stable oscillations form, we examine the stability of the fixed point of equations (\[rDelay\]) and (\[sDelay\]). By considering the system in the symmetric case where $\alpha=\beta=\gamma=1$, we can derive analytical results which provide a qualitative understanding of the system in general. The fixed point of the system in the symmetric case is $(r,s)=(\tfrac{1}{3},\tfrac{1}{3})$.

As $m \rightarrow \infty$, the trinomial distribution (\[tri\]) becomes increasingly well approximated by a bivariate normal distribution [@Gri01] having the same means $\mu_R = m r, \mu_S = m s$, variances $\sigma_R^2 = m r(1-r), \sigma_S^2 = m s(1-s) $ and correlation $\rho = - (m r s)/(\sigma_R \sigma_S)$. By defining the parameter, $$z = \frac{(x-\mu_R)^2}{\sigma_R^2} + \frac{2 \rho (x-\mu_R)(y-\mu_S)}{\sigma_R \sigma_S} + \frac{(y-\mu_S)^2}{\sigma_S^2}$$ we can write this bivariate normal density as: $$\label{bivnorm}
\phi(x,y)dx dy = \frac{\exp \left[-\frac{z(x,y)}{2(1-\rho^2)}\right]}{2\pi \sigma_R \sigma_S \sqrt{1-\rho^2}} dx dy.$$ We can derive analytical approximations for the functionals $p_R(\bar{r}_m, \bar{s}_m)$ and $p_R(\bar{r}_m, \bar{s}_m)$, defined in (\[pfunc1\]) and (\[pfunc2\]), by integrating this density over the domains $\mathcal{D}_\omega$. In the symmetric case this task may be achieved by introducing new variables [@Tou15] $(X,Y)$, related to $(x,y)$ as follows $$\begin{aligned}
x &=  \frac{m}{3} + X - \frac{Y}{\sqrt{3}} \\
y  &=  \frac{m}{3} + \frac{2Y}{\sqrt{3}}.\end{aligned}$$ We write the transformed density 