User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/08/10

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Warning Signals

Alternate models

  • Writing down the change-point model correctly now, starting with the simulation. Done. Implemented in sde_likelihood.R
  • Implementing Neyman-Pearson comparisons for all three models.

Linearizing directly from the saddle node model

Dynamics:

[math]\displaystyle{ \frac{K e n^2}{n^2 + h^2} - e n - a }[/math]

Slope (alpha) at equilibrium comes from the derivative:

[math]\displaystyle{ \frac{1}{n} \left( \frac{K e n^2}{n^2 + h^2} - e n - a \right) -\frac{2 K e n^3}{\left( n^2 + h^2 \right)^2} }[/math]
[math]\displaystyle{ \implies -\frac{2 K e n^3}{\left( n^2 + h^2 \right)^2} }[/math]

is the slope evaluated at n equal to equilbrium (which is the root of some messy cubic).