User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/11

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 * style="background-color: #EEE"|[[Image:owwnotebook_icon.png|128px]] Stochastic Population Dynamics
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 * style="background-color: #F2F2F2" align="center"|  |Main project page


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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:

\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} $$

git push origin branchname will do the trick.
 * 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

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 straight-forward linear noise approximation.


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