Modeling Movement in Spatial Capture-Recapture (SCR) Models

Andy Royle U.S. Geological Survey, Patuxent Wildlife Research

Center

Angela Fuller U.S. Geological Survey, NY Cooperative Fish &

Wildlife Research Unit, Cornell University

Chris Sutherland Cornell University

Capture-recapture

Spatial capture-recapture: a hierarchical CR model

Uses of SCR models

Spatial models of density

Resource selection

Transience/dispersal

Key idea: can study these things using encounter history data, don’t need telemetry!

Overview

Background: Capture-recapture Models

Models for estimating population size, N, from individual encounter history data – usually obtained from an array of traps or similar devices

Occasion

individual 1 2 3 4 5

-----------------------

1 1 0 1 0 1

2 0 1 0 0 0

3 0 1 1 1 0

4 0 0 1 0 1

5 0 1 0 0 0

… .. .. ..

…

Individual encounter history data

𝑦𝑖 ,𝑘 = encounter of individual i in sample occasion k 𝑦𝑖 ,𝑘 ~ Bern(𝑝𝑖 ,𝑘 ); i=1,2,…,N CR models: all about modeling

variation in 𝑝𝑖 ,𝑘 Behavioral response time effects individual heterogeneity Summarized by Otis et al. (1978)

N unknown

New technologies produce vast quantities of encounter history data

Camera traps

DNA sampling

Scat picked up by searching space

Urine on scent sticks or in snow

Tissue samples from treed individuals

Hair snares

Acoustic sampling (whales, birds, bats)

All studies produce spatially explicit encounter information (e.g., trap locations, or encounter location)

Classical capture-recapture models do not accommodate the spatial attribution of encounters or traps. (fish bowl sampling)

Spatial attribution of CR data

Cannot model spatial effects:

Type of trap

Baited or not

How long traps are operational during a sample period

Habitat type around trap

Trap specific environmental conditions

Behavioral response at the trap level

Loss of spatial information

2. Spatial

Capture-

Recapture

Models

Spatial capture-recapture: an extension of CR to make use of encounter location data in order to study spatial aspects of animal populations

Encounter location provides information about spatial processes including:

(a) spatial variation in animal density

(b) resource selection

(c) animal movement

Spatial Capture-Recapture (SCR)

Spatial distribution of organisms is naturally described by point process models (and has been for many decades in plant ecology)

For capture-recapture systems, describe distribution of individuals by a point process (Efford, 2004; Oikos)

𝒔𝑖 = activity center or home range center for individual i

Describe Pr(encounter in trap) conditional on where an individual lives 𝒔 𝑖

Getting SPACE into capture-recapture: Spatial point process model

Biological process: How individuals are distributed in space – a model for “activity centers” or home range centers

{ 𝒔1, 𝒔2,…, 𝒔𝑁 } = realization of a point process

𝒔 𝑖 ~ Uniform(S); S = state-space of point process

-or-

Pr(s) ∝ exp (𝛽0 + 𝛽1*Cov(s)) [=D(s), intensity function or “animal density”]

Observation model (trap and individual specific encounter)

y ij| 𝒔 𝑖 ~ Bern( p(x j, 𝒔 𝑖) )

x j = trap location

SCR: A hierarchical model

linked by allowing

probability of encounter

to depend on s:

p(xj, 𝒔𝑖) =

p0*exp (−𝑑𝑖𝑠𝑡(𝒙, 𝒔)2/𝜎2)

[or some other function]

SCR encounter probability model

Decreasing function of distance between traps and activity centers:

p(x, 𝒔) = p0*exp (−𝑑𝑖𝑠𝑡(𝑥, 𝒔)2/𝜎 2)

x = trap location

s = home range center

En

co

un

ter

Pro

ba

bilit

y

SCR models are basically GLMMs (but N is unknown)

MLE based on marginal likelihood:

Borchers, D.L. and M.G. Efford. 2008. Spatially explicit maximum likelihood methods for capture-recapture studies. Biometrics 64:377-385

R package ‘secr ’ (M.G. Efford)

Bayesian Analysis by MCMC (data augmentation)

Royle, J.A. and K.Y. Young . 2008. A hierarchical model for spatial capture-recapture data. Ecology 89:2281-2289.

Inference for SCR models

3. The

Promise of

SCR

SCR models provide a probabilistic characterization of where animals live

Ecologists can address questions of spatial population ecology from ordinary encounter history data Spatial patterns in density Resource selection or space usage Dispersal/transience Landscape connectivity (Chris Sutherland’s poster)

Historically CR has been used to estimate N or Density but nothing about spatial ecology

SCR: a unified framework for inference about density and spatial population processes

SCR: A Paradigm Shift

Can model explicit factors that influence density

log(D(s)) = beta0 + beta1*Elev(s)

SCR: Density models

Elevation (100 – 800 m)

NY black bear study (thanks: Cat Sun)

Royle, J. A., Chandler, R. B., Sun, C. C., & Fuller, A. K. (2013). Integrating

resource selection information with spatial capture–recapture. Methods in

Ecology and Evolution, 4(6), 520-530.

• Explicit covariates

• Flexible wiggly surfaces:

GAMs and splines

(Borchers et al.; also in

‘secr’ package)

• Posterior distributions of

individual 𝒔𝑖

• “Small area” estimates

Density surface estimated from NY bear study (C. Sun)

Bears/100

km^2

SCR models: downscaling N

Two scales of movement

“local movement” about static s (space usage or resource selection)

Dynamic movement: Transience or dispersal, dynamic point process for activity centers s

Modeling Movement with SCR

Pr(encounter) affected by local movement about 𝒔𝑖

p(x, 𝒔) =“probability animal moved to/uses x and is encountered”

Observed encounters are the outcome of some movement process and “trap effectiveness”

SCR has an implicit movement model

Local movement

SCR derived from a Poisson Cluster Process

Parent nodes : 𝒔 𝑖 for i=1,..,N

Offspring : “use locations” u ~ k(u|𝒔)

Encounter process : think of traps as randomly thinning the offspring -- can only observe u in the vicinity of trap locations x

SCR and Movement

Formal correspondence between 𝑝(𝑥|𝑠) and k(u|𝑠)

Movement outcomes : “iid” draws from k(u|𝒔)

Encounter locations provide information about parameters of k(u|𝒔)

SCR and Movement

Modeling Resource Selection

Estimated Resource Selection:

“probability of using a pixel relative to

a pixel of mean elevation”

Resource selection: How individuals use space affects encounter probability at trap locations

Model covariates in k(u|𝒔) :

k(u|𝒔) ∝ exp (𝛼 ∗ 𝑧 𝒖 − 𝑑𝑖𝑠𝑡(𝒖, 𝒔))

Royle, J. A., Chandler, R. B., Sun, C. C., & Fuller, A. K. (2013).

Integrating resource selection information with spatial

capture–recapture. Methods in Ecology and Evolution, 4(6),

520-530.

Johnson, D.S., Thomas, D.L., Ver Hoef, J.M. & Christ, A.

(2008) A general framework for the analysis of animal

resource selection from telemetry data. Biometrics, 64,

968–976.

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Can we generalize the static SCR model? Dynamic point process models that describe biological features:

Transient individuals – no home range, “just passing through”

Transient space usage – seasonal shifts in home range to accommodate resource changes

Dispersal

Dynamic point process models

2-d random walk transience/dispersal models

𝒔 𝑖 , 𝑡 |𝒔 𝑖 , 𝑡 − 1 ~ BVN(𝒔 𝑖 , 𝑡 − 1 , 𝜏 2 𝐈)

.

Temporal scale: t = long period, year [dispersal]

t = short period, day [transience]

Simple model can explain transience or dispersal

Dynamic models

SCR and d i sper sa l :

Schaub , M . , & Roy l e , J . A . (2013 ) . E st ima t i ng t r ue i n ste a d o f appar e n t sur v i va l u s i ng spat i a l Co r mack –Jo l l y –S eb er mode l s . Method s i n Eco l ogy and Evo l u t i o n

Dynamic point processes

𝒔 𝑖, 𝑡 |𝒔 𝑖, 𝑡 − 1 ~ BVN(𝒔 𝑖, 𝑡 − 1 , 𝜏2𝐈)

traps A trajectory of

one transient

individual

illustrating how

exposure to

traps changes

over time

p 0 ~ dun i f ( 0 ,1 )

s i g m a.sc r < - s q r t ( 1 / ( 2 *a lpha1) )

a l pha1~ dnorm(0 , .1 )

ps i~ dun i f (0 ,1 )

s ig m a.a r ~ dun i f ( 0 ,5 )

t au< - 1 / ( s i gma.a r *s i gma.a r )

f o r ( i i n 1 : M) {

z [ i ] ~ dbe r n (ps i )

s [ i , 1 ,1 ]~ dun i f ( x l im [1 ] , x l im [2 ] )

s [ i , 2 ,1 ]~ dun i f ( y l im [1 ] , y l im [2 ] )

f o r ( j i n 1 : J ) {

d [ i , j , 1 ]< - pow( pow( s [ i , 1 ,1 ] -X [ j , 1 ] ,2 ) + pow( s [ i , 2 ,1 ] -X [ j , 2 ] ,2 ) ,0 .5 )

y [ i , j ,1 ] ~ db in ( p [ i , j , 1 ] ,1 )

p [ i , j , 1 ]< - z [ i ] *p0*exp ( - a l pha1*d [ i , j , 1 ] *d [ i , j , 1 ] )

}

f o r (k i n 2 : K ) {

s [ i , 1 ,k ] ~ dnor m(s [ i , 1 ,k -1 ] , tau )

s [ i , 2 ,k ] ~ dnor m(s [ i , 2 ,k -1 ] , tau )

f o r ( j i n 1 : J ) {

d [ i , j , k ]< - pow( pow( s [ i , 1 , k ] -X [ j , 1 ] ,2 ) + pow( s [ i , 2 , k ] -X [ j , 2 ] ,2 ) ,0 .5 )

y [ i , j , k ] ~ db in ( p [ i , j , k ] , 1 )

p [ i , j , k ]< - z [ i ] * p0 *exp ( - a l pha1* d [ i , j , k ] *d [ i , j , k ] )

}

}

}

N< - s um(z [ ] )

D< - N / a r ea

}

Bayesian analysis of the transience model: BUGS/JAGS

Prior distributions

State model

Observation model

Observation model t = 1

Initial state

What if you don’t care about movement but just want to estimate density? (or have very sparse data so cannot effectively fit explicit movement models)

Is the density estimation robust to misspecification by the static 𝒔𝑖 model?

Robustness to misspecification

Simulated populations with 50% transients, moving according to a Gaussian random walk with parameter 𝜎𝑟𝑤

2

D = 1.2 individuals/unit area

T = 5 sampling occasions (“weeks”)

Misspecified by a “no transience” model

p(x, 𝒔) = p0*exp (−𝑑𝑖𝑠𝑡(𝑥, 𝒔)2/𝜎𝑠𝑐𝑟2 )

Results:

NO Bias in estimating D across a range of simulation conditions (𝜎𝑟𝑤 /𝜎𝑠𝑐𝑟 = 0.5 to 4.0)

Effective “sigma” of the encounter probability model is biased to represent composite 𝜎𝑠𝑐𝑟 and 𝜎𝑟𝑤

SIMULATION STUDY RESULTS

SCR models are not just about estimating density

You can fit models of transience and dispersal from encounter history data -- “paradigm shift”

Unified framework for modeling density and movement and other spatial processes

But if you don’t care about movement, static model is robust for estimating N under Markovian dispersal/transience models

SUMMARY: WHY SHOULD YOU CARE?