User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/13

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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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Goals

 * van Kampen expansion of LPA model, maximize noise in L while keeping A robust?

\begin{align} \frac{d \langle \xi^2 \rangle}{d t} &= 2 \frac{\partial \alpha_{1,0}}{\partial \phi} \langle \xi^2 \rangle + 2 \frac{\partial \alpha_{1,0} }{\partial \psi} \langle \xi \eta \rangle + \alpha_{2,0} \\ \frac{d \langle \xi \eta \rangle}{d t} &= \left( \frac{\partial \alpha_{1,0}}{\partial \phi} + \frac{\partial \beta_{1,0} }{\partial \psi} \right)\langle \xi \eta \rangle+ \frac{\partial \alpha_{1,0} }{\partial \psi} \langle \eta^2 \rangle \\ \frac{d \langle \xi^2 \rangle}{d t} &= 2 \frac{\partial \beta_{1,0}}{\partial \psi} \langle \eta^2 \rangle + \beta_{2,0} \end{align} $$
 * Indiv. heterogeneity issue: dividing larval class into a smaller, highly-cannibal and remainder of more weakly cannibalistic.
 * Pulse dynamics -- interspike interval, magnitude?

Potential Data

 * 1995 Dennis et al. -- 5 replicate populations (2-cycle dataset), 5 age-cycles.
 * eight-year chaotic time series
 * Strawbridge daily samples (single replicate), occasional disease in lines though.

Literature

 * Caswell 2009 Sensitivity of density dependent dynamics
 * Kendall (yet unpublished)
 * Vindenes 2008 Individual heterogeneity & demographic stochasticity.
 * Ludwig 1996 Failure of diffusion approximation in small populations.

Next Steps

 * 1) Analytic description of large demographic noise systems
 * 2) Comparisons to beetle datasets
 * 3) Paper Outline.  Skype meeting May 28th.


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