Eutrophication: An Introductory Model

Anderson's model was written in Dynamo (see appendix D) and published in Toward Global Equilibrium. The model included state variables to keep track of

a pool of accumulated nutrients,

a biotic population,

the decaying detritus, and

the oxygen in the Hypolimnion.

Anderson selected carbon as the nutrient because it is thought to be the limiting nutrient in some limnetic systems. Figure 1 shows a Stella model that will lead us part way toward Anderson's Model. The diagram shows three of the four stocks that appear in Anderson's model. Each stock is measured in micrograms (mg) of carbon per liter of lake water. The unit of time is years, so all flows are measured in mg/year per liter of lake water.

Figure 1. Initial conditions for a portion of Anderson's eutrophication model.
(All units refer to a liter of lake water.)

The initial conditions are written onto the flow diagram following the equilibrium diagram format explained in chapter 5. Except for the small flow of nutrients into the lake, the system would be in equilibrium. For example:

Stocks	In Flows (in mg/yr)	Out Flows (in mg/yr)
carbon in the nutrient pool	carbon respiration: 0.12 detritus decay: 1.2	carbon fixation: 1.32
carbon in the biomass	carbon fixation: 1.32	carbon respiration: 0.12 biomass death: 1.20
carbon in the Detritus	biomass death: 1.20	detritus decay: 1.20

But there is a small flow of nutrients into the lake. Anderson sets the "nutrients flowing into the lake" at 0.001 mg/yr per liter of lake water. This value was taken as typical for a lake experiencing natural eutrophication. The total carbon amounts to 55.55 mg with almost all of it stored in the nutrient pool. Figure 1 shows 3 mg stored in the Detritus and a minute amount stored in the living biomass.

The largest flow feeding carbon into the nutrient pool is detritus decay. The decay is normally 40%/yr of the carbon stored in the detritus, but the actual flow can be adjusted up or down depending on the oxygen concentration in the Hypolimnion. The equation for the flow would be written in Stella as follows:

detritus_decay=carbon_in_Detritus*detritus_decay_rate*oxygen_limitation_factor

Anderson's model assigns a stock variable to simulate the oxygen concentration in the hypolimnion. To keep the introductory model simple, let's assume that the oxygen concentration will remain constant at 9 mg per liter. Figure 1 shows the oxygen limitation factor is at 1.0. This is a dimensionless multiplier that adjusts the detritus decay up or down in proportion to the oxygen concentration. If oxygen were to increase to 10% above the "normal" value of 9 mg, for example, detritus decay would be 10% faster than indicated by the 40%/yr detritus decay rate.

Figure 1 shows that the flow of carbon back into the nutrient pool is also governed by the oxygen concentration in the Hypolimnion. The Stella equation for this flow would be:

carbon_respired_back_into_pool=carbon_in_biomass*respiration rate*oxygen_limitation_factor

The normal respiration rate is 40%/yr, the same as the detritus decay rate. Figure 1 indicates that there is ten times less carbon stored in the living biomass than in the detritus, so the carbon respired back to the pool is ten times smaller than the flow from detritus decay.

Carbon is removed from the nutrient pool by the carbon fixation by growing biomass. Anderson assumes that "aquatic primary producers fix about eight times their own mass per year," so the biomass fixation factor is set at 800%/yr. But the actual fixation is controlled by the carbon in the nutrient pool. In the Figure 1 illustration, the 52.55 mg of carbon is at 55% of the normal carbon concentration, so the fixation is only 55% of the value expected from the 800%/yr fixation factor. The relevant Stella equations would be:

carbon_fixation_by_growing_biomass=carbon_in_biomass*biomass_fixation_factor*carbon_limitation_factor
carbon_limitation_factor=carbon_in_nutrient_pool/normal_carbon_as_nutrient

Once carbon is fixed in the living biomass, it can return to the nutrient pool through respiration or pass to the detritus with the death of the biomass. The biomass death rate is set at 400%/yr. This high death rate suggests that the "average life" of carbon fixed in the living biomass" is extremely short--only around 1 day.

This is an extremely short time constant given the long time horizon for the model. The short time constant has two, important implications:

First, it explains why the carbon stored in the biomass is such a minute fraction of the total carbon in the lake.

Second, it alerts us to potential problems when simulating the model over a 200 year time hyorizon.

Recall the discussion of DT in chapter 11 of Modeling the Environment and the warning that we face unusual numerical challenges with models that have long time horizons and extremely short time constants. You should brace yourself for such problems if you undertake some of the exercises with this model.