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

TL;DR

This paper proves that for a specific 2-dimensional Ricker map with delay d=1, local stability of the equilibrium guarantees its global stability, confirming a long-standing conjecture using rigorous computational and analytical methods.

Contribution

The paper establishes the conjecture that local stability implies global stability for the 2D Ricker map with delay d=1, combining computer-aided and analytical techniques.

Findings

Confirmed the conjecture for d=1 case.

Used rigorous computer-aided calculations.

Provided analytical proof of global stability.

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.