Emergence of edge state in suspension of self-propelled particles

We numerically study a model convection system of a suspension of self-propelled particles, motivated by recent experimental findings of localized and bistable bioconvection pattern, being distinct from classical Rayleigh–Bénard convection. Linear stability analysis of the model system reveals that the trivial noncovection state is stabilized by an increase of self-propelled speed in the vertical direction. Through numerical simulations, we found a nonlinear convection state even when the nonconvection state is stable. Applying ideas and tools developed in wall-bounded flows, we numerically identified an edge state, which is an unstable solution on a basin boundary in the model dynamical systems.

💡 Research Summary

This paper presents a numerical investigation into the onset of convection patterns in a suspension of self-propelled particles (SPPs), motivated by experimental observations of localized and bistable bioconvection in swimming microorganisms. The authors develop a two-dimensional model within a doubly periodic domain, extending the classical Rayleigh–Bénard convection framework by incorporating a term representing the vertical self-propelled speed of particles, characterized by the Péclet number (Pe). A prescribed sinusoidal equilibrium density profile allows for independent control over the self-motility effects.

The core findings are tripartite. First, linear stability analysis of the trivial, non-convective state reveals that increasing the self-propelled speed (Pe) stabilizes this base state. The neutral stability curve shifts to higher values of the average density (Rayleigh number, Ra), indicating that upward swimming counteracts the buoyancy-driven instability that would typically initiate convection.

Second, despite this linear stability, direct numerical simulations in a parameter regime where the base state is linearly stable (e.g., Ra=0.4, Pr=2.5, Pe=1) uncover a coexisting, stable nonlinear convective state. This Upper Branch (UB) solution forms a spatially localized density pattern, demonstrating that the system is bistable—a phenomenon not possible in standard Rayleigh–Bénard convection.

Third, to dissect the dynamical structure of this bistability, the authors employ methodologies from dynamical systems theory, well-established in studying transition in wall-bounded shear flows. Using a bisection method on a family of initial conditions interpolating between the base state and the UB state, they identify a critical initial condition that neither decays nor grows. This condition converges to an unstable stationary solution lying precisely on the boundary separating the basins of attraction of the two stable states. This solution, termed the Lower Branch (LB) or “edge state,” is invariant and its stable manifold constitutes the basin boundary. The LB solution is also localized but with a smaller amplitude than the UB solution.

In conclusion, the study demonstrates that self-propulsion can qualitatively alter convection dynamics, enabling bistability and localization even when the homogeneous state is linearly stable. The identification of an edge state provides a unifying dynamical systems framework for understanding the subcritical, localized transitions observed in this bioconvection model, drawing a direct analogy to phenomena in shear flow turbulence. This work bridges fluid dynamics and active matter physics, offering a theoretical foundation for complex pattern formation in living systems.