A discrete-time model for breathing is given by

$$\begin{aligned} V_{n+1} &=\alpha C_{n-k}, \\ C_{n+1}-C_{n} &=\gamma-\beta V_{n+1}, \end{aligned}$$

where $V_{n}$ is the volume of each breath in time step $n$ and $C_{n}$ is the concentration of carbon dioxide in the blood at the end of time step $n$. The parameters $\alpha, \beta$ and $\gamma$ are all positive. Briefly explain the biological meaning of each of the above equations.

Find the steady state. For $k=0$ and $k=1$ determine the stability of the steady state.

For general (integer) $k>1$, by seeking parameter values when the modulus of a perturbation to the steady state is constant, find the range of parameters where the solution is stable. What is the periodicity of the constant-modulus solution at the edge of this range? Comment on how the size of the range depends on $k$.

This can be developed into a more realistic model by changing the term $-\beta V_{n+1}$ to $-\beta C_{n} V_{n+1}$ in (2). Briefly explain the biological meaning of this change. Show that for both $k=0$ and $k=1$ the new steady state is stable if $0<a<1$, where $a=\sqrt{\alpha \beta \gamma}$.