Trail-Mediated Self-Interaction

A number of microorganisms leave persistent trails while moving along surfaces. For single-cell organisms, the trail-mediated self-interaction will influence its dynamics. It has been discussed recently [Kranz $\mathit{\text{et}}\mathit{\text{al.}}$ Phys. Rev. Lett. $\mathbf{\text{117}}$, 8101 (2016)] that the self-interaction may localize the organism above a critical coupling ${\chi }_c$ to the trail. Here we will derive a generalized active particle model capturing the key features of the self-interaction and analyze its behavior for smaller couplings $\chi < \chi_c$. We find that fluctuations in propulsion speed shift the localization transition to stronger couplings.