# Presentation: 015310

##  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

## Model Parameters and Assumptions

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

## A Probabilistic Model

- Pr(detection history 1001) =

- 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 |

- Naive

## 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 0_1_
-                      *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 0_1_
-                      *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-*p**t*1)(1-*p**t*2)
***t*: probability of colonization in *t*+1 given absence in *t*

## Modeling

- P(10 00 11 01)  =

## 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**)*

## 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*, *p**t*)
- Low-AIC model relationships:
- Initial occupancy, **1 = f(distance band)
- P(colonization), ***t *= f(distance*time)
- P(extinction), ***t *= f(distance)
- P(detection), *p**t* = f(distance*time)

## Slide 25

## Slide 26

## Slide 27

## Gamma(1976)

## Slide 29

## 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

## 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 

## Purple Heron Colony Extinction Probabilities

- Extinction west = east = 0.137  0.03

## 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

##  Purple Heron Colony Dynamics

## 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