One important limitation of traditional capture-recapture methods is
that density cannot be explicitly estimated. SCR explicitly estimates the
surveyed area whilst traditional CR derived estimates of density by ad hoc estimation
of the effective surveyed area. Multiple methods can be used to estimate this
including buffering by the mean maximum distance move (MMDM), or half MMDM or
by an estimated female home range radius if such data is available. All of
which would return three potentially vastly different estimates of density for
the same N. Additionally, it is
assumed that no animals are able to move across the buffer established by this method. 

Spatially-explicit capture-recapture (SECR, Borchers and Efford, 2008) or spatial
capture-recapture (SCR, Royle et al., 2013b) models are
increasingly used for monitoring animal populations worldwide and have been
shown to produce more precise estimates than conventional mark-recapture
methods (Sollmann et al., 2011).

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Bias in N also arises from unmodelled
heterogeneity in capture probability, p.
In non-spatial CR p remains constant,
i.e. all individuals are assumed to be equally detectable. Detectability of
individuals is clearly related to their location relative to traps and, even
between two individuals in the same location the rate at which they move around
the environment and thus encounter traps. Models of heterogeneity (Mh) in SCR account for
heterogeneity through considering the processes which generated the data rather
than the resulting data itself (Royle et al., 2013b)RG1 .

Royle et al. (2013b) identify four
key limitations of traditional CR methods:

1.     Density
estimation is conducted ad hoc

2.     Heterogeneity
in detection probability cannot be explicitly estimated

3.     Trap-level
covariates are not considered

4.     The spatial
processes underlying the data are not considered

5.     Individuals must
be uniquely identifiable

Such limitations were not unrecognised by the scientific community;
however, methods with which to approach these problems are a relatively recent
development. 

What is SCR?

Efford (2004) developed a
method for using the spatial information inherent in capture-recapture studies and
these methods were subsequently formalised to explicitly model density either
through maximum likelihood estimation (MLE, Borchers and Efford, 2008) or in a
Bayesian framework (Royle and Young, 2008).

SCR models
exploit the spatial information, such as the juxtaposition between the
locations of individuals and traps, inherent to capture-recapture surveys. The
distribution (or location) of individuals in space is described by a point
process model. The points represent the activity centres of individuals (si ; i = 1,2,…N) which are latent, or unobserved, variables in the
model. We gather information about them by observing animals with detectors
(traps, cameras, area searches, acoustic detectors, DNA sampling…). These
realised locations, ui,
represent a thinned point process model where the thinning is determined by how
the locations are observed, such as the location of detectors. All possible
locations of si, i.e. all
animals that could be captured, are represented by the state-space, . Therefore, N is the number of si
in  and density can be expressed as (Royle
et al., 2013b)

.

So, as SCR models link individuals with
space, so too do they define N in
terms of the surveyed area. Consequently, density is explicitly modelled and
population size, as a function of density, can be estimated for a specific
state-space, or survey area. The main limitation of traditional
capture-recapture was the inability to define the survey area, or state-space.
By formally associating with , SCR allows us to gain meaningful ecological inference from
capture-recapture surveys by explicitly modelling density.RG2 

The model which concerns
us in spatially explicit density estimation is that which relates the thinned point
process we observe, to the actual distribution of animals in space, N. Recall that our variables are the
unobserved activity centres, s, the
realised locations of individuals, u,
in addition to our response variable, y,
which is what we record upon capture. Relating our encounter model, movement
model and point process model we have (Royle
et al., 2013b, p.42) 
 

.

Where s is the distribution of activity
centres in the state space, , which may or may not be homogenous
depending on whether density varies across space, u|s models the
locations of animals given their activity centre and y|u describes how the
observed data arise given the locations of animals. Additionally, note that u is the realisation of a movement
model which most SCR models do not quantify, there y|s is modelled. In
fixed detector location designs the observation model is therefore (Royle
et al., 2013b, p.45)

.

This is the
encounter model, that is the probability of detecting an individual given the
distance between s and the detector
and can also be written as (Royle
et al., 2013b, p. 43)

.

It has
parameters p0 (Borchers
and Efford (2008) use g0), which is the capture probability when si is at the tap location xj, and  is a spatial scale parameter determining how
rapidly capture probability declines with distance. Often is a spatial scale
parameter determining how rapidly capture probability declines with distance.
Often the encounter model is yij|si = Bernoulli (pij). Where yij is the observed data when
individual recognition is possible. For count or detection data, density
estimation of unmarked individuals is possible through specifying observed data
as n(y) giving us the model n(y)|y.

Looking more
closely at s and recalling that it
is the distribution of animals across space, recalling that this point process
describes N, we can characterise it
as s|, where  is the intensity of this process or population density. Therefore, N ~ Poisson , where  is the area of the state-space or in the case
of implementing Bayesian analysis

N ~ Binomial , where  ,

and M is a large integer used for data augmentation (Royle
et al., 2013b, p. 42).

Bringing this
together our simple SCR model has a population size which is Possion
distributed, where activity centres are uniformly distributed and capture
probability is a function of the distance between the location of these
activity centres and that of our detectors. 
Or can be described by (Royle
et al., 2013b, p. 43)

,

,

.

This simple
model can then be modified as our situation requires, examples of which will be
found in my thesis.

Whilst the work
discussed above shows the many advancements in capture-recapture modelling by
the spatial framework, SCR models, like all models are not without limitations
and assumptions. Key assumptions of (but not unique to) spatially-explicit
density estimation include (Royle
et al., 2013b):

1.    
Demographic
closure: no recruitment, migration or mortality

2.    
Geographic
closure

3.    
Random
distribution of activity centres, si

4.    
Detection
declines as a function of distance: of an activity centre from a detector

5.    
Encounters
are independent between and within individuals

6.    
Marks
are not lost or misidentified

Other
limitations which may be confronted in conducting SCR include trap saturation
limiting recaptures, dealing with partially marked or unmarked populations,
low-density populations, appropriate spacing of detectors relative to animal
movements, and a lack of stationarity and symmetry of animal home ranges.
Nonetheless spatially explicit methods offer the scope to address these issues,
particularly through Bayesian inference, and solutions can be used to gain
further insight into animal populations. An increasing body of literature
investigates the robustness of SCR to departures from these assumptions, some
of which are pertinent to my research and are discussed below in addition to
other advances in modelling techniques such as continuous time models.
Additional research is required in some areas, such as the independence of
encounters between individuals (i.e. for social species), which I will explore
in the fourth chapter of my thesis.

By explicitly including spatial elements in our
population models we can thereby explicitly model factors central to many
ecological questions using one framework. For example, resource selection (Royle et al., 2013c), sex-specific movement and detectability (Sollmann et al., 2011), landscape connectivity (Fuller et al., 2016;
Royle et al., 2013a)
and population dynamics (Chandler and Clark,
2014; Ergon and Gardner, 2014; Gardner et al., 2010; Schaub and Royle, 2014;
Whittington and Sawaya, 2015)