Regensburg 2025 – scientific programme

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BP: Fachverband Biologische Physik

BP 10: Focus Session: Nonlinear Dynamics in Biological Systems I (joint session DY/BP)

BP 10.2: Talk

Tuesday, March 18, 2025, 10:00–10:15, H43

Exceptional Points and Stability in Nonlinear Models of Population Dynamics having PT symmetry — •Alexander Felski — Max Planck Institute for the Science of Light, Erlangen, Germany

Nonlinearity and non-Hermiticity, for example due to environmental gain-loss processes, are a common occurrence throughout numerous areas of science. For the latter, parity-time-reflection (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 foundational nonlinear models 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.

Keywords: Exceptional points; Population dynamics; PT symmetry; Non-Hermitian systems