patch occupancy dynamics: estimation and modeling using “presence-absence” data
TRANSCRIPT
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Patch Occupancy Dynamics: Estimation and Modeling Using
“Presence-absence” Data
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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)
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Patch Occupancy: Motivation
• Extensive monitoring programs
• Incidence functions and metapopulations
• Disease modeling
• Surveys of geographic range and temporal changes in range
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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)
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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
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• 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
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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
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A Probabilistic Model
• Pr(detection history 1001) =
4321 11ψ iiiii pppp
kj
kjk p ψ11ψ4
1
• Pr(detection history 0000) =
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Modeling
• P(10 00 11 01) =
424133231
212*21
)1)(1(
)1)(1)(1(
pppp
p
)1( 12111 pp
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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
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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.)
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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
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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)
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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
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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)
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Gamma(1976)
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Gamma(1996)
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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)
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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)
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Purple Heron Study Areas
West:disturbance
Central:DISTURBANCE
East:protected
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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?
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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
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Purple Heron Colonization Probabilities
0.0
0.2
0.4
0.6
0.8
1.0
Years
Co
lon
iza
tio
n P
r
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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
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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-
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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)
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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
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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
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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
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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