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

Huichao Xing; Jun Wu; Conghua Wang; Huichao Xing; Jun Wu; Conghua Wang

doi:10.3934/math.2026545

AIMS Mathematics

2026, Volume 11, Issue 5: 13216-13232. doi: 10.3934/math.2026545

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Research article

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

1.
Faculty of Education, Yunnan Normal University, Kunming, Yunnan 650500, China
2.
Chongqing Municipal University Key Laboratory of Optimization Techniques and Intelligent Vehicle, National Center for Applied Mathematics in Chongqing, Chongqing Normal University, Chongqing 401331, China

Received: 03 April 2026 Revised: 24 April 2026 Accepted: 29 April 2026 Published: 12 May 2026
MSC : 34K18, 34K20, 92C42

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.
- synNotch relay,
- delay differential equations,
- saddle-node bifurcation,
- Hopf bifurcation
Citation: Huichao Xing, Jun Wu, Conghua Wang. Bistability and delay-induced oscillations in a density-gated synNotch model[J]. AIMS Mathematics, 2026, 11(5): 13216-13232. doi: 10.3934/math.2026545

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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.

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