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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 • Components : • Observations: đ?‘Śđ?‘– =count of birds at point i, “dataâ€? – Observation model [đ?‘Śđ?‘– | đ?‘ đ?‘– ] âˆź Multinomial đ?‘ đ?‘–đ?‘Ą , đ?œ‹ đ?‘? • đ?‘ đ?‘–đ?‘Ą = local population size • p = probability of encountering an individual • đ?œ™ = probability that an individual in the grid population is available at the point

• state variable: outcome of ecological process of interest – Process model • đ?‘ đ?‘– ~ đ?‘ƒđ?‘œđ?‘–đ?‘ đ?‘ đ?‘œđ?‘› đ?œ† • Log đ?œ†đ?‘– = Îą + đ?›˝đ?‘Ľ;

Conserving birds and their habitats.


Hierarchical Model Metapopulation parameters

Local parameters

Data

đ?‘ 1 ,đ?‘?1

đ?‘Ś1

đ?›ź, β, ∑đ?œ™ , ∑đ?‘? đ?‘ 2 ,đ?‘?2

‌

đ?‘ đ?‘› ,đ?‘?đ?‘›

đ?‘Ś2

‌

đ?‘Śđ?‘›

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Generalized Multinomial Mixture Model • Model đ?‘€đ?‘– ~ đ?‘ƒđ?‘œđ?‘–đ?‘ đ?‘ đ?‘œđ?‘› đ?œ† đ?‘ đ?‘–đ?‘Ą âˆź Binomial đ?‘€đ?‘– , đ?œ™ đ?‘Śđ?‘–đ?‘—đ?‘Ą âˆź đ?‘€đ?‘˘đ?‘™đ?‘Ąđ?‘–đ?‘›đ?‘œđ?‘šđ?‘–đ?‘Žđ?‘™ đ?‘ đ?‘–đ?‘Ą , đ?œ‹ đ?‘?

• Parameters

• đ?‘€đ?‘– - latent variable, avg. abundance in a grid • đ?œ™ – probability that an individual in the grid population is available at the point • p – detection probability

Conserving birds and their habitats.


Observation model • Multinomial cell probabilities • For removal sampling study with 3 intervals, the multinomial cell probabilities are đ?œ‹ = đ?‘?, 1 − đ?‘? đ?‘?, 1 − đ?‘? 2 đ?‘?, (1 − đ?‘?)3 đ?œ‹1 = Pr(1xx) = đ?‘?1 đ?œ‹2 = Pr(01x) = (1- đ?‘?1 ) đ?‘?2 đ?œ‹3 = Pr(001) = (1- đ?‘?1 ) (1- đ?‘?2 ) đ?‘?3 đ?œ‹0 = Pr(000) = (1 − đ?‘?)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 • 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?‘’ −đ?œ† đ?œ†đ?‘€đ?‘Ą đ?‘Ą đ?‘€đ?‘– !

=

exp(−3)35 5∗4∗3∗2∗1

= 0.1

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Robust Design • Primary sampling periods • 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 • ���� = count at site i during secondary period j within primary period t • Transects = 35 • Points = 16 primary periods • 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)

• 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...

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