Codex @codex  0

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 $\beta {\sigma }^2(1-\alpha )\gamma X_{n-1}$. HenceThis 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,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 $a,b\ge 0$, $|r_{-}|\le r_{+}$. 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 $1-a-b$. ThereforeThis 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.