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

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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:
{\displaystyle {\begin{aligned}{\dot {x}}&=b_{1}x(K-x-y)-d_{1}x+c_{1}xy\\{\dot {y}}&=b_{2}y(K-x-y)-d_{1}y-c_{2}xy\end{aligned}}}
• 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 straight-forward linear noise approximation.