Long-time states after reciprocal chemotaxis-induced instability

Determine the eventual steady states reached by the two-species chemotactic reaction-diffusion Selkov model after the reciprocal chemotaxis-induced instability, including whether additional linear stabilizing terms such as hyperdiffusion are required and whether the resulting states can be chaotic.

Background

The paper identifies a novel instability in the reciprocal chemotaxis regime, particularly when the product of the chemotactic couplings is sufficiently large and positive. In the linear analysis, the preferred wavevector can diverge, and the direct numerical simulations show signatures of this instability. However, the simulations do not reach a steady state, so the asymptotic behavior of the unstable system is unresolved.

The authors suggest that additional stabilizing terms, such as hyperdiffusion, may be necessary for a systematic analysis. They also raise the possibility that the eventual dynamics could produce chaotic steady or long-time states, making the characterization of the post-instability regime an open problem.

References

However, from either our linear stability analysis or DNS studies, we are unable to comment on the eventual steady states that the systems ultimately reach through this chemotaxis induced instability, as even our DNS studies do not appear to reach a steady state. It is possible that additional linear stabilizing terms, e.g., hyperdiffusion terms are required for systematic studies of these instabilities. The resulting steady states may also be chaotic. More work is needed to determine the nature of the resulting steady states.

— Chemotaxis-induced linear instabilities and pattern formation in a reaction-diffusion model  (2609.01159 - Karmakar et al., 1 Sep 2026) in Section Conclusion and outlook