@article{Xing2026, 
author = {Huichao Xing and Jun Wu and Conghua Wang},
title = {Bistability and delay-induced oscillations in a density-gated synNotch model},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {13216-13232},
keywords = {synNotch relay, delay differential equations, saddle-node bifurcation, Hopf bifurcation},
url = {https://www.sciopen.com/article/10.3934/math.2026545},
doi = {10.3934/math.2026545},
abstract = {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.}
}