In a discrete-time model, adults and larvae of a population at time $n$ are represented by $a_{n}$ and $b_{n}$ respectively. The model is represented by the equations

$$\begin{aligned} a_{n+1} &=(1-k) a_{n}+\frac{b_{n}}{1+a_{n}} \\ b_{n+1} &=\mu a_{n} \end{aligned}$$

You may assume that $k \in(0,1)$ and $\mu>0$. Give an explanation of what each of the terms represents, and hence give a description of the population model.

By combining the equations to describe the dynamics purely in terms of the adults, find all equilibria of the system. Show that the equilibrium with the population absent $(a=0)$ is unstable exactly when there exists an equilibrium with the population present $(a>0)$.

Give the condition on $\mu$ and $k$ for the equilibrium with $a>0$ to be stable, and sketch the corresponding region in the $(k, \mu)$ plane.

What happens to the population close to the boundaries of this region?

If this model was modified to include stochastic effects, briefly describe qualitatively the likely dynamics near the boundaries of the region found above.