Spatial Features of Growth, Genetic Diversity and Weather Extremes in Population Models

Simone Vincenzi EU Marie Curie Fellow University of California Santa Cruz Polytechnic of Milan

SIAM meeting, San Diego 2013

People  

Giulio  De  Leo  

Alain  Crivelli  

Marc  Mangel  

Dusan  Jesensek  

Slovenian  field  crew  

Hans  Skaug  

Gianluigi  Rossi  

Population ecology in the next decade

Past environments

Evolutionary history

Mean traits

Plasticity

Genetic variation

Climate change Novel environment

Individual fitness

Evolution

Population performance

Population size

Persistence

Environmental change Habitat fragmentation

Model system Marble trout

100 km

Slovenia

Methods

•  Random effects models using Automatic Differentiation Model Builder (ADMB)

•  Multi-state models for movement

•  Stochastic simulations of genetically-explicit models

•  Methods for modeling weather extremes

Topics

1.  Space, density and growth 2.  Space and genetic structure 3.  Space and weather extremes

Topics

1.   Space and growth 2.  Space and genetic structure 3.  Space and weather extremes

Marble trout •  Resident stream-living

salmonid

•  High plasticity of body size, 1 up to 20-25 kg

•  Spawning in November

•  Hatching in March

•  Maximum age 6 to 8 (in the studied streams)

•  Low movement

Marble trout populations 3 basins

30-1000 fish in each population

Isolated for 1000s of yrs

High among-population genetic differentiation

Extremely low within-population genetic variability

Baca Idrijca Soca

10 km

Fragmentation 500 in 1996

600 in 1998

5 km

Do physical barriers influence growth?

Observations for growth in length

Somatic growth trajectories

Gacnik more than 8,000 unique fish sampled since 1998 Zakojska more than 1,500 unique fish sampled since 1996

Von Bertalanffy growth function

dWdt

= aW (t)2/3 ! bW (t)

W (t) = !L(t)3

dLdt

= q ! kL

1)

2)

3)

4)

Parameters historically estimated at the population level

Random effect vB growth function •  Fixed effect Maximum likelihood •  Random effect “shrinkage” •  Not considering autocorrelation leads to biased

estimates

x = covariates u,v = individual random effects

Automa,c  differen,a,on  model  builder  -­‐  ADMB  

•  Developed  for  fisheries  problem  •  Es2ma2on  of  highly-­‐complex  nonlinear  models  •  Might  be  seen  as  an  alterna2ve  to  Bugs,  Jags  •  Very  fast  and  extremely  flexible  •  Clear  measure  of  convergence  •  ADMB-­‐RE  module  •  Empirical  Bayes  method  •  Random  effects  allow  to  borrow  strength    

Fournier, D.A., et al. 2012. AD Model Builder: Using automatic differentiation for statistical inference of highly parameterized complex nonlinear models.Optimization Methods and Software 27:233–249

Automa,c  differen,a,on  model  builder  -­‐  ADMB  

•  Developed  for  fisheries  problem  •  Es2ma2on  of  highly-­‐complex  nonlinear  models  •  Might  be  seen  as  an  alterna2ve  to  Bugs,  Jags  •  Very  fast  and  extremely  flexible  •  Clear  measure  of  convergence  •  ADMB-­‐RE  module  •  Empirical  Bayes  method  •  Random  effects  allow  to  borrow  strength    

Fournier, D.A., et al. 2012. AD Model Builder: Using automatic differentiation for statistical inference of highly parameterized complex nonlinear models.Optimization Methods and Software 27:233–249

Borrowing  strength  

Hindcasting Zakojska Gacnik

R2 >0.95

Relationship between REs

Real validation •  We should have standard for prediction in

ecology

Real validation •  Can I predict future trajectories with just one

observation?

Real validation

fish 1 fish 2 fish 3 fish 4

•  Can I predict future trajectories with just one observation?

Real validation •  Can I predict future trajectories with just one

observation?

“missing” data

Real validation •  Can I predict future trajectories with just one

observation?

•  Compare model predictions with those based on mean observed length at age

Prediction R2

MAE

“missing” data

Zakojska

Zakojska, other sample

Zakojska, other sample

3903

Let’s do a background check Change sector age 13

1 3

vB  growth  func,on  by  sector  

W = worst B = best

W

B

vB  growth  func,on  by  sector  

W = worst B = best

B

W

Topics

1.  Space and growth 2.   Space and genetic structure 3.  Space and weather extremes

Population structure

P = average number of pairwise differences between two individuals sampled from different or the same populations

Different formulations depending on molecular marker and number of loci

Biallelic single locus 5-20 microsatellites >200k SNPs

Global FST = 0.6 (pairwise 0.3-0.8)

Fumagalli, L et al. 2002. Extreme genetic differentiation among the remnant populations of marble trout (Salmo marmoratus) in Slovenia. Molecular Ecology 11:2711–2716

10 km

Population structure

Consequences of fragmentation for genetic structure

Multi-state model

a = number recaptured in A b = number recapture in B

But, an individual has to:

–  Remain in study area (survive with probability S) –  Move to B with probability y or remain in A with probability 1- y –  Be captured in A with probability pa or in B with probability pb

A B

only if pa = pb

is proportional movement

Consequences of fragmentation for genetic structure

0.14

0.17

0.11

0.10 Software MARK

Consequences of fragmentation for genetic structure

0.14

0.17

0.11

0.10

Let’s simulate the demographic and genetic dynamics

20 loci 3 alleles neutral 250 fish 100 yrs

Consequences of fragmentation for genetic structure

Locus 1

Locus 2

Global FST computed every 5 years

Sectors as populations

20 loci 3 alleles neutral 250 fish 100 yrs

Consequences of fragmentation for genetic structure

FST increases and then stabilizes

A genetic structure can quite rapidly develop

0.14

0.17

0.11 0.10

Topics

1.  Space and growth 2.  Space and genetic structure 3.   Space and weather extremes

Max F

low

Time (yrs)

100-yr flood

Climate change and extreme events

Catastrophes

Climate change

increased intensity, altered frequency and seasonality of catastropic events

Marble trout and floods

Major flood

Medium flood

YEAR STREAM BASIN 99 00 01 02 03 04 05 06 07 08 09 10

Huda

Zakojska Baca

Gorska

Lipovesck Soca

Zadlascica

Trebuscica

Studenc Idrijca

Idrijca

Gatsnick

Svenica

9 10

1

Stream discharge

10 km

Block maxima

Extremes in time series

•  Increasing  interest  and  new  methods    •  Mo2veted  by  climate  change  and  recent  stock  market  crashes  

•  Time-­‐trends  

•  Spa2al  correla2on  of  extremes  – Regional  extremes  – Predic2on  of  extremes  at  unobserved    loca2ons  

Extremes in time series

•  Increasing  interest  and  new  methods    •  Mo2veted  by  climate  change  and  recent  stock  market  crashes  

•  Time-­‐trends  

•  Spa2al  correla2on  of  extremes  – Regional  extremes  – Predic2on  of  extremes  at  unobserved    loca2ons  

GEV

Trends in GEV (MLE estimation) Non-homogenous trends

Stream discharge

10 km

Next step: spatial correlation of extremes

Points to bring home (thank you) Ecology

•  Spatial scale, habitat structure and density have profound effects on demography, life histories and genetic structure of individuals and populations

•  We need long-term (individual) data

Mathematical and statistical methods

•  ADMB-RE allow fast and reliable estimation of parameters of complex models

•  Diversity of methods needed