Survival-augmented Ricker map

The population iterationcombines annual survival with density-dependent recruitment. Its positive equilibrium iswhich exists when $k>1-r$.

At the upper multiplier-$-1$ threshold of $x\mapsto sx+kx{e}^{-\gamma x}$, set $u=\gamma (x-x_{\ast })$. The second iterate is $u-2(s^2-s/2+1/6)u^3+O(u^4)$. Since its cubic damping coefficient is positive for $0\le s<1$, this endpoint is locally asymptotically stable despite the inconclusive linear multiplier.

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.