A neglected flower garden contains $M_{n}$ marigolds in the summer of year $n$. On average each marigold produces $\gamma$ seeds through the summer. Seeds may germinate after one or two winters. After three winters or more they will not germinate. Each winter a fraction $1-\alpha$ of all seeds in the garden are eaten by birds (with no preference to the age of the seed). In spring a fraction $\mu$ of seeds that have survived one winter and a fraction $\nu$ of seeds that have survived two winters germinate. Finite resources of water mean that the number of marigolds growing to maturity from $S$ germinating seeds is $\mathcal{N}(S)$, where $\mathcal{N}(S)$ is an increasing function such that $\mathcal{N}(0)=0, \mathcal{N}^{\prime}(0)=1, \mathcal{N}^{\prime}(S)$ is a decreasing function of $S$ and $\mathcal{N}(S) \rightarrow N_{\max }$ as $S \rightarrow \infty$

Show that $M_{n}$ satisfies the equation

$$M_{n+1}=\mathcal{N}\left(\alpha \mu \gamma M_{n}+\nu \gamma \alpha^{2}(1-\mu) M_{n-1}\right)$$

Write down an equation for the number $M_{*}$ of marigolds in a steady state. Show graphically that there are two solutions, one with $M_{*}=0$ and the other with $M_{*}>0$ if

$$\alpha \mu \gamma+\nu \gamma \alpha^{2}(1-\mu)>1$$

Show that the $M_{*}=0$ steady-state solution is unstable to small perturbations in this case.