The number of a certain type of annual plant in year $n$ is given by $x_{n}$. Each plant produces $k$ seeds that year and then dies before the next year. The proportion of seeds that germinate to produce a new plant the next year is given by $e^{-\gamma x_{n}}$ where $\gamma>0$. Explain briefly why the system can be described by

$$x_{n+1}=k x_{n} e^{-\gamma x_{n}}$$

Give conditions on $k$ for a stable positive equilibrium of the plant population size to be possible.

Winters become milder and now a proportion $s$ of all plants survive each year $(s \in(0,1))$. Assume that plants produce seeds as before while they are alive. Show that a wider range of $k$ now gives a stable positive equilibrium population.