Exceptional Points and Stability in Nonlinear Models of Population Dynamics having $\mathcal{PT}$ symmetry

TL;DR

This paper explores how exceptional points influence stability and phase transitions in nonlinear population models with $\\mathcal{PT}$ symmetry, revealing new insights into their global and local dynamical behaviors.

Contribution

It introduces the analysis of exceptional points in nonlinear population dynamics models, connecting symmetry properties to stability changes and phase transitions beyond physics.

Findings

Exceptional points coincide with abrupt stability changes in nonlinear models.

Global symmetry properties determine the location of exceptional points.

Local symmetries around stationary states relate to stability and phase transitions.

Abstract

Nonlinearity and non-Hermiticity, for example due to environmental gain-loss processes, are a common occurrence throughout numerous areas of science and lie at the root of many remarkable phenomena. For the latter, parity-time-reflection ($\mathcal{PT}$) symmetry has played an eminent role in understanding exceptional-point structures and phase transitions in these systems. Yet their interplay has remained by-and-large unexplored. We analyze models governed by the replicator equation of evolutionary game theory and related Lotka-Volterra systems of population dynamics. These are foundational nonlinear models that find widespread application and offer a broad platform for non-Hermitian theory beyond physics. In this context we study the emergence of exceptional points in two cases: (a) when the governing symmetry properties are tied to global properties of the models, and, in contrast, (b) when these symmetries emerge locally around stationary states--in which case the connection between the linear non-Hermitian model and an underlying nonlinear system becomes tenuous. We outline further that when the relevant symmetries are related to global properties, the location of exceptional points in the linearization around coexistence equilibria coincides with abrupt global changes in the stability of the nonlinear dynamics. Exceptional points may thus offer a new local characteristic for the understanding of these systems. Tri-trophic models of population ecology serve as test cases for higher-dimensional systems.