# 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) \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: \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).