Survival and Invasion Dynamics in Cell Populations: An Analytical Framework for Threshold Behavior in Nonlinear Age-Structured Models

Abstract. Cell populations invade through a combination of proliferation and motility. Proliferation depends on the internal timing of cell division: how long cells take to complete the cell cycle. This timing varies substantially within (and across) cell types, creating age structure where cells at different times since their last division have different propensities to divide. Classical mathematical models of cell spreading treat division as memoryless and predict exponential cell-cycle-time distributions. Lineage tracing instead reveals peaked, gamma-like distributions, indicating a maturation delay followed by a fertility window. This mismatch between model assumptions and experimental observations motivates models that incorporate age structure consistent with observed cell-cycle timing, while remaining analytically tractable. We introduce a moment-hierarchy framework that tracks the time since last division, with age resetting to zero at division. The framework yields explicit formulae for steady-state age distributions, cell-cycle-time distributions, and invasion speeds. For age-independent rates, we recover the classical Fisher–KPP; for specific age-dependent rates, we obtain cell-cycle-time distributions consistent with experimental data. Three main principles emerge. First, the framework yields explicit bounds on the population carrying capacity and determines the parameter conditions under which the population persists. Second, assuming homogeneous division rates at the invasion front overestimates invasion speeds: the minimal speed is lower when division is age-dependent. Third, the condition for population survival coincides with the condition for invasion. Relevance to Life Sciences. Our study addresses a fundamental question in wound healing, tissue regeneration, and tumor invasion: how the timing of cell division regulates whether populations can persist and how fast they spread. We link single-cell division timing to population-level outcomes by deriving explicit formulae for division-time distributions, survival thresholds, and invasion speeds. These formulae enable direct parameter estimation from lineage-tracing data and, as demonstrated by fitting division-time measurements from cells under control and perturbed conditions, show that interventions that prolong the cell cycle can shift populations toward extinction. We show that division timing determines invasion speed, providing a mechanistic basis for interpreting competitive outcomes in collective cell migration. Mathematical Content. We formulate an age-structured transport–renewal equation for a motile and proliferating cell population, where cell age tracks the time since last division and resets to zero upon division. Cell division and death are modelled as distinct processes, each dependent on cell age and population density. We derive a generalized moment system that connects the total population to its age-weighted moments. This reduction preserves the essential structure of the model while providing a finite-dimensional, tractable system. We classify models by their rate functions—constant, exponentially decaying, or gamma-like—representing age-homogeneous, early-biased, and delayed division. For each case, this framework yields explicit expressions for the population survival condition, the steady-state age and cell-cycle-time distributions, and the minimal invasion speed, all derived without closure approximations.