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).

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)