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 (Formula presented.) which shows that for (Formula presented.), and (Formula presented.) all orbits of the planar Ricker map converge to a fixed point. We establish that for 0 <2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (Formula presented.)). Our approach bypasses and improves on methods that rely on monotonicity, which require (Formula presented.). 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.