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MBW:A Dynamic Global Model for Planktonic Foraminifera

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This is a summary and extension of A Dynamic Global Model for Planktonic Foraminifera by I. Fraile, M. Schulz, S. Mulitza, and M. Kucera , Biogeosciences Discussions, 4323-4384 (26 November 2007). All information comes from this source. Article reviewed and extended by Mariah Walton and Nick Levine.

Summary

In 'A Dynamic Global Model for Planktonic Foraminifera', Fraile et al. (2007) introduce a model for four species of planktic foraminifera, which are significant for their frequent use as paleocean proxies. The model is roughly based on Michaelis-Menten kinetics, and models foram population growth as a function of optimal temperature and food concentration. The biotic and abiotic factors that influence foram growth rates were determined using a 2-dimensional global mixed-layer ocean ecosystem model. Simulated outputs were then compared to actual samples to validate the model.

Study Features:

  • To predict the flux in food and nutrient availability, an ocean ecosystem model was implemented that uses hydrographic data derived from a previously described ocean circulation model. The ecosystem model determines nutrient distributions as well as the abundances of different planktonic organisms.
  • The foram growth model uses an ordinary differential equation, with parameter values obtained from the ocean ecosystem model output. The temperature dependence of foram abundances assumes a Gaussian distribution of the foram species within its temperature range.
  • The PLAFOM model of foram biomass concentrations uses a modified Michaelis-Menten kinetics equation.
  • This model was designed for 5 marine planktonic foraminifera taxa, including one species that has 2 morphological variants.

The model, however, still remains fairly new and untested. To better establish model sensitivities to the key parameters (Temperature and food Concentration), we looked at the isolated model and set our own inputs. By establishing different scenarios of food magnitude and variability, as well as different magnitudes and variabilities of T, we were able to determine that the model responds strongly to both variables, but in different ways. Temperature plays a larger role in governing the seasonal shape of the foram population, while food largely drives the overall magnitude.

Biological System

Foraminifera (forams) are single celled protists that live in ocean. They can live on the ocean floor (benthic forams), or in the upper mixed layer of the ocean (planktonic forams). Forams build microscopic shells out of calcium carbon or particulate grains, allowing us to identify them even after they're dead. We care about them because they play an important role in the carbon cycle, but perhaps more importantly they are the most frequently used proxy for past ocean environments. Much of this proxy work requires assumptions about species preferences for different environments, ideally based on observations (i.e. we need to know where things like to live now, and why, before we can understand why they were living in set places in the past). But most proxy records, and even some modern observations, are highly limited in their temporal resolution, frequently representing several year averages.

In their paper 'A dynamic global model for plankonic foraminifera' Fraile et al model four well studied species of forams in an effort to bridge the gaps in our knowledge of global foram distributions on a seasonal as well as long term scale. These species are shown below. N. pachyderma has two forms, distinguished by the way their shells coil (sinistral=left coiling, dextral=right coiling). These two forms have different temperature preferences, which is why they are modeled separately.


Fraile forams.jpg


In modeling foram concentrations, the authors considered foram growth rates as a function of food availability, growth efficiency, and food preference (e.g. phytoplankton vs. detritus). Foram growth is governed by temperature and food availability. Species typically have an optimal temperature, and range of temperatures around that optimum at which they can still survive. For food, forams can be carvnivorous, herbivorous, and omnivorous depending on the species. Fraile et al break these food categories down into zooplankton (Z), small phytoplankton (SP), diatoms (D), and organic detritus (DR) which is simply falling organic material such as dead phytoplankton. This act of consumption is called 'grazing'. These species of forams only live a few days to a month, which means they are fairly sensitive to changes in any of these factors.

Foram mortality comes from predation (either cannibalism or consumption by a higher trophic level), respiration (death), and competition (between different species).

Mathematical Model

Fraile's model, called PLAFORM, tracks changes in foram biomass through time as a function of temperature and food availability. Each species has prescribed constants, most noteably for food preference and consumption efficiency, optimal temperature range, and maximum growth rate. Growth rates were determined using a modified version of the Michaelis-Menten equation. For another example of applications of this model, see Modelling the Tryptophan Operon. Temperature and food concentrations are obtained from an already validated ecosystem model (Moore et al 2002a). This ecosystem model is two dimensional, only looking at the vertically integrated mixed layer of the ocean. Foram concentrations are calculated on the level of a single grid cell and represent this integrated mixed layer, with no mixing of populations between cells.

Fraile model.jpg

As shown in the diagram above, foram growth and mortality are calculated separately. Growth is dependent overall on temperature (through alpha), but also on food preference and food concentration. Mortality is chiefly based on foram population itself, as no foram-specific predators are known. Fraile's model incorporates competition between different species, and also has differing growth rates and preferences when a species favored food is not available.


Fraile eqns.jpg

Model Validation

To validate their model, Fraile et al collected surface and core-top samples of forams in different locations throughout the course of a year, and compared these samples to modeled biomass (based on the observed temperatures at each location). The model seemed to capture very basic trends, and relative magnitudes of different species, but was not very accurate on a site by site basis. It appears to do better, however, at capturing general spatial distributions. Below are the results for N. pachyderma (sinistral).

Fraile fig2.jpg

Fraile fig7.jpg

Sensitivities

In an effort to expand the model above we have simulated the model in MATLAB (code found below), looking at only one species--N. pachyderma (sin.). The numerous figures shown in the original paper prove the model is, for the most part, valid. But how sensitive is the model to changes in food supply and temperature?

In an effort to explore this we studied the sensitivities of the model to set inputs on a case by case basis. When looking at food source (C) we used the lhsdesign function in MATLAB to simulate random values (You can read more about Latin Hypercube sampling here). We changed the magnitude of C to simulate both a food surplus and a food shortage. When analyzing temperature (T), we used both random and normally distributed inputs to evaluate the model's sensitivity to temperature. Normally distributed inputs were chosen to resemble the characteristics of true temperature patterns during a normal year using a sine function.

Case 1: Random C’s (using the lhsdesign function) and Normally Distributed T’s

Picture 1c.jpg When using a normally distributed input for temperature (T), as shown above, we see a strong relationship between the foraminifera carbon concentration (F) and temperature. Temperature gives the model its shape where the randomness in the food supply (C) introduces noise. (Note: we use a C of average manitude so as not to represent a food abundance or a food shortage).

Case 2: Random C’s and Random T’s

Picture 2a.jpg In this case, we see the same noise introduced into the foraminifera carbon concentration (F) by the randomness in the food source (C). (Note: we use a C of average magnitude so as not to represent a food abundance or a food shortage). However, this example further proves that the shape of the F is driven by the temperature input.

Case 3: Food Surplus and Normally Distributed T’s

Picture 3a.jpg Here we maintain the normally distributed temperature to see what affect a food surplus (very high magnitude C) will have on the foraminifera carbon concentration (F). Increasing C results in a much cleaner, less noisy F. It is also interesting to note the magnitude of F is of the same order as C. We still see that the normally distributed temperature continues to drive the shape of the model.

Case 4: Food Shortage and Normally Distributed T’s

Picture 4a.jpg In this example we create a food shortage (C) and maintain a normally distributed temperature. As expected, this results in a very noisy foraminifera carbon concentration (F) but again the magnitude of C and F are of the same order. The temperature input continues to drive the shape of the model.

  • Note: When the food supply is on the order of 1E-6, there is not enough food to drive the model.

Case 5: Random C's with Normally Distributed T's (of larger magnitudes)

Picture 6a.jpg Here we see that as temperature deviates from the optimal temperature (i.e. of significantly larger magnitude) it no longer drives the shape of the curve. This is due to the fact that as the temperature is increased, a critical point is reached in the model and the relative abundance of the species is reduced until the species cannot survive.

Conclusions

In summary, both T and C are important, but generate different model responses. There is a direct relationship between the food supply and the foraminifera carbon concentration in that the magnitude of the food supply drives the magnitude of the foram concentration. Additionally, the precision of the foraminifera carbon concentration is rooted in the magnitude of the food source. The more food, the smoother the model. If you assume that temperature remains in the optimal range, the shape of the foram carbon concentration is driven by the temperature.

  • As an aside, the same conclusions can be made if you plot random temperatures against the food surplus and food shortage scenarios. To limit the length of the wiki, and the fact that the MATLAB code is below, we leave this up to the reader to confirm.

Based on these cases, we cannot state that either the temperature or food supply is more sensitive than the other (this is like comparing apples to oranges), but we can say that food appears to be more important for setting the population magnitude, and temperature for driving seasonal trends. In the real ocean, food availability tends to follow temperature, as phytoplankton need light, which drives temperature. The result is a amplified seasonal trend, with both C and T (and hence F) following the same sinusoidal curve. This is an important implication for paleoceanography work, as it means that foram abundance can only be used as a very rough temperature proxy, because biomass magnitude is heavily influenced by food supply.


Model Applications

In a follow-up study, Fraile et al. (2009) tested the ability of their PLAFOM ecosystem model to predict global climate conditions during the Last Glacial Maximum (LGM), which occurred between 19k-26k years before present. Considering the extensive use of forams for reconstructing past climate conditions and in particular for predicting sea surface temperatures, this study is particularly relevant for testing the validity of their foram biomass model. The study used the global coupled Community Climate System Model v3 (CCSM3) to force the foram model, using both modern climate conditions and conditions present during the LGM. The model was validated using foraminifera abundances derived from ocean sediment cores collected from sites across the globe. Model sensitivity to changes in nutrient distributions during the LGM as a result of sea level lowering was also tested using 2 different models.

The model was relatively insensitive to changes in ocean nutrient concentrations, indicating that the modeled ocean temperatures during the LGM were robust even when nutrient concentrations were not well constrained. In general, all foram species shifted their latitudinal ranges southward during the LGM, which would be expected given the cooler sea surface temperatures. However, core data suggested that certain species became less abundant during the LGM rather than shifting their distributions, thereby introducing error into the model. Regardless, the overall biomass-weighted mean temperatures predicted by the model for each species were not greatly changed by these discrepancies. A key finding of this study was a somewhat large discrepancy in the timing of peak biomass flux during the LGM compared with modern seasonal dynamics in foram abundances. This shift in the timing of maximum species production was most pronounced in polar species, where climate changes are most pronounced. The effects of species changes in the seasonal timing of peak biomass production could be enough to mask changes in sea surface temperature modeled for the LGM. This work therefore demonstrates the importance of modeling seasonality changes in order to accurately predict sea surface temperatures using forams, and implies that interpretation of paleorecords based on modern foram reconstructions may incorrectly estimate ocean sea surface temperatures.

References and External Links

1. I. Fraile, M. Schulz, S. Mulitza, M. Kucera. " A Dynamic Global Model for Planktonic Foraminifera" Biogeosciences Discussions, 4323-4384 (26 November 2007). <http://www.biogeosciences-discuss.net/4/4323/2007/bgd-4-4323-2007.pdf>.

2. Latin hypercube sampling." Wikipedia, the Free Encyclopedia. Web. 17 Apr. 2010. <http://en.wikipedia.org/wiki/Latin_hypercube>.

3. Foraminifera."Wikipedia, the Free Encyclopedia. Web. 17 Apr. 2010. <http://en.wikipedia.org/wiki/Foraminifera>.

4. I. Fraile, M. Schulz, M. Kucera. 2008. "Predicting the global distribution of planktonic foraminifera using a dynamic ecosystem model". Biogeosciences, 5: 891-911.

5. I. Fraile, M. Schultz, S. Mulitza, U. Merkel, M. Prange, A. Paul. 2009. "Modeling the seasonal distribution of planktonic foraminifera during the Last Glacial Maximum". Paleoceanography 24:15 pages.doi:10.1029/2008PA001686


Related Pages

For another example of a growth model, please review: MBW:Nonlinear_Control_for_Algae_Growth_Models and Nonlinear_Control_For_Algae_Growth_Models_Part_II.

MATLAB Code

%Defines constants for the model
sigmaT=4;
Topt=3.8;
Tmax=30;
k=1;
p_SP=.3;
p_D=.7;
p_Z=0;
p_DR=0;
Gmax_SP=1.08;
Gmax_Z=2.16;
g=.66;
pl=1;
rl=.06;
b=4000;
GGE=.3;
k=1;
n=303.15;
F=zeros([365,1]);
F0=.03;

%Here we will define a random C of normal magnitude
C=lhsdesign(365,4)/10;

%Here we define a random C of significantly larger magnitude
%C=lhsdesign(365,4)*1000;


%Here we define T norally distributed
%x1=zeros([365,1]);
%x1=-pi/2;
%for i=2:365
%x1(i)=x1(i-1)+pi/180;
%end

%Ts=zeros([365,1]);
%for i=1:365
%Ts(i)=2*(sin(x1(i))+1);
%end

% Here we define T to be random 
Ts = rand([365,1])*5;

Fp=ones([365,1]);
pre=ones([365,1]);
res=ones([365,1]);
comp=ones([365,1]);
M=ones([365,1]);
alphaT=ones([365,1]);
TG=zeros([365,1]);
t=ones([365,1]);
for i=1:365
t(i)=i;
end

Fp(1)=max((F0-.01),0);
pre(1)=pl*exp(-b*(1/(Ts(1)+273.15)-1/(Tmax+273.15)))*Fp(1)^2;
res(1)=rl*Fp(1);
comp(1)=0;
M(1)=pre(1)+res(1)+comp(1);
alphaT(1)=(n*exp(-.5*(((Ts(1)-Topt))/sigmaT)^2))^(1/k);
if ((F0>0) && (C(1,1)>0))
TG(1)=p_SP*(Gmax_SP*alphaT(1)*F0*C(1,1)/(C(1,1)+g));
end
if ((F0>0) && (C(1,2)>0))
TG(1)=TG(1)+p_D*(Gmax_SP*alphaT(1)*F0*C(1,2)/(C(1,2)+g));
end
if ((F0>0) && (C(1,3)>0))
TG(1)=TG(1)+p_DR*(Gmax_SP*alphaT(1)*F0*C(1,3)/(C(1,3)+g));
end
if ((F0>0) && (C(1,4)>0))
TG(1)=TG(1)+p_Z*(Gmax_Z*alphaT(1)*F0*C(1,4)/(C(1,4)+g));
end
F(1)=F0+GGE*TG(1)-M(1);

for i=2:365;
Fp(i)=max((F(i-1)-.01),0);
pre(i)=pl*exp(-b*(1/(Ts(i)+273.15)-1/(Tmax+273.15)))*Fp(1)^2;
res(i)=rl*Fp(i);
comp(i)=0;
M(i)=pre(i)+res(i)+comp(i);
alphaT(i)=(n*exp(-.5*((Ts(i)-Topt)/sigmaT)^2))^(1/k);

if ((F0>0) && (C(i,1)>0))
TG(i)=p_SP*(Gmax_SP*alphaT(i)*F0*C(i,1)/(C(i,1)+g));
end
if ((F0>0) && (C(i,2)>0))
TG(i)=TG(i)+p_D*(Gmax_SP*alphaT(i)*F0*C(i,2)/(C(i,2)+g));
end
if ((F0>0) && (C(i,3)>0))
TG(i)=TG(i)+p_DR*(Gmax_SP*alphaT(i)*F0*C(i,3)/(C(i,3)+g));
end
if ((F0>0) && (C(i,4)>0))
TG(i)=TG(i)+p_Z*(Gmax_Z*alphaT(i)*F0*C(i,4)/(C(i,4)+g));
end

F(i)=F(i-1)+GGE*TG(i)-M(i);
end

subplot(2,2,4);
scatter(Ts, F);
xlabel('Temperature');
ylabel('Foraminifera Carbon Concentration');
title('F vs.T');

subplot(2,2,2);
hold on
scatter(C(:,1),F,'b+');
scatter(C(:,3),F,'g');
xlabel('Food Source');
ylabel('Foraminifera Carbon Concentration');
title('F vs.C');
hold off

z=1:1:365;
subplot(2,2, [1 3]);
hold on
[Z,Y,X]=plotyy(z,F,z,Ts);
set(get(Z(2),'Ylabel'),'String','Temperature');
set(X,'LineStyle','.');
set(Z, 'xlim',[0 365]);
xlabel('Number of Days');
ylabel('Foraminifera Carbon Concentration');
title('F and T vs # of Days');
hold off