Consider a biological system in which concentrations $x(t)$ and $y(t)$ satisfy

$$\frac{d x}{d t}=f(y)-x \quad \text { and } \quad \frac{d y}{d t}=g(x)-y$$

where $f$ and $g$ are positive and monotonically decreasing functions of their arguments, so that $x$ represses the synthesis of $y$ and vice versa.

(a) Suppose the functions $f$ and $g$ are bounded. Sketch the phase plane and explain why there is always at least one steady state. Show that if there is a steady state with

$$\frac{\partial \ln f}{\partial \ln y} \frac{\partial \ln g}{\partial \ln x}>1$$

then the system is multistable.

(b) If $f=\lambda /\left(1+y^{m}\right)$ and $g=\lambda /\left(1+x^{n}\right)$, where $\lambda, m$ and $n$ are positive constants, what values of $m$ and $n$ allow the system to display multistability for some value of $\lambda$ ?

Can $f=\lambda / y^{m}$ and $g=\lambda / x^{n}$ generate multistability? Explain your answer carefully.