| Abstract: |
| Motivated by a surface based defense strategy of the leaf beetle \emph{Hemisphaerota cyanea}, we analyze a biomathematical model for surface tension driven fluid transport in networks of interconnected droplets. Curvature induced pressure differences generate volume exchange through narrow channels, producing competitive growth and extinction dynamics within the droplet ensemble. The model takes the form of a graph based system of ordinary differential equations and incorporates power law rheology to capture non Newtonian biological fluids.\
In the shear thickening regime, solution nonuniqueness challenges the interpretation of biologically meaningful behavior. We establish that positivity of droplet volumes is preserved, ensuring physical relevance despite the lack of uniqueness. Moreover, the long term dynamics are shown to be simple and robust: all trajectories converge to equilibria. An energy functional linked to surface area provides a Lyapunov structure that orders steady states and clarifies long term outcome selection. |
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