Survival and invasion dynamics in cell populations: an analytical framework for threshold behaviour in nonlinear age-structured models

TL;DR

This paper develops an analytical framework for age-structured cell populations where division timing depends on the cell’s age, not a memoryless rate. By separating division and death and retaining age dependence, the authors derive an integro-differential reduction and a moment-hierarchy that yield explicit steady-state age distributions, cell-cycle-time distributions, and invasion speeds. They classify five biologically motivated cases, revealing how gamma-like maturation delays constrain survival and carrying capacity, and how age structure lowers invasion speeds relative to classical Fisher–KPP theory. A key finding is the equivalence between the conditions for population persistence and positive invasion speed, providing a unified view of persistence and spread. The framework connects measurable division-timing statistics to population-level dynamics and offers testable predictions for experiments and applications in wound healing and tumor invasion.

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, by contrast, reveals peaked, gamma-like distributions that indicate a maturation delay leading to a fertility window. This gap motivates a modelling framework that incorporates age-dependent cell division rates while retaining analytical tractability. We address this through a moment-hierarchy framework that tracks time since cell 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 classical Fisher--KPP. Three fundamental principles emerge. First, age structure systematically reduces a population's carrying capacity and narrows the viable parameter range for positive steady states. Second, classical linear theory overestimates invasion speeds; the true minimal speed is slower when division is age-dependent. Third, the parameter condition for population survival is identical to the condition for a positive invasion speed.

Figures (7)

• Figure 1: (a) Distribution of intermitotic times (IMTs) from lineage-tracing data (grey bars) with fitted exponential (red), Erlang (green), and gamma mixture (blue) models. (b) Schematic of the cell cycle. Cells progress through four main phases: Gap 1 (G1, growth and preparation for DNA synthesis), Synthesis (S, DNA replication), Gap 2 (G2, preparation for mitosis), and Mitosis (M, cell division; here mitosis and cell division are used interchangeably). In our model, cell age (defined as time since last division) resets to zero upon mitosis, providing a coarse-grained representation of cell-cycle progression. The intermitotic time (IMT) is measured from one mitosis to the next (M$\to$M). Experimental IMT data reproduced with permission, courtesy of Richard L. Mort (Lancaster University) and Matthew J. Ford (University of Cambridge), provided by Christian A. Yates (University of Bath). Cell-cycle stages adapted from jiang_2025 (BioRender).
• Figure 1: Numerical steady-state population sizes $\bar{P}$ plotted against threshold conditions (Table \ref{['tab:comparison-Pc']}) for each non-spatial model (Cases 1--5). Dashed lines indicate the analytical upper bounds $\bar{P}_c$. Each panel corresponds to a different model case: (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; and (e) Case 5.
• Figure 1: Steady-state age distributions across five model cases with different division and death structures. Panels (a)-(e) show steady-state distributions $F(a)$ for Cases 1-5: numerical solutions (coloured lines with shading) compared to analytical (black dashed lines). Red triangles mark the mean population age for each case. Panel (f) compares mean population age $\bar{a}_{\mathrm{pop}}$ and mean division age $\bar{a}_{\mathrm{div}}$ across all cases.
• Figure 1: Cell–cycle-time distributions across five model cases. Panels (a--e) show numerical CCTDs (solid) with fitted gamma distributions (dashed) for Cases 1--5; the fitted shape $k$ and scale $\theta$ parameters are indicated. Squares mark mean division ages. Panel (f) overlays all cases after normalisation to isolate shape differences: exponential (Case 1, $k=1$), hypoexponential (Case 2, $k<1$), and bell–shaped (Cases 3--5, $k>1$). Panel (h) summarises the fitted gamma parameters and mean division ages.
• Figure 1: Travelling wave profiles for different model cases. Each panel shows the evolution of the population density $P(z)$ as a function of the travelling wave coordinate $z = x - ct$. The numerically estimated wave speed $c_\mathrm{est}$ approximates the theoretical minimal wave speed $c_{\min}$. The wave speed predicted by the linear theory $c_\mathrm{lin}$ is also provided. Parameters: $\alpha = 0.01$, $\mu = 0.005$, $\kappa = 3 \times 10^{-4}$; $\beta$ satisfies $\beta = r \times (\text{invasion condition})$ with $r=5$ and invasion conditions from Table \ref{['tab:comparison-Pc']}, which places each case at the same relative distance above its extinction threshold. Case 5 additionally requires $\gamma < \mu \alpha e$ to ensure non-negative death rates. We set $\gamma = 1 \times 10^{-4}$.

Theorems & Definitions (2)

• Lemma 2.1
• Proof 1