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

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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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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:
 * $$ \frac{K e n^2}{n^2 + h^2} - e n - a $$

Slope (alpha) at equilibrium comes from the derivative:
 * $$ \frac{1}{n} \left( \frac{K e n^2}{n^2 + h^2} - e n - a \right) -\frac{2 K e n^3}{\left( n^2 + h^2 \right)^2} $$
 * $$ \implies -\frac{2 K e n^3}{\left( n^2 + h^2 \right)^2} $$

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


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