editorial note: These notes pre-date the formal start of my online laboratory notebook, Feb 2 2010: The Lab Notebook Goes Open and were adapted from a LaTeX document in which I kept notes on this topic during my summer at IIASA. Lacking a proper notebook then, documents like this one were updated periodically and occassionally branched into new ones. The post date represents the last time the LaTeX document was edited in the course of that research.
\[\begin{aligned} \mathrm{d}n_1 = r n_1 \left(1 - \frac{n_1 + C(x_1, x_2) n_2}{K(x_1) } \right) \mathrm{d}t + \frac{1}{\sqrt{K_o} } \sqrt{r n_1 \left(1 + \frac{n_1 + C(x_1, x_2) n_2}{K(x_1) } \right) } \mathrm{d}W_1\end{aligned}\]
\[\begin{aligned} p &= \frac{n_1}{n}, \quad q = 1-p = \frac{n_2}{n} \quad n = n_1 + n_2 \\ \mathrm{d}p &= \frac{\mathrm{d}n_1}{n}, \quad \mathrm{d}q = \frac{\mathrm{d}n_2}{n} \\ \mathrm{d}n &= \mathrm{d}n_1 + \mathrm{d}n_2 = n (\mathrm{d}p + \mathrm{d}q) \end{aligned}\]
\[\begin{aligned} \mathrm{d}p(p,n) = \alpha_1(p,n) \mathrm{d}t + \sqrt{ \alpha_2(p,n) } \mathrm{d}W_p \\ \mathrm{d}n(p,n) = \beta_1(p,n) \mathrm{d}t + \sqrt{ \beta_2(p,n) } \mathrm{d}W_n\end{aligned}\]
\[\begin{aligned} f_i(p) = r p \left(1 - n(p) \frac{p + c_{12}(1-p) }{k_i } \right) \\ g_i(p) = r p \left(1 + n(p) \frac{p + c_{12}(1-p) }{k_i } \right) \end{aligned}\]
\[\begin{aligned} \alpha_1 &= f_1(p) \\ \alpha_2 &= \frac{ g_1(p) }{K_0 n(p) } \\ \beta_1 &= n f_1(p) + n f_2(1-p) \\ \beta_2 &= n*\frac{g_1(p)+g_2(1-p)}{K_0 } \end{aligned}\]
\[\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \sigma_p &= 2 \partial_p \alpha_1(p,n) \sigma_p +2\partial_n\alpha_1(p,n) \langle p n \rangle + \alpha_2(p,n) \\ \frac{\mathrm{d}}{\mathrm{d}t} \sigma_n&= 2 \partial_n \beta_1(p,n) \sigma_n +2\partial_p\beta_1(p,n) \langle p n \rangle + \beta_2(p,n) \\ \frac{\mathrm{d}}{\mathrm{d}t} \langle p n \rangle &= \partial_n \alpha_1(p,n) \sigma_n +\partial_p \alpha_1 \langle pn \rangle + \partial_p\beta_1 (p,n) \sigma_p + \partial_n \beta_1(p,n) \langle p n \rangle\end{aligned}\]
\[\begin{aligned} \partial_p \alpha_1 &= \\ \partial_n \alpha_1 &= \\ \partial_p \beta_1 &= \\ \partial_n \beta_1 &=\end{aligned}\]