Appendix S1 Stationary Probability Density
Horsthemke and Lefever (1984) equations 6.10 and 6.11 describe how to compute the
stationary probability density for X. In focusing on the fast dynamics of X, we
assume that values of the slow variables U and M are constant for long enough to
make the stationary density meaningful. Further, we assume that the input
disturbance is zero. These stationary densities are a heuristic, illustrating the effect of
stochastic recycling over a period of time when input disturbance and changes in U
and M are negligible.
We write the fast dynamics as
(1) dX  f ( X )dt  g ( X )dW
where f(X) is
(2) f ( X )  cU  ( s  h) X  rMR ( X )
and
(3) g ( X )  MR ( X )
With these definitions, the stationary density (using Ito stochastic calculus) is
(4) p s ( X ) 
 2
Q
exp  2
g (X )

2

f (u ) 
du 
X 0 g 2 (u )

X
where the normalization constant Q is computed as
(5) Q 1  
b2
b1
 1
2
exp( 2
 2

 g (X )

X
X0

f (u )
du ) dx
2
g (u ) 
The stationary density is not defined unless Q is finite. If the stationary density is
defined, then we can compute the mean of the stationary density as
L2
(6)
E( X ) 
 Xp ( X )dX
s
L1
where L1 and L2 are some convenient limits that give good coverage of ps(X) while
avoiding underflow errors. Then the variance is
(7)
L2 2

var( X )    X p s ( X )dX   E ( X ) 2
 L1

and the standard deviation (convenient because it has the same units as X) is the
square root of var(X).
For ecologically-interesting cases, the stationary density cannot always be computed,
because the exponential of the integral in equation 4 causes overflow errors. We
addressed this problem in two ways, as follows.
(1) We computed indexes of the mean and variance by the following pseudocode (i)
compute the log of ps(X) assuming Q=1, (ii) center log(ps(X)) by subtracting the
median, (iii) truncate the centered log(ps(X)) at [-700,700], the limits for
exponentiation on our computer, (iv) exponentiate to get ps(X), (v) integrate to
compute Q, (vi) repeat steps i - iv using the estimated value of Q, (vii) use the
resulting estimate of ps(X) to estimate the integrals in expressions 6 and 7. We
experimented with different centering and truncation rules, and with various
approaches to estimating Q. We saw no particular advantage to the various
approaches that we tried. All integrals were computed on a 500-point mesh using the
Matlab 7.0 routine QuadT.m.
(2) We used Monte Carlo simulation to compute the mean and standard deviation of
X. For each estimate of the mean and standard deviation, we simulated equations 1-3
(setting H=1) from each of 1000 different initial values of X for 1080 time steps. The
initial value of U was set at its deterministic steady state. The initial value of M was
set at its deterministic steady state if there was one equilibrium, or the oligotrophic
steady state if the system was bistable. The 1000 final values of X (one from each of
the 1000 initial values) were used to calculate the mean and standard deviation.
To illustrate results of the two methods, we compare means and standard deviations of
the stationary distribution across a range of initial values of sediment phosphorus (M).
The turnover time of M is longer than that of X, by a ratio of about 830 in the
oligotrophic steady state and about 9.3 in the eutrophic steady state (Carpenter 2005).
Thus it may be reasonable to consider a stationary distribution of X for a given value
of M. Perhaps it is more reasonable to think of such a stationary distribution in
relation to a gradient of a parameter that can be viewed as a constant for a given lake.
Results across a range of a parameter value are presented in the main text.
Deterministic equilibria show the bistability of the system at intermediate values of M
(colored lines in Figures S1-1 and S1-2). Indices of the mean and standard deviation
of X computed by Fokker-Planck equations (Method 1) are presented in Figure S1-1.
The mean is close to the deterministic mean, for M below about 400 g m-2. As M
rises above 400 g m-2, the mean index rises to approach the eutrophic steady state
value for X. The standard deviation index of X rises as M increases. However, there
is a notable jump in the standard deviation index between M values of about 300 and
500 g m-2. Thus the standard deviation of X increases notably at values of M lower
than those that trigger the regime shift.
Stationary means and standard deviations computed by Monte Carlo simulation
(Method 2) are presented in Figure S1-2. There are slight differences from the
indexes shown in the previous figure. We attribute these differences to the centering
and truncation necessary to compute the integrals of the Fokker-Planck equations.
The mean pulls away from the oligotrophic attractor when M is about 300 g m-2, and
crosses the unstable equilibrium when M is slightly above 400 g m-2. The standard
deviation rises sharply between M values of about 200 and 300 g m-2. As was the
case with Method 1, the standard deviation of X increases substantially at values of M
lower than those that trigger the regime shift.
Reference for Appendix S1
Carpenter, S.R. (2005). Eutrophication of aquatic ecosystems: bistability and soil
phosphorus. Proc. Natl. Acad. Sci. USA 102: 10002-10005.
Horsthemke, W. and Lefever, R. (1984). Noise-Induced Transitions. SpringerVerlag, NY.
Figure S1-1 Equilibria of X (g m-2) and indexes of the mean and standard deviation,
computed by method 1, over a range of M (g m-2). Note log scale for the y axis.
Cyan line shows the oligotrophic attractors, magenta line shows the unstable
equilibria, and green line shows the eutrophic attractor. Blue circles show the mean
of the stationary distribution and red triangles show the standard deviation.
Figure S1-2 Equilibria of X (g m-2) and mean and standard deviation computed by
Monte Carlo (Method 2), over the same range of M (g m-2) as in Figure S1-1. Note
log scale for the y axis. Cyan line shows the oligotrophic attractors, magenta line
shows the unstable equilibria, and green line shows the eutrophic attractor. Blue
circles show the mean of the stationary distribution and red triangles show the
standard deviation.