Coupled interplays between proliferation and hydrodynamic interactions modulate the transport of chemotactic entities

Abstract

This study investigates the stability properties of two interacting chemotactic populations by introducing an extended mathematical model that simultaneously incorporates competitive hydrodynamic interactions and kinetic growth dynamics. Using a standard plane wave analysis, we demonstrate that the interplay between hydrodynamic and kinetic interactions governs the system’s stability. Specifically, the system remains stable for long wave vectors when strong-strong kinetic interactions are coupled with either weak-weak or strong-weak hydrodynamic interactions. Conversely, stability for short wave vectors is achieved when weak-weak kinetic interactions are paired with similar hydrodynamic configurations. To explore localized dynamics, we perform a generalized linear stability analysis using spatially localized functions. This reveals the emergence of rapid oscillations that modulate both stationary and non-stationary wave patterns. Modulated structures are sensitive to the choice of the envelope function: waves initiated with Gaussian-type profiles are more prone to instability than those generated with secant-type functions. Numerical simulations further illustrate the formation of stable non-uniformly distributed structures. The generalized perturbation framework presented here highlights the delicate balance between hydrodynamics and competitive interactions occurring within biological tissues during invasion, showcasing new insights into collective transport, pattern formation, and strategies for organ repair.

The dynamics of localized unstable modes at short wave vectors indicate that instability grows with self-induced hydrodynamic flows in the \(WW_p-SW_h\) regime. In other words, self-induced flows act to extend instability across the parameter domain. In addition, instability remains localized along kinetic competition parameters, highlighting the extensive role played by those factors to contain the instability when self-induced hydrodynamics flows become dominant in the medium