# User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/04/11

Stochastic Population Dynamics Main project page
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## Effective warning signals

• Predicting variance of variance by direct calculation -- still need to crunch some math for the expected convergence.
• Still, the approach should be able to do more than describe single points as unexpected deviates.
• Essentially the same as the phyolgenentic problem -- one rate vs two rates. Model selection approaches?
• So far theory is essentially built on a model selection between linear models.
• Calculate the eigenvalue directly rather than ratio of eigenvalue to noise:
• Estimate the eigenvalue from the correlation function and from power spectrum, rather than the lag-1 autocorrelation, or variance.
• Proper signal processing techniques for detecting bifurcations?

### Coding Progress

• Added a proper autocorrelation function calculation, log transform and linear regression gives the eigenvalue and the variance.
• Tested using the Langevin model

$\displaystyle{ x_{t+1} = x_t \left(1- \frac{\kappa}{\gamma} \Delta t \right) + \Delta t \sqrt{\frac{2 K_B T}{\gamma} } \xi_t }$

Whose correlation function is given by

$\displaystyle{ \frac{K_BT}{\gamma} e^{-\kappa t / \gamma} }$