Hierarchical Abundance Models Rob Sparks, David Pavlacky, David Hanni Rocky Mountain Bird Observatory

Conserving Birds & Their Habitats

Objectives • Estimate Abundance at the grid level • IMBCR data

• Understand mechanisms influencing abundance • Create spatially explicit maps

2 Conserving birds and their habitats.

Hierarchical Model â&#x20AC;˘ Components : â&#x20AC;˘ Observations: đ?&#x2018;Śđ?&#x2018;&#x2013; =count of birds at point i, â&#x20AC;&#x153;dataâ&#x20AC;? â&#x20AC;&#x201C; Observation model [đ?&#x2018;Śđ?&#x2018;&#x2013; | đ?&#x2018; đ?&#x2018;&#x2013; ] â&#x2C6;ź Multinomial đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;Ą , đ?&#x153;&#x2039; đ?&#x2018;? â&#x20AC;˘ đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;Ą = local population size â&#x20AC;˘ p = probability of encountering an individual â&#x20AC;˘ đ?&#x153;&#x2122; = probability that an individual in the grid population is available at the point

â&#x20AC;˘ state variable: outcome of ecological process of interest â&#x20AC;&#x201C; Process model â&#x20AC;˘ đ?&#x2018; đ?&#x2018;&#x2013; ~ đ?&#x2018;&#x192;đ?&#x2018;&#x153;đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ?&#x153;&#x2020; â&#x20AC;˘ Log đ?&#x153;&#x2020;đ?&#x2018;&#x2013; = Îą + đ?&#x203A;˝đ?&#x2018;Ľ;

Conserving birds and their habitats.

Hierarchical Model Metapopulation parameters

Local parameters

Data

đ?&#x2018; 1 ,đ?&#x2018;?1

đ?&#x2018;Ś1

đ?&#x203A;ź, Î˛, â&#x2C6;&#x2018;đ?&#x153;&#x2122; , â&#x2C6;&#x2018;đ?&#x2018;? đ?&#x2018; 2 ,đ?&#x2018;?2

â&#x20AC;Ś

đ?&#x2018; đ?&#x2018;&#x203A; ,đ?&#x2018;?đ?&#x2018;&#x203A;

đ?&#x2018;Ś2

â&#x20AC;Ś

đ?&#x2018;Śđ?&#x2018;&#x203A;

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Generalized Multinomial Mixture Model â&#x20AC;˘ Model đ?&#x2018;&#x20AC;đ?&#x2018;&#x2013; ~ đ?&#x2018;&#x192;đ?&#x2018;&#x153;đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018; đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ?&#x153;&#x2020; đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;Ą â&#x2C6;ź Binomial đ?&#x2018;&#x20AC;đ?&#x2018;&#x2013; , đ?&#x153;&#x2122; đ?&#x2018;Śđ?&#x2018;&#x2013;đ?&#x2018;&#x2014;đ?&#x2018;Ą â&#x2C6;ź đ?&#x2018;&#x20AC;đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x153;đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x17D;đ?&#x2018;&#x2122; đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;Ą , đ?&#x153;&#x2039; đ?&#x2018;?

â&#x20AC;˘ Parameters

â&#x20AC;˘ đ?&#x2018;&#x20AC;đ?&#x2018;&#x2013; - latent variable, avg. abundance in a grid â&#x20AC;˘ đ?&#x153;&#x2122; â&#x20AC;&#x201C; probability that an individual in the grid population is available at the point â&#x20AC;˘ p â&#x20AC;&#x201C; detection probability

Conserving birds and their habitats.

Observation model â&#x20AC;˘ Multinomial cell probabilities â&#x20AC;˘ For removal sampling study with 3 intervals, the multinomial cell probabilities are đ?&#x153;&#x2039; = đ?&#x2018;?, 1 â&#x2C6;&#x2019; đ?&#x2018;? đ?&#x2018;?, 1 â&#x2C6;&#x2019; đ?&#x2018;? 2 đ?&#x2018;?, (1 â&#x2C6;&#x2019; đ?&#x2018;?)3 đ?&#x153;&#x2039;1 = Pr(1xx) = đ?&#x2018;?1 đ?&#x153;&#x2039;2 = Pr(01x) = (1- đ?&#x2018;?1 ) đ?&#x2018;?2 đ?&#x153;&#x2039;3 = Pr(001) = (1- đ?&#x2018;?1 ) (1- đ?&#x2018;?2 ) đ?&#x2018;?3 đ?&#x153;&#x2039;0 = Pr(000) = (1 â&#x2C6;&#x2019; đ?&#x2018;?)3

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Process model • Prior Distribution • One important assumption is that a prior distribution is specified based on how you think species are distributed • Spatial distribution is governed by the Poisson or Neg. binomial • How many birds occur at each site within the study area • Site abundance is a function of the mechanisms governing the distribution 7 Conserving birds and their habitats.

Likelihood ℒ 𝜆, 𝜙, 𝑝 𝑦𝑖𝑡 = 𝑅

𝑖=1

𝑀𝑖 =max 𝑦𝑖𝑡

−𝜆 𝑀𝑡 𝑡 𝑀𝑖 ! 𝑒 𝜆 𝑀 𝑡−𝑦 𝑦𝑖1 𝑦𝑖2 𝑦𝑖3 𝑖 ( 𝜙𝜋 𝜙𝜋2 𝜙𝜋3 𝜙𝜋4 )∗ 𝑦𝑖1 ! 𝑦𝑖2 ! 𝑦𝑖3 ! 𝑦𝑖0 ! 1 𝑀𝑖 !

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Poisson probability density function â&#x20AC;˘ For example number of birds at a given site x=5 and the mean abundance ,lambda =3 the result is a probability of 0.1 đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x153;&#x2020; đ?&#x153;&#x2020;đ?&#x2018;&#x20AC;đ?&#x2018;Ą đ?&#x2018;Ą đ?&#x2018;&#x20AC;đ?&#x2018;&#x2013; !

=

exp(â&#x2C6;&#x2019;3)35 5â&#x2C6;&#x2014;4â&#x2C6;&#x2014;3â&#x2C6;&#x2014;2â&#x2C6;&#x2014;1

= 0.1

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Robust Design â&#x20AC;˘ Primary sampling periods â&#x20AC;˘ Secondary sampling periods time

1 1

2

Primary Periods

3

1

2 2

3 3

1

2

3

Secondary Periods

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Trade time for space Primary Periods = 16 points, (Population is open) Secondary Periods = 3 intervals, (Population is closed)

1 1

2

Primary Periods

3

Secondary Periods

1

2 2

3 3

1

2

... 3

. . . Conserving birds and their habitats.

GRWA Data from Tonto NF â&#x20AC;˘ đ?&#x2018;Śđ?&#x2018;&#x2013;đ?&#x2018;&#x2014;đ?&#x2018;Ą = count at site i during secondary period j within primary period t â&#x20AC;˘ Transects = 35 â&#x20AC;˘ Points = 16 primary periods â&#x20AC;˘ Min intervals = 3 secondary periods

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Grace’s Warbler Covariates • Abundance • Land Fire dataset for habitat covariates • Ponderosa Pine woodland and savanna • Conifer Oak woodland • DEM 30 meter • elevation • Availability • Start time in minutes

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• Detection • Julian Date

Conserving birds and their habitats.

Model Selection Model Conf Oak+Elevation Conf Oak+Elevation+Ponderosa Pine Elevation Ponderosa Pine+Elev Ponderosa Pine+Conf Oak Conf Oak Ponderosa Pine Null

k

AIC

Delta AIC

7 8 6 7 7 6 6 3

294.7 296.6 298.2 300.2 301.7 310.5 312.4 322.5

0 1.9 3.5 5.4 7.0 15.8 17.7 27.7

Model Selection for abundance covariates

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Goodness of fit

(Bootstrap replicates =1000)

â&#x20AC;˘ p = 0.58

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Conifer Oak effects

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Management Tool • Fire Polygons • 2002-2010 • Target restoration efforts

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Spatially Explicit

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Pygmy Nuthatch Model • Abundance (AZ,CO,SD,WY,ID,MT) • Land Fire dataset for habitat covariates • Ponderosa Pine • DEM 30 meter (elevation) • Wildfire

• Detection • Julian Date • Start Time

• Availability • # of snags Conserving birds and their habitats.

Preliminary Results

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Preliminary Results

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Applications •

Model Distribution/Relationships to inform management • Habitat • Climate Change • Bark Beetle • Fire

Develop Priority Conservation Areas • Overlaying predicted distribution • Oil and Gas • Wind Power • Climate Change

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References • Royle, J. A. (2004) Generalized estimators of avian abundance from count survey data. Animal Biodiversity and Conservation 27, pp. 375–386. • Chandler, R. B., J. A. Royle, and D. I. King. 2011. Inference about density and temporary emigration in unmarked populations. Ecology 92:1429-1435.

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Questions?

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Hierarchical Abundance Models

Our objectives were to 1) model the effects of landscape structure on bird abundance, 2) predict the distribution of the species in the Bird...

Hierarchical Abundance Models

Our objectives were to 1) model the effects of landscape structure on bird abundance, 2) predict the distribution of the species in the Bird...