Critical Slowing Down along the Separatrix of Lotka-Volterra Model of Competition

TL;DR

This paper investigates how the time to reach stable states in the Lotka-Volterra competition model diverges near the separatrix, revealing critical slowing down akin to phase transitions.

Contribution

It systematically analyzes transient behaviors and identifies critical slowing down near the separatrix in the Lotka-Volterra competition model.

Findings

Time to reach stable fixed point diverges logarithmically near the separatrix.

Critical slowing down observed near the saddle point.

Metastable behavior identified before reaching the stable fixed point.

Abstract

The Lotka-Volterra model of competition has been studied by numerical simulations using the Runge-Kutta-Fehlberg algorithm. The stable fixed points, unstable fixed point, saddle node, basins of attraction, and the separatices are found. The transient behaviours associated with reaching the stable fixed point are studied systematically. It is observed that the time of reaching the stable fixed point in any one of the basins of attraction, depends strongly on the initial distance from the separatrix. As the initial point approached the separatrix, this time was found to diverge logarithmically. The divergence of the time, required to reach the stable fixed point, indicates the critical slowing down near the critical point in equilibrium phase transition. A metastable behaviour was also observed near the saddle fixed point before reaching the stable fixed point.