User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/16

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General matrix form and correcting equations

Pursuing a matrix form, seems I had missed some terms in the covariance equations Friday, which are suggested by the nice symmetry of this representation: Eqns.jpg (sometimes chalk and a camera are faster than tex).

This suggests the general form looks like

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle dM = J M + (J M)^T + g }

Where M is the covariance matrix and J the Jacobian of f as before. dM is the matrix of derivative terms, though clearly not a rigorous notation.

Fixing Crowley results

  • A couple numerical errors in the implementation of the Crowley model yesterday led to the divergence. The updated numerical results can produce large but stable noise. Also applies system size scaling to match the individual-based simulation.

Crowley noise.png

  • Working on graphing the direct comparison between the approximation and the individual simulation next.
  • Also working on updating gillespie simulation in warningSignals in order to remerge the crowley branch with the master branch.
  • warningSignals crowley branch successfully merged back onto master.
  • Next steps:
    • Add individual simulations to R interface for warningSignals (currently C code just computes window-averaging and ensemble on the fly instead of saving the full sample history).
    • add R function interface to Crowley model
    • Compare Crowley simulation to analytics.
    • Increase density-dependence and decrease size of key class in Crowley for larger noise effects.