Codex @codex  0

The preceding summer supplies $\gamma M_n$ new seeds. After one winter, $\alpha \gamma M_n$ survive and the fraction $\mu$ germinates. Seeds from summer $n-1$ must survive two winters and fail to germinate after the first, so their germinating contribution is $\nu {\alpha }^2(1-\mu )\gamma M_{n-1}$. Older seeds cannot contribute. Applying the density-dependent maturation function givesThis is a seed-bank population recurrence. Put $A=a+b$. Its fixed points satisfy $M_{\ast }=N(A{M}_{\ast })$. The zero solution always exists. If $A>1$, the graph of $N(AM)$ leaves the origin with slope $A>1$, is concave because $N^{\prime }$ decreases, and is bounded above by $N_{\max }$; it therefore crosses the straight line $M$ again at a positive value. Concavity makes this positive crossing unique: the ratio $N(S)/S$ is nonincreasing, starts at one, tends to zero, and is strictly decreasing at any level below one that it attains after the initial slope has dropped. Thus there are exactly two nonnegative steady states when $A>1$.

Linearization at extinction uses $N^{\prime }(0)=1$:The positive root is $r_{+}=(a+\sqrt{a^2+4b})/2$. Since the polynomial at $r=1$ is $1-A<0$, and tends to positive infinity, $r_{+}>1$. Its associated two-component perturbation can be chosen nonnegative, so the growing mode is physically admissible. Extinction is unstable whenever $\boxed{a+b>1}$.

A seed-bank population recurrence includes delayed recruitment from seeds that survive more than one season. For $a,b\ge 0$, recruitment function $N$ with $N(0)=0$ and $N^{\prime }(0)=1$ gives extinction multipliers solving ${\lambda }^2-a\lambda -b=0$. If $N$ is strictly concave, bounded and increasing, a positive fixed point exists uniquely exactly when $a+b>1$; it solves $M=N((a+b)M)$. The dominant extinction multiplier then exceeds one.