A field contains $X_{n}$ seed-producing poppies in August of year $n$. On average each poppy produces $\gamma$ seeds, a number that is assumed not to vary from year to year. A fraction $\sigma$ of seeds survive the winter and a fraction $\alpha$ of those germinate in May of year $n+1$. A fraction $\beta$ of those that survive the next winter germinate in year $n+2$. Show that $X_{n}$ satisfies the following difference equation:

$$X_{n+1}=\alpha \sigma \gamma X_{n}+\beta \sigma^{2}(1-\alpha) \gamma X_{n-1}$$

Write down the general solution of this equation, and show that the poppies in the field will eventually die out if

$$\sigma \gamma[(1-\alpha) \beta \sigma+\alpha]<1$$