Exceptional points and stability in nonlinear models of population dynamics having <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> symmetry

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 (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">PT</a:mi></a:math>) 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 local linear non-Hermitian model and an underlying global nonlinear system becomes tenuous. We further outline 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. Tritrophic models of population ecology serve as test cases for higher-dimensional systems.