Modeling Taihu Lake Pollution with Differential Equations: Predict Future Concentrations

We are asked to build a mathematical model for the pollutant concentration in Taihu Lake, one of China’s largest freshwater lakes, based on weekly monitoring data from five points after imposing a production quota on upstream paper mills.

Task: Assuming the weekly discharge flow and concentration remain unchanged after the quota, use the monitoring data to establish an appropriate mathematical model and predict pollutant concentrations after week 20 and week 30.

The model can be expressed as a differential equation that relates the rate of change of concentration to the inflow concentration, outflow, lake volume, and flow rate, assuming these quantities are constant.

Key variables:

C(t): pollutant concentration at time t (weeks)

Q: weekly discharge flow (assumed constant)

V: total lake volume (constant)

C_in: concentration of incoming pollutant (assumed constant)

t: time in weeks

With these definitions, the differential equation takes the form:

dC/dt = (Q/V) * (C_in - C)

Assumptions underlying the model (as identified by ChatGPT) include:

Constant inflow rate.

Constant inflow concentration.

Perfect mixing: pollutant is uniformly distributed throughout the lake.

Constant lake volume (steady state).

Neglect of natural purification processes such as biodegradation or sedimentation.

Neglect of other pollution sources besides the upstream paper mills.

Constant outflow rate with outflow concentration equal to the lake’s concentration.

Using the weekly concentration data, the model can be calibrated and then employed to forecast concentrations at week 20 and week 30.