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

TL;DR

The paper investigates how non-Hermitian, PT-symmetric structures interact with nonlinear population dynamics modeled by replicator and Lotka-Volterra systems. It analyzes PT-symmetric deformations of rock-paper-scissors and generalized Lotka-Volterra models, identifying exceptional points in the linearized dynamics around the coexistence equilibrium and connecting these to global stability via Lyapunov constructions. It shows that EPs occur at $|λ_{EP}|=1$ and, when global PT symmetry is preserved, signal abrupt global destabilization, while explicit symmetry breaking can decouple EPs from global stability; extending to tri-trophic networks reveals a consistent dimensional pattern. The work thus provides a framework for using EPs as a local diagnostic in nonlinear population dynamics, bridging non-Hermitian physics and ecological models with potential implications for stability control in real-world ecosystems.

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.

Figures (9)

• Figure 1: Eigenvalues of the $\mathcal{PT}$-symmetric payoff matrix $A_\lambda$ as a function of the deformation strength $\lambda$. EPs (black dots) arise at $\lambda_\text{EP}^\pm = \pm 1$, marking transitions in the nature of the coexistence equilibrium $\Bar{\mathbf{x}}_\text{c}$ from a center to a saddle point.
• Figure 2: Schematic visualization of the stationary states of the RPS dynamics and their nature on the 2-simplex as a function of the deformation strength $\lambda$. Shown are the globally unstable (red), and globally or regionally stable (blue) regimes, with exemplary dynamics in the top. The bifurcation of additional boundary equilibria is illustrated in dependence of $\lambda$ as dashed lines.
• Figure 3: RPS dynamics on the 2-simplex at exemplary values of the deformation strength $\lambda$. In (a) and (d) the dynamics are globally unstable. In (b) a regionally stable environment exist around the coexistence point. In (c) the dynamics are globally stable. High (low) velocities are indicated in white (black).
• Figure 4: Pitchfork bifurcation of the fixed boundary equilibrium $\Bar{\mathbf{x}}_\text{r}$ (a) without and (b) with defect. The globally unstable regime of deformation strengths $\lambda$ is indicated in red, while the globally or regionally stable regime is indicated in blue.
• Figure 5: Eigenvalues of the payoff matrix $A_\lambda^\delta$ as a function of the deformation strength $\lambda$. EPs arise at $\lambda_\text{EP}^\pm = \pm 1$ (black and red dots) as a result of passive $\mathcal{PT}$-symmetry breaking. The stability transition (black circles) of the deformed model does not coincide with the EPs due to the broken global symmetry.