Appendix 1. Lake ecosystem dynamics.

Difference equations for the dynamics of pollutant in the water (P) and pollutant in the sediment (M) are

Pt+1 = Pt + lt exp[zt ς - (ς2/2)] - (s + h) Pt + r Mt f(Pt)


Mt+1 = Mt + s Pt - b Mt - r Mt f(Pt)


f(P) = Pq / (mq + Pq).


 The parameters are mean input rate of P (lt); 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 zt, 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

Xt+1 = Xt+ a - (s + h) Xt + r Yt g(Xt)


Yt+1 = Yt + s Xt - b Yt - r Yt g(Xt)


g(X) = Xq / (1 + Xq).


A detailed analysis of the fast variable for this model (setting Yt = 1) is presented by Carpenter et al. (1999).