= Solution
First-season recruits in year $n+1$ contribute $\alpha\sigma\gamma X_n$. Seeds produced in year $n-1$ that survive two winters, fail first-season germination and germinate in their second season contribute $\beta\sigma^2(1-\alpha)\gamma X_{n-1}$. Hence
$$
\boxed{X_{n+1}=aX_n+bX_{n-1},\quad a=\alpha\sigma\gamma,\quad b=\beta\sigma^2(1-\alpha)\gamma.}
$$
This treats recruitment from older seeds as absent; otherwise the two-season recurrence would need further terms. The <characteristic polynomial> is $r^2-ar-b$. For distinct roots,
$$
\boxed{X_n=A r_+^n+B r_-^n,\qquad r_\pm=\frac{a\pm\sqrt{a^2+4b}}2,}
$$
with constants determined by two initial values. For $b=0,a>0$, the recurrence is first order after the initial step; for $a=b=0$ it gives $X_{n+1}=0$. These describe the zero-root degeneracies without interpreting $0^0$ in the formula.
Since $a,b\ge0$, $|r_-|\le r_+$. The positive root is less than one exactly when $a+b<1$: the <polynomial> is nonpositive at zero, has only one positive root, and its value at one is $1-a-b$. Therefore
$$
\boxed{\sigma\gamma[\alpha+(1-\alpha)\beta\sigma]<1\ \Longrightarrow\ X_n\longrightarrow0.}
$$
This is the decay condition for the <two-season seed-bank recurrence>. The deterministic model describes expected or averaged abundance; literal eventual integer extinction requires that interpretation. A corresponding branching model with geometrically decaying expected abundance becomes extinct <almost surely>, since the <probability> of any surviving plants is at most the expected number.
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