Using statistical models to study
climate-disturbance-plant
interactions
Andrew Latimer –
amlatimer@ucdavis.edu
Adam Wilson – adam.wilson@yale.edu
Cory Merow – cory.merow@gmail.com
Mediterranean climate
• Key features: warm temperate, with cool wet winters
and hot dry summers.
Climate change effects?
The obvious ones:
• Warmer winter -> more growth
• Hotter summer -> more fire
Complications
May raise growth rate while hammering
other parts of life cycle (e.g. survival)
Fire-plant feedbacks can
produce rapid shift
-> fuel starvation
-> flammability
Feedback example – Sierra forest
• When feedbacks positive, slight shift in
disturbance regime can cause state change
Moderate
fire
(-)
e.g. Collins et al.
(2009) Ecosystems
Less fuel,
milder fire
Intense fire
(+)
More shrubs,
intense fire
e.g. Stevens et al.
(2013) Can.J.For.Res.
Photos: Jens Stevens
Fire-plant interaction example: South African fynbos
No fire
(fire
suppressed)
Fire
No fire
(plant
growth
suppressed)
Population size
More frequent fire can eliminate
populations
8-year intervals
between fires
4-year intervals
between fires
Stevens & Beckage, Oecologia 2010
Climate
Growth,
Survival etc.
Ignition
and spread
Disturbance
Fire
Regrowth of
Fuel,
flammability
Plant species,
communities
Challenges – ecological data
• Scales of process can be large
– Field data collection is almost always not
• Interpolation and extrapolation (worse in
projections)
Current
Future
Environment
Challenges – environmental data
• Highly multivariate environmental measurements
– Sensor outputs
– Model outputs
– Heterogeneous data sources
• Multivariate responses
– Community: Many species, life stages
– Individual: phenotypes, genotypes
• What’s important and why?
– Q: Are the factors on both sides sparse?
Outline
1)
2)
3)
4)
A little more introduction
Fire return times
Biomass regrowth
Demography
Study system for this talk
Cape Floristic Region of South Africa
• Fynbos shrubland interfacing with karroo desert
• Evolutionary radiation -> very high plant diversity and
endemism
• Diversity concentrated in ~30 lineages over 0.5-30 MY
CFR climate change
Historical patterns
Recent change (1950-2000)
Total Spring/Summer Precipitation
Change
per
Year (mm)
(mm)
Change
per
decade
High : 10.97
Low : -10.06
Climate change projections
CMIP5 multimodel average
2081-2100 -- RCP8.5
Stippling: ≥ 8/11 models agree on sign
Downscaled projections: Adam Wilson
Thomas et al. (2004) Nature.
Midgley et al. (2006). Diversity & Distributions.
Example 1: Fire occurrence
What affects fire frequency?
Climate
Growth,
Survival etc.
Ignition
and spread
Disturbance
Fire
Regrowth of
fuel
Plant species,
communities
Cape region fire data
Cape Nature Fire Database (1927-2005)
18°0'0"E
18°30'0"E
19°0'0"E
19°30'0"E
20°0'0"E
20°30'0"E
21°0'0"E
21°30'0"E
22°0'0"E
22°30'0"E
23°0'0"E
31°30'0"S
32°0'0"S
34°30'0"S
34°30'0"S
34°0'0"S
34°0'0"S
33°30'0"S
33°30'0"S
33°0'0"S
33°0'0"S
32°30'0"S
32°30'0"S
32°0'0"S
31°30'0"S
Fire_Code
SWBG/12/1988/01
Reserve_NME Swartberg
Date
12/1998
Perimeter
38,326 meters
Area
3,606 hectares
Veld_Age
8 years
Cause
Unknown
18°0'0"E
0 1020
40
18°30'0"E
60
19°0'0"E
80 100
Kilometers
19°30'0"E
1:2,500,000
20°0'0"E
20°30'0"E
21°0'0"E
21°30'0"E
22°0'0"E
22°30'0"E
23°0'0"E
Thanks to
Helen
DeKlerk
Fire data preparation
•
•
•
•
Overlay ~2km x 2km grid on reserve areas
Seasonal time resolution (3 months)
Cell considered “burned” if >50% area burned
Voxels of weather data:
– 80 seasons x 2611 cells = 208,880 voxels
Issues…
• Very multivariate weather data
• Approach: reduce dimensionality by biological
intuition, hypothesis
• Chris Wikle:
“Sensible science-based parameterizations or
dimension reductions”
Issue: highly multivariate environmental data
Mean Precipitation in December (mm)
−31
80
70
−32
60
50
−33
40
30
−34
20
10
18
20
22
24
Weather station data
Daily, point locations
Included in WorldClim (50)
Available Stations (523)
Bayesian kriging
with covariates
(Adam Wilson)
Gridded weather data
Daily, ~2km grid
1.290
1.280
f(ppt.year.lag1)
1.282
f(ppt.year.mean)
1.280
1.270
150
Meetings (many…)
Soil moisture model
1.278
200
-32
1.284
50-year average maximum drought length
-1
0
1
2
-1
0
ppt.year.mean
-33
1
2
ppt.year.lag1
1.29
1.31
1.0
1.25
1.27
f(gdd.grseas.mean)
1.6
1.4
50
-34
1.2
f(ppt.year.anom)
1.8
100
-1.5
-0.5
0.5 1.0 1.5 2.0
-2
-1
0
0
20
22
24
Indices -- i.e. hypotheses
Seasonal to yearly
Lon
Choice of methods
ppt.year.anom
gdd.grseas.mean
Modeling
1
Fire history data
Timeline for one grid cell
6 years
14 years
?
?
1980
(beginning
of record)
Other hypothetical fire histories:
.
fires
2000
(end
of record)
Nonparametric survival model
Zi = observed time between a pair of fires in cell I
Pit : P(Zi > t | Zi > t-1) i.e. probability of
“surviving” season t without a fire
Climate: Long-term mean
Probit(P) = XTβ + e
Weather: anomalies in each
grid cell * season
Climate index (AAO)
Random effects: season, sub-region
Note Alan Gelfand’s 2013 paper adding spatial random effects
Dealing with censoring
• Left censoring: no previous fire observed in record
Unobserved pit predicted from XTβ
{Unobserved fire times} ~ Multinom(1-pit, t in [1950-1979])
Gives length distribution of unobserved fire intervals
# Note may inflate fire frequency because a few cells
will have failed to burn in these 30 years
?
Wilson et al. (2010) Ecological Modelling
Cumulative fire probabilities
Wilson et al. (2010) Ecological Modelling
Expected fire return times
Wilson et al. (2010) Ecological Modelling
Importance of large-scale atmospheric
circulation patterns
• Importance of AAO brings us back to complex
multivariate problem (why AAO??)
• But in this case pretty clear how it works
• Note trends in ozone depletion (ozone hole)
associated with positive AAO
– Manatsa et al. (2013) Nature Geosci.
Abram, et al. (2014) Nature Climate Change doi:10.1038/nclimate2235
Part 2: modeling postfire regrowth
Climate
Growth,
Survival etc.
Ignition
and spread
Disturbance
Fire
Regrowth of
fuel
Plant species,
communities
Remote sensing:
Watching plants grow from space
• MODIS terra NDVI data
Model
• Functional form that can match recovery pattern
• Parameters
– αi : minimum NDVI
– γi : difference between min and max NDVI
– λi: recovery rate parameter
– Αi : amplitude of seasonal variation
Wet, coastal
~4 years
Dry, interior
~8 years
Wilson et al. in prep
Spatial variation in regrowth rates
Regrowth rate based on
Threshold NDVI value
(95% of max NDVI)
Wilson et al. in prep
Tying this back to demographic models
Factors related to regrowth rate
Wilson et al. in prep
Projected change in recovery time
Years
longer
shorter
Comparing recovery time
to observed fire return intervals
Some correspondence
Slowest growing areas
may be burning too often
Conclusions on fire
• Climate change likely to shorten fire return times
• And also increase regrowth rates in many areas
– But: Lower warm-season precipitation in the west,
hotter summers
• Big shifts possible
– Frequent fire in slow-recovery areas
– Fuel starvation and fynbos contraction (?)
Example 3: Demography
Tradeoff:
-- More biological detail
-- MUCH less data
Climate
Growth,
Survival etc.
Ignition
and spread
Disturbance
Fire
Regrowth of
fuel
Plant species,
communities
Focal species: Protea repens
Data: Protea Atlas Project
Tony Rebelo of SANBI
Goal: model population performance
across region
• Measure across gradients
• Demographic rates:
– Growth & Fecundity
– Mortality (dead adults)
– Recruitment (seedlings per parent)
Issue: recruitment sites limited by
fire occurrence
Analysis issues
• Misalignment
– Growth and seed production measured at
different sites from seedling recruitment
• Missing some steps in life cycle
– Seedling emergence and survival
Scaling issues raised by Jim and Alan
Integral projection model (IPM)
n t+1 = A n t
nt+1 (z) =
t
z
z'
nt(z)
nt+1(z’)
K(z’,z)
ò
all
sizes
K(z', z) nt (z)dz = time
= size at t
= size at t+1
= size distribution of individuals at t
= size distribution of individuals at t+1
= projection kernel
(Matrix)
(IPM )
Life History: Perennial shrub
nt+1 (z') = ò
“kernel”
W
[ P(z', z) + F(z', z)]
state
vector
nt (z) dz
P(z’,z) = (survival) *
(growth)
F(z’,z) =
(mean # flowers/plant) *
(mean # seeds/flower)
(establishment probability)*
(offspring size)
mean = b0 + b1size + b2 size + b3 Rain + b4Temp
2
0.06
0.08
4
0.04
2
0.02
8
6
2
8
4
0.00
6
0.06
Size (t)
8
Size (t)
0.04
0.06
0.08
0.00
8
0.02
8
6
6
0.08
4
4
Size (t+1)
0.04
2
2
Size (t)
4
6
8
0.00
2
0.02
Size (t)
0.02
0.04
4
0.04
4
0.06
6
0.08
8
0.06
6
Size (t+1)
Size (t+1)
0.02
2
0.00
8
6
8
6
4
Size (t+1)
6
4
4
Size (t+1)
2
0.00
2
2
2
0.08
8
Vital Rate Regression: Growth
Size (t)
Maps of mean model-estimated demographic rates
Jump into the bog of elasticity
Merow et al. 2014 Ecography
Mean modeled pop growth rate
If we treat this as species distribution model
And assume presence predicted where λ ≥ 1.0
Merow et al. 2014 Ecography
Predictions
Forecasts
Merow et al. 2014 Ecography
Conclusions
• Multivariate data hard to escape
– Not really enough to rely on intuitions
– Still left with the question: what are the
interesting parts of big data sets
• Building some mechanism into model isn’t
enough to reliably get at key transitions
– Problems of underfitting and extrapolation
– Can get individual processes but not their
interactions
Acknowledgments
UC Davis:
Jens Stevens, Melis Akman
Other US:
John Silander (Uconn)
Alan Gelfand (Duke)
South Africa
Tony Rebelo (SANBI)
Jasper Slingsby (SAEON)
Helen de Klerk (CapeNature)
Field crew members: Rene Wolmarans,
Bianca Lopez, Lisa Nupen
Funding Sources:
NSF Dimensions of Biodiversity grant
DEB 10-45985
UC Davis Dept of Plant Sciences
IPM models: Danish Research Council
(via Signe Normand)
Predicting community response to
extreme drought event
• Resample 3 kinds of plots
– Long environmental gradients (spatial variation)
– Time series data (15 years in 80 plots at one site)
– Experimental water manipulations
• What kinds of responses can we predict?
– Abundance of different kinds of species
– Shifts in plant type or function
• Which kinds of data predict best?
Vital rate models
Growth
Average growth/year
Survival
% Survival
~
~
Environment
Size + Environment
Fecundity
Flowering Probability
# Flowers/Individual
Seeds/Flower
=
Germination
=
~ Size + Environment
~ Size + Environment
74
1.1%
Offspring size
Size
~ Environment
Forecasts
Merow et al. 2014 Ecography
Predictions – uncertainty
Map of collection sites
Jane Carlson, Nicholls State U.
Common garden experiment
• Collected seed from 19 source populations across
gradients for a widespread species (Protea repens)
• Planted ~800 seedlings in common garden
• Measured leaf traits, working on wood traits
Genetic variation in growth rate
90
height (cm)
Plant
Height_cm_2013
80
70
60
50
40
30
0
2
4
julmint
6
Mean daily minimum temperature in winter (C)
8
Anysberg: small stature probably
reflects both genetic and plastic responses
Eight-year old P repens
Anysberg, interior dry CFR
Photos: Adam Wilson
Big evolutionary questions
• What drives high diversity?
– Ecological speciation
– Isolation on habitat islands – archipelago-like patterns
– Climatic and geological stability
Photo:
Adam Wilson
Wilson et al. in prep
Demographic model
Seed release
Germination
& first-year survival
Germinant
& seedling
Pre-Repro
Reproductive
Large
Repro
Seedling
survival
Growth &
Juv. Surv.
Recruitment
Adult
survival
Adult success
Data
Predictor variables
Mean annual precip.
Min. July temp.
Max. January temp.
Precipitation seasonality
Winter soil moisture days
fecundity/
% High fertility soil
% Fine texture soil
% Acidic soil
Feedback example – Sierra forest
• When feedbacks positive, slight shift in
disturbance regime can cause state change
Moderate
fire
(-)
e.g. Collins et al.
(2009) Ecosystems
Less fuel,
milder fire
Intense fire
(+)
More shrubs,
intense fire
e.g. Stevens et al.
(2013) Can.J.For.Res.
Photos: Jens Stevens
Example vital rate model:
seed production
Seedheadsi,j,t ~ Poisson(λi,j,t)
log(λi,j,t) = β0 + β1*sizei,j,t +
β2*size2i,j,t + β3*Precipj,t
Mediterranean climate features
temperature
precipitation
Very dry!
Less dry
Priors