A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds

Abstract

A quadratic Lyapunov function is demonstrated for the non-invertible planar Ricker map which shows that for , and all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of ). Our approach bypasses and improves on methods that rely on monotonicity, which require . We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle.