Patch Occupancy Dynamics: Estimation and Modeling Using

“Presence-absence” Data

Patch Occupancy: The Problem

• Conduct “presence-absence” (detection-nondetection) surveys

• Estimate what fraction of sites (or area) is occupied by a species when species is not always detected with certainty, even when present (p < 1)

Patch Occupancy: Motivation

• Extensive monitoring programs

• Incidence functions and metapopulations

• Disease modeling

• Surveys of geographic range and temporal changes in range

Key Design Issue: Replication

• *Temporal replication: repeat visits to sample units

• Spatial replication: randomly selected subsample units within each sample unit

• Replicate visits occur within a relatively short period of time (e.g., a breeding season)

Data Summary: Detection Histories

• A detection history for each visited site or sample unit– 1 denotes detection– 0 denotes nondetection

• Example detection history: 1 0 0 1– Denotes 4 visits to site– Detection at visits 1 and 4

• The detection process is independent at each site

• No heterogeneity that cannot be explained by covariates

• Sites are closed to changes in occupancy state between sampling occasions

Model Parameters and Assumptions

i -probability site i is occupied

pij -probability of detecting the species in site i at time j, given species is present

Model Parameters and Assumptions

A Probabilistic Model

• Pr(detection history 1001) =

4321 11ψ iiiii pppp

kj

kjk p ψ11ψ4

1

• Pr(detection history 0000) =

A Probabilistic Model

• The combination of these statements forms the model likelihood

• Maximum likelihood estimates of parameters can be obtained

• However, parameters cannot be site specific without additional information (covariates)

• Suggest non-parametric bootstrap be used to estimate SE

Software

• Windows-based software:– Program PRESENCE (Darryl MacKenzie)– Program MARK (Gary White)

• Fit both predefined and custom models, with or without covariates

• Provide maximum likelihood estimates of parameters and associated standard errors

• Assess model fit

Example: Anurans at Maryland Wetlands (Droege and Lachman)

• FrogwatchUSA (NWF/USGS)• Volunteers surveyed sites for 3-minute periods

after sundown on multiple nights• 29 wetland sites; piedmont and coastal plain• 27 Feb. – 30 May, 2000• Covariates:

– Sites: habitat ([pond, lake] or [swamp, marsh, wet meadow])

– Sampling occasion: air temperature

Example: Anurans at Maryland Wetlands (Droege and Lachman)

• American toad (Bufo americanus)– Detections at 10 of 29 sites

• Spring peeper (Hyla crucifer)– Detections at 24 of 29 sites

Example: Anurans at Maryland Wetlands (B. americanus)

Model AIC

(hab)p(tmp) 0.00 0.50 0.13

(.)p(tmp) 0.42 0.49 0.14

(hab)p(.) 0.49 0.49 0.12

(.)p(.) 0.70 0.49 0.13

ψ̂ ψ̂ˆES

Naive 0.34ψ̂

Patch Occupancy as a State Variable: Modeling Dynamics

• Patch occupancy dynamics• Model changes in occupancy over time• Parameters of interest:

t = t+1/ t = rate of change in occupancy t = P(absence at time t+1 | presence at t) =

patch extinction probability t = P(presence at t+1 | absence at t) =

patch colonization probability

Pollock’s Robust Design: Patch Occupancy Dynamics

• Sampling scheme: 2 temporal scales– Primary sampling periods: long intervals

between periods such that occupancy status can change

– Secondary sampling periods: short intervals between periods such that occupancy status is expected not to change

Robust Design Capture History

• History : 10 00 11 01 primary(i) secondary(j)

• 10, 01, 11 = presence• Interior ‘00’ =

Patch occupied but occupancy not detected, or Patch not occupied (=locally extinct) yet

recolonized later

Robust Design Detection History

• History : 10 00 11 01 primary(i) secondary(j)

• Parameters: – 1-t: probability of survival from t to t+1– p*t: probability of detection in primary

period t – p*t = 1-(1-pt1)(1-pt2) t: probability of colonization in t+1 given

absence in t

Modeling

• P(10 00 11 01) =

424133231

212*21

)1)(1(

)1)(1)(1(

pppp

p

)1( 12111 pp

Parameter Relationships: Alternative Parameterizations

• Standard parameterization: (1, t, t)

• P(occupied at 2 | 1, 1, 1) =

• Alternative parameterizations: (1, t, t), (1, t, t), (t, t), (t, t)

11112 )1()1(

1

111

1

21

)1(1

Main assumptions

• All patches are independent (with respect to site dynamics) and identifiable

– Independence violated when subpatches exist within a site

• No colonization and extinction between secondary periods

– Violated when patches are settled or disappear between secondary periods => breeding phenology, disturbance

• No heterogeneity among patches in colonization and extinction probabilities except for that associated with identified patch covariates

– Violated with unidentified heterogeneity (reduce via stratification, etc.)

Software

• PRESENCE: Darryl MacKenzie– Open models have been coded and used for a

few sample applications.– Darryl is writing HELP files to facilitate use.

• MARK: Gary White – Implementation of one parameterization of the

open patch-dynamics model based on the MacKenzie et al. ms

Example Applications

• Tiger salamanders (Minnesota farm ponds and natural wetlands, 2000-2001; Melinda Knutson)– Estimated p’s were 0.25 and 0.55– Estimated P(extinction) = 0.17; Naïve estimate = 0.25

• Northern spotted owls (California study area, 1997-2001; Alan Franklin)– Potential breeding territory occupancy– Estimated p range (0.37 – 0.59); Estimated =0.98– Inference: constant P(extinction), time-varying

P(colonization)

Example: Range Expansion by House Finches in Eastern NA

• Released at Long Island, NY, 1942• Impressive expansion westward• Data from NA Breeding Bird Survey

– Conducted in breeding season– >4000 routes in NA– 3-minute point counts at each of 50 roadside stops at

0.8 km intervals for each route

• Occupancy analysis: based on number of stops at which species detected – view stops as geographic replicates for route

House Finch Range Expansion: Modeling

• 26 100-km “bands” extending westward from NY• Data from every 5th year, 1976-2001

• Model parameterization: (1, t, t, pt)

• Low-AIC model relationships:– Initial occupancy, 1 = f(distance band)

– P(colonization), t = f(distance*time)

– P(extinction), t = f(distance)

– P(detection), pt = f(distance*time)

Gamma(1976)

Gamma(1996)

Purple Heron, Ardea purpurea, Colony Dynamics

• Colonial breeder in the Camargue, France

• Colony sizes from 1 to 300 nests

• Colonies found only in reed beds; n = 43 sites

• Likely that p < 1

breeds in May => reed stems grown

small nests ( 0.5 m diameter ) with brown color (similar to reeds)

Purple Heron Colony Dynamics

• Two surveys (early May & late May) per year by plane (100 m above ground) covering the entire Camargue area, each lasting one or two days

• Since 1981 (Kayser et al. 1994, Hafner & Fasola 1997)

• Study area divided in 3 sub-areas based on known different management practices of breeding sites (Mathevet 2000)

Purple Heron Study Areas

West:disturbance

Central:DISTURBANCE

East:protected

Purple Heron Colony Dynamics: Hypotheses

• Temporal variation in extinction\colonization probabilities more likely in central (highly disturbed) area.

• Extinction\colonization probabilities higher in central (highly disturbed) area?

Purple Heron Colony Dynamics:Model Selection

Model AICc np 2 df P

[g*t, g*t] 405.6 114 - - -

[g*t, t] 352.5 76 40.6 38 0.36

[g*t, g] 357.1 60 81.8 54 0.009

[g*t, ] 356.9 60 80.2 54 0.012

[t, t] 348.5 38 109.5 76 0.006

[w=e(.) c(t), t] 308.0 39 78.4 75 0.38

[g, t] 310.4 22 108.8 92 0.11

LRT [g*t, t] vs [g, t] : 254 = 80.5, P = 0.011

Purple Heron Colonization Probabilities

0.0

0.2

0.4

0.6

0.8

1.0

Years

Co

lon

iza

tio

n P

r

Purple Heron Colony Extinction Probabilities

Extinction west = east = 0.137 0.03

0.0

0.2

0.4

0.6

0.8

1.0

Years

Ex

tin

ctio

n P

r

central

Purple Heron Colony Dynamics

• Is colonization of sites in the west or east a function of extinction in central?

• Linear-logistic models coded in SURVIV:

w = e(a + b c)/(1+e(a + b c))

e = e(a + b c)/(1+e(a + b c))

a = intercept parameter

b = slope parameter

= 1-

Purple Heron Colony Dynamics Model Selection

Model AICc np 2 df P

[w=e(.) c(t), t] 308.0 39 78.4 75 0.38

[, w=f(c)] 315.2 41 80.0 73 0.27

[, e=f(c)] 319.1 41 86.7 73 0.13

Intercept = -0.29 0.50 (-1.27 to 0.69)Slope = -3.59 0.61 (-4.78 to –2.40)

Purple Heron Colony Dynamics

0.0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

Extinction central area

Co

lon

iza

tio

n w

est

are

a

Purple Heron Colony Dynamics

0

0.2

0.4

0.6

0.8

1

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000

Years

Co

lon

iza

tio

n P

r

log-lin

time

Conclusions

• “Presence-absence” surveys can be used for inference when repeat visits permit estimation of detection probability

• Models permit estimation of occupancy during a single season or year

• Models permit estimation of patch-dynamic rate parameters (extinction, colonization, rate of change) over multiple seasons or years

Occupancy Modeling Ongoing and Future Work

• Heterogeneous detection probabilities– Finite mixture models– Detection probability = f(abundance), where abundance

~ Poisson

• Multiple-species modeling– Single season– Multiple seasons

• Hybrid models: presence-absence + capture-recapture

• Study design optimization