (not shown) reveal no problems.
Reed beds clearly enable the smaller Black rail to survive in wetlands where the Virginia rail occurs, giving occupancy estimates almost as high as sites without the big rail.
Wetland area is an important determinant of occupancy for the Virginia rail: a coefficient value of 1.5 is high for a standardised covariate in a logistic regression. We’ll do a plot of probability of occupancy against log(area):
range(railSims$logArea) # +/- 2.5
xx <- seq(-2.5, 2.5, length=101)
toPlot <- matrix(NA, 101, 3)
colnames(toPlot) <- c("est", "lo", "up")
for(i in 1:101){
psi <- plogis(mc2sps3$a0 + mc2sps3$aArea * xx[i])
toPlot[i, ] <- c(mean(psi), hdi(psi))
}
plot(xx, toPlot[, 1], type='n', ylim=range(toPlot),
xlab="standardised log area", ylab="Occupancy")
polygon(c(xx, rev(xx)), c(toPlot[,2], rev(toPlot[,3])),
col='skyblue', border=NA)
lines(xx, toPlot[,1], lwd=2)

You can download a ZIP file with all the code here.
Febri Widodo emailed me with a question about calculation of the Species Interaction Factor (SIF) (MacKenzie et al, 2004) from the output of
wiqid::occ2sps
. That’s a mess, but it’s very easy with Bayesian MCMC output. Here’s the calculation based on themc2sps1
output:SIF <- mc2sps1$psiB[,2] / (mc2sps1$psiA*mc2sps1$psiB[,2] +
(1-mc2sps1$psiA)*mc2sps1$psiB[,1])
postPlot(SIF)
Here’s the plot
Mike,
Thanks again for the blog posts, they’ve been extremely helpful.
For the species-interaction model with covariates on psiA (area) and psiB (reed), how might the structure of the model differ if reed was a continuous covariate?
Joshua,
Good question! Suppose we have a continuous variable
reediness
which goes from 0 to 100. We simulate some random values and standardize withset.seed(27)
reediness <- runif(nSites, 0, 100)
reedinessS <- standardize(reediness)
In the JAGS code, in the "sites" loop, we need
logit(psiBA[i]) <- b0 + bRd * reediness[i] # when sps A present
zB[i] ~ dbern(ifelse(zA[i]==1, psiBA[i], psiBa))
[Edit: the first argument to
ifelse
should be a condition, so corrected it tozA[i]==1
.]The first line there calculates
psiBA[i]
for each site, depending on the value ofreediness[i]
, and the second line usesifelse
to selectpsiBA[i]
orpsiBa
depending on the value ofzA[i]
.psiBa
is the same for all sites.Then we need priors for our new parameters:
b0 ~ dunif(-5, 5)
bRd ~ dunif(-5, 5)
psiBa ~ dbeta(1, 1) # when sps A absent
In the R code, we need to change
jagsData
to includereediness=reedinessS
andwanted
to includeb0
,bRd
andpsiBa
.The rest of the code should then work ok. Of course,
reediness
is fiction, so we'd expectbRd
to be near zero, but with only 160 sites, spurious correlations can easily arise.Mike,
Thank you!!!!!
Mike,
One more quick question. I also wondered how the 2-species model structure might change if a categorical factor was included in either the “no-interaction” or “interaction” models. In this case, “Time” is the categorical factor.
For the no-interaction model: I assumed it would look like the following:
And for the Interaction Model, I assumed it would look something like the following (I’m much less confident in this
structure though!):
Any assistance on this would be much appreciated, Thanks!
I edited your post to include some formatting which you can’t include and corrected typos (italicized).
I need to run things to see if they work! Let’s assume the first 40 sites for the Rails data were surveyed in (say) January, the next 40 in March, then June and September, and we think occupancy changes through the year. So our
Time
covariate will be:Time <- rep(1:4, each=40)
Your first model works fine (after corrections) and gives 4 values for each of
psiA
andpsiB
.The second model won’t work: you have
psiB[1]
andpsiB[2]
inside thenCat
loop, so will get an error saying you are redefining a node. Take those outside thenCat
loop will prevent the error, butpsiB
will not change withTime
.psiB
needs to be a matrix with different values for eachTime
and for eachzA
. Change the ecological model tozB[i] ~ dbern(psiB[zA[i] + 1, Time[i]])
and in the
nCat
loop havepsiB[1, i] ~ dbeta(1, 1)
psiB[2, i] ~ dbeta(1, 1)
Do remember in all this that our sample sizes are getting pretty small; ok if your original sample is huge.
A colleague emailed me asking “how [can we] tweak the model to deal with NA data?”
NAs in the detection data are not a problem, provided the two species have the same pattern of NAs. If the analysis used the detection matrices,
DHA
andDHB
above, everything should just work.If you are using the aggregated data,
yA
andyB
, it’s a tad more complicated. You need to includena.rm=TRUE
in the call torowSums
. Also the number of occasions is no longer the same for all sites, so you need to do:n <- rowSums(!is.na(DHA))
and in the JAGS code use
n[i]
instead ofn
.JAGS doesn't allow NAs in covariates. If these are survey-level covariates (eg, survey date) and the NAs correspond to NAs in the detection matrices, the values don't matter, so just plug in 0 or the mean of the other values or whatever.
Got it, thanks very much Mike for the quick answer 🙂