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