Bistability and delay-induced oscillations in a density-gated synNotch model

Synthetic signaling circuits provide a versatile framework for programming contact-dependent cell behaviors, yet their collective dynamics are strongly shaped by cell density and processing delay. In this work, we studied a reduced delay differential model motivated by synNotch-type signaling with density-dependent attenuation and adaptive inhibition. The analysis was organized along two complementary routes. First, the equilibrium structure was characterized through a density-driven saddle-node bifurcation analysis, which identified the emergence of a three-equilibrium region and, when the outer-branch trace condition is satisfied, a low/high bistable subinterval. Second, the local dynamics around positive equilibria were examined through a delay-induced Hopf bifurcation analysis, which determined the onset of oscillatory behavior and the associated stability switching. Numerical simulations confirmed the predicted branch structure and the delay-dependent stability switching on the upper equilibrium branch. These results provided a compact dynamical description of how density and intracellular processing time jointly regulated state selection and rhythmic activity in contact-mediated synthetic signaling systems.