A mathematical model with state-dependent delay for cell population dynamics
Abstract
We consider an age-structured partial differential equation model for cell population dynamics in which the cell cycle duration depends on the total number of quiescent cells. Using the method of characteristics, the model is reduced to a state-dependent delay differential equation. We analyze the stability properties of the resulting equation. By constructing a Lyapunov function, we obtain a sufficient condition for the global asymptotic stability of the trivial equilibrium, corresponding to cell extinction. We further prove the existence of a unique positive equilibrium emerging from a transcritical bifurcation of the trivial equilibrium. The analysis of the positive equilibrium reveals the possible occurrence of a Hopf bifurcation and provides criteria for stability switches. Numerical simulations illustrate the theoretical results. This modeling framework is motivated by periodic hematological diseases, characterized by pathological oscillations in blood cell populations, and captures the influence of the physiological state on cell cycle durations.
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BibTeX
@article{adimy2026mathematical,
title = {{A mathematical model with state-dependent delay for cell population dynamics}},
author = {Mostafa Adimy and Nohayla Alaoui and My Lhassan Hbid},
journal = {Discrete and Continuous Dynamical Systems - Series S},
year = {2026},
month = sep,
doi = {10.3934/dcdss.2026243},
url = {https://inria.hal.science/hal-05776523v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A mathematical model with state-dependent delay for cell population dynamics
AU - Mostafa Adimy
AU - Nohayla Alaoui
AU - My Lhassan Hbid
PY - 2026
DA - 2026/09/23
JO - Discrete and Continuous Dynamical Systems - Series S
DO - 10.3934/dcdss.2026243
UR - https://inria.hal.science/hal-05776523v1
AB - We consider an age-structured partial differential equation model for cell population dynamics in which the cell cycle duration depends on the total number of quiescent cells. Using the method of characteristics, the model is reduced to a state-dependent delay differential equation. We analyze the stability properties of the resulting equation. By constructing a Lyapunov function, we obtain a sufficient condition for the global asymptotic stability of the trivial equilibrium, corresponding to cell extinction. We further prove the existence of a unique positive equilibrium emerging from a transcritical bifurcation of the trivial equilibrium. The analysis of the positive equilibrium reveals the possible occurrence of a Hopf bifurcation and provides criteria for stability switches. Numerical simulations illustrate the theoretical results. This modeling framework is motivated by periodic hematological diseases, characterized by pathological oscillations in blood cell populations, and captures the influence of the physiological state on cell cycle durations.
SN - 1937-1632
SN - 1937-1179
ER -