Jobim2011 o gimenez

Modèles à structure cachée pour la
dynamique des populations

Olivier Gimenez
Centre d’Ecologie Fonctionnelle et Evolutive - Montpellier

Population dynamics in the wild

• Conservation: what are the reasons of a population decline
and how to stop it?
• Harvesting: how many individuals can be harvested in a
sustainable way?
• Pest control: what is the most efficient (cheapest) way to
get rid of an alien species?
• Evolutionary ecology: to understand the evolution of life
histories.


Investigating process in natural populations

Long-term individual monitoring datasets

Methodological issues when moving from lab
to natural conditions
Issue 1: detectability < 1
Issue 2: individual heterogeneity (IH)


Investigating process in the wild

Long-term monitoring

Issue 2: individual heterogeneity

Issue of detectability < 1

How to reliably estimate demographic
parameters in the wild?

Individuals may be seen or not

If they’re not... Are they breeding? Are they
on the study site? Are they dead?

Individually mark and monitor individuals:
capture-recapture (CR) data

Incomplete registration
Laplace, P.S., 1786. Sur les naissances les mariages et les morts. Histoire de
l’Académie Royale des Sciences. Année 1783, p. 693

"POPULATION dans le Royaume, compris l’île de Corse,
suivant l'ordre des généralités, pendant l'année 1783."

Why bother with p < 1?

Capture-
recapture
approach

Naïve
approach
with p = 1


1.0
0.8

Survival Capture-
0.6

recapture
approach
0.4

Naïve
0.2

2 4 6 8 10 12 14 approach
Age with p = 1

Bias in survival and rate of senescence
(Gimenez et al. 2008 Am. Nat.)


1.0
0.8
0.6

Capture-
recapture
Survival
0.4

approach
0.2

Naïve
0.0

-4 -2 0 2 4
approach
with p = 1
Body mass

Bias in shape of selection
(Gimenez et al. 2008 Am. Nat.)

Investigating evolution in the wild

Investigating evolution in the wild (Grant,
Reznick, ...)

Long-term monitoring

Issue 2: individual heterogeneity (IH)

Issue of individual heterogeneity

Simple CR models assume homogeneity

From a statistical point of view, IH can cause
bias in parameter estimates – see later on

Issue of individual heterogeneity

Standard CR models assume homogeneity

From a statistical point of view, IH can cause
bias in parameter estimates

From a biological point of view, IH is of
interest – individual quality

What is individual quality?

Quality varies among individuals within a
population
High quality individuals have greater fitness
than low quality ones
⇒ Among-individual heterogeneity that
is positively correlated to fitness
Wilson & Nussey 2009 TREE

What is individual quality?
Quality varies among individuals within a population
High quality individuals have greater fitness than low
quality ones
⇒ Among-individual heterogeneity that is
positively correlated to fitness (Wilson & Nussey 2009)

Why is it so important?
Natural selection can occur if individuals
vary in phenotype and fitness
A response to selection depends on this
variation having a genetic basis
IH may lead to flawed inference

Accounting for individual heterogeneity

CR models do not cope that well with quality
Accounting for individual heterogeneity
If you’re a biologist, rely on empirical
measures (mass, gender, age, experience, etc.)
How to incorporate this information?

If you’re a statistician, intrinsic property of
individuals
How to filter out the signal from noisy observations?

Capture-recapture models

Introduction: CR data
How to account for individual variation
Case study 1: estimating abundance
Case study 2: detecting trade-offs

Can quality have a genetic basis or is it a
consequence of environmental effects?
Case study 3: quantifying heritability

Perspectives

Common marking methods
• Ear tags for mammals / leg bands for birds.

• Passive integrated transponder (PIT) tags.

Marking by camera-trap / photo-identification
lynx whales

Marking by noninvasive genetic sampling
• Individuals are uniquely identified using
microsatellite profiling on hair, dung, … samples
bear (hair) bat (droppings)
wolf (dung)

orang-utan (hair) elephant (dung)

Capture-recapture data

An encounter history: hi = (1 0 1)

Modelling CR data

φ

1 0 1

Survival probability φ

Modelling CR data

φ

1 0 1

1− p

Detection probability p

Modelling CR data

φ φ

1 0 1

1− p

Modelling CR data

φ φ

1 0 1

1− p p

Modelling CR data

φ φ

1 0 1 Pr (hi ) = φ (1 − p ) φ p

1− p p

Survival probability φ
Detection probability p

Modelling CR data

A probabilistic framework

Pr (hi ) = φ (1 − p ) φ p

Central role of likelihood (frequentist / bayesian)

L = ∏ Pr (hi )
i

How to account for IH in
φi

Cas 1: estimation de l’abondance

L’exemple de la recolonisation du loup (Canis lupus)
dans les Alpes
Sarah Cubaynes & Lucile Marescot

- Cubaynes et al. (2010). Importance of accounting for detection heterogeneity when
estimating abundance: the case of French wolves. Conservation Biology. 24:621-626.
- Marescot et al. (2011). Capture-recapture population growth rate as a robust tool against
detection heterogeneity for population management. Ecological Applications.


L’exemple de la recolonisation du loup (Canis lupus)
dans les Alpes


Echantillonnage (ONCFS) Séquençage ADN (LECA)

Des données génétiques de capture-recapture


Echantillonnage (ONCFS) Séquençage ADN (LECA)

Des données génétiques de capture-recapture
Les individus dominants sont plus facilement « capturés » (marquage
du territoire)

La population est un mélange de 2 classes d’individus :
« facilement capturables » et « difficilement capturables »

L’information à laquelle on aimerait accéder

Les états :
• Vivant et facilement capturable (L)
• Vivant et difficilement capturable (H)
• Mort (†)

Les informations dont on dispose…

Les états :
• Vivant et facilement capturable (L)
• Vivant et difficilement capturable (H)
• Mort (†)

Les données : Présence (1) / Absence (0)

Modèle de Markov caché (Pradel 2005 Bcs)

États initiaux : L H †
Π = (1 − π π 0)

Transition entre états (survie) :
L H †
L φ 0 1−φ
Φ=
H 0 φ 1−φ
† 0 0 1

L’hétérogénéité de capture

Lien entre observations et états :

Pas
détecté Détecté

L 1 − pL pL
B = H 1 − pH pH
† 1 0

pL : probabilité de capture des individus facilement capturables
pH : probabilité de capture des individus difficilement capturables

Probability of a capture history

Under homogeneity, the capture
history ‘101’ has probability

Pr(101 = φ ⋅ (1− p) ⋅φ ⋅ p
)

φ is survival
p is detection for all individuals

Probability of a capture history

Under heterogeneity:

( ) ( )
Pr(101 = π ⋅φ ⋅ 1− pL ⋅φ ⋅ pL + (1− π ) ⋅φ ⋅ 1− pH ⋅φ ⋅ pH
)
π is the probability that the individual
belongs to state L
pL is the detection for lowly detectable
individuals
pH is the detection for highly detectable
individuals

In brief: 2-step analysis for estimating abundance

Model selection procedure (AIC)

1) Patterns of temporal variation in survival and detection
Constant, seasonal or annual?

2) Patterns of individual variation in detection
Homogeneity vs Heterogeneity
Plug in parameters of
the model best
supported by the data
Abundance estimation with heterogeneity

ˆ ≈ m + π × u + (1 − π )× u
N
ˆ ˆ
pH
ˆ pL
ˆ pH
ˆ

Is detection heterogeneity needed?

Model AIC
φ(.), p(state,season) 2022.28
φ(.), p(season) 2150.09

Homogeneity vs. Heterogeneity
detection probability
1.0

1.0
Homogeneity Heterogeneity
0.8

0.8
0.6

0.6
0.4

0.4
0.2

0.2
0.0

0.0

Summer Autumn Winter Spring Summer Autumn Winter Spring

Homogeneity vs. Heterogeneity
detection probability
1.0

1.0
• Detectability strongly Heterogeneity
differs between classes
0.8

0.8
H individual = dominant
L individual = young +
0.6

0.6
subordinate
0.4

0.4
0.2

0.2
0.0

0.0

Summer Autumn Winter Spring Summer Autumn Winter Spring

Seasonal variation in abundance

N
111

25
64
10 29
0

Overall growth of the population
Marked seasonal variations

Ignoring detection heterogeneity leads
to strong bias in abundance estimation
Heterog.
Homog.

N

64 [29 ; 111]

33 [17 ; 54]

Underestimation by 50% of abundance

Ignoring detection heterogeneity leads
to bias in survival estimation

0.83 (0.06)
0.9

0.68 (0.08)
φ
0.8
survie annuelle

Homogeneity vs.
heterogeneity in
0.7

detection
0.6

CJS H

underestimation by12% for survival

M. Buoro (thèse, co-dir. E. Prévost1)

1 UMR INRA/UPPA Ecobiop, Saint Pée s/ Nivelle, France
Photo: Paul Nicklen (National Geographic)

Assessing life-history tradeoffs in the wild

• Natural selection favours individuals that
maximize their fitness

• Limited resource acquisition: strategy of
resource allocation

• Trade-off between traits related to
fitness

• Issue of detectability, again

State-space modelling of CR data
(Gimenez et al. 2007)
Dynamic process model

Hidden states

Xi,t-1

Xi,t


Hidden states

Xi,t-1

φi,t

survival
Xi,t


Hidden states

Xi,t-1

φi,t

survival
Xi,t

State equation

Dynamic process model Observation

Hidden states Observations

Xi,t-1 Yi,t-1

φi,t

survival
Xi,t Yi,t


Hidden states

Xi,t-1 Yi,t-1

detection

Xi,t Yi,t Pt

Observation equation


Hidden states Observations

Xi,t-1 Yi,t-1

φi,t detection

survival
Xi,t Yi,t Pt

State equation Observation equation

Atlantic salmon life cycle

Reproduction Development of
juveniles
Freshwater

Migration to stream Migration to sea
Growth at sea

Sea

Atlantic salmon life cycle

Development
of juveniles
Freshwater

Sea

Juveniles Autumn

1st year of life

Migrants Spring

Freshwater

Sea

Juveniles Autumn

1st year of life

Migrants Residents Spring

Freshwater

Sea

Juveniles Autumn

1st year of life


Freshwater
Sexual maturation Autumn
(males)
2nd year of life

Migrants Spring

Sea

Juveniles Autumn

1st year of life


Freshwater
Sexual maturation Autumn
(males)
2nd year of life

Migrants Spring

Adults

Sea

Juveniles Autumn
Winter survival 1st year

Résidents + 1
Migrants
an
Spring

Freshwater
Is there a tradeoff between
migration and winter survival?

State-space model Juveniles

Migrants Residents


Juveniles
marked in
autumn

Migrants
recapture in
spring


Migrants Residents


Juveniles
Migration Migration
probability choice
marked in
autumn

Migrants
recapture in
spring


Migrants Residents

Probabilistic reaction norm

Juveniles
Migration Migration
Size
probability choice
marked in
autumn

Migrants
recapture in
spring


Migrants Residents


Juveniles
Migration Migration
Size
probability choice
marked in
autumn

Migrants
Survival Migrants
probability Survivor
recapture in
spring


Migrants Residents

Observation
Selective mortality

Juveniles
Migration Migration
Size
probability choice
marked in
autumn

Migrants
Random Survival Migrants
recapture in
effect
spring


Migrants Residents


Juveniles
Migration Migration
Size
probability choice
marked in
autumn

Detection
probability

Migrants
Random Survival Migrants
recapture in
effect
spring

α1 T0
β1
κ

α2 T0.1 S0.1
β2
Φ1 Lf
Φ2 T1 S1

T1.1 pL1
Ψmâle

S1.1 S1.2
T1.2.1 T1.2.2

S1.1.1 S1.1.2 S1.2.1 S1.2.2 IV1
T2.2 Pr.det T1.2.2det

pP1
A1
T1.2capt T2.2capt T1.3.1 T1.3.2a T1.3.2b T1.3.3

pA1
pC1

S3 S2.1 S2.2 S2.3

δ2
Φ3 Φ3
δ1
Processus d’observation
Processus d’observation des
des smolts 2+ (marqués
pL2 smolts 2+ (marqués au stade
au stade tacon 1+) au
tacon 0+) au printemps 2007
printemps 2007 i in 1:1851

j in 1:286
IV2 pP2

Results (1)

1.0
Migration Probability
0.2 0.4 0.6 0.8

Size-dependent
probabilistic reaction
norm for age at
0.0

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 migration
Size (mm)

Buoro, M., Prévost, E. and O. Gimenez (2010). Investigating evolutionary
trade-offs in wild populations of Atlantic salmon (Salmo salar): incorporating
detection probabilities and individual heterogeneity. Evolution. 64: 2629-2642

Results (1)

1.0
0.2 0.4 0.6 0.8

Size-dependent
norm for age at
0.0

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 migration
Size (mm)

Juveniles longer than 100 mm in autumn has a probability to migrate close to 1.

Results (1)

1.0
0.2 0.4 0.6 0.8

Size-dependent
norm for age at
0.0

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 migration
Size (mm)

A juvenile of 90 mm has 50% of chance of migrating to the sea at 1year of age.

Results (1)

1.0
0.2 0.4 0.6 0.8

Size-dependent
norm for age at
0.0

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 migration
Size (mm)

A juvenile of 90 mm has 50% of chance of migrating to the sea at 1year of age.
Juveniles shorter than 60 mm in autumn has a probability to migrate almost null.

Results (2)

1.0
0.2 0.4 0.6 0.8

Size-dependent
norm for age at
0.0

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 migration
Size (mm)
Selective mortality
1.0

Migrants
0.8

Survival cost in
Survival Probability

deciding to stay an
0.6

Residents extra year in
0.4

freshwater
0.2
0.0

Heritability in the wild
Animal models: mixed models incorporating
genetic, environmental and other factors.
Capture-recapture models: assess demographic
parameters with p < 1 and individual variability.
The idea of combining Animal and Capture-
recapture models is in the air (O’Hara et al. 2008;
Cam 2009).

The idea is in the air (O’Hara et al. 2008)

Cam 2009).
For (demographic) parameters strongly related to
fitness, we expect low heritability. But, predictions
not so clear in wild populations

Cam 2009).
For (demographic) parameters strongly related to
fitness, we expect low heritability. But, predictions
not so clear in wild populations
M2R BioStat J. Papaïx & S. Cubaynes’s PhD (co-dir.
C. Lavergne)

Introducing the threshold model
• Survival is related to a continuous underlying latent
variable li,t which, given Xi,t = 1, satisfies
1 if li,t > κ
X i,t +1 = 
0 if li,t ≤ κ
• where κ is a threshold value, and li,t is usually
referred to as the liability

Liability

li,t = Xi,t +1 Xi,t = 1

Introducing the threshold model
• Survival is related to a continuous underlying latent
variable li,t which, given Xi,t = 1, satisfies
1 if li,t > κ
X i,t +1 = 
0 if li,t ≤ κ
• where κ is a threshold value, and li,t is usually
referred to as the liability with

(
li,t ~ N µi ,t , σ ε2 )
• For identifiability reasons, κ = 0 and σε = 1 without
loss of generality

Plug in the animal model
It can be shown that φi,t = F µi ,t ( ) where F is
the c.d.f. of a N(0,1)
Then, we write:
F (φi,t ) = probit (φi,t ) = µi ,t = η + bt + ei + ai
−1

Then, we write:
−1

mean survival

Then, we write:
−1

mean survival

yearly effect

(
bt ~ N 0, σ t
2
)

Then, we write:
−1

mean survival
non-genetic effect

yearly effect
(
ei ~ N 0, σ e2 )
(
bt ~ N 0, σ t
2
)

Then, we write:
−1

mean survival
non-genetic effect

yearly effect (
ei ~ N 0, σ e2 )
(
bt ~ N 0, σ t2 ) additive genetic effect

(a1,K, aN ) ~ MN (0,σ 2
a )
A

Calculating heritability

- Capture-recapture animal models (CRAMs)

- Decomposing components of variance in survival

- Heritability = contribution of genetic variance to the total

σ a2
h2 = 2 2 2
σ t + σ e + σ a +1

Case study on blue tits in Corsica
Mark-recapture data Social pedigree

• Blue tits – Study site in
Corsica. 654 individuals,
• 1979 – 2007 ⇒ 29 years of 218 fathers (sires),
monitoring)! 215 mothers (dams),
12 generations.

Résultats
Julien Papaïx

Papaïx, J., S. Cubaynes, M. Buoro, A. Charmantier, P. Perret and O. Gimenez
(2010). Combining capture–recapture data and pedigree information to assess
heritability of demographic parameters in the wild. Journal of Evolutionary
Biology. 23: 2176-2184

Additive genetic variance

Posterior median = 0.110,
95% credible interval = [0.006; 0.308]

Is survival heritability important in blue tits?

Model selection

Conclusions
CR methodology is catching up with ‘p=1’ world
(medicine)

IH needs to be accounted for, otherwise
Bias in abundance estimation (PhD S. Cubaynes)
Obscur life-history tradeoffs (PhD M. Buoro)

Recent statistical methods: hidden-Markov model
and state-space models - cope with IH when p < 1

If possible, biological view – measure quality

Perspectives - Methods
Continue efforts in developing methods to properly
account for individual heterogeneity

Estimation des états cachés – Viterbi ou autre

Semi-chaîne de Markov pour modéliser la durée de
séjour dans un état (R. Choquet, collab. Y. Guédon)

Fit and compare models (PhD S. Cubaynes)
Is heritability important in blue tits (model selection)?
Shall we go for discrete or continuous heterogeneity?
Speed up estimation (algorithms; random effects)?

Software implementation

Implementation issues: software

Program E-SURGE
Hidden Markov models, individual
covariates, mixtures and individual
random effects

Implementation issues: software

Program E-SURGE
Hidden Markov models, individual
covariates, mixtures and individual
random effects

R. Choquet & E. Nogué

Perspectives - Biology
Consider other demographic parameters (dispersal
and breeding probabilities e.g.);
→ A. Charmantier, B. Doligez, E. Cam, B. Sheldon

Fixed vs. dynamic individual heterogeneity:
→ E. Cam and S. Tuljapurkar

From individuals to species
→ E. Papadatou’s post-doc & S. Cubaynes’s PhD
→ Museum for community ecology aspects

Integrating evolutionary and demography views:
→ S. Servanty’s post-doc and M. Gamelon’s PhD

Estimating abundance in open populations

Standard capture-recapture models provide
estimates of survival and detection probabilities
An estimate of abundance N is obtained as:

ˆ=n
N
ˆ
p

Estimating abundance in open population


Number of
individuals
ˆ=n
N
detected
ˆ
p

Estimating abundance in open population


ˆ=n
N
ˆ
p
Estimated
detection
probability

What if heterogeneity in detection?
The number of counted individuals can be split
into two quantities
Newly marked (u) and previously marked (m)

n=u+m

into two quantities

n=u+m

Number of previously marked individuals
with probability pH

into two quantities

n=u+m

Number of unmarked individuals, made of:
• π×u individuals with low capturability pL and
• (1-π)×u individuals with high capturability pH

Abundance with detection heterogeneity
An estimate of abundance N accounting for
heterogeneity is obtained as:

ˆ ≈ m + π × u + (1 − π ) × u
N
ˆ ˆ
ˆH
p pˆL pˆH

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