Understanding natural populations with dynamic
models
Edmund M. HartUniversity of Vermont
The beginning
Charles Elton 1900-1991
A. J. Nicholson1895-1969
The beginning
H. G. Andrewartha 1907-1992
L. Charles Birch1918-2009The logarithm of the average population size per month for
several years in the study of Thrips imaginis
The unanswered question
H. G. Andrewartha 1907-1992
Charles Elton 1900-1991
L. Charles Birch1918-2009
A. J. Nicholson1895-1969
How can we fit experimental and observational data to population dynamic models in order to understand what regulates populations?
First principles
1 1t
t t
N B DrN N
N B D
First principles
1 1 1
ln tt
t t t
NN B DrN N N
1 1t t t tN N r N
First principles
1 1t t t tN N r N
( , , , ...)tr f N environment competitors etc
Mathematical FrameworkThree basic types of population growth
Random Walk
Exponential Growth
Logistic Growth (Ricker form shown)
20 (0, )tr N
20 1 exp( ) 0t tr r N c Ν( ,σ )
20 (0, )tr r N
Mathematical FrameworkRandom walk Density dependent
Exponential
Mathematical FrameworkRandom walk Density dependent
Exponential
Mathematical FrameworkVertical shift
)()( 1 ttt zgNfr
Mathematical FrameworkLateral shift
)( 1 ttt zNfr
Testing hypotheses
Two methods: Carry out experiments and test how
populations change over parameter space
Fit models to observational data
Experimental approach
How can expected changes in the mean and variance of an environmental factor caused by climate change alter population processes in aquatic communities?
Experimental approach
Climate change in New England
Experimental approach
Experimental approach
Surface response7 Levels of Water Variation7 Levels of Water mean depthFully crossed for 49 tubs
Means (cm): 6.6,9.9,13.2, 16.5,19.8, 23.1, 26.4
Coeffecients of Variation (C.V.): 0,.1,.2,.3,.4,.5,.6
~1.5 m
Experimental approachMean Water Level
Wat
er C
.V.
Low water level, high CV
Low water level, low CV
High water level, high CV
High water level, low CV
Experimental approach
Experimental approach
Experimental approach
Midges
Mosquitoes
Experimental approach
β1 (p<0.05) R2=0.27
β2 (p<0.05)β3 (p<0.05) R2=0.49
0 1 2 3 *mn mny MWL WCV MWL WCV
Experimental approach
2[ 1]~ ( , )tjk jk jk t jk rr N X
jk
jk
~ ( , )j BB MVN U
jk
jk
[ 1]t jkX
B
U
Growth rate, same as r0
Strength of density dependence
Log abundance
Grand mean
Effect of mean water level
Effect of water level CV
A vector of 0’s of length 2
A 2x2 variance covariance matrix
Experimental approachEstimates of the Gompertz logistic (GL) parameters for each treatment combination for growth rate and density dependence in Culicidae and Chironomidae. Darker squares indicate either higher population growth rate or stronger density dependence.
Growth rate Density dependence
Experimental approachGrowth rate Density dependence
Experimental approach
• The mean and variance of pond hydrological process impacts larval abundance in opposing directions
• Abundances change due to alterations in population dynamic parameters
Changes in intrinsic rate of increase in mosquitoes probably due to female oviposition choice
Density dependent effects in midges most likely caused by competition for space
Observational approach Using monitoring data, how
can we understand what controls toxic algal bloom population dynamics in Missisquoi Bay?
Observational approach
Observational approach
Observational approachMicrocystis Anabaena
Observational approachThe nutrients The competitors
Chlorophyceae (green algae)
TP TN
TP
TN
SRP
Bacillariophyceae (diatoms)
Cryptophyceae
Observational approachToxic algal blooms in Missisquoi Bay
2003 - 2006• Data is from the Rubenstein
Ecosystems Science Laboratory’s toxic algal bloom monitoring program
• Data from dominant taxa (Microcystis 2003-2005, Anabaena 2006)
• Averaged across all sites within Missisquoi bay for each year
• Included only sites that had ancillary nutrient data
Observational approach
1 2 1 1 2( , ... ) ( , ... ) ( 1 1 ... 1 )t t t t d t t t d t t t dr f N N N g E E E h C C C
1 1t t t tN N r N
Observational approachExogenous drivers
1 2 1 1 2( , ... ) ( , ... ) ( 1 1 ... 1 )t t t t d t t t d t t t dr f N N N g E E E h C C C
)exp()( 10 cNrNf tdt 1( )t d t dg E E 1( 1 ) 1t d t dh C C
Ricker logistic growth Linear Linear
Observational approach
)exp()( 10 cNrNf tdt 1( )t d t dg E E 1( 1 ) 1t d t dh C C
dttt EcNrr 110 )exp(
)exp( 110 dttt EcNrr
)1exp( 110 dttt CcNrr
1 2 1 1 2( , ... ) ( , ... ) ( 1 1 ... 1 )t t t t d t t t d t t t dr f N N N g E E E h C C C
Observational approachWe fit 29 different models from the following:
Assessed model fit with AICc (AIC + 2K(K+1)/n-K-1)
ttt EcNrr 110 )exp(
)exp( 110 ttt EcNrr
)1exp( 1110 ttt CcNrr
1110 )exp( ttt EcNrr
)exp( 1110 ttt EcNrr
)exp(10 cNrr tt 0rrt
Random walk / exponential growth
Density dependent (endogenous factors)
CompetitorsEnvironmental factors
tt Err 10
0 1 1t tr r E
Observational approach
Growth rates of toxic algal blooms in Missisquoi Bay 2003 - 2006
Observational approach
Growth rates of toxic algal blooms in Missisquoi Bay 2003 - 2006
Observational approach
2004 Microcystis
Observational approach2003 Microcystis 2005 Microcystis
2004 Microcystis 2006 Anabaena
Julian Day
Julian Day
Growth Rate
Microcystis
(cells/ml)182 2.54 3667.88188 0.65 46381.51195 0.23 89095.14
203 -1.28111960.5
4210 -0.45 31070.73217 -0.19 19824.80224 0.52 16395.25231 -0.05 27626.31238 0.52 26363.80247 -0.48 44301.53252 0.47 27541.29259 -0.99 43930.60267 -0.01 16324.47273 -0.93 16104.06280 0.35 6366.31
Julian Day
Growth Rate
Microcystis
(cells/ml)182 2.54 3667.88188 0.65 46381.51195 0.23 89095.14
203 -1.28111960.5
4210 -0.45 31070.73217 -0.19 19824.80224 0.52 16395.25231 -0.05 27626.31238 0.52 26363.80247 -0.48 44301.53252 0.47 27541.29259 -0.99 43930.60267 -0.01 16324.47273 -0.93 16104.06280 0.35 6366.31
Observational approachJulian Day
Microcystis (cells/ml)
182 3667.883188 46381.514195 89095.144203 111960.543210 31070.727217 19824.800224 16395.252231 27626.305238 26363.801247 44301.534252 27541.291259 43930.596267 16324.465273 16104.062280 6366.310287 9052.005
Model AICc ∆AICc AIC weight
R2
33.1 0 0.63 0.8
38.3 5.2 0.04 0.71
38.4 5.3 0.04 0.64
38.9 5.8 0.03 0.7
38.9 5.8 0.03 0.7
Observational approach
1110 )exp( ttt TNcNrr
ttt TPcNrr 110 )exp(
t
ttt TP
TNcNrr 110 )exp(
)exp(10 cNrr tt
1110 )exp( ttt SRPcNrr
t
ttt TP
TNNr 08.0)8.10exp(28.0 1
Model AICc ∆AICc AIC weight
R2
78.8 0 0.21 0.18
81.2 2.4 0.06 -
81.4 2.6 0.06 0.13
81.6 2.8 0.05 0.12
81.7 2.9 0.05 0.04
Decline phase dynamics
)exp( 110 ttt TNcNrr
0rrt
)*1.3305.7exp(12.0 1 ttt TNNr
)exp(10 cNrr tt
)exp( 110 ttt TPcNrr
)exp( 1110 ttt CrcNrr
* Cr = Cryptophyceae
Two phase growth
Growth rates of toxic algal blooms in Missisquoi Bay 2003 - 2006
0 1 1
0 1 1
exp( ) , 5
exp( ), 5
tt
tt
t t
TNr N c tTPr
r N c TN t
Observational approachPartial residual plot of bloomphase growth rate modelPopulation size and N:P on bloom phase data
Observational approach
• Toxic algal blooms have two distinct dynamic phases, a pattern observed across years and genera.
• N:P important in the bloom phase, but not the decline, i.e. nutrients don’t always matter.
• Capturing the dynamics of a bloom are important. i.e. if correlating N:P with populations, depending when samples are taken you may get different results
Conclusions• Populations can be understood from both
experimental and observational data
• Population dynamic models provide a deeper understanding of changes in abundance and correlation with environmental variables.
• Dynamic models showed how climate change alters different aspects of population processes depending on the taxa and its life history, which in turn drive abundance.
• Dynamic models of observational data elucidated relationships between environmental covariates and population growth rates that otherwise are missed by simple regression on abundances.
AcknowledgementsCommittee MembersNick GotelliAlison BrodySara CahanBrian Beckage
Jericho forestDavid BrynnDon Tobi
Undergraduate field assistantsChris GravesCyrus Mallon (University of Groningen)
Co-Authors on the plankton manuscriptNick GotelliRebecca GorneyMary Watzin
My faithful field companion,Tuesday. General helper and protector from squirrels and the occasional bear
FundingVermont EPSCoRNSF