Local stability implies global stability for the 2-dimensional Ricker map

Abstract

Consider the difference equation $x_{k+1}=x_k e^{α-x_{n-d}}$ where $α$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.