User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/15

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Large Intrinsic Noise

Modified Crowley model

Realizing Arbitrarily large demographic noise systems??

Consider the Crowley model from last week which I'd implemented as an individual birth-death model: (x is the better competitor, y the better colonist)

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 \begin{align} \dot x &= b_1 x (K - x - y) - d_1 x + c_1 x y = \alpha_1(x,y) \\ \dot y &= b_2 y (K - x - y) - d_1 y - c_2 x y = \beta_1(x,y) \end{align} }

I've implemented the linear noise approximation for this model as a system of coupled ODEs:

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 \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} }

And solved numerically (R code, links directly to this version and can run stand-alone from the package) using parameter values matching the individual based simulation (C code from warning_signals package).

Diverge.png

As this simulation clearly shows, even though I've started the population at the expected abundances and the average dynamics are stable, the variance term for the colonist diverges. With weaker c_2 it is easy to have equilibrium variances and have the expected variance of the competitor be larger than that of the colonist (despite smaller abundance).