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There are $k{x}_n$ seeds and their germinating proportion is $e^{-\gamma x_n}$, giving the survival-augmented Ricker map with zero survival. A positive equilibrium point satisfies $e^{-\gamma x_{\ast }}=1/k$, soIt exists when $k>1$ and is strictly linearly stable when $|1-\log k|<1$. Thus the usual strict linear-stability interval is $\boxed{1<k<e^2}$.

With winter survival, surviving adults and new plants add:Writing $L=\log [k/(1-s)]$, the linearization of a dynamical system has multiplier $1-(1-s)L$. Hence the strict linear-stability interval becomesIts lower endpoint is less than $1$. Its upper endpoint exceeds $e^2$: for $t=1-s\in (0,1)$, $\log t+2/t$ is decreasing in $t$ and equals $2$ at $t=1$.

The nonlinear stability at the Ricker flip threshold needs a separate check. The upper endpoints have multiplier $-1$, so the strict linear test alone does not decide stability there. They are in fact locally asymptotically stable, with algebraic rather than geometric convergence. To see this, put $u=\gamma (x-x_{\ast })$ at $L=2/(1-s)$. The local map isand its second iterate is $u-2(s^2-s/2+1/6)u^3+O(u^4)$. The cubic coefficient in parentheses is positive for every $s\in [0,1)$. Thus, if “stable” includes nonlinear endpoint stability, the upper inequalities can be replaced by $\le$; the lower endpoints have no positive equilibrium.

At the positive equilibrium,The fixed point is strictly linearly stable whenThe upper endpoint is also locally asymptotically stable, as the nonlinear stability at the Ricker flip threshold shows; hence the full local asymptotic-stability interval includes that endpoint. The lower endpoint has no positive equilibrium.