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Note that on the plots of the steady state values of $s$ and $g$ as a function of $\rho_0$, the $s_0$ curve has a global maximum. Because of the form of $f_{1,2}$, there is no closed form solution for the coordinates of the maximum. However, there is an easy closed form solution if $k_1 = k_2 = k$ and $\delta_1 = \delta_2 = \delta$, as is the case for our biologically estimated parameters. This solution is $(\rho_0,s)=(k,k/2)$.

Thus, it is natural to look for an approximate solution when $k_{1,2}$ and $\delta_{1,2}$ are each slightly detuned from equality. The basic idea is to expand everything in a power series in a small parameter $\epsilon$.

Take the steady state formula for $s$ from our manuscript, differentiate it, and set it equal to zero to look for the critical point. This is our governing equation. Then expand everything in a power series. Without loss of generality, we can do this as 


$$k_1 = k + \epsilon K, \quad k_2 = k - \epsilon K$$

$$\delta_1 = \delta + \epsilon \Delta, \quad \delta_2 = \delta - \epsilon \Delta$$

$$\rho_0 = \rho_{00} + \epsilon \rho_{01}$$.

Substituting the power series into the equation for $s^\prime$ and solving at $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ yields

$$\rho_{00} = k$$

and 

$$\rho_{01} = K + \frac{\Delta k}{\delta}$$

Thus, the maximum solitarious density occurs at

$$\rho_0 \approx k + K + \frac{\Delta k}{\delta}$$.

Substituting back into the original formula for $s$ and keeping through $\mathcal{O}(\epsilon)$ gives us the maximum steady state solitarious density, which is

$$s_{max} \approx \frac{k}{2} + \frac{\Delta k}{2\delta}$$

I've checked how good this approximation is vis-a-vis the parameter sensitivities I discussed in yesterday's email. That is, I've set $k = 65$ and $\delta = 0.25$. I've let the deviation $K$ be as large as $0.3 k$ and the deviation $\Delta$ be as large as $0.3 \delta$ -- that is, I've considered up to 30\% deviation from the mean value.s Comparing the exact (numerical) values for the critical point's coordinates to the approximate values, you get up to 20\% error for the $\rho_0$ coordinate and 15\% error for the $s$ coordinate.


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