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The science
Mass balance of a well-mixed lake
For a lake of volume V with throughflow Q, pollutant load L and first-order decay k: V dC/dt = L - QC - kVC. In million m³, tonnes and mg/L (= t per million m³) no unit factors appear.
Exponential relaxation
C(t) = Ceq + (C0 - Ceq) e-t/τ with Ceq = L / (Q + kV) and response time τ = V / (Q + kV). After the reduction year the load drops to L(1 - r), the equilibrium moves down, and the lake approaches it with the same τ. A big, slowly flushed lake needs several τ before prevention shows in the water.
Why prevention beats clean-up
The equilibrium is proportional to the load, so the reduction needed to meet a limit is r = 1 - Clim(Q + kV)/L (third graph). Cutting the source works on every year of the future; flushing and decay are fixed by the lake.
Two independent numerical checks
1. A fixed-step fourth-order Runge-Kutta integration (0.02 yr, load switched exactly on a step boundary) is compared with the piecewise analytic solution at every step.
2. The cloud of orange spheres is a separate stochastic model, not the ODE: each sphere is a mass quantum q; new quanta arrive as a Poisson process with rate L/q and every quantum leaves with probability 1 - e-(Q+kV)Δt/V per step. Starting empty, the number of quanta stays Poisson with mean V C(t)/q, so |N - μ| < 4√μ must hold; the page tests this live.
Honest simplifications
One fully mixed box ignores stratification, sediment release, seasonal flow and temperature. Decay is a single constant. The sphere positions are decorative (random walk with a drift toward the outlet); only their number is physical. Fish are drawn far larger than real fish, and the fish “health index” 1/(1+(C/1.6Clim)3) is an illustrative curve, not toxicology. Water colour is a visual cue for C/Clim, not an optical model. The page is an educational model, not a decision tool.
Things to try
Compare “Strict prevention” with “Act too late”; set the reduction to 0 and watch the equilibrium; in the slow-flushed lake note how long τ is; raise k and see decay replace flushing. Use the Underwater camera to watch the quanta accumulate near the outfall.