Difference equations for the dynamics of pollutant in the water (P) and pollutant in the sediment (M) are
P_{t+1} = P_{t} + l_{t} exp[z_{t} ς - (ς^{2}/2)] - (s + h) P_{t }+ r M_{t }f(P_{t}) |
(A1.1) |
M_{t+1} = M_{t} + s P_{t} - b M_{t} - r M_{t} f(P_{t}) |
(A1.2) |
f(P) = P^{q} / (m^{q} + P^{q}). |
(A1.3) |
The parameters are mean input rate of P (l_{t}); standard deviation of the logarithm of inputs, ς; proportions of P lost to sedimentation (s) and hydrologic outflow (h) at each time step; proportions of M recycled to the water (r) or buried permanently at each time step (b); the P level at which the recycling rate is half maximal (m); and an exponent (q) that controls the steepness of the recycling curve. Random disturbances to the input are introduced by z_{t}, which is a normal random variate with standard deviation = 1.
In the Market Manager and Governing Board models, we used a nondimensional version of the model. This nondimensional version is formed by defining X = P/m, Y = M/m, a = l/m, yielding
X_{t+1} = X_{t}+ a - (s + h) X_{t} + r Y_{t} g(X_{t}) |
(A.1.4) |
Y_{t+1} = Y_{t} + s X_{t} - b Y_{t} - r Y_{t} g(X_{t}) |
(A.1.5) |
g(X) = X^{q} / (1 + X^{q}). |
(A.1.6) |
A detailed analysis of the fast variable for this model (setting Y_{t} = 1) is presented by Carpenter et al. (1999).