Demographic Noise
Crowley Model
 Exploring the extent to which demographic stochasticity in a key class can reverberate through the entire population dynamics.
 Starting with a simple test case of a modified Crowley model:
 [math]\displaystyle{
\begin{align}
\dot x &= b_1 x (K  x  y)  d_1 x + c_1 x y \\
\dot y &= b_2 y (K  x  y)  d_1 y  c_2 x y
\end{align}
}[/math]
 Model is implemented on a branch of the warningSignals package. Model in crowley.c, Depends on modified gillespie.c library in this package.
 Branching necessary as implementation simplifies the gillespie function dependencies, but these changes would break compatibility with warning_signals.c and its R functions. Once warning_signals.c is updated to the new interface format I can remerge the branches.
 compile with make crowley for the moment, will add R interface soon.
 Branches need to be pushed to the github server separately, see the github directions, but essentially
git push origin branchname
will do the trick.
Parameters for small class
 Values: b_1 = 0.11, b_2 = 0.6, d_1 = d_2 = c_1 = 0.1, c_2 = 4, K = 10000.
 Expected sizes: E(x) = 500, E(y) = 4500, (analytic)
 Variation (from simulation) SD(x) ≈ 120, SD(y) ≈ 1000
 For K = 1000, means are 1/10th and extinction very common within 500 time units, and possible in either group. Extinctions of smaller population not uncommon for K = 10000 in 5000 time units.
 Freezing dynamics of x (i.e. b_1 = .00, d_1 = .00, c_1 = .00) greatly decreases fluctuations in Y (more than an order of magnitude, SD(y) ≈ 70).
 Analytic description should be straightforward linear noise approximation.
