A gene product with concentration $g$ is produced by a chemical $S$ of concentration $s$, is autocatalysed and degrades linearly according to the kinetic equation

$$\frac{d g}{d t}=f(g, s)=s+k \frac{g^{2}}{1+g^{2}}-g$$

where $k>0$ is a constant.

First consider the case $s=0$. Show that if $k>2$ there are two positive steady states, and determine their stability. Sketch the reaction rate $f(g, 0)$.

Now consider $s>0$. Show that there is a single steady state if $s$ exceeds a critical value. If the system starts in the steady state $g=0$ with $s=0$ and then $s$ is increased sufficiently before decreasing back to zero, show that a biochemical switch can be achieved to a state $g=g_{2}$, whose value you should determine.