Conservation Ecology: Ecological and Social Dynamics in Simple Models of Ecosystem Management: Appendix 1

Appendix 1. Lake ecosystem dynamics.

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 - (ς²/2)] - (s + h) P_t+ r M_tf(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).