Global Well-Posedness and Stability of Nonlocal Damage-Structured Lineage Model with Feedback and Dedifferentiation

A nonlocal transport–reaction system is proposed to model the coupled dynamics of stem and differentiated cell populations, structured by a continuous damage variable. The framework incorporates bidirectional transitions via differentiation and dedifferentiation, with nonlocal birth operators encoding damage redistribution upon division and Hill-type feedback regulation dependent on total populations. Global well-posedness of solutions in $C([0,\infty );L^1([0,\infty )\times L^1\left([0,\infty )\right))$ is established by combining the contraction mapping principle for local existence with a priori $L^1$ bounds for global existence, ensuring uniqueness and nonnegativity. Integration yields balance laws for total populations, reducing to a finite-dimensional autonomous ordinary differential equation (ODE) system under constant death rates. Linearization reveals a bifurcation threshold separating extinction, homeostasis, and unbounded growth. Under compensatory feedback, Dulac’s criterion precludes periodic orbits, and the Poincaré–Bendixson theorem confines bounded trajectories to equilibria or heteroclinics. Uniqueness implies global asymptotic stability. A scaling invariance for steady states under uniform feedback rescaling is identified. The analysis extends structured population theory to feedback-regulated compartments with nonlocal operators and reversible dedifferentiation, providing explicit stability criteria and linking an infinite-dimensional structured model to tractable low-dimensional reductions.