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

Abstract

Consider the difference equation $x_{k+1}=x_k{e}^{\alpha -x_{n-d}}$ where $\alpha$ 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.

for associated C++ program, mathematica worksheet and output, see http://www.math.u-szeged.hu/~krisztin/ricker

References in corpus (1)

• Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system

Cited by in corpus (2)

• Iterative oscillation tests for differential equations with several non-monotone arguments
• Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation