CASE STUDY: Phosphorus Cycle

Source:

Consider a Spherical Cow: A Course in Environmental Problem Solving, John Harte, University Science Books, 1988.

Overview:

Phosphorus is one the key elements found in nature, and it plays an important role in the metabolism of most living organisms. The movement of phosphorus throughout ecosystems has profound effects not only on the survival and wellbeing of organisms, but also on the economic well-being of the community. Phosphorus is found in several forms: dissolved in water, as phosphates (found in many detergents and fertilizers), and as a part of the organs of living animals and plants.

We wish to build a computational model of the flux (movement) of inorganic phosphorus in a lake in which phosphorus flows in closed cycle (in other words, no phosphorus comes in or goes out of the lake). We are going to assume that the phosphorus in the lake has reached a steady state: the levels of phosphorus remain the same over time in each of the reservoirs. In this fictional lake, we will have three reservoirs: phosphorus in living biomass (animals and plants), phosphorus in dead organic matter (detritus), and "unattached" or free inorganic phosphorus dissolved in the water. We are interested in calculating the concentration of phosphorus per each liter of reservoir.

The initial parameters that are described in the Building the Model section should represent the steady state. Suppose, however, that some inorganic phosphorus is suddenly added to the lake? Suppose the amount added is relatively small compared to the amount already present? How will the amount of phosphorus in each of the three major compartment of the lake (inorganic phosphorus, biomass phosphorus, and detritus phosphorus) behave after this perturbation?

Building the Model:

The three reservoirs needed for this model are described above. Initial values for each are as follows:

Phosphorus in living biomass: 0.2 micromoles per liter

Phosphorus in dead organic matter: 1.0 micromoles per liter

Inorganic phosphorus in lake: 0.1 micromoles per liter

In this model, inorganic phosphorus is taken up (absorbed) by living biomass (plants and animals) dependent on an uptake rate. An average uptake rate for biomass is 2.5 times the amount of inorganic phosphorus and times the amount of phosphorus already in the living biomass. This rate is in units of uptake/micromoles/day.

The phosphorus in living biomass stays there for a certain amount of time known as the residence time. Residence time is defined as the amount of time that a substance will "reside" in a reservoir before it moves on to the next reservoir. The amount of phosphorus that moves from biomass into dead organic matter depends on the residence time and the amount of phosphorus in living matter. For this model, we'll define the residence time as 0.25 per day. In other words, the phosphorus stays in a living organism roughly for 6 hours before it moves on.

Eventually, dead organic matter will decay or decompose. Once this occurs, the phosphorus that was fixed in the matter will become "free", and will dissolve in the water of the lake. The decay rate will describe how quickly organic phosphorus becomes inorganic phosphorus. For this model, we'll set the decay rate at 0.05 per day (some amount of matter decays every 72 minutes).

Run this model for enough time (time units in days) to make sure you see the steady state. At this point, we wish to perturb the system, see if we can throw it out of kilter. At some early time in the simulation (time = 0, time = 0.5 day, your choice). We wish to insert a small amount of phosphorus -- about 0.02 micromoles per liter -- into the cycle to investigate the effects of small amounts of phosphorus on the steady state system. Once you have a perturbed system, you may wish to investigate the effect of various amounts of added phosphorus on the ecosystem.

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