Two-season seed-bank recurrence ($X_{n+1}=a{X}_n+b{X}_{n-1}$)

A seed-producing population with first-season and second-season recruitment leads to nonnegative recurrence coefficients $a,b$. Its characteristic roots are $(a\pm \sqrt{a^2+4b})/2$. The nonnegative dominant root is below one precisely when $a+b<1$, giving deterministic decay of every finite initial population. This is a mean-population model; literal integer extinction requires a stochastic interpretation or a separate integer-population argument.