A maximization program expressed by per capita variales - Ramsey model
Take the differential equation for the natural resource. I will write $S \equiv sL$ where lowercase is the per capita magnitude. So

$$\dot{S}=\left(1-S\right)S-\gamma AK$$

$$\implies \frac {d(sL)}{dt} = (1-sL)sL - \gamma AkL$$

$$\implies \dot sL + s\dot L = sL - s^2L^2 -\gamma AkL$$

Divide by $L$ to get

$$ \dot s + sn = s - s^2L -\gamma Ak$$

and using $L=e^{nt}$ and re-arranging we get

$$ \dot s = (1-n) s - s^2e^{nt} -\gamma Ak$$

Setting this equal to zero, we have a quadratic polynomial in $s$

$$\dot s = 0 \implies e^{nt}s^2 - (1-n) s + \gamma Ak=0$$

We see that the leading term increases as $t$ increases. The only way to maintain equality with zero, and constant $s$, is if $k$ decreases as time passes (and this will be only temporary since $k$ cannot turn negative). 

So it appears that you cannot have a "steady state in per capita terms" for all variables here, consumption, capital and the natural resource.
Most probably you can maintain consumption and capital at constant steady state values while the per capita, at least, natural resource depletes to extinction (but you have to show that this is optimal with respect to intertemporal utility maximization).