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Population dynamics of toxic
   algal blooms in Lake
         Champlain

   Edmund M. Hart, Nicholas J. Gotelli,
     Rebecca Gorney, Mary Watzin
If you bang your head against
     a wall long enough…
…sometimes you break
     through.
The problem…

Toxic algal blooms in Missisquoi Bay
           2003 - 2006
The problem…

Growth rates of toxic algal blooms in Missisquoi Bay
                  2003 - 2006
The Question…
What controls toxic algal bloom
population dynamics in Missisquoi
Bay?
The Lake
The Algae
Microcystis         Anabaena
The data
                                               Microcystis 2003

                                                                                                           Tyler
                             Julian   Microcystis
                                                    Julian    Highgate Chapma Route 78 Highgate            Place
                             Day      (cells/ml)
                                                    Day       Cliffs   n Bay    Access Springs Alburg Boatdock
                                182   3667.883            182 747.8851 1509.895 10350.7 2063.053 NA        NA
                                188 46381.514
•   Data is from the Rubenstein                           188 NA       NA       NA       NA       NA       NA
                                195 89095.144             195 128876.4 195970 11626.42 19907.8 NA          NA
    Ecosystems Science Laboratory’s toxic
                                203 111960.543            203 NA       NA       111960.5 NA       NA       NA
    algal bloom monitoring program
                                210 31070.727             210 26196.89 60016.66 30515.1 7554.263 NA        NA
•   Data from dominant taxa (Microcystis
                                217 19824.800             217 26749.99 10106.43 5350.629 37092.14 NA       NA
                                224 16395.252
    2003-2005, Anabaena 2006) 27626.305
                                231
                                                          224 20330.28 18108.55 17739.17 9403.008 NA       NA
                                                          231 29417.31 14473.77 24029.86 42584.29 NA       NA
•   Averaged across all sites within 26363.801
                                238                       238 29852.44 32075.16 32581.51 10946.1 NA        NA
    Missisquoi bay for each year247 44301.534             247 38663.23 31373.18 40378.36 66791.37 NA       NA
                                252 27541.291
•   Included only sites that had ancillary
                                259 43930.596
                                                          252 24605.86 8037.793 16097.45 16343.4 72621.95 NA
                                                          259 28372.98 141275 13656.03 23240.47 13108.5 NA
    nutrient data               267 16324.465             267 13770.97 17851.11 25768.67 5911.274 18320.3 NA
                                273 16104.062             273 19411.78 14821.51 8386.386 25066.42 12834.23 NA
                                280   6366.310            280 3067.318 11353.27 6735.426 9277.938 1397.603 NA
                                287   9052.005            287 NA       NA       3493.938 NA       NA       14610.07
Ancillary data
     The nutrients                          The competitors




                           Chlorophyceae (green algae)   Bacillariophyceae (diatoms)



TP   SRP             TN




           TN

           TP                                    Cryptophyceae
Mathematical Framework
Population models take on a general
form of
            N t f ( Nt 1 )

            Basic types include:

   Random Walk                             Nt   Nt   1       Norm(0,    2
                                                                            )

   Exponential Growth                      Nt   r0 N t   1    Νorm( 0 ,σ 2 )

   Logistic Growth   (Ricker form                             Nt
   shown)                           Nt   N t 1 exp r0 1                Νorm( 0,σ 2 )
                                                              K
Mathematical Framework
                             Density dependent
 Random walk




               Exponential
Mathematical Framework
Typically we analyze growth rates
                                     Nt
          Nt      Nt 1 exp r0 1
                                     K




                                    Nt
               ln( NNt t 1 ) r0 1
                                    K




           rt      r0 N t 1 exp( c)
Mathematical Framework
                            Density dependent
Random walk




              Exponential
Mathematical Framework
          Exogenous drivers
  rt   f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d )
Mathematical Framework
                        Exogenous drivers
                rt   f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d )



f ( N t d ) r0 N t 1 exp( c)            f ( Et d )      1   Et   d         f (C1t d )          C1t
                                                                                               1     d

Ricker logistic growth                        Linear                              Linear
Mathematical Framework
                        Exogenous drivers
                rt   f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d )



f ( N t d ) r0 N t 1 exp( c)            f ( Et d )      1   Et   d                f (C1t d )   C1t
                                                                                               1     d




                                 rt   r0 N t 1 exp( c)               1   Et   d

                                rt    r0 N t 1 exp( c            1   Et d )
                                rt    r0 N t 1 exp( c            1   C1t d )
A naïve analysis
               For each year fit the following models
Random walk /        Density dependent         Environmental factors                        Competitors
exponential growth   (endogenous factors)

     rt   r0          rt   r0 N t 1 exp( c)   rt   r0 N t 1 exp( c)       1   Et       rt    r0 N t 1 exp( c   C1t 1 )
                                                                                                               1


                                              rt   r0 N t 1 exp( c)       1   Et   1

                                              rt   r0 N t 1 exp( c    1   Et )

                                              rt   r0 N t 1 exp( c    1   Et 1 )
                                              rt   r0   1   Et

                                              rt   r0   1   Et


           Assessed model fit with AICc (AIC + 2K(K+1)/n-K-1)
A naïve analysis
     Microcystis 2004
A naïve analysis

     Microcystis 2004

                                                               AICc   ∆AICc   AIC      R2
          Model                                                               weight

                                         TN t                  29.7   0       0.33     0.4
                rt         r0        1
                                         TPt                                           5

                                                               30.5   0.81    0.22     0.4
                rt         r0       1TPt   1                                           1

                                                               32.2   2.45    0.10     0.4
          rt         r0     N t 1 exp( c)       TN t
                                                1          1                           9

                                                        TN t 32.3     2.57    0.09     0.4
           rt         r0        N t 1 exp( c)       1                                  8
                                                        TPt
                                                               33.3   3.58    0.05     0.4
           rt         r0        N t 1 exp( c)       1TPt   1                           4
A Problem

Autocorrelation plot for Microcystis 2004
A solution?
  Detrending!
   Microcystis 2004
A solution?
    Probably not…                  Microcystis 2004



•   Need to have evidence to
    assume an environmental
    change results in shifting
    carrying capacity.

•   Can introduce spurious
    corellations
Another solution?
                                           Step detrending!




Figure 1: Total counts of Soay sheep on the island of Hirta, showing two hypotheses for the apparent trend in the
average number of sheep (dotted lines). A, Step trend. B, Linear trend. From Am Nat 168(6):784-795.
Another solution?
                    Step detrending!
                        Microcystis 2004




 Too short,
only 5 points!
Another solution!
                       Time series “stitching”
                               Julian       Growth Microcystis
Julian   Microcystis
                               Day          Rate       (cells/ml)
Day      (cells/ml)
                                        182       2.54     3667.88
     182   3667.883                     188       0.65 46381.51
     188 46381.514                      195       0.23 89095.14
     195 89095.144                      203      -1.28 111960.54
     203 111960.543                     210      -0.45 31070.73
     210 31070.727                      217      -0.19 19824.80
     217 19824.800                      224       0.52 16395.25
     224 16395.252                      231      -0.05 27626.31
     231 27626.305                      238       0.52 26363.80
     238 26363.801                      247      -0.48 44301.53
     247 44301.534                      252       0.47 27541.29
     252 27541.291                      259      -0.99 43930.60
     259 43930.596
                                        267      -0.01 16324.47
     267 16324.465
                                        273      -0.93 16104.06
     273 16104.062
                                        280       0.35     6366.31
     280   6366.310
     287   9052.005
Squint real hard
                                         I can see it
Toxic algal blooms in Missisquoi Bay    because I’m
           2003 - 2006                      always
                                        squinting to
                                       keep an eye
                                       out for ninjas!
Phase portraits
    2004 Microcystis
Phase portraits
   2003 Microcystis    2005 Microcystis




2004 Microcystis         2006 Anabaena
A naïve analysis revisited
                For each year fit the following models
Random walk /        Density dependent         Environmental factors                        Competitors
exponential growth   (endogenous factors)

     rt   r0          rt   r0 N t 1 exp( c)   rt   r0 N t 1 exp( c)       1   Et       rt    r0 N t 1 exp( c   C1t 1 )
                                                                                                               1


                                              rt   r0 N t 1 exp( c)       1   Et   1


 Do all this again                            rt   r0 N t 1 exp( c    1   Et )

 but with our two                             rt   r0 N t 1 exp( c    1   Et 1 )
 new series!                                  rt   r0   1   Et

                                              rt   r0   1   Et


           Assessed model fit with AICc (AIC + 2K(K+1)/n-K-1)
Bloom phase dynamics

                                                                        AICc   ∆AICc   AIC      R2
                       Model                                                           weight

                                                         TN t           33.1   0       0.63     0.8
                        rt    r0    N t 1 exp( c)    1
                                                         TPt
                                                                        38.3   5.2     0.04     0.71
                         rt    r0   N t 1 exp( c)        TPt
                                                         1


                                                                        38.4   5.3     0.04     0.64
                         rt    r0    N t 1 exp( c)

                                                                        38.9   5.8     0.03     0.7
                        rt    r0    N t 1 exp( c)    1TN t      1


                                                                        38.9   5.8     0.03     0.7
                        rt    r0    N t 1 exp( c)    1   SRPt       1




                                               TN t
   rt   0.28   N t 1 exp( 10 .8) 0.08
                                               TPt
Decline phase dynamics

                                                                     AICc    ∆AICc     AIC         R2
                   Model                                                               weight


                       rt         r0   N t 1 exp( c        TN t )    78.8    0         0.21        0.18
                                                       1


                                                                     81.2    2.4       0.06        -
                  rt             r0

                                                                     81.4    2.6       0.06        0.13
                   rt            r0    N t 1 exp( c    TPt )
                                                       1


                                                                     81.6    2.8       0.05        0.12
                  * rt            r0   N t 1 exp( c    1   Crt 1 )

                            rt    r0   N t 1 exp( c)                 81.7    2.9       0.05        0.04


                                                                            * Cr = Cryptophyceae




    rt   0.12   N t 1 exp( 7.05 33 .1* TN t )
The problem revisited…

   Growth rates of toxic algal blooms in Missisquoi Bay
                     2003 - 2006
Is it N:P then?
No, but what can we say then?

• 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.

• Once a bloom starts, you can’t really do anything about
  it.

              And one more thing about N:P
A final thought on N:P

             Population size and N:P   Partial residual plot of bloom
Smith 1983   on bloom phase data       phase growth rate model
Thanks!
• VT EPSCoR

• My collaborators
      – Nicholas Gotelli
      – Rebecca Gorney
      – Mary Watzin


• The EPSCoR
 complex systems group




perfunctory comic to keep you entertained during questions
Mathematical Framework
   Environmental Factors
       Effect on growth rate
         rt    r0 N t 1 exp( c)   1   X

    Effect on density dependence

          rt   r0 N t 1 exp( c    1   X)
Plankton Time Series Analysis




           A naïve approach                             Complex population dynamic approach


Using a complex population dynamics modeling approach we parse four years of
plankton time series into two distinct phases, bloom phase and decline
phase, each with distinct dynamics. This method provides a far superior fit to
traditional statistical correlative methods.

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Population dynamics of toxic algal blooms in Lake Champlain

  • 1. Population dynamics of toxic algal blooms in Lake Champlain Edmund M. Hart, Nicholas J. Gotelli, Rebecca Gorney, Mary Watzin
  • 2. If you bang your head against a wall long enough…
  • 4. The problem… Toxic algal blooms in Missisquoi Bay 2003 - 2006
  • 5. The problem… Growth rates of toxic algal blooms in Missisquoi Bay 2003 - 2006
  • 6. The Question… What controls toxic algal bloom population dynamics in Missisquoi Bay?
  • 9. The data Microcystis 2003 Tyler Julian Microcystis Julian Highgate Chapma Route 78 Highgate Place Day (cells/ml) Day Cliffs n Bay Access Springs Alburg Boatdock 182 3667.883 182 747.8851 1509.895 10350.7 2063.053 NA NA 188 46381.514 • Data is from the Rubenstein 188 NA NA NA NA NA NA 195 89095.144 195 128876.4 195970 11626.42 19907.8 NA NA Ecosystems Science Laboratory’s toxic 203 111960.543 203 NA NA 111960.5 NA NA NA algal bloom monitoring program 210 31070.727 210 26196.89 60016.66 30515.1 7554.263 NA NA • Data from dominant taxa (Microcystis 217 19824.800 217 26749.99 10106.43 5350.629 37092.14 NA NA 224 16395.252 2003-2005, Anabaena 2006) 27626.305 231 224 20330.28 18108.55 17739.17 9403.008 NA NA 231 29417.31 14473.77 24029.86 42584.29 NA NA • Averaged across all sites within 26363.801 238 238 29852.44 32075.16 32581.51 10946.1 NA NA Missisquoi bay for each year247 44301.534 247 38663.23 31373.18 40378.36 66791.37 NA NA 252 27541.291 • Included only sites that had ancillary 259 43930.596 252 24605.86 8037.793 16097.45 16343.4 72621.95 NA 259 28372.98 141275 13656.03 23240.47 13108.5 NA nutrient data 267 16324.465 267 13770.97 17851.11 25768.67 5911.274 18320.3 NA 273 16104.062 273 19411.78 14821.51 8386.386 25066.42 12834.23 NA 280 6366.310 280 3067.318 11353.27 6735.426 9277.938 1397.603 NA 287 9052.005 287 NA NA 3493.938 NA NA 14610.07
  • 10. Ancillary data The nutrients The competitors Chlorophyceae (green algae) Bacillariophyceae (diatoms) TP SRP TN TN TP Cryptophyceae
  • 11. Mathematical Framework Population models take on a general form of N t f ( Nt 1 ) Basic types include: Random Walk Nt Nt 1 Norm(0, 2 ) Exponential Growth Nt r0 N t 1 Νorm( 0 ,σ 2 ) Logistic Growth (Ricker form Nt shown) Nt N t 1 exp r0 1 Νorm( 0,σ 2 ) K
  • 12. Mathematical Framework Density dependent Random walk Exponential
  • 13. Mathematical Framework Typically we analyze growth rates Nt Nt Nt 1 exp r0 1 K Nt ln( NNt t 1 ) r0 1 K rt r0 N t 1 exp( c)
  • 14. Mathematical Framework Density dependent Random walk Exponential
  • 15. Mathematical Framework Exogenous drivers rt f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d )
  • 16. Mathematical Framework Exogenous drivers rt f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d ) f ( N t d ) r0 N t 1 exp( c) f ( Et d ) 1 Et d f (C1t d ) C1t 1 d Ricker logistic growth Linear Linear
  • 17. Mathematical Framework Exogenous drivers rt f ( N t 1 , N t 2 ... N t d , Et , Et 1... Et d , C1t 1 C1t 2...C1t d ) f ( N t d ) r0 N t 1 exp( c) f ( Et d ) 1 Et d f (C1t d ) C1t 1 d rt r0 N t 1 exp( c) 1 Et d rt r0 N t 1 exp( c 1 Et d ) rt r0 N t 1 exp( c 1 C1t d )
  • 18. A naïve analysis For each year fit the following models Random walk / Density dependent Environmental factors Competitors exponential growth (endogenous factors) rt r0 rt r0 N t 1 exp( c) rt r0 N t 1 exp( c) 1 Et rt r0 N t 1 exp( c C1t 1 ) 1 rt r0 N t 1 exp( c) 1 Et 1 rt r0 N t 1 exp( c 1 Et ) rt r0 N t 1 exp( c 1 Et 1 ) rt r0 1 Et rt r0 1 Et Assessed model fit with AICc (AIC + 2K(K+1)/n-K-1)
  • 19. A naïve analysis Microcystis 2004
  • 20. A naïve analysis Microcystis 2004 AICc ∆AICc AIC R2 Model weight TN t 29.7 0 0.33 0.4 rt r0 1 TPt 5 30.5 0.81 0.22 0.4 rt r0 1TPt 1 1 32.2 2.45 0.10 0.4 rt r0 N t 1 exp( c) TN t 1 1 9 TN t 32.3 2.57 0.09 0.4 rt r0 N t 1 exp( c) 1 8 TPt 33.3 3.58 0.05 0.4 rt r0 N t 1 exp( c) 1TPt 1 4
  • 21. A Problem Autocorrelation plot for Microcystis 2004
  • 22. A solution? Detrending! Microcystis 2004
  • 23. A solution? Probably not… Microcystis 2004 • Need to have evidence to assume an environmental change results in shifting carrying capacity. • Can introduce spurious corellations
  • 24. Another solution? Step detrending! Figure 1: Total counts of Soay sheep on the island of Hirta, showing two hypotheses for the apparent trend in the average number of sheep (dotted lines). A, Step trend. B, Linear trend. From Am Nat 168(6):784-795.
  • 25. Another solution? Step detrending! Microcystis 2004 Too short, only 5 points!
  • 26. Another solution! Time series “stitching” Julian Growth Microcystis Julian Microcystis Day Rate (cells/ml) Day (cells/ml) 182 2.54 3667.88 182 3667.883 188 0.65 46381.51 188 46381.514 195 0.23 89095.14 195 89095.144 203 -1.28 111960.54 203 111960.543 210 -0.45 31070.73 210 31070.727 217 -0.19 19824.80 217 19824.800 224 0.52 16395.25 224 16395.252 231 -0.05 27626.31 231 27626.305 238 0.52 26363.80 238 26363.801 247 -0.48 44301.53 247 44301.534 252 0.47 27541.29 252 27541.291 259 -0.99 43930.60 259 43930.596 267 -0.01 16324.47 267 16324.465 273 -0.93 16104.06 273 16104.062 280 0.35 6366.31 280 6366.310 287 9052.005
  • 27. Squint real hard I can see it Toxic algal blooms in Missisquoi Bay because I’m 2003 - 2006 always squinting to keep an eye out for ninjas!
  • 28. Phase portraits 2004 Microcystis
  • 29. Phase portraits 2003 Microcystis 2005 Microcystis 2004 Microcystis 2006 Anabaena
  • 30. A naïve analysis revisited For each year fit the following models Random walk / Density dependent Environmental factors Competitors exponential growth (endogenous factors) rt r0 rt r0 N t 1 exp( c) rt r0 N t 1 exp( c) 1 Et rt r0 N t 1 exp( c C1t 1 ) 1 rt r0 N t 1 exp( c) 1 Et 1 Do all this again rt r0 N t 1 exp( c 1 Et ) but with our two rt r0 N t 1 exp( c 1 Et 1 ) new series! rt r0 1 Et rt r0 1 Et Assessed model fit with AICc (AIC + 2K(K+1)/n-K-1)
  • 31. Bloom phase dynamics AICc ∆AICc AIC R2 Model weight TN t 33.1 0 0.63 0.8 rt r0 N t 1 exp( c) 1 TPt 38.3 5.2 0.04 0.71 rt r0 N t 1 exp( c) TPt 1 38.4 5.3 0.04 0.64 rt r0 N t 1 exp( c) 38.9 5.8 0.03 0.7 rt r0 N t 1 exp( c) 1TN t 1 38.9 5.8 0.03 0.7 rt r0 N t 1 exp( c) 1 SRPt 1 TN t rt 0.28 N t 1 exp( 10 .8) 0.08 TPt
  • 32. Decline phase dynamics AICc ∆AICc AIC R2 Model weight rt r0 N t 1 exp( c TN t ) 78.8 0 0.21 0.18 1 81.2 2.4 0.06 - rt r0 81.4 2.6 0.06 0.13 rt r0 N t 1 exp( c TPt ) 1 81.6 2.8 0.05 0.12 * rt r0 N t 1 exp( c 1 Crt 1 ) rt r0 N t 1 exp( c) 81.7 2.9 0.05 0.04 * Cr = Cryptophyceae rt 0.12 N t 1 exp( 7.05 33 .1* TN t )
  • 33. The problem revisited… Growth rates of toxic algal blooms in Missisquoi Bay 2003 - 2006
  • 34. Is it N:P then?
  • 35. No, but what can we say then? • 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. • Once a bloom starts, you can’t really do anything about it. And one more thing about N:P
  • 36. A final thought on N:P Population size and N:P Partial residual plot of bloom Smith 1983 on bloom phase data phase growth rate model
  • 37. Thanks! • VT EPSCoR • My collaborators – Nicholas Gotelli – Rebecca Gorney – Mary Watzin • The EPSCoR complex systems group perfunctory comic to keep you entertained during questions
  • 38. Mathematical Framework Environmental Factors Effect on growth rate rt r0 N t 1 exp( c) 1 X Effect on density dependence rt r0 N t 1 exp( c 1 X)
  • 39. Plankton Time Series Analysis A naïve approach Complex population dynamic approach Using a complex population dynamics modeling approach we parse four years of plankton time series into two distinct phases, bloom phase and decline phase, each with distinct dynamics. This method provides a far superior fit to traditional statistical correlative methods.