Home › Occupancy Estimation Two Species

Occupancy Estimation Two Species

Contents – Index

Occupancy Estimation Two Species

Parameterization

Two versions of the 2-species occupancy models are included in Program MARK.  The first is the original parametrization developed in the MacKenzie et al. (2006) book.  This parametrization is difficult to use because of a multinomial constraint on the r parameters.  Further, this parametrization does not work well with covariates.  It is identified in MARK as 113. 2SpecOccup: Two species Occupancy Estimation  

The parametrization for the MacKenzie et al. (2006) 2-species model is:

psiAB  Probability of both species being present
psiA  Probability of occupancy for species A, regardless of occupancy status of species B
psiB  Probability of occupancy for species B, regardless of occupancy status of species A
pA  Probability of detection for species A, given species B is absent
pB  Probability of detection for species B, given species A is absent
rAB  Probability of detecting both species, given both species are present
rAb  Probability of detecting species A, but not B, given both species are present
raB  Probability of detecting species B, but not A, given both species are present
rab  Probability of detecting neither, given both species are present.  This parameter is obtained by subtraction: 1 – rAB – rAb – raB, so is not in the PIMs.

Because rab is obtained by subtraction, the multinomial logit  (MLogit link function) is often required to obtain a numerical solution.    

The second parametrization is termed the conditional occupancy model and is stable when covariates are included.  Also, this parameterization assumes that Species A is the dominant species, and Species B the subordinate species.  The data type is identified in MARK as 137. 2SpecConOccup: Two species Conditional Occupancy Estimation.  The parametrization for this 2-species conditional occupancy model (Richmond et al. 2010) follows.

psiA  Probability of occupancy for species A
psiBA  Probability of occupancy for species B, given species A is present
psiBa  Probability of occupancy for species B, given species A is absent
pA  Probability of detection for species A, given species B is absent
pB  Probability of detection for species B, given species A is absent
rA  Probability of detection for species A, given both species are present
rBA  Probability of detection for species B, given both species are present and species A is detected
rBa  Probability of detection for species B, given both species are present and species A is not detected

Some of the parameters of the McKenzie et al. parametrization can be defined in terms of the conditional :

  psiB = psiA*psiBA + (1 – psiA) psiBa

  psiAB = psiA*psiBA

  psiB and psiAB are included as a derived parameter for this data type.

The short labels for the 9 derived parameters for the conditional data type are: 1) “PsiA”, 2) “PsiBA”, 3) “PsiBa”, 4) “PsiB”, 4) “PsiAB”, 6) “SIF”, 7) “A only”, 8) “B only”, and 9) “Unoccupied”. The following diagram from Larissa Bailey helps explain the 9 derived parameters of the Richmond conditional parameterization.

So, derived parameters #1-3 are seen in the above figure: #1 psi(A) is the unconditional probability that species A is present at the site, but #2 psi(BA) = probability that B is present given A is present and #3 psi(Ba) = probability that B is present given A is not present are conditional on the presence or absence of A, respectively.  The remaining 6 derived parameters within each season are derived from these three.

Moving from left to right at the bottom of the figure:
#9 Unoccupied, is the first product on the left, derived from (1 – psi(A)) (1 – psi(Ba)), is the probability that neither A or B are present at the site.
#8 Species B Only, is the second product on the left, derived from (1 – psi(A)) psi(Ba), is the probability that the site is occupied by B only.
#7 Species A Only, is the third product on the left, derived from psi(A) (1 – psi(BA)), is the probability that the site is occupied by A only. 
#5 Both Species Occupancy, is the final/fourth product from the left, derived from psi(A) psi(BA), is the probability of both species being present at the site.
#4 Species B Occupancy, is the sum of Species B Only and Both Species Occupancy, derived from (1 – psi(A)) psi(Ba) +  psi(A) psi(BA), is the probability the site is occupied by species B.
#6 is the SIF (Species Interaction Factor), defined below for the Richmond model, as SIF = (psiA psiBA) / (psiA(psiA*psiBA + (1 – psiA)psiBa).

The following graphic from Brittainy Mosher provides a visual definition of the derived parameters.

Species Interaction Factor

Both parameterizations provide estimates of the species interaction factor, SIF.  The estimate of SIF for each attribute group is provided as a derived parameter.  For the McKenzie et al. (2006) model, 

  SIF = psiAB/(psiA*psiB) .

For the conditional occupancy model, SIF = (psiA psiBA) / (psiA(psiA*psiBA + (1 – psiA)psiBa).

Encounter Histories

The encounter histories for the 2-species occupancy model are provided as a pair for each occasion.  The first member of the pair is the status of species A: 1 = detected; 0 = not detected; and . (dot) = not searched.  The same codes are used for the second member of the pair for species B.

As an example, the encounter history 11001001.11. means that both A and B were detected at time 1, neither were detected at time 2, only A was detected at time 3, and only B was detected at time 4.  Species A was not searched for at time 5, but B was detected.  Species A was detected at time 6, but B was not searched for.